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a/0dE1T4oBgHgl3EQfkwTA/content/tmp_files/2301.03278v1.pdf.txt b/0dE1T4oBgHgl3EQfkwTA/content/tmp_files/2301.03278v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ebcd35a97990a07bc30b3790f1272683c465097 --- /dev/null +++ b/0dE1T4oBgHgl3EQfkwTA/content/tmp_files/2301.03278v1.pdf.txt @@ -0,0 +1,1799 @@ +MNRAS 000, 1–14 (0000) +Preprint 10 January 2023 +Compiled using MNRAS LATEX style file v3.0 +MulGuisin, a Topological Clustering Algorithm, and Its +Performance as a Cosmic Structure Finder +Young Ju1,2, Inkyu Park1,2⋆, Cristiano G. Sabiu1,2 and Sungwook E. Hong,3,4 +1Department of Physics, University of Seoul, 163 Seoulsiripdae-ro, Dongdaemun-gu, Seoul 02504, Republic of Korea +2Natural Science Research Institute, University of Seoul, 163 Seoulsiripdae-ro, Dongdaemun-gu, Seoul 02504, Republic of Korea +3Korea Astronomy and Space Science Institute, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea +4Astronomy Campus, University of Science and Technology, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea +10 January 2023 +ABSTRACT +We introduce a new clustering algorithm, MulGuisin (MGS), that can find galaxy clusters using topological informa- +tion from the galaxy distribution. This algorithm was first introduced in an LHC experiment as a Jet Finder software, +which looks for particles that clump together in close proximity. The algorithm preferentially considers particles with +high energies and merges them only when they are closer than a certain distance to create a jet. MGS shares some +similarities with the minimum spanning tree (MST) since it provides both clustering and graph-based topology in- +formation. Also, similar to the density-based spatial clustering of applications with noise (DBSCAN), MGS uses the +ranking or the local density of each particle to construct clustering. In this paper, we compare the performances of +clustering algorithms using some controlled data and some realistic simulation data as well as the SDSS observation +data, and we demonstrate that our new algorithm find clusters most efficiently and it defines galaxy clusters in a way +that most closely resembles human vision. +Key words: large-scale structure of Universe, galaxies: clusters: general, methods: statistical, software: data analysis +1 INTRODUCTION +In the standard ΛCDM cosmology paradigm, structures in +the universe grow in a hierarchical manner (e.g., White & +Rees 1978; Fall & Efstathiou 1980; Blumenthal et al. 1984). +It means that smaller structures of matter start forming ear- +lier, and the more massive structures form by the merging and +accretion of smaller structures at later epoch of the universe. +Therefore, understanding cosmic structures in the universe +at various scales is crucial for understanding the nature of +our universe. For example, numerous statistics of large-scale +structures, such as topological analyses (e.g., Gott et al. 1986; +Park & Gott 1991; Park & Kim 2010; Appleby et al. 2017, +2018), Alcock-Paczynski tests (e.g., Alcock & Paczynski 1979; +Ballinger et al. 1996; Li et al. 2014; Park et al. 2019), and +small-scale redshift-space distortion (RSD; e.g., Sheth 1996; +DeRose et al. 2019; Tonegawa et al. 2020), have been used to +constraint cosmological parameters such as the matter den- +sity parameter (Ωm) and the equation-of-state parameter of +dark energy (ωde). +While numerous statistics of cosmic structures have been +used to understand the evolution and structure formation +⋆ E-mail: icpark@uos.ac.kr +of our universe, the exact definition of cosmic structures re- +mains unclear. This is mainly because the matter distribution +on large scale is continuous, and therefore, there exists no +specific discrete boundary for each structure. Also, the mem- +bership for a certain structure might change if one considers +other properties than just position, such as dynamics, mass, +and so on (e.g., Serra & Diaferio 2013; Gifford et al. 2013). +Due to this ambiguity, numerous clustering algorithms have +been proposed and used in the astronomical community. For +example, Knebe et al. (2011) compared the various properties +of dark matter (DM) halos found by 17 different halo-finding +algorithms run on the same cosmological N-body simulation. +Galaxy clustering algorithms have been used as essential +tools for identifying galaxy clusters or super-clusters, as well +as for investigating the large structure of the universe, includ- +ing the filament structures. The most commonly used galaxy +clustering algorithms in astronomy research are the friends- +of-friends (FoF; Davis et al. 1985) and the minimum span- +ning tree (MST; Borůvka 1926). These algorithms were in- +troduced in the 1980s and have been widely used as standard +galaxy clustering algorithms. In recent years, with the rapid +development of machine learning (ML) technology, clustering +algorithms such as DBSCAN (Density-based Spatial Cluster- +ing of Applications with Noise; Ester et al. 1996) have also +been applied to galaxy clustering. These clustering softwares, +© 0000 The Authors +arXiv:2301.03278v1 [astro-ph.IM] 9 Jan 2023 + +2 +Y. Ju et al. +including the ML based one, show comparable performance +in galaxy clustering and produce consistent clustering results. +However, the results of clustering do not always represent the +clusters that the human eye can find. There are cases where a +distribution clearly contains a cluster but it is not recognized +as such by clustering algorithms, and there are cases where +it is clearly divided into two clusters, visually, but appears as +a single lump to the software. +As such, we have explored the possibility of developing an +algorithm that creates galaxy clusters in a way that more +closely resembles how the human eye and brain identify pat- +terns. One approach we considered was to adapt jet-finding +software used in high-energy particle physics research, with a +particular focus on the MulGuisin (MGS) algorithm as a po- +tentially suitable software for galaxy clustering. MulGuisin +(ᄆ +ᅮ +ᆯᄀ +ᅱᄉ +ᅵ +ᆫ) is a Korean word for a ghost that lives in water +and is a figure that often appears in old Korean stories. The +MGS algorithm started with the idea that the ghosts hiding +in the water could be found in the order of height by simply +draining the water from the lake. +Initially, we copied the MGS software from the A Toroidal +LHC Apparatus (ATLAS) Jet-Finding library released in the +early 1990s and developed it into a 3D galaxy clustering al- +gorithm. We then made several sample galaxy distributions +to check the performance of MGS and compared the results +to those produced by other standard clustering algorithms +such as FoF, MST, and DBSCAN. As a result, it was found +that MGS had characteristics that other algorithms could not +show, and it was also found that the cluster results created by +MGS were most similar to the cluster results that the human +eye found. +Since the clusters created by MGS show different shapes +when compared with clusters formed by other algorithms, +and the number of clusters and the size distribution of clus- +ters are quite different from those of classical algorithms, us- +ing MGS makes a big difference in searching halos and super- +clusters, and can yield a different interpretation for the large +scale structures of the universe. Therefore, we anticipate that +this new algorithm will be used as a new methodology in +galaxy cluster research and furthermore used to create new +interpretations in cosmology studies. +The structure of this paper is as follows. In Section 2, we +introduce the MulGuisin clustering algorithm, as well as FoF, +MST, and DBSCAN as benchmark clustering algorithms for +comparison. In Section 3, we describe both controlled random +data and realistic galaxy distribution data that we will use +for the performance test. We apply the above four clustering +algorithms to the data and compare their performances in +Section 4, and we summarize our results in Section 5. +2 METHODS +A halo or galaxy cluster is a group of galaxies held together by +gravity. Finding such cluster structures in galaxy distribution +data is very important for astronomical research because it +provides a tool to study super-clusters, filaments, and even +bigger the large scale structure of the universe. +A cluster can be defined as a concentration of points or +cells in a localized volume. The task of cluster identification +has been extensively studied in the field of computational +science, and a wide range of clustering algorithms have been +developed for this purpose. Because different algorithms have +different strengths and weaknesses, it is important for re- +searchers to carefully select the algorithm that best suits their +specific research purpose. +In this section, we first introduce our MulGuisin clustering +algorithm and introduce two clustering tools that are widely +used in the field of astronomy and a newly developed clus- +tering program through machine learning. +2.1 MulGuisin galaxy clustering algorithm +The MulGuisin (MGS) clustering algorithm was first intro- +duced as a jet finder for the Large Hadron Collider (LHC) +physics in the ATLAS Collaboration (Bosman et al. 1998). +The algorithm is neither a variant of the conventional cone +algorithm nor a variant of the kT algorithm that is used in +various collider experiments as the standard tools for finding +jets. Although it has shown some improvements in jet recon- +struction performance, such as optimized jet orientation and +jet energy resolution, but has not been used as a standard +jet-finding tool for LHC experiments. +Fig. 1 shows how the MGS algorithm works. The MGS +algorithm first finds the most massive point from the input +data and names it a cluster seed. Then it finds the second +massive point and decides whether the point should belong +to the first cluster or stand alone as the seed of a new cluster. +This decision is made by checking how close the test point +is to any neighboring clusters, for which we introduce a pa- +rameter called linking length (ℓMST). That is, if the distance +between the test point and the closest point in the cluster +is less than this parameter, the test point is attached to the +cluster, otherwise, it becomes the seed of a new cluster. The +algorithm then finds the next massive point and repeats the +above process until there are no more points left to test. At +this stage, all points are converted into clusters. Of course, +some points do not belong to any cluster and remain. +Fig. 2 is an illustration to explain how the MGS algorithm +creates galaxy clusters. In the figure, the points are sorted in +order according to their mass and number. And according to +this order, they become new cluster seeds or stick to exist- +ing clusters. After going through the process, a cluster forms +a tree-like structure that is sequentially connected accord- +ing to the order of mass. The points in a cluster then form +branches and nodes, and from the characteristic structure of +such tree shape, one may able to study the topology of the +galaxy cluster. +2.2 Benchmark Algorithms +In order to compare the performance of the MGS algorithm +with those of other standard clustering algorithms, we select +three benchmark algorithms, mostly based on their popular- +ity in the astronomical community, mathematical clarity, and +versatility. They are the friends-of-friends (FoF), minimum +spanning tree (MST), and the density-based spatial cluster- +ing of applications with noise (DBSCAN). Here we briefly +introduce each package and describe how they make clus- +ters.1 +1 Note that running our MGS algorithm from scratch may take +a longer time than the above benchmark algorithms, especially +MNRAS 000, 1–14 (0000) + +MulGuisin Clustering Algorithm +3 +Figure 1. Schematic flow chart to describe how MulGuisin (MGS) algorithm works +Figure 2. Diagram showing how the MulGuisin (MGS) algorithm works to identify clusters. Each gray circle represents a galaxy, and the +size of the circle denotes its local density, with the number specifying the galaxies ranking in descending order of density. In this specific +example, 21 galaxies are grouped into 3 clusters and 2 isolated galaxies. +MNRAS 000, 1–14 (0000) + +galaxies +No +Yes +clusters +No +Yes1st cluster +8 +5 +10 +numberofchildrene3 +12 +1st seed +19 +linking distance +isolatedgalaxy +3rd cluster +Oth generation +3rd seed +2nd cluster +2ndseed +3 +1st generation +13 +9 +16 +14 +2nd generation +b +15 +4th generation +21 +isolatedgalaxy4 +Y. Ju et al. +2.2.1 Friends-of-Friends (FoF) +The friends-of-friends (FoF) algorithm is a commonly used +technique for identifying clusters in astrophysical data +(Huchra & Geller 1982; Tago et al. 2008; Duarte & Mamon +2014; Tempel et al. 2016). This algorithm has a single free +parameter, the linking length (ℓFoF), which determines the +distance threshold for linking two data points. Points that are +within this distance of each other are considered to be con- +nected, and all connected points are grouped together into a +single cluster. +One limitation of the FoF algorithm is that it can be diffi- +cult to choose an appropriate linking length. Different values +of this parameter can result in clusters of different shapes +or numbers, making it challenging to determine the optimal +value (Tago et al. 2008).2 In this study, we use the Halotools +implementation of the FoF algorithm (Hearin et al. 2017) to +identify clusters in our datasets by applying various ℓFoF. +Unless otherwise noted, we assume all FoF groups containing +two or more members as clusters. +2.2.2 Minimum Spanning Tree (MST) +Galaxy data can be represented as a graph, with each galaxy +represented as a node and the distance between two galaxies +represented as an edge. The minimum spanning tree (MST) +algorithm is a method for constructing a unique network from +this data by connecting all nodes with minimum edges. Unlike +other clustering algorithms, the MST does not require the use +of a free parameter such as a linking length to construct the +entire network. However, the MST connects all nodes and +may not produce clusters with shapes that accurately reflect +those of the original clusters. +Nevertheless, MST has been used in cosmology to study +the large-scale structure of the universe (Barrow et al. 1985; +Krzewina & Saslaw 1996; Naidoo et al. 2020). In this study, +we use the MiSTree package (Naidoo 2019) to construct MSTs +from our galaxy data. Then, we find clusters from the single +MST tree by cutting nodes longer than the linking length +(ℓMST). Similar to the FoF case, we apply various values of +ℓMST and assume all tree segments containing two or more +members as clusters. +2.2.3 Density-based Spatial Clustering of Applications with +Noise (DBSCAN) +The use of machine learning (ML) techniques is widespread +in astronomy, as they enable the identification of patterns in +data using algorithms. ML algorithms can be classified based +on the type of data they are applied to, and one type, called +when the number of data points is large. We found that most of +the MGS calculation time, for a large number of data points, is +taken in constructing the Voronoi tesselation and calculating the +local density for each point. If we separate MGS into a density +calculation and a tree building part, we found that the tree building +takes a similar time to the benchmark algorithms. +2 Note that the appropriate choice linking length for identifying +DM halos from the DM particles in the N-body simulations is well +known (ℓFoF ≃ 0.2⟨dparticle⟩) (More et al. 2011). However, the +optimal choice of linking length in general clustering problems is +not well known. +unsupervised ML, is used with unlabeled data. Clustering al- +gorithms, a subcategory of unsupervised ML algorithms, are +used to group together data points with similar properties. +One popular clustering algorithm is DBSCAN (density-based +spatial clustering of applications with noise), which has been +applied in a variety of contexts (Ester et al. 1996; Sander +et al. 2017). +DBSCAN is a density-based clustering algorithm that +groups together data points based on their local density. In +this algorithm, each cluster is identified by defining its core, +which consists of high-density points within a certain dis- +tance. The definition of core requires two free parameters, +min_samples and eps, which determine the minimum num- +ber of neighbors a point must have within a given radius in +order to be considered as the core. Then, other points that are +directly reachable from some core points within eps are also +considered part of the cluster, while other points are labeled +as noise. +In this study, we use the scikit-learn package (Pedregosa +et al. 2011) to implement the DBSCAN algorithm and iden- +tify clusters in our data by applying various eps (or, the “link- +ing length” in DBSCAN (ℓDBSCAN)). Unless otherwise noted, +we assume min_samples = 3. +2.3 A Simple 2D Toy Model Test +To see how the shape of the clusters generated by the MGS +algorithm differs from the results of other clustering algo- +rithms, we created simple simulation data and compared the +results. We first assume that there are 5 clusters in 2D space, +and consider the case where each cluster contains 50 galaxies +equally. The width of the galaxy distribution of each cluster +was fixed to 10. The coordinates of the two-dimensional space +span from 0 to 100 on both the X- and Y-axes, and the posi- +tion of each cluster is set to have three different distributions, +from far away from each other to all close together, as shown +in Fig. 3 column (a) in rows (1), (2) and (3). +As shown in Fig. 3, both the MGS and MST algorithms +correctly find 5 clusters when the distances among the clus- +ters are sufficiently far apart. However, when the clusters get +closer together, MST can’t differentiate between the clus- +ters and starts recognizing them as one big cluster. Even +for the cases where clusters are attached to each other as +shown in Fig. 3 (3), MGS still recognizes four among five +true clusters like the human eyes can distinguish each clus- +ter, whereas MST recognizes 4 adjacent clusters as one huge +cluster. These differences can create serious differences in re- +sults when studying the number and mass distributions of +clusters. +Note that, although we leave its details as future works, +the tree structures made by the MGS algorithm have a non- +negligible number of long nodes connecting two distant points +in the cluster, while the MST algorithm connects only rea- +sonably nearby points. This is because the MGS algorithm +connects data points based on their local density, not only the +distance between the points. Therefore, if two highly dense +points are within the linking length, then they would be con- +nected in the MGS but may not be in the MST. +In the next section, we will compare the performance of +MGS and other algorithms with more realistic 3D data. +MNRAS 000, 1–14 (0000) + +MulGuisin Clustering Algorithm +5 +0 +20 +40 +60 +80 +100 +(1) +LL = 10 +0 +20 +40 +60 +80 +100 +(2) +0 +20 +40 +60 +80 +100 +(a) +0 +20 +40 +60 +80 +100 +(3) +0 +20 +40 +60 +80 +100 +(b) +0 +20 +40 +60 +80 +100 +(c) +Figure 3. A simple 2D toy model test of the MGS algorithm by comparing it with the MST algorithm. (a) Input distributions of 5 +clusters with different degrees of separation from each other ((1)–(3)). Background color denotes the galaxy number distribution we used +for generating the galaxies. (b) Clusters found by MGS and their tree structures. (c) Clusters found by MST and their tree structures. +3 DATA +Our final goal is to apply the MGS algorithm described in +Section 2 to the galaxy clusters or other large-scale struc- +tures of the universe. However, since some inconsistencies ex- +ist between various clustering algorithms for finding clusters +or other large-scale structures (e.g., see Knebe et al. 2011, +and references therein), we cannot compare the MGS clus- +ters found in the realistic data with their “truth”. +Therefore, we apply two types of data sets in this section +to compare the performance between MGS and other bench- +mark algorithms. The first sets, called the “controlled ran- +dom data” (D1–D3), are those that we design all properties +of clusters, including their positions and member galaxy dis- +tributions. Since we already know the true information of +each cluster, we can test which algorithms predict the true +clusters better in which conditions. The next sets, called the +“realistic data” (D4), are the observational and simulation +data sets of galaxies around z ≃ 0, and we focus on com- +paring the properties of predicted clusters in each algorithm. +Table 1 summarizes the data sets we use in this work. +3.1 Controlled Random Data (D1–D3) +3.1.1 Different Spatial Dispersion (D1) +We use controlled, simulated data to evaluate the perfor- +mance of the MGS algorithm in comparison to other clus- +tering algorithms. These data are generated randomly and +allow us to control the shape and distribution of clusters to +test the algorithms under different conditions. The first set +of data consists of 100 galaxies per cluster and 50 clusters +within a 3-dimensional cubic volume of space with a side +length 200 h−1Mpc. The cluster center positions are chosen +randomly, and the galaxies in each cluster are distributed ac- +cording to a Gaussian distribution with a variable standard +deviation (σ) that controls the spatial dispersion. The D1-LD +data has a low spatial dispersion (σ = 1 h−1Mpc), leading to +well-separated clusters, while the D1-HD data has a higher +MNRAS 000, 1–14 (0000) + +6 +Y. Ju et al. +Data set +Description +D1-LD +50 randomly positioned clusters, each of which contains 100 galaxies randomly spread by the 3D Gaussian +distribution with standard deviation σ = 1 h−1Mpc. The total number of galaxies is 5,000. +D1-HD +Same as D1-LD, but with the greater standard deviation σ = 10 h−1Mpc. +D2-NA +Same as D1-HD, but the number of galaxies in each cluster follows an exponential random distribution. +The total number of galaxies is 7,041. +D2-LA +Same as D2-NA, but adding uniformly randomly distributed noisy galaxies to the entire box to increase +the total galaxy number density 1.5 times of D2-NA.The total number of galaxies is 12,041. +D2-HA +Same as D2-LA, but adding more noisy galaxies so that the total galaxy number is twice D2-NA. The +total number of galaxies is 17,041. +D3-HOD +500 randomly positioned clusters with a mass distribution similar to the Press-Schechter mass function. +The number of galaxies for each cluster follows HODa for massive halos (M ⩾ 1013 h−1M⊙). The +galaxies are spread by the NFW profile with the concentration parameter to 10. The total number of +galaxies is 50,257. +D4-SDSS +Volume-limited sample of the KIAS-VAGCb with absolute r-band magnitude Mr − 5 log h < −20. +D4-HR4 +Four lightcone data of mock galaxy catalogs from the Horizon Run 4 simulationc with a similar condition +to D4-SDSS. +a Kravtsov et al. (2004). b Choi et al. (2010a). c Kim et al. (2015); Hong et al. (2016). +Table 1. Name and description of galaxy data sets that we use in this analysis. The box size of all controlled data (D1–D3) is +(200 h−1Mpc)3. +X +100 +75 +50 +25 +0 +25 +50 +75 +100 +Y +100 +75 +50 +25 +0 +25 +50 +75 +100 +Z +0 +25 +50 +75 +100 +125 +150 +175 +200 +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +D1-LD +D1-HD +X +100 +75 +50 +25 +0 +25 +50 +75 +100 +Y +100 +75 +50 +25 +0 +25 +50 +75 +100 +Z +0 +25 +50 +75 +100 +125 +150 +175 +200 +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +D1-LD +D1-HD +X +0 +50 +100 +150 +200 +Y +0 +50 +100 +150 +200 +Z +0 +50 +100 +150 +200 +D3-HOD +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +D2-NA +D2-LA +D2-HA +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +D2-NA +D2-LA +D2-HA +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +D2-NA +D2-LA +D2-HA +Figure 4. Three-dimensional galaxy distributions of controlled data sets used in this paper. Top: D1-LD(left), D1-HD(middle), and D3- +HOD(right). Bottom: D2-NA(left), D2-LA(middle), and D2-HA(right), with noisy additional galaxies shown as yellow dots. See Table 1 +for details. +MNRAS 000, 1–14 (0000) + +MulGuisin Clustering Algorithm +7 +spatial dispersion (σ = 10 h−1Mpc), resulting in clusters that +are closer together. Upper left and middle panels of Fig. 4 +show the distribution of galaxies in the D1-LD and D1-HD +data sets. +3.1.2 Additional Noisy Galaxies (D2) +We generate additional controlled data sets that are similar +to D1 but with slightly different characteristics. We again +place 50 cluster centers at the same positions as D1, but this +time we use an exponential distribution to generate a variable +number of galaxies for each cluster: +P(Ngal) = +� +� +� +1 +∆N exp +� +−Ngal − N0 +∆N +� +if Ngal > N0 +0 +otherwise +. +(1) +Here, we set N0 and ∆N as 50 and 100, respectively, so +that the minimum number of galaxies per cluster and the +total number of galaxies roughly match with D1. The galax- +ies are spatially distributed according to a Gaussian function +centred on the cluster’s centre with a standard deviation of +σ = 10 h−1Mpc, as was done for D1-HD, and we call this +new controlled data D2-NA. The lower left panel of Fig. 4 +shows the distribution of galaxies in the D2-NA data. The +total number of galaxies in this data set is 7,041. +In addition to D2-NA, we introduce two more data sets +that are created by adding unclustered galaxies, which are +sampled uniformly in the entire box. We add these ‘noisy’ +galaxies so as to test how the algorithms are affected by the +background density. The lower middle panel of Fig. 4 shows +the D2-LA data, where we add 5,000 galaxies (yellow dots) +to increase the galaxy number density by 1.5 times compared +to D2-NA. On the other hand, the lower right panel shows +the D2-HA data, where we add 10,000 galaxies to make the +total galaxy number density twice that of D2-NA. +3.1.3 HOD-based Mock Galaxies (D3-HOD) +We generate a third set of controlled, simulated data to cre- +ate a more complex environment for testing the performance +of the MGS algorithm. We use an analytic formula to model +the distribution of galaxies in this data set. First, we create +500 cluster center positions by sampling uniform random dis- +tribution within a (200 h−1Mpc)3 box. Then, we obtain the +normalized version of Press-Schechter halo mass function at +z = 0(Press & Schechter 1974), with a concordance ΛCDM +cosmology to the Planck 2015 data (Planck Collaboration +et al. 2016), for massive halos Mhalo > 1013 h−1M⊙ using the +Colossus package (Diemer 2018). We then obtain masses for +each of the 500 clusters by randomly sampling for the mass +function.3 +We then use this information to generate a distribution +of the number of galaxies using a halo occupation distribu- +tion (HOD) model. The mean halo occupation is typically as- +sumed to follow a power law at massivehalo masses (Berlind +3 Note that neither the positions nor the mass distribution of clus- +ters in D3-HOD follows the estimation from the standard cosmol- +ogy. However, here we focus only on providing complex environ- +ments, and therefore, such differences do not affect our motivation. +See Section 3.2 for realistic data sets instead. +& Weinberg 2002; Kravtsov et al. 2004): +Navg(Mcluster) = +� +� +� +�Mcluster +M1 +�α +if Mhalo > Mmin +0 +otherwise +, +(2) +where α, Mmin, and M1 correspond to the power-law in- +dex, cutoff halo mass where halo cannot contain galaxies, +and the mass scale containing a single galaxy at the given +condition of galaxy sample. Here, we use α = 0.87 and +Mmin = 1013 h−1M⊙ by following Kravtsov et al. (2004). We +set M1 = 1011 h−1M⊙ so that the minimum number of galax- +ies for each cluster is set as 50. Also, for simplicity, we calcu- +late the actual number of galaxies at each cluster by applying +the ceiling to Navg. +Next, we use the Colossus package to create an Navarro- +Frenk-White (NFW) profile (Navarro et al. 1996) +ρ(x) = Mcluster +4πR3 +vir +�� +ln(1 + cs) − +cs +1 + cs +� +x(x + c−1 +s )2�−1 +, +(3) +where x ≡ r/Rvir. The concentration parameter for the NFW +profile cs is fixed as 10, and the virial radii Rvir is determined +by the cluster mass accordingly. We then randomly distribute +the galaxies according to this profile, and the resulting data +set consists of 50,257 galaxies. The upper right panel of Fig. 4 +shows the distribution of galaxies in D3-HOD. +3.2 Realistic Data: SDSS & Horizon Run 4 (D4) +In the previous subsection, we described a set of controlled +data catalogues for which we can carefully control the prop- +erties of the clusters. Such data are useful for testing the +performance of MGS over other benchmark algorithms by +comparing the properties of identified clusters with the in- +put truth. However, the true distribution of galaxies in the +universe differs from these controlled random data in the fol- +lowing ways. First, unlike those in the controlled random data +with low noise levels, the boundaries of clusters in the uni- +verse are often not clearly defined (e.g., Serra & Diaferio 2013; +Gifford et al. 2013). Also, the spatial distribution of galax- +ies in each cluster may not follow spherical symmetry (e.g., +Limousin et al. 2013, for a good review). Furthermore, the +redshift-space distortion elongates spherical clusters in real +space, which may require that we separate linking lengths +between the radial and tangential directions (Farrens et al. +2011; Tempel et al. 2016). +Therefore, it is necessary to adopt a realistic galaxy dis- +tribution for a fair performance test of the MGS algorithm. +However, unlike for the case of the controlled random data +where we know the answer, we may only study the dif- +ference between the cluster properties from the MGS and +other benchmark algorithms. Here we use observational data +and four corresponding sets of mock simulation data — the +volume-limited KIAS-Value Added Galaxy Catalog (KIAS- +VAGC) of the Sloan Digital Sky Survey (SDSS) Main Galaxy +Sample with r-band absolute magnitude Mr − 5 log h < −20 +(Choi et al. 2010a) and the lightcone mock galaxy samples +from the Horizon Run 4 simulation (Kim et al. 2015; Hong +et al. 2016). +MNRAS 000, 1–14 (0000) + +8 +Y. Ju et al. +3.2.1 Volume-limited KIAS-VAGC (D4-SDSS) +The KIAS Value-Added Galaxy Catalog (KIAS-VAGC; Choi +et al. 2010a) is an upgraded version of the New York Uni- +versity Value-Added Galaxy Catalog (NYU-VAGC; Blanton +et al. 2005), which is part of the Sloan Digital Sky Survey +(SDSS) Data Release 7 (Abazajian et al. 2009), by adding +some missing redshifts to improve spectroscopic complete- +ness. This catalog has been widely used in numerous studies, +including cosmic voids statistics (Pan et al. 2012; Hoyle et al. +2012), largest structures of universe (Park et al. 2012), frac- +tion of barred galaxies (Lee et al. 2012a), and the properties +of active galactic nuclei (AGN; Lee et al. 2012b; Hwang et al. +2012; Bae & Woo 2014). +Most of the KIAS-VAGC galaxies were observed with the +apparent r-band magnitude limit r = 17.6. It means that, in +terms of absolute magnitude, the catalog contains less bright +galaxies at lower redshifts, while only very bright galaxies +could be seen at higher redshifts. Therefore, for a fair com- +parison between galaxies over a wide redshift range, we ap- +ply a “volume-limited” selection by selecting galaxies brighter +than a certain absolute r-band magnitude (Choi et al. 2010b). +Here, we use Mr − 5 log h < −20. By combining with the +given apparent r-band magnitude limit, such absolute mag- +nitude cutoff naturally provides the upper redshift bound of +our volume-limited sample (z < 0.107; left panel of Fig. 5). +We also apply the lower redshift bound z > 0.02, by consider- +ing the incompleteness of the galaxy sample below the given +redshift. +In addition to the volume-limited selection in the redshift- +magnitude plane, we also apply a sky selection for simplifica- +tion. Specifically, we select galaxies within the SDSS Survey +coordinate −33.5◦ < η < 36.5◦ and −48◦ < λ < 51◦, in or- +der to maximize the sky area with a simple geometry, and +to avoid issues arising from a complicated boundary (right +panel of Fig. 5). +3.2.2 Horizon Run 4 (D4-HR4) +The Horizon Run 4 simulation (HR4; Kim et al. 2015) is an +extremely large cosmological N-body simulation that uses +6, 3003 DM particles within a periodic cube with a comoving +volume V = (3.15 h−1cGpc)3. It assumes a vanilla ΛCDM +cosmological model in concordance with the Wilkinson Mi- +crowave Anisotropy Probe (WMAP) 5th-year result (Dunkley +et al. 2009). Among 2,001 timesteps between z = 100 to 0, +75 coarse timesteps with mean time difference ∆t = 0.18 Gyr +are chosen between z = 12 to 0 to build a merging tree of +FoF halos. The FoF linking length is 0.2 times the particle +mean separation, and we identify halos only whose mass is +greater than M min +halo = 2.7 × 1011 h−1M⊙. +The mock galaxies are then produced by so-called the +most bound halo particle (MBP)-galaxy abundance match- +ing method (Hong et al. 2016). We find MBPs for all halos in +the merging tree and adopt their positions and peculiar ve- +locities as those of corresponding mock galaxies. The “mass” +of mock galaxies, which is used as a proxy of stellar mass or +luminosity, is defined as the mass of their hosting halos. For +satellite halos, we identify their MBPs at the timestep just +before the infall event and trace them until they are totally +absorbed toward their central halo by tidal disruption. For +estimating the tidal disruption timescale tmerge), we adopt a +modified model of Jiang et al. (2008), +tmerge +tdyn += (0.94ϵ0.60 + 0.60)/0.86 +ln[1 + (Mhost/Msat)] +�Mhost +Msat +�α +, +(4) +where ϵ, Mhost, Msat, tdyn are the circularity of the satellite’s +orbit, the mass of central and satellite halos, and the orbital +period of virialized objects, respectively. We adopt α = 1.5 +for a better match of the galaxy two-point correlation func- +tion (2pCF) at scales less than 1 h−1Mpc at a given spatial +resolution of the HR4 (Zehavi et al. 2011; Park et al. 2019). +Then the mass of survived satellite galaxies is defined as the +mass of their hosting halos just before the infall. +After producing snapshot mock galaxy catalogs for coarse +timesteps, we then produce lightcone mock galaxy catalogs +up to z = 1.5. The all-sky lightcone DM particle data of the +HR4 were created during the simulation by stacking the co- +moving shells at the corresponding redshifts. Then, we com- +pare the IDs of the galaxy MBPs at each coarse timestep +snapshots and those of DM particles at the lightcone data +with the coarse comoving shells. If the MBP ID of a given +mock galaxy matches that of a particle in the lightcone data, +we assign a galaxy in the lightcone data. Here, we adopt +the position and peculiar velocity from the particle at the +lightcone data, while the galaxy “mass” comes from the mock +galaxy at the nearest snapshot. +After creating the all-sky lightcone mock galaxy catalog, +we cut it in a similar way to the volume-limited KIAS-VAGC +sample. First, we apply the redshift space distortion (RSD) +for each mock galaxy for a fair comparison with observation, +by using real-space positions and peculiar velocities. Then, +we apply the same redshift range 0.02 < z < 0.107 and set +the lower bound of galaxy “mass,” so that the galaxy number +density of the HR4 lightcone data is identical to that of KIAS- +VAGC. After that, we create four non-overlapping subsets +from it with the same angular geometry as our SDSS Survey +coordinate selection. +During the analysis, we found that the fiber collision in the +fiber-fed spectroscopic observations affects various clustering +statistics (Zehavi et al. 2002; Guo et al. 2012; Reid et al. 2014; +Tonegawa et al. 2020). Therefore, for a fair comparison, our +HR4 mock galaxy catalogs also need to follow the same fiber +collision condition as the KIAS-VAGC. To do so, we select +pairs of mock galaxies whose angular distance is less than +55 arcseconds and keep only one from each pair by random +selection. Because SDSS observations were partially overlap- +ping, some close-pairs have both redshifts. In order to reflect +this, we only fiber-collide 60% of the close pairs. +4 RESULTS +We test MGS and the other algorithms using the 3 controlled +data and observation data. We run 4 algorithms with various +linking-length and find out the number of clusters. The same +process is repeated by changing linking-length and we check +the tendency of the number of clusters. +We use the three controlled data sets and observation data +to evaluate the performance of the MGS algorithm and com- +pare it to other clustering algorithms. We run each of the +four algorithms with different values of the linking length +and count the number of clusters identified by each algo- +rithm. We repeat this process for a range of linking lengths +MNRAS 000, 1–14 (0000) + +MulGuisin Clustering Algorithm +9 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Redshift +23 +22 +21 +20 +19 +18 +17 +r +5logh +150 +100 +50 +0 +50 +100 +150 + [degree] +60 +40 +20 +0 +20 +40 +60 + [degree] +Figure 5. Selection of the volume-limited sample of the KIAS Value-Added Galaxy Catalog (KIAS-VAGS) used in this study (red boxes). +Left: Volume-limited selection in the redshift vs. absolute r-band magnitude plane with Mr − 5 log h < −20. Right: Sky selection in SDSS +Survey coordinates (η, λ). +30 +35 +40 +45 +50 +Number of Clusters +D1-LD +MGS +MST +FoF +DBSCAN +2 +4 +6 +8 +10 +Linking-length +0 +10 +20 +30 +40 +50 +Number of Clusters +D1-HD +Figure 6. The number of clusters as a function of linking length +for D1-LD (top panel) and D1-HD (bottom). Each of the 4 clus- +tering algorithms are indicated using different colors and symbols. +Note that MST and DBSCAN show considerable overlap. The Hor- +izontal dash shows the original number of clusters, which is 50. +and analyze the trends in the number of clusters identified +by each algorithm. This allows us to assess the sensitivity of +the algorithms to the choice of linking length and to compare +their performance in identifying clusters in the different data +sets. +4.1 Results with Controlled Data +Fig. 6 shows the number of clusters identified by each algo- +rithm as a function of the linking length for the controlled +data set 1. The top panel shows the results for the D1- +LD data, which consists of well-separated clusters. The al- +gorithms are expected to identify 50 clusters in this data set. +All four algorithms perform well in identifying the clusters, +but the MGS algorithm stands out for its ability to accu- +rately identify the correct number of clusters. In particular, +for large linking lengths, the FoF and DBSCAN algorithms +identify fewer than 50 clusters, because they connect neigh- +boring clusters and merge them into a single cluster. +The bottom panel of Fig. 6 shows the results for the D1- +HD data, which has a higher level of spatial dispersion and +some clusters that are close to each other. For small link- +ing lengths, the algorithms identify fewer than 50 clusters +because the linking length is not sufficient to connect the +galaxies in these clusters. As the linking length increases, the +behavior of the algorithms becomes more distinct. The MGS +algorithm continues to accurately identify the correct number +of clusters, while the other algorithms identify fewer clusters +due to the merging of originally separate clusters. The MGS +algorithm is able to track the structure of the clusters and +identify their boundaries, leading to more accurate results in +this type of data. +Fig. 7 shows the results for controlled data set 2, which +includes the D2-NA data with no additional galaxies and the +D2-LA and D2-HA data with additional galaxies. The top +panel shows the number of clusters identified by each algo- +rithm for the D2-NA data. When this data was generated, +the minimum number of galaxies per cluster was set to 50. +In the region of small linking lengths, all algorithms iden- +tify fewer than 50 clusters because the linking length is too +small to connect the galaxies in the clusters. As a result, +the clusters identified by the algorithms have fewer than 50 +member galaxies, and are therefore not considered as true +clusters. For larger linking lengths, particularly those larger +than 5, the difference between the MGS algorithm and the +other algorithms becomes more pronounced. The MGS al- +gorithm continues to accurately identify the correct number +of clusters, while the other algorithms identify fewer clusters +due to the merging of originally separate clusters. +The behavior of the algorithms with additional galaxies is +even more distinct. The middle panel of Fig. 7 shows the re- +sults for the D2-LA data, where the other three algorithms +identify only a single cluster for very large linking lengths. +As the linking length increases, the algorithms merge several +clusters into a single giant cluster, resulting in a significantly +lower number of clusters than the original data. This rapid +MNRAS 000, 1–14 (0000) + +10 +Y. Ju et al. +10 +20 +30 +40 +50 +60 +Number of Clusters +D2-NA +d += 10.43 +Nmin = 50 +MGS +MST +FoF +DBSCAN +10 +20 +30 +40 +50 +60 +Number of Clusters +D2-LA +d += 8.73 +Nmin = 56 +2 +4 +6 +8 +10 +12 +14 +Linking-length +0 +10 +20 +30 +40 +50 +60 +Number of Clusters +D2-HA +d += 7.77 +Nmin = 59 +Figure 7. Same as Fig. 6, but with D2-NA(top), D2-LA(middle), +and D2-HA(bottom). The vertical dashed line is mean-separation +of data (⟨d⟩). Since each data set has a different overall number +density, we assign different minimum number of member galaxies +(Nmin) to define clusters. +increment of a single giant cluster is called “percolation,” and +it is known to occur at linking length similar to the mean- +separation (ℓ ≃ ⟨d⟩) for the ideal random Poisson graph (Dall +& Christensen 2002). Fig. 7 clearly shows that such percola- +tion occurs at ℓ ≃ ⟨d⟩ for all three benchmark algorithms. +Note that, however, the percolation occurs at the low- +est linking length in FoF, while both MST and DBSCAN +share a similar value of linking length at percolation. This +is because FoF does not have an additional consideration +for limiting the cluster boundary that exists in the other +two algorithms (minimize the number of edges in MST, and +core definition in DBSCAN). Fig. 8 shows the 3D distribu- +tions of clustering results from various algorithms at linking +length ℓ = 11 h−1Mpc, which is longer than the mean sep- +aration ⟨d⟩ = 8.73 h−1Mpc. As expected, three benchmark +algorithms show percolation (blue color), while our MGS al- +gorithm successfully reconstructs most of the true clusters. +Note that only one giant cluster is found in the FoF algo- +rithm, while both MST and DBSCAN have two additional +small clusters (green and orange colors). +The behavior of the MGS algorithm for the D2-HA data +is slightly different. In the region of small linking lengths, +the MGS algorithm accurately identifies the correct number +of clusters. However, for larger linking lengths, particularly +ℓ > 13 h−1Mpc, the MGS algorithm identifies additional clus- +ters that were not present in the original data. These “fake” +clusters are not true clusters and are not representative of the +underlying structure of the data. This behavior highlights the +ability of the MGS algorithm to identify clusters in data with +a complex distribution of galaxies but also underscores the +importance of choosing an appropriate linking length to avoid +identifying false clusters. +Fig. 9 shows the number of clusters for controlled 3 data +with a more complex environment than D1–D2. At ℓ ≳ +⟨d⟩/2 ≈ 3 h−1Mpc, the number of clusters using FoF and +DBSCAN decreases as the linking length increases, resulting +in the percolation at ℓ ≳ ⟨d⟩.The MST shows a flat curve +when the linking length is larger than ∼ 8 h−1Mpc. This is +because MST connects all galaxies with minimal edge first, +and then we cut off the links with linking length. Therefore, +if there were no links longer than 8 h−1Mpc in the original +tree, then cutting the links with any longer linking length +than 8 h−1Mpc would not change the result. So, the number +of clusters using MST shows a constant value. +In contrast, the number of clusters identified by the MGS +algorithm slowly decreases as the linking length increases. +This is because the clusters in this data set are close to each +other and are easily merged by the algorithm for large linking +lengths. However, the MGS algorithm is able to identify clus- +ters based on density, which allows it to retain the structure +of the clusters even for large linking lengths. This is the main +advantage of the MGS algorithm compared to the other three +algorithms, which are not able to accurately identify clusters +in complex data sets. +4.2 Results with Observational and Cosmological +Simulation Data +Fig. 10 shows the results of the four algorithms applied to +both KIAS-VAGC observational data and four sets of HR4 +lightcone data. We track the number of detected clusters +changing with both linking lengths and with the minimum +number of member galaxies from 2 to 5. +For all four clustering algorithms, the HR4 simulation re- +sults match well with the observations within cosmic vari- +ance, especially for n ⩾ 5 at ℓ ≳ ⟨dparticle⟩ = 0.5 h−1Mpc. +On the other hand, HR4 tends to underestimate the num- +ber of clusters for a smaller minimum number of member +galaxies and/or smaller linking length ℓ ≲ 0.5 h−1Mpc. This +may mean that, despite the agreement with the observation in +terms of 2pCF below 1 h−1Mpc-scale, some disagreements ex- +ist between HR4 and observation in terms of the higher-order +statistics in smaller scales than the particle mean separation +scale. +One notable feature of the MGS algorithm is that it does +not create a single giant cluster for large linking lengths. In- +stead, the algorithm identifies a number of smaller clusters, +even for large linking lengths. This is in contrast to the other +three algorithms, which all create a single giant cluster for +large linking lengths. This difference highlights the ability of +the MGS algorithm to accurately identify clusters in data +with a complex distribution of galaxies. +Fig. 11 shows the number of member galaxies for the 1st +MNRAS 000, 1–14 (0000) + +MulGuisin Clustering Algorithm +11 +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +X +0 +50 +100 +150 +200 +Y +0 +50 +100 +150 +200 +Z +0 +50 +100 +150 +200 +X +100 +50 +0 +50 +100 +Y +100 +50 +0 +50 +100 +Z +0 +50 +100 +150 +200 +MGS +MST +FoF +DBSCAN +Figure 8. 3D distribution of clustering results from MGS and other benchmark algorithms in D2-LA with linking length ℓ = 11 h−1Mpc, +which is longer than the mean-separation ⟨d⟩ = 8.73 h−1Mpc. Color indicates the cluster membership. MGS finds 49 clusters among 50 +true clusters, while other algorithms connect most of galaxies and finally make a giant cluster (blue color). +to 4th largest clusters identified by each algorithm in D4- +SDSS and D4-HR4. Similar to Fig. 10, both results from the +simulation and observation data match well with each other. +The top left panel of the figure shows the shape of the largest +cluster for each algorithm. As the linking length increases +over certain value, the largest cluster identified by the FoF, +MST, and DBSCAN algorithms contains all of the galaxies in +the data, while the MGS algorithm identifies a cluster with +only a portion of the galaxies. This indicates that the MGS +algorithm is able to identify multiple clusters even for large +linking lengths, while the other algorithms merge all of the +galaxies into a single giant cluster. +The main difference between the MGS algorithm and the +other three algorithms becomes particularly clear when ex- +amining the number of member galaxies in the 2nd to 4th +largest clusters (upper right and bottom panels of Fig. 11). As +the linking length increases, the number of member galaxies +in these clusters identified by the FoF, MST, and DBSCAN +algorithms decreases to zero. This is because the first largest +cluster identified by these algorithms took all the galaxies +in the data, leaving no galaxies to be considered for further +clustering. In contrast, the MGS algorithm is able to identify +multiple clusters even with large linking lengths as the largest +cluster does not monopolize all galaxies. This demonstrates +the ability of the MGS algorithm to accurately identify clus- +ters in data with a complex distribution of galaxies. +Fig. 12 shows the 30 largest clusters found by the MGS +algorithm in the D4-SDSS data with a linking length ℓMGS = +10 h−1Mpc. While such a large choice of linking length makes +a single giant cluster in all other three benchmark algorithms +(see Fig. 11), none of the 30 clusters suffer percolation. All +30 largest clusters are well-separated and have some even dis- +tribution of galaxies in the XY-plane (that is, the tangential +plane). On the other hand, most of the clusters have some- +what elongated features in the line-of-sight direction, which +clearly shows the Finger-of-God effect due to the RSD. There- +fore, although this needs further inspection, we consider that +the 30 largest clusters found by the MGS algorithm could be +MNRAS 000, 1–14 (0000) + +12 +Y. Ju et al. +0 +2 +4 +6 +8 +10 +12 +14 +Linking-Length +0 +100 +200 +300 +400 +500 +Number of cluster +D3-HOD +MGS +MST +FoF +DBSCAN +Figure 9. Same as Figs. 6–7, but with D3-HOD. +100 +101 +102 +103 +104 +Number of clusters +MGS +n +2 +n +3 +n +4 +n +5 +MST +100 +101 +Linking length(h +1Mpc) +100 +101 +102 +103 +104 +Number of clusters +FoF +100 +101 +Linking length(h +1Mpc) +DBSCAN +Figure 10. Same as Figs. 6, 7 & 9, but with D4-SDSS (thick +lines) and D4-HR4. For D4-HR4, the average values and the ranges +between minimums and the maximums of 4 data samples are drawn +as thin lines and error bars. Results from each clustering algorithm +are shown on different panels, while the color indicates the different +choices of the minimum number of member galaxies to identify +clusters. +the actual large structures similar to galaxy (super)clusters +in real space. +5 CONCLUSIONS +The MulGuisin (MGS) algorithm is a powerful technique for +identifying clusters in data from astrophysical simulations +and observations. It consistently produces results closer to +those inferred from human visual inspection. In comparison +to other clustering algorithms, such as the friends-of-friends +(FoF) algorithm, the minimum spanning tree (MST) algo- +100 +101 +102 +103 +104 +105 +Number of galaxies +1st cluster +MGS +MST +FoF +DBSCAN +2nd cluster +10 +1 +100 +101 +Linking length (h +1Mpc) +100 +101 +102 +103 +104 +105 +Number of galaxies +3rd cluster +10 +1 +100 +101 +Linking length (h +1Mpc) +4th cluster +Figure 11. The number of member galaxies for the 1st, 2nd, 3rd +and 4th largest clusters in D4-SDSS (thick lines) and in D4-HR4 +as a function of linking length. For D4-HR4, the average values +and the ranges between minimums and the maximums of 4 data +samples are drawn as thin lines and error bars. Color indicates the +different clustering algorithms. +rithm, and the DBSCAN algorithm, the MGS algorithm has +several advantages. The MGS algorithm is able to take into +consideration the local density and is able to accurately iden- +tify clusters even in complex data sets with a large number +of galaxies. In contrast, the FoF, MST, and DBSCAN algo- +rithms often merge clusters into a single giant cluster for large +linking lengths, losing the ability to accurately identify indi- +vidual clusters. This characteristic of the MGS algorithm is +particularly important for analyzing data from astrophysical +simulations and observations. +In this proof of concept work we have shown that the jet- +finding algorithm MGS can be applied to mock Galaxy data +resulting in reliable cluster identification. However the iden- +tification of clusters in real observation is a difficult issue due +to survey incompleteness, selection effects, redshift-space dis- +tortions, etc. In future work we will test MGS in the presence +of realistic observational systematic effects. +MGS also provides auxiliary topological information such +as the number and length of connections for each galaxy. In +future work we will explore the use of this enhanced enhanced +information in testing or constraining cosmological models. +ACKNOWLEDGEMENTS +The authors thank Changbom Park, Dongsu Bak, and Ena +Choi for helpful discussions. This research was supported by +Basic Science Research Program through the National Re- +search Foundation of Korea(NRF) funded by the Ministry +of Education(grant number) S.E.H. was supported by the +project ᄋ +ᅮᄌ +ᅮᄀ +ᅥᄃ +ᅢᄀ +ᅮᄌ +ᅩᄅ +ᅳ +ᆯ ᄋ +ᅵᄋ +ᅭ +ᆼᄒ +ᅡ +ᆫ ᄋ +ᅡ +ᆷᄒ +ᅳ +ᆨᄋ +ᅮᄌ +ᅮ ᄋ +ᅧ +ᆫᄀ +ᅮ (“Under- +MNRAS 000, 1–14 (0000) + +MulGuisin Clustering Algorithm +13 +200 +100 +0 +100 +200 +X (h +1Mpc) +200 +100 +0 +100 +200 +Y (h +1Mpc) +0 +100 +200 +300 +400 +500 +Z (h +1Mpc) +200 +100 +0 +100 +200 +Y (h +1Mpc) +200 +100 +0 +100 +200 +X (h +1Mpc) +0 +100 +200 +300 +400 +500 +Z (h +1Mpc) +X (h +1Mpc) +200 +100 +0 +100 +200 +Y (h +1Mpc) +200 +100 +0 +100 +200 +Z (h +1Mpc) +0 +100 +200 +300 +400 +500 +Figure 12. Top 30 largest clusters (colors) found by the MGS in the D4-SDSS galaxies (gray dots). The observer is located at the +origin. The linking length is ℓMGS = 10 h−1Mpc, where all D4-SDSS galaxies fall into a single giant cluster in all other three benchmark +algorithms (see Fig. 11). Note that, even in such a large linking length, none of the 30 largest clusters suffers percolation. +standing Dark Universe Using Large Scale Structure of the +Universe”), funded by the Ministry of Science. C.G.S is sup- +port via the Basic Science Research Program from the Na- +tional Research Foundation of South Korea (NRF) funded +by the Ministry of Education (2018R1A6A1A06024977 and +2020R1I1A1A01073494). +This work was supported by the Supercomputing Cen- +ter/Korea Institute of Science and Technology Information, +with supercomputing resources including technical support +(KSC-2013-G2-003), and the simulation data were trans- +ferred through a high-speed network provided by KRE- +ONET/GLORIAD. +Funding for the SDSS and SDSS-II has been provided by +the Alfred P. Sloan Foundation, the Participating Institu- +tions, the National Science Foundation, the US Department +of Energy, the National Aeronautics and Space Administra- +tion, the Japanese Monbukagakusho, the Max Planck Society, +and the Higher Education Funding Council for England. The +SDSS website is http://www.sdss.org/. +The SDSS is managed by the Astrophysical Research Con- +sortium for the Participating Institutions. The Participating +Institutions are the American Museum of Natural History, +Astrophysical Institute Potsdam, University of Basel, Uni- +versity of Cambridge, Case Western Reserve University, Uni- +versity of Chicago, Drexel University, Fermilab, the Institute +for Advanced Study, the Japan Participation Group, Johns +Hopkins University, the Joint Institute for Nuclear Astro- +physics, the Kavli Institute for Particle Astrophysics and Cos- +mology, the Korean Scientist Group, the Chinese Academy of +Sciences (LAMOST), Los Alamos National Laboratory, Max +Planck Institute for Astronomy (MPIA), the Max Planck In- +stitute for Astrophysics (MPA), New Mexico State Univer- +sity, Ohio State University, University of Pittsburgh, Uni- +versity of Portsmouth, Princeton University, the US Naval +Observatory, and the University of Washington. +MNRAS 000, 1–14 (0000) + +14 +Y. Ju et al. +DATA AVAILABILITY +The up-to-date MulGuisin algorithm can be downloaded at +https://github.com/youngju20/Mulguisin. +REFERENCES +Abazajian K. N., et al., 2009, ApJS, 182, 543 +Alcock C., Paczynski B., 1979, Nature, 281, 358 +Appleby S., Park C., Hong S. E., Kim J., 2017, ApJ, 836, 45 +Appleby S., Chingangbam P., Park C., Hong S. E., Kim J., Gane- +san V., 2018, ApJ, 858, 87 +Bae H.-J., Woo J.-H., 2014, ApJ, 795, 30 +Ballinger W. E., Peacock J. A., Heavens A. F., 1996, MNRAS, 282, +877 +Barrow J. D., Bhavsar S. P., Sonoda D. H., 1985, Monthly Notices +of the Royal Astronomical Society, 216, 17 +Berlind A. A., Weinberg D. H., 2002, The Astrophysical Journal, +575, 587 +Blanton M. R., et al., 2005, AJ, 129, 2562 +Blumenthal G. R., Faber S. M., Primack J. R., Rees M. J., 1984, +Nature, 311, 517 +Borůvka O., 1926, Plàce mor. přírodově d. spol. v Brně III (3), pp +37–58 +Bosman M., Park I., Corbal M., Costanzo D., Lami S., Paoletti +R., Azuelos G., Strahl K., 1998, ATLAS Software, 98, 038 +Choi Y.-Y., Han D.-H., Kim S. S., 2010a, Journal of Korean As- +tronomical Society, 43, 191 +Choi Y.-Y., Park C., Kim J., Gott J. Richard I., Weinberg D. H., +Vogeley M. S., Kim S. S., SDSS Collaboration 2010b, ApJS, +190, 181 +Dall J., Christensen M., 2002, Phys. Rev. E, 66, 016121 +Davis M., Efstathiou G., Frenk C. S., White S. D. M., 1985, ApJ, +292, 371 +DeRose J., et al., 2019, ApJ, 875, 69 +Diemer B., 2018, The Astrophysical Journal Supplement Series, +239, 35 +Duarte M., Mamon G. A., 2014, Monthly Notices of the Royal +Astronomical Society, 440, 1763 +Dunkley J., et al., 2009, ApJS, 180, 306 +Ester M., Kriegel H.-P., Sander J., Xu X., et al., 1996, in kdd. pp +226–231 +Fall S. M., Efstathiou G., 1980, MNRAS, 193, 189 +Farrens S., Abdalla F. B., Cypriano E. S., Sabiu C., Blake C., 2011, +Mon. Not. Roy. Astron. Soc., 417, 1402 +Gifford D., Miller C., Kern N., 2013, ApJ, 773, 116 +Gott J. Richard I., Melott A. L., Dickinson M., 1986, ApJ, 306, +341 +Guo H., Zehavi I., Zheng Z., 2012, ApJ, 756, 127 +Hearin A. P., et al., 2017, The Astronomical Journal, 154, 190 +Hong S. E., Park C., Kim J., 2016, ApJ, 823, 103 +Hoyle F., Vogeley M. S., Pan D., 2012, MNRAS, 426, 3041 +Huchra J. P., Geller M. J., 1982, The Astrophysical Journal, 257, +423 +Hwang H. S., Park C., Elbaz D., Choi Y. Y., 2012, A&A, 538, A15 +Jiang C. Y., Jing Y. P., Faltenbacher A., Lin W. P., Li C., 2008, +ApJ, 675, 1095 +Kim J., Park C., L’Huillier B., Hong S. E., 2015, Journal of Korean +Astronomical Society, 48, 213 +Knebe A., et al., 2011, MNRAS, 415, 2293 +Kravtsov A. V., Berlind A. A., Wechsler R. H., Klypin A. A., Got- +tlober S., Allgood B., Primack J. R., 2004, The Astrophysical +Journal, 609, 35 +Krzewina L. G., Saslaw W. C., 1996, Monthly Notices of the Royal +Astronomical Society, 278, 869 +Lee G.-H., Park C., Lee M. G., Choi Y.-Y., 2012a, ApJ, 745, 125 +Lee G.-H., Woo J.-H., Lee M. G., Hwang H. S., Lee J. C., Sohn J., +Lee J. H., 2012b, ApJ, 750, 141 +Li X.-D., Park C., Forero-Romero J. E., Kim J., 2014, ApJ, 796, +137 +Limousin M., Morandi A., Sereno M., Meneghetti M., Ettori S., +Bartelmann M., Verdugo T., 2013, Space Sci. Rev., 177, 155 +More S., Kravtsov A. V., Dalal N., Gottlöber S., 2011, ApJS, 195, +4 +Naidoo K., 2019, Journal of Open Source Software, 4, 1721 +Naidoo K., Whiteway L., Massara E., Gualdi D., Lahav O., Viel +M., Gil-Marún H., Font-Ribera A., 2020, Monthly Notices of +the Royal Astronomical Society, 491, 1709 +Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563 +Pan D. C., Vogeley M. S., Hoyle F., Choi Y.-Y., Park C., 2012, +MNRAS, 421, 926 +Park C., Gott J. R. I., 1991, ApJ, 378, 457 +Park C., Kim Y.-R., 2010, ApJ, 715, L185 +Park C., Choi Y.-Y., Kim J., Gott J. Richard I., Kim S. S., Kim +K.-S., 2012, ApJ, 759, L7 +Park H., Park C., Sabiu C. G., Li X.-d., Hong S. E., Kim J., +Tonegawa M., Zheng Y., 2019, ApJ, 881, 146 +Pedregosa F., et al., 2011, Journal of Machine Learning Research, +12, 2825 +Planck Collaboration et al., 2016, A&A, 594, A13 +Press W. H., Schechter P., 1974, ApJ, 187, 425 +Reid B. A., Seo H.-J., Leauthaud A., Tinker J. L., White M., 2014, +MNRAS, 444, 476 +Sander J., Ester M., et al., 2017, ACM Transactions on Database +Systems, 42, 1 +Serra A. L., Diaferio A., 2013, ApJ, 768, 116 +Sheth R. K., 1996, MNRAS, 279, 1310 +Tago E., Einasto J., Saar E., Tempel E., Einasto M., Vennik J., +Müller V., 2008, Astronomy & Astrophysics, 479, 927 +Tempel E., Kipper R., Tamm A., Gramann M., Einasto M., Sepp +T., Tuvikene T., 2016, Astronomy & Astrophysics, 588, A14 +Tonegawa M., Park C., Zheng Y., Park H., Hong S. E., Hwang +H. S., Kim J., 2020, ApJ, 897, 17 +White S. D. M., Rees M. J., 1978, MNRAS, 183, 341 +Zehavi I., et al., 2002, ApJ, 571, 172 +Zehavi I., et al., 2011, ApJ, 736, 59 +This paper has been typeset from a TEX/LATEX file prepared by +the author. +MNRAS 000, 1–14 (0000) + diff --git a/0dE1T4oBgHgl3EQfkwTA/content/tmp_files/load_file.txt b/0dE1T4oBgHgl3EQfkwTA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..76edd67405837f0917978cc4903bbbcb54558367 --- /dev/null +++ b/0dE1T4oBgHgl3EQfkwTA/content/tmp_files/load_file.txt @@ -0,0 +1,1248 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf,len=1247 +page_content='MNRAS 000, 1–14 (0000) Preprint 10 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='0 MulGuisin, a Topological Clustering Algorithm, and Its Performance as a Cosmic Structure Finder Young Ju1,2, Inkyu 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 776 Daedeok-daero,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Yuseong-gu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Daejeon 34055,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Republic of Korea 10 January 2023 ABSTRACT We introduce a new clustering algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' MulGuisin (MGS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' that can find galaxy clusters using topological informa- tion from the galaxy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This algorithm was first introduced in an LHC experiment as a Jet Finder software, which looks for particles that clump together in close proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The algorithm preferentially considers particles with high energies and merges them only when they are closer than a certain distance to create a jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' MGS shares some similarities with the minimum spanning tree (MST) since it provides both clustering and graph-based topology in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Also, similar to the density-based spatial clustering of applications with noise (DBSCAN), MGS uses the ranking or the local density of each particle to construct clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In this paper, we compare the performances of clustering algorithms using some controlled data and some realistic simulation data as well as the SDSS observation data, and we demonstrate that our new algorithm find clusters most efficiently and it defines galaxy clusters in a way that most closely resembles human vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Key words: large-scale structure of Universe, galaxies: clusters: general, methods: statistical, software: data analysis 1 INTRODUCTION In the standard ΛCDM cosmology paradigm, structures in the universe grow in a hierarchical manner (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', White & Rees 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Fall & Efstathiou 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Blumenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' It means that smaller structures of matter start forming ear- lier, and the more massive structures form by the merging and accretion of smaller structures at later epoch of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Therefore, understanding cosmic structures in the universe at various scales is crucial for understanding the nature of our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' For example, numerous statistics of large-scale structures, such as topological analyses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Gott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Park & Gott 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Park & Kim 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2017, 2018), Alcock-Paczynski tests (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Alcock & Paczynski 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Ballinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2019), and small-scale redshift-space distortion (RSD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Sheth 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' DeRose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Tonegawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2020), have been used to constraint cosmological parameters such as the matter den- sity parameter (Ωm) and the equation-of-state parameter of dark energy (ωde).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' While numerous statistics of cosmic structures have been used to understand the evolution and structure formation ⋆ E-mail: icpark@uos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='kr of our universe, the exact definition of cosmic structures re- mains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This is mainly because the matter distribution on large scale is continuous, and therefore, there exists no specific discrete boundary for each structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Also, the mem- bership for a certain structure might change if one considers other properties than just position, such as dynamics, mass, and so on (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Serra & Diaferio 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Gifford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Due to this ambiguity, numerous clustering algorithms have been proposed and used in the astronomical community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' For example, Knebe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' (2011) compared the various properties of dark matter (DM) halos found by 17 different halo-finding algorithms run on the same cosmological N-body simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Galaxy clustering algorithms have been used as essential tools for identifying galaxy clusters or super-clusters, as well as for investigating the large structure of the universe, includ- ing the filament structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The most commonly used galaxy clustering algorithms in astronomy research are the friends- of-friends (FoF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 1985) and the minimum span- ning tree (MST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Borůvka 1926).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' These algorithms were in- troduced in the 1980s and have been widely used as standard galaxy clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In recent years, with the rapid development of machine learning (ML) technology, clustering algorithms such as DBSCAN (Density-based Spatial Cluster- ing of Applications with Noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Ester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 1996) have also been applied to galaxy clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' These clustering softwares, © 0000 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='03278v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='IM] 9 Jan 2023 2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' including the ML based one, show comparable performance in galaxy clustering and produce consistent clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' However, the results of clustering do not always represent the clusters that the human eye can find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' There are cases where a distribution clearly contains a cluster but it is not recognized as such by clustering algorithms, and there are cases where it is clearly divided into two clusters, visually, but appears as a single lump to the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' As such, we have explored the possibility of developing an algorithm that creates galaxy clusters in a way that more closely resembles how the human eye and brain identify pat- terns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' One approach we considered was to adapt jet-finding software used in high-energy particle physics research, with a particular focus on the MulGuisin (MGS) algorithm as a po- tentially suitable software for galaxy clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' MulGuisin (ᄆ ᅮ ᆯᄀ ᅱᄉ ᅵ ᆫ) is a Korean word for a ghost that lives in water and is a figure that often appears in old Korean stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The MGS algorithm started with the idea that the ghosts hiding in the water could be found in the order of height by simply draining the water from the lake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Initially, we copied the MGS software from the A Toroidal LHC Apparatus (ATLAS) Jet-Finding library released in the early 1990s and developed it into a 3D galaxy clustering al- gorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We then made several sample galaxy distributions to check the performance of MGS and compared the results to those produced by other standard clustering algorithms such as FoF, MST, and DBSCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' As a result, it was found that MGS had characteristics that other algorithms could not show, and it was also found that the cluster results created by MGS were most similar to the cluster results that the human eye found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Since the clusters created by MGS show different shapes when compared with clusters formed by other algorithms, and the number of clusters and the size distribution of clus- ters are quite different from those of classical algorithms, us- ing MGS makes a big difference in searching halos and super- clusters, and can yield a different interpretation for the large scale structures of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Therefore, we anticipate that this new algorithm will be used as a new methodology in galaxy cluster research and furthermore used to create new interpretations in cosmology studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The structure of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In Section 2, we introduce the MulGuisin clustering algorithm, as well as FoF, MST, and DBSCAN as benchmark clustering algorithms for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In Section 3, we describe both controlled random data and realistic galaxy distribution data that we will use for the performance test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We apply the above four clustering algorithms to the data and compare their performances in Section 4, and we summarize our results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2 METHODS A halo or galaxy cluster is a group of galaxies held together by gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Finding such cluster structures in galaxy distribution data is very important for astronomical research because it provides a tool to study super-clusters, filaments, and even bigger the large scale structure of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' A cluster can be defined as a concentration of points or cells in a localized volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The task of cluster identification has been extensively studied in the field of computational science, and a wide range of clustering algorithms have been developed for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Because different algorithms have different strengths and weaknesses, it is important for re- searchers to carefully select the algorithm that best suits their specific research purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In this section, we first introduce our MulGuisin clustering algorithm and introduce two clustering tools that are widely used in the field of astronomy and a newly developed clus- tering program through machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='1 MulGuisin galaxy clustering algorithm The MulGuisin (MGS) clustering algorithm was first intro- duced as a jet finder for the Large Hadron Collider (LHC) physics in the ATLAS Collaboration (Bosman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The algorithm is neither a variant of the conventional cone algorithm nor a variant of the kT algorithm that is used in various collider experiments as the standard tools for finding jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Although it has shown some improvements in jet recon- struction performance, such as optimized jet orientation and jet energy resolution, but has not been used as a standard jet-finding tool for LHC experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 1 shows how the MGS algorithm works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The MGS algorithm first finds the most massive point from the input data and names it a cluster seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Then it finds the second massive point and decides whether the point should belong to the first cluster or stand alone as the seed of a new cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This decision is made by checking how close the test point is to any neighboring clusters, for which we introduce a pa- rameter called linking length (ℓMST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' That is, if the distance between the test point and the closest point in the cluster is less than this parameter, the test point is attached to the cluster, otherwise, it becomes the seed of a new cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The algorithm then finds the next massive point and repeats the above process until there are no more points left to test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' At this stage, all points are converted into clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Of course, some points do not belong to any cluster and remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2 is an illustration to explain how the MGS algorithm creates galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In the figure, the points are sorted in order according to their mass and number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' And according to this order, they become new cluster seeds or stick to exist- ing clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' After going through the process, a cluster forms a tree-like structure that is sequentially connected accord- ing to the order of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The points in a cluster then form branches and nodes, and from the characteristic structure of such tree shape, one may able to study the topology of the galaxy cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2 Benchmark Algorithms In order to compare the performance of the MGS algorithm with those of other standard clustering algorithms, we select three benchmark algorithms, mostly based on their popular- ity in the astronomical community, mathematical clarity, and versatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' They are the friends-of-friends (FoF), minimum spanning tree (MST), and the density-based spatial cluster- ing of applications with noise (DBSCAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Here we briefly introduce each package and describe how they make clus- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='1 1 Note that running our MGS algorithm from scratch may take a longer time than the above benchmark algorithms, especially MNRAS 000, 1–14 (0000) MulGuisin Clustering Algorithm 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Schematic flow chart to describe how MulGuisin (MGS) algorithm works Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Diagram showing how the MulGuisin (MGS) algorithm works to identify clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Each gray circle represents a galaxy, and the size of the circle denotes its local density, with the number specifying the galaxies ranking in descending order of density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In this specific example, 21 galaxies are grouped into 3 clusters and 2 isolated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' MNRAS 000, 1–14 (0000) galaxies No Yes clusters No Yes1st cluster 8 5 10 numberofchildrene3 12 1st seed 19 linking distance isolatedgalaxy 3rd cluster Oth generation 3rd seed 2nd cluster 2ndseed 3 1st generation 13 9 16 14 2nd generation b 15 4th generation 21 isolatedgalaxy4 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='1 Friends-of-Friends (FoF) The friends-of-friends (FoF) algorithm is a commonly used technique for identifying clusters in astrophysical data (Huchra & Geller 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Tago et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Duarte & Mamon 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Tempel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This algorithm has a single free parameter, the linking length (ℓFoF), which determines the distance threshold for linking two data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Points that are within this distance of each other are considered to be con- nected, and all connected points are grouped together into a single cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' One limitation of the FoF algorithm is that it can be diffi- cult to choose an appropriate linking length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Different values of this parameter can result in clusters of different shapes or numbers, making it challenging to determine the optimal value (Tago et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2 In this study, we use the Halotools implementation of the FoF algorithm (Hearin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2017) to identify clusters in our datasets by applying various ℓFoF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Unless otherwise noted, we assume all FoF groups containing two or more members as clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2 Minimum Spanning Tree (MST) Galaxy data can be represented as a graph, with each galaxy represented as a node and the distance between two galaxies represented as an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The minimum spanning tree (MST) algorithm is a method for constructing a unique network from this data by connecting all nodes with minimum edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Unlike other clustering algorithms, the MST does not require the use of a free parameter such as a linking length to construct the entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' However, the MST connects all nodes and may not produce clusters with shapes that accurately reflect those of the original clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Nevertheless, MST has been used in cosmology to study the large-scale structure of the universe (Barrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Krzewina & Saslaw 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Naidoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In this study, we use the MiSTree package (Naidoo 2019) to construct MSTs from our galaxy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Then, we find clusters from the single MST tree by cutting nodes longer than the linking length (ℓMST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Similar to the FoF case, we apply various values of ℓMST and assume all tree segments containing two or more members as clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='3 Density-based Spatial Clustering of Applications with Noise (DBSCAN) The use of machine learning (ML) techniques is widespread in astronomy, as they enable the identification of patterns in data using algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' ML algorithms can be classified based on the type of data they are applied to, and one type, called when the number of data points is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We found that most of the MGS calculation time, for a large number of data points, is taken in constructing the Voronoi tesselation and calculating the local density for each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' If we separate MGS into a density calculation and a tree building part, we found that the tree building takes a similar time to the benchmark algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2 Note that the appropriate choice linking length for identifying DM halos from the DM particles in the N-body simulations is well known (ℓFoF ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2⟨dparticle⟩) (More et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' However, the optimal choice of linking length in general clustering problems is not well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' unsupervised ML, is used with unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Clustering al- gorithms, a subcategory of unsupervised ML algorithms, are used to group together data points with similar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' One popular clustering algorithm is DBSCAN (density-based spatial clustering of applications with noise), which has been applied in a variety of contexts (Ester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' DBSCAN is a density-based clustering algorithm that groups together data points based on their local density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In this algorithm, each cluster is identified by defining its core, which consists of high-density points within a certain dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The definition of core requires two free parameters, min_samples and eps, which determine the minimum num- ber of neighbors a point must have within a given radius in order to be considered as the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Then, other points that are directly reachable from some core points within eps are also considered part of the cluster, while other points are labeled as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In this study, we use the scikit-learn package (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2011) to implement the DBSCAN algorithm and iden- tify clusters in our data by applying various eps (or, the “link- ing length” in DBSCAN (ℓDBSCAN)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Unless otherwise noted, we assume min_samples = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='3 A Simple 2D Toy Model Test To see how the shape of the clusters generated by the MGS algorithm differs from the results of other clustering algo- rithms, we created simple simulation data and compared the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We first assume that there are 5 clusters in 2D space, and consider the case where each cluster contains 50 galaxies equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The width of the galaxy distribution of each cluster was fixed to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The coordinates of the two-dimensional space span from 0 to 100 on both the X- and Y-axes, and the posi- tion of each cluster is set to have three different distributions, from far away from each other to all close together, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 3 column (a) in rows (1), (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 3, both the MGS and MST algorithms correctly find 5 clusters when the distances among the clus- ters are sufficiently far apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' However, when the clusters get closer together, MST can’t differentiate between the clus- ters and starts recognizing them as one big cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Even for the cases where clusters are attached to each other as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 3 (3), MGS still recognizes four among five true clusters like the human eyes can distinguish each clus- ter, whereas MST recognizes 4 adjacent clusters as one huge cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' These differences can create serious differences in re- sults when studying the number and mass distributions of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Note that, although we leave its details as future works, the tree structures made by the MGS algorithm have a non- negligible number of long nodes connecting two distant points in the cluster, while the MST algorithm connects only rea- sonably nearby points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This is because the MGS algorithm connects data points based on their local density, not only the distance between the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Therefore, if two highly dense points are within the linking length, then they would be con- nected in the MGS but may not be in the MST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In the next section, we will compare the performance of MGS and other algorithms with more realistic 3D data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' MNRAS 000, 1–14 (0000) MulGuisin Clustering Algorithm 5 0 20 40 60 80 100 (1) LL = 10 0 20 40 60 80 100 (2) 0 20 40 60 80 100 (a) 0 20 40 60 80 100 (3) 0 20 40 60 80 100 (b) 0 20 40 60 80 100 (c) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' A simple 2D toy model test of the MGS algorithm by comparing it with the MST algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' (a) Input distributions of 5 clusters with different degrees of separation from each other ((1)–(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Background color denotes the galaxy number distribution we used for generating the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' (b) Clusters found by MGS and their tree structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' (c) Clusters found by MST and their tree structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 3 DATA Our final goal is to apply the MGS algorithm described in Section 2 to the galaxy clusters or other large-scale struc- tures of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' However, since some inconsistencies ex- ist between various clustering algorithms for finding clusters or other large-scale structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', see Knebe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2011, and references therein), we cannot compare the MGS clus- ters found in the realistic data with their “truth”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Therefore, we apply two types of data sets in this section to compare the performance between MGS and other bench- mark algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The first sets, called the “controlled ran- dom data” (D1–D3), are those that we design all properties of clusters, including their positions and member galaxy dis- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Since we already know the true information of each cluster, we can test which algorithms predict the true clusters better in which conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The next sets, called the “realistic data” (D4), are the observational and simulation data sets of galaxies around z ≃ 0, and we focus on com- paring the properties of predicted clusters in each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Table 1 summarizes the data sets we use in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='1 Controlled Random Data (D1–D3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='1 Different Spatial Dispersion (D1) We use controlled, simulated data to evaluate the perfor- mance of the MGS algorithm in comparison to other clus- tering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' These data are generated randomly and allow us to control the shape and distribution of clusters to test the algorithms under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The first set of data consists of 100 galaxies per cluster and 50 clusters within a 3-dimensional cubic volume of space with a side length 200 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The cluster center positions are chosen randomly, and the galaxies in each cluster are distributed ac- cording to a Gaussian distribution with a variable standard deviation (σ) that controls the spatial dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The D1-LD data has a low spatial dispersion (σ = 1 h−1Mpc), leading to well-separated clusters, while the D1-HD data has a higher MNRAS 000, 1–14 (0000) 6 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Data set Description D1-LD 50 randomly positioned clusters, each of which contains 100 galaxies randomly spread by the 3D Gaussian distribution with standard deviation σ = 1 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The total number of galaxies is 5,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' D1-HD Same as D1-LD, but with the greater standard deviation σ = 10 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' D2-NA Same as D1-HD, but the number of galaxies in each cluster follows an exponential random distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The total number of galaxies is 7,041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' D2-LA Same as D2-NA, but adding uniformly randomly distributed noisy galaxies to the entire box to increase the total galaxy number density 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='5 times of D2-NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='The total number of galaxies is 12,041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' D2-HA Same as D2-LA, but adding more noisy galaxies so that the total galaxy number is twice D2-NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The total number of galaxies is 17,041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' D3-HOD 500 randomly positioned clusters with a mass distribution similar to the Press-Schechter mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The number of galaxies for each cluster follows HODa for massive halos (M ⩾ 1013 h−1M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The galaxies are spread by the NFW profile with the concentration parameter to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The total number of galaxies is 50,257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' D4-SDSS Volume-limited sample of the KIAS-VAGCb with absolute r-band magnitude Mr − 5 log h < −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' D4-HR4 Four lightcone data of mock galaxy catalogs from the Horizon Run 4 simulationc with a similar condition to D4-SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' a Kravtsov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' b Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' (2010a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' c Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Name and description of galaxy data sets that we use in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The box size of all controlled data (D1–D3) is (200 h−1Mpc)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='D2-NA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='D2-LA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='D2-HA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Three-dimensional galaxy distributions of controlled data sets used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Top: D1-LD(left), D1-HD(middle), and D3- HOD(right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Bottom: D2-NA(left), D2-LA(middle), and D2-HA(right), with noisy additional galaxies shown as yellow dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' See Table 1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' MNRAS 000, 1–14 (0000) MulGuisin Clustering Algorithm 7 spatial dispersion (σ = 10 h−1Mpc), resulting in clusters that are closer together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Upper left and middle panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 4 show the distribution of galaxies in the D1-LD and D1-HD data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2 Additional Noisy Galaxies (D2) We generate additional controlled data sets that are similar to D1 but with slightly different characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We again place 50 cluster centers at the same positions as D1, but this time we use an exponential distribution to generate a variable number of galaxies for each cluster: P(Ngal) = � � � 1 ∆N exp � −Ngal − N0 ∆N � if Ngal > N0 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' (1) Here, we set N0 and ∆N as 50 and 100, respectively, so that the minimum number of galaxies per cluster and the total number of galaxies roughly match with D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The galax- ies are spatially distributed according to a Gaussian function centred on the cluster’s centre with a standard deviation of σ = 10 h−1Mpc, as was done for D1-HD, and we call this new controlled data D2-NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The lower left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 4 shows the distribution of galaxies in the D2-NA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The total number of galaxies in this data set is 7,041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In addition to D2-NA, we introduce two more data sets that are created by adding unclustered galaxies, which are sampled uniformly in the entire box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We add these ‘noisy’ galaxies so as to test how the algorithms are affected by the background density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The lower middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 4 shows the D2-LA data, where we add 5,000 galaxies (yellow dots) to increase the galaxy number density by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='5 times compared to D2-NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' On the other hand, the lower right panel shows the D2-HA data, where we add 10,000 galaxies to make the total galaxy number density twice that of D2-NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='3 HOD-based Mock Galaxies (D3-HOD) We generate a third set of controlled, simulated data to cre- ate a more complex environment for testing the performance of the MGS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We use an analytic formula to model the distribution of galaxies in this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' First, we create 500 cluster center positions by sampling uniform random dis- tribution within a (200 h−1Mpc)3 box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Then, we obtain the normalized version of Press-Schechter halo mass function at z = 0(Press & Schechter 1974), with a concordance ΛCDM cosmology to the Planck 2015 data (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2016), for massive halos Mhalo > 1013 h−1M⊙ using the Colossus package (Diemer 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We then obtain masses for each of the 500 clusters by randomly sampling for the mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='3 We then use this information to generate a distribution of the number of galaxies using a halo occupation distribu- tion (HOD) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The mean halo occupation is typically as- sumed to follow a power law at massivehalo masses (Berlind 3 Note that neither the positions nor the mass distribution of clus- ters in D3-HOD follows the estimation from the standard cosmol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' However, here we focus only on providing complex environ- ments, and therefore, such differences do not affect our motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2 for realistic data sets instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' & Weinberg 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Kravtsov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2004): Navg(Mcluster) = � � � �Mcluster M1 �α if Mhalo > Mmin 0 otherwise , (2) where α, Mmin, and M1 correspond to the power-law in- dex, cutoff halo mass where halo cannot contain galaxies, and the mass scale containing a single galaxy at the given condition of galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Here, we use α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='87 and Mmin = 1013 h−1M⊙ by following Kravtsov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We set M1 = 1011 h−1M⊙ so that the minimum number of galax- ies for each cluster is set as 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Also, for simplicity, we calcu- late the actual number of galaxies at each cluster by applying the ceiling to Navg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Next, we use the Colossus package to create an Navarro- Frenk-White (NFW) profile (Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 1996) ρ(x) = Mcluster 4πR3 vir �� ln(1 + cs) − cs 1 + cs � x(x + c−1 s )2�−1 , (3) where x ≡ r/Rvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The concentration parameter for the NFW profile cs is fixed as 10, and the virial radii Rvir is determined by the cluster mass accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We then randomly distribute the galaxies according to this profile, and the resulting data set consists of 50,257 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The upper right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 4 shows the distribution of galaxies in D3-HOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2 Realistic Data: SDSS & Horizon Run 4 (D4) In the previous subsection, we described a set of controlled data catalogues for which we can carefully control the prop- erties of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Such data are useful for testing the performance of MGS over other benchmark algorithms by comparing the properties of identified clusters with the in- put truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' However, the true distribution of galaxies in the universe differs from these controlled random data in the fol- lowing ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' First, unlike those in the controlled random data with low noise levels, the boundaries of clusters in the uni- verse are often not clearly defined (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Serra & Diaferio 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Gifford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Also, the spatial distribution of galax- ies in each cluster may not follow spherical symmetry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Limousin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2013, for a good review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Furthermore, the redshift-space distortion elongates spherical clusters in real space, which may require that we separate linking lengths between the radial and tangential directions (Farrens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Tempel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Therefore, it is necessary to adopt a realistic galaxy dis- tribution for a fair performance test of the MGS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' However, unlike for the case of the controlled random data where we know the answer, we may only study the dif- ference between the cluster properties from the MGS and other benchmark algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Here we use observational data and four corresponding sets of mock simulation data — the volume-limited KIAS-Value Added Galaxy Catalog (KIAS- VAGC) of the Sloan Digital Sky Survey (SDSS) Main Galaxy Sample with r-band absolute magnitude Mr − 5 log h < −20 (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2010a) and the lightcone mock galaxy samples from the Horizon Run 4 simulation (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' MNRAS 000, 1–14 (0000) 8 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='1 Volume-limited KIAS-VAGC (D4-SDSS) The KIAS Value-Added Galaxy Catalog (KIAS-VAGC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2010a) is an upgraded version of the New York Uni- versity Value-Added Galaxy Catalog (NYU-VAGC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Blanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2005), which is part of the Sloan Digital Sky Survey (SDSS) Data Release 7 (Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2009), by adding some missing redshifts to improve spectroscopic complete- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This catalog has been widely used in numerous studies, including cosmic voids statistics (Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Hoyle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2012), largest structures of universe (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2012), frac- tion of barred galaxies (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2012a), and the properties of active galactic nuclei (AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Hwang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Bae & Woo 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Most of the KIAS-VAGC galaxies were observed with the apparent r-band magnitude limit r = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' It means that, in terms of absolute magnitude, the catalog contains less bright galaxies at lower redshifts, while only very bright galaxies could be seen at higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Therefore, for a fair com- parison between galaxies over a wide redshift range, we ap- ply a “volume-limited” selection by selecting galaxies brighter than a certain absolute r-band magnitude (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2010b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Here, we use Mr − 5 log h < −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' By combining with the given apparent r-band magnitude limit, such absolute mag- nitude cutoff naturally provides the upper redshift bound of our volume-limited sample (z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='107;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We also apply the lower redshift bound z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='02, by consider- ing the incompleteness of the galaxy sample below the given redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In addition to the volume-limited selection in the redshift- magnitude plane, we also apply a sky selection for simplifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Specifically, we select galaxies within the SDSS Survey coordinate −33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='5◦ < η < 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='5◦ and −48◦ < λ < 51◦, in or- der to maximize the sky area with a simple geometry, and to avoid issues arising from a complicated boundary (right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2 Horizon Run 4 (D4-HR4) The Horizon Run 4 simulation (HR4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2015) is an extremely large cosmological N-body simulation that uses 6, 3003 DM particles within a periodic cube with a comoving volume V = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='15 h−1cGpc)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' It assumes a vanilla ΛCDM cosmological model in concordance with the Wilkinson Mi- crowave Anisotropy Probe (WMAP) 5th-year result (Dunkley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Among 2,001 timesteps between z = 100 to 0, 75 coarse timesteps with mean time difference ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='18 Gyr are chosen between z = 12 to 0 to build a merging tree of FoF halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The FoF linking length is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2 times the particle mean separation, and we identify halos only whose mass is greater than M min halo = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='7 × 1011 h−1M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The mock galaxies are then produced by so-called the most bound halo particle (MBP)-galaxy abundance match- ing method (Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We find MBPs for all halos in the merging tree and adopt their positions and peculiar ve- locities as those of corresponding mock galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The “mass” of mock galaxies, which is used as a proxy of stellar mass or luminosity, is defined as the mass of their hosting halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' For satellite halos, we identify their MBPs at the timestep just before the infall event and trace them until they are totally absorbed toward their central halo by tidal disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' For estimating the tidal disruption timescale tmerge), we adopt a modified model of Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' (2008), tmerge tdyn = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='94ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='60 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='60)/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='86 ln[1 + (Mhost/Msat)] �Mhost Msat �α , (4) where ϵ, Mhost, Msat, tdyn are the circularity of the satellite’s orbit, the mass of central and satellite halos, and the orbital period of virialized objects, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We adopt α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='5 for a better match of the galaxy two-point correlation func- tion (2pCF) at scales less than 1 h−1Mpc at a given spatial resolution of the HR4 (Zehavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Then the mass of survived satellite galaxies is defined as the mass of their hosting halos just before the infall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' After producing snapshot mock galaxy catalogs for coarse timesteps, we then produce lightcone mock galaxy catalogs up to z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The all-sky lightcone DM particle data of the HR4 were created during the simulation by stacking the co- moving shells at the corresponding redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Then, we com- pare the IDs of the galaxy MBPs at each coarse timestep snapshots and those of DM particles at the lightcone data with the coarse comoving shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' If the MBP ID of a given mock galaxy matches that of a particle in the lightcone data, we assign a galaxy in the lightcone data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Here, we adopt the position and peculiar velocity from the particle at the lightcone data, while the galaxy “mass” comes from the mock galaxy at the nearest snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' After creating the all-sky lightcone mock galaxy catalog, we cut it in a similar way to the volume-limited KIAS-VAGC sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' First, we apply the redshift space distortion (RSD) for each mock galaxy for a fair comparison with observation, by using real-space positions and peculiar velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Then, we apply the same redshift range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='02 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='107 and set the lower bound of galaxy “mass,” so that the galaxy number density of the HR4 lightcone data is identical to that of KIAS- VAGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' After that, we create four non-overlapping subsets from it with the same angular geometry as our SDSS Survey coordinate selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' During the analysis, we found that the fiber collision in the fiber-fed spectroscopic observations affects various clustering statistics (Zehavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Tonegawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Therefore, for a fair comparison, our HR4 mock galaxy catalogs also need to follow the same fiber collision condition as the KIAS-VAGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' To do so, we select pairs of mock galaxies whose angular distance is less than 55 arcseconds and keep only one from each pair by random selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Because SDSS observations were partially overlap- ping, some close-pairs have both redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In order to reflect this, we only fiber-collide 60% of the close pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 4 RESULTS We test MGS and the other algorithms using the 3 controlled data and observation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We run 4 algorithms with various linking-length and find out the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The same process is repeated by changing linking-length and we check the tendency of the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We use the three controlled data sets and observation data to evaluate the performance of the MGS algorithm and com- pare it to other clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We run each of the four algorithms with different values of the linking length and count the number of clusters identified by each algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We repeat this process for a range of linking lengths MNRAS 000, 1–14 (0000) MulGuisin Clustering Algorithm 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='25 Redshift 23 22 21 20 19 18 17 r 5logh 150 100 50 0 50 100 150 [degree] 60 40 20 0 20 40 60 [degree] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Selection of the volume-limited sample of the KIAS Value-Added Galaxy Catalog (KIAS-VAGS) used in this study (red boxes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Left: Volume-limited selection in the redshift vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' absolute r-band magnitude plane with Mr − 5 log h < −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Right: Sky selection in SDSS Survey coordinates (η, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 30 35 40 45 50 Number of Clusters D1-LD MGS MST FoF DBSCAN 2 4 6 8 10 Linking-length 0 10 20 30 40 50 Number of Clusters D1-HD Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The number of clusters as a function of linking length for D1-LD (top panel) and D1-HD (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Each of the 4 clus- tering algorithms are indicated using different colors and symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Note that MST and DBSCAN show considerable overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The Hor- izontal dash shows the original number of clusters, which is 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' and analyze the trends in the number of clusters identified by each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This allows us to assess the sensitivity of the algorithms to the choice of linking length and to compare their performance in identifying clusters in the different data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='1 Results with Controlled Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 6 shows the number of clusters identified by each algo- rithm as a function of the linking length for the controlled data set 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The top panel shows the results for the D1- LD data, which consists of well-separated clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The al- gorithms are expected to identify 50 clusters in this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' All four algorithms perform well in identifying the clusters, but the MGS algorithm stands out for its ability to accu- rately identify the correct number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In particular, for large linking lengths, the FoF and DBSCAN algorithms identify fewer than 50 clusters, because they connect neigh- boring clusters and merge them into a single cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 6 shows the results for the D1- HD data, which has a higher level of spatial dispersion and some clusters that are close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' For small link- ing lengths, the algorithms identify fewer than 50 clusters because the linking length is not sufficient to connect the galaxies in these clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' As the linking length increases, the behavior of the algorithms becomes more distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The MGS algorithm continues to accurately identify the correct number of clusters, while the other algorithms identify fewer clusters due to the merging of originally separate clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The MGS algorithm is able to track the structure of the clusters and identify their boundaries, leading to more accurate results in this type of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 7 shows the results for controlled data set 2, which includes the D2-NA data with no additional galaxies and the D2-LA and D2-HA data with additional galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The top panel shows the number of clusters identified by each algo- rithm for the D2-NA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' When this data was generated, the minimum number of galaxies per cluster was set to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In the region of small linking lengths, all algorithms iden- tify fewer than 50 clusters because the linking length is too small to connect the galaxies in the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' As a result, the clusters identified by the algorithms have fewer than 50 member galaxies, and are therefore not considered as true clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' For larger linking lengths, particularly those larger than 5, the difference between the MGS algorithm and the other algorithms becomes more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The MGS al- gorithm continues to accurately identify the correct number of clusters, while the other algorithms identify fewer clusters due to the merging of originally separate clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The behavior of the algorithms with additional galaxies is even more distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 7 shows the re- sults for the D2-LA data, where the other three algorithms identify only a single cluster for very large linking lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' As the linking length increases, the algorithms merge several clusters into a single giant cluster, resulting in a significantly lower number of clusters than the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This rapid MNRAS 000, 1–14 (0000) 10 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 10 20 30 40 50 60 Number of Clusters D2-NA d = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='43 Nmin = 50 MGS MST FoF DBSCAN 10 20 30 40 50 60 Number of Clusters D2-LA d = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='73 Nmin = 56 2 4 6 8 10 12 14 Linking-length 0 10 20 30 40 50 60 Number of Clusters D2-HA d = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='77 Nmin = 59 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 6, but with D2-NA(top), D2-LA(middle), and D2-HA(bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The vertical dashed line is mean-separation of data (⟨d⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Since each data set has a different overall number density, we assign different minimum number of member galaxies (Nmin) to define clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' increment of a single giant cluster is called “percolation,” and it is known to occur at linking length similar to the mean- separation (ℓ ≃ ⟨d⟩) for the ideal random Poisson graph (Dall & Christensen 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 7 clearly shows that such percola- tion occurs at ℓ ≃ ⟨d⟩ for all three benchmark algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Note that, however, the percolation occurs at the low- est linking length in FoF, while both MST and DBSCAN share a similar value of linking length at percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This is because FoF does not have an additional consideration for limiting the cluster boundary that exists in the other two algorithms (minimize the number of edges in MST, and core definition in DBSCAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 8 shows the 3D distribu- tions of clustering results from various algorithms at linking length ℓ = 11 h−1Mpc, which is longer than the mean sep- aration ⟨d⟩ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='73 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' As expected, three benchmark algorithms show percolation (blue color), while our MGS al- gorithm successfully reconstructs most of the true clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Note that only one giant cluster is found in the FoF algo- rithm, while both MST and DBSCAN have two additional small clusters (green and orange colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The behavior of the MGS algorithm for the D2-HA data is slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In the region of small linking lengths, the MGS algorithm accurately identifies the correct number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' However, for larger linking lengths, particularly ℓ > 13 h−1Mpc, the MGS algorithm identifies additional clus- ters that were not present in the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' These “fake” clusters are not true clusters and are not representative of the underlying structure of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This behavior highlights the ability of the MGS algorithm to identify clusters in data with a complex distribution of galaxies but also underscores the importance of choosing an appropriate linking length to avoid identifying false clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 9 shows the number of clusters for controlled 3 data with a more complex environment than D1–D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' At ℓ ≳ ⟨d⟩/2 ≈ 3 h−1Mpc, the number of clusters using FoF and DBSCAN decreases as the linking length increases, resulting in the percolation at ℓ ≳ ⟨d⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='The MST shows a flat curve when the linking length is larger than ∼ 8 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This is because MST connects all galaxies with minimal edge first, and then we cut off the links with linking length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Therefore, if there were no links longer than 8 h−1Mpc in the original tree, then cutting the links with any longer linking length than 8 h−1Mpc would not change the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' So, the number of clusters using MST shows a constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In contrast, the number of clusters identified by the MGS algorithm slowly decreases as the linking length increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This is because the clusters in this data set are close to each other and are easily merged by the algorithm for large linking lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' However, the MGS algorithm is able to identify clus- ters based on density, which allows it to retain the structure of the clusters even for large linking lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This is the main advantage of the MGS algorithm compared to the other three algorithms, which are not able to accurately identify clusters in complex data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='2 Results with Observational and Cosmological Simulation Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 10 shows the results of the four algorithms applied to both KIAS-VAGC observational data and four sets of HR4 lightcone data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' We track the number of detected clusters changing with both linking lengths and with the minimum number of member galaxies from 2 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' For all four clustering algorithms, the HR4 simulation re- sults match well with the observations within cosmic vari- ance, especially for n ⩾ 5 at ℓ ≳ ⟨dparticle⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='5 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' On the other hand, HR4 tends to underestimate the num- ber of clusters for a smaller minimum number of member galaxies and/or smaller linking length ℓ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='5 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This may mean that, despite the agreement with the observation in terms of 2pCF below 1 h−1Mpc-scale, some disagreements ex- ist between HR4 and observation in terms of the higher-order statistics in smaller scales than the particle mean separation scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' One notable feature of the MGS algorithm is that it does not create a single giant cluster for large linking lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In- stead, the algorithm identifies a number of smaller clusters, even for large linking lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This is in contrast to the other three algorithms, which all create a single giant cluster for large linking lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This difference highlights the ability of the MGS algorithm to accurately identify clusters in data with a complex distribution of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 11 shows the number of member galaxies for the 1st MNRAS 000, 1–14 (0000) MulGuisin Clustering Algorithm 11 X 100 50 0 50 100 Y 100 50 0 50 100 Z 0 50 100 150 200 X 100 50 0 50 100 Y 100 50 0 50 100 Z 0 50 100 150 200 X 0 50 100 150 200 Y 0 50 100 150 200 Z 0 50 100 150 200 X 100 50 0 50 100 Y 100 50 0 50 100 Z 0 50 100 150 200 MGS MST FoF DBSCAN Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 3D distribution of clustering results from MGS and other benchmark algorithms in D2-LA with linking length ℓ = 11 h−1Mpc, which is longer than the mean-separation ⟨d⟩ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='73 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Color indicates the cluster membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' MGS finds 49 clusters among 50 true clusters, while other algorithms connect most of galaxies and finally make a giant cluster (blue color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' to 4th largest clusters identified by each algorithm in D4- SDSS and D4-HR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 10, both results from the simulation and observation data match well with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The top left panel of the figure shows the shape of the largest cluster for each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' As the linking length increases over certain value, the largest cluster identified by the FoF, MST, and DBSCAN algorithms contains all of the galaxies in the data, while the MGS algorithm identifies a cluster with only a portion of the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This indicates that the MGS algorithm is able to identify multiple clusters even for large linking lengths, while the other algorithms merge all of the galaxies into a single giant cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The main difference between the MGS algorithm and the other three algorithms becomes particularly clear when ex- amining the number of member galaxies in the 2nd to 4th largest clusters (upper right and bottom panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' As the linking length increases, the number of member galaxies in these clusters identified by the FoF, MST, and DBSCAN algorithms decreases to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This is because the first largest cluster identified by these algorithms took all the galaxies in the data, leaving no galaxies to be considered for further clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In contrast, the MGS algorithm is able to identify multiple clusters even with large linking lengths as the largest cluster does not monopolize all galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This demonstrates the ability of the MGS algorithm to accurately identify clus- ters in data with a complex distribution of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 12 shows the 30 largest clusters found by the MGS algorithm in the D4-SDSS data with a linking length ℓMGS = 10 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' While such a large choice of linking length makes a single giant cluster in all other three benchmark algorithms (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 11), none of the 30 clusters suffer percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' All 30 largest clusters are well-separated and have some even dis- tribution of galaxies in the XY-plane (that is, the tangential plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' On the other hand, most of the clusters have some- what elongated features in the line-of-sight direction, which clearly shows the Finger-of-God effect due to the RSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' There- fore, although this needs further inspection, we consider that the 30 largest clusters found by the MGS algorithm could be MNRAS 000, 1–14 (0000) 12 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 Linking-Length 0 100 200 300 400 500 Number of cluster D3-HOD MGS MST FoF DBSCAN Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Same as Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 6–7, but with D3-HOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 100 101 102 103 104 Number of clusters MGS n 2 n 3 n 4 n 5 MST 100 101 Linking length(h 1Mpc) 100 101 102 103 104 Number of clusters FoF 100 101 Linking length(h 1Mpc) DBSCAN Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Same as Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 6, 7 & 9, but with D4-SDSS (thick lines) and D4-HR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' For D4-HR4, the average values and the ranges between minimums and the maximums of 4 data samples are drawn as thin lines and error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Results from each clustering algorithm are shown on different panels, while the color indicates the different choices of the minimum number of member galaxies to identify clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' the actual large structures similar to galaxy (super)clusters in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 5 CONCLUSIONS The MulGuisin (MGS) algorithm is a powerful technique for identifying clusters in data from astrophysical simulations and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' It consistently produces results closer to those inferred from human visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In comparison to other clustering algorithms, such as the friends-of-friends (FoF) algorithm, the minimum spanning tree (MST) algo- 100 101 102 103 104 105 Number of galaxies 1st cluster MGS MST FoF DBSCAN 2nd cluster 10 1 100 101 Linking length (h 1Mpc) 100 101 102 103 104 105 Number of galaxies 3rd cluster 10 1 100 101 Linking length (h 1Mpc) 4th cluster Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The number of member galaxies for the 1st, 2nd, 3rd and 4th largest clusters in D4-SDSS (thick lines) and in D4-HR4 as a function of linking length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' For D4-HR4, the average values and the ranges between minimums and the maximums of 4 data samples are drawn as thin lines and error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Color indicates the different clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' rithm, and the DBSCAN algorithm, the MGS algorithm has several advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The MGS algorithm is able to take into consideration the local density and is able to accurately iden- tify clusters even in complex data sets with a large number of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In contrast, the FoF, MST, and DBSCAN algo- rithms often merge clusters into a single giant cluster for large linking lengths, losing the ability to accurately identify indi- vidual clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This characteristic of the MGS algorithm is particularly important for analyzing data from astrophysical simulations and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In this proof of concept work we have shown that the jet- finding algorithm MGS can be applied to mock Galaxy data resulting in reliable cluster identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' However the iden- tification of clusters in real observation is a difficult issue due to survey incompleteness, selection effects, redshift-space dis- tortions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In future work we will test MGS in the presence of realistic observational systematic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' MGS also provides auxiliary topological information such as the number and length of connections for each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' In future work we will explore the use of this enhanced enhanced information in testing or constraining cosmological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors thank Changbom Park, Dongsu Bak, and Ena Choi for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This research was supported by Basic Science Research Program through the National Re- search Foundation of Korea(NRF) funded by the Ministry of Education(grant number) S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' was supported by the project ᄋ ᅮᄌ ᅮᄀ ᅥᄃ ᅢᄀ ᅮᄌ ᅩᄅ ᅳ ᆯ ᄋ ᅵᄋ ᅭ ᆼᄒ ᅡ ᆫ ᄋ ᅡ ᆷᄒ ᅳ ᆨᄋ ᅮᄌ ᅮ ᄋ ᅧ ᆫᄀ ᅮ (“Under- MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 1–14 (0000) MulGuisin Clustering Algorithm 13 200 100 0 100 200 X (h 1Mpc) 200 100 0 100 200 Y (h 1Mpc) 0 100 200 300 400 500 Z (h 1Mpc) 200 100 0 100 200 Y (h 1Mpc) 200 100 0 100 200 X (h 1Mpc) 0 100 200 300 400 500 Z (h 1Mpc) X (h 1Mpc) 200 100 0 100 200 Y (h 1Mpc) 200 100 0 100 200 Z (h 1Mpc) 0 100 200 300 400 500 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Top 30 largest clusters (colors) found by the MGS in the D4-SDSS galaxies (gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The observer is located at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The linking length is ℓMGS = 10 h−1Mpc, where all D4-SDSS galaxies fall into a single giant cluster in all other three benchmark algorithms (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Note that, even in such a large linking length, none of the 30 largest clusters suffers percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' standing Dark Universe Using Large Scale Structure of the Universe”), funded by the Ministry of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='S is sup- port via the Basic Science Research Program from the Na- tional Research Foundation of South Korea (NRF) funded by the Ministry of Education (2018R1A6A1A06024977 and 2020R1I1A1A01073494).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' This work was supported by the Supercomputing Cen- ter/Korea Institute of Science and Technology Information, with supercomputing resources including technical support (KSC-2013-G2-003), and the simulation data were trans- ferred through a high-speed network provided by KRE- ONET/GLORIAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Funding for the SDSS and SDSS-II has been provided by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Sloan Foundation, the Participating Institu- tions, the National Science Foundation, the US Department of Energy, the National Aeronautics and Space Administra- tion, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The SDSS website is http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The SDSS is managed by the Astrophysical Research Con- sortium for the Participating Institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' The Participating Institutions are the American Museum of Natural History,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Astrophysical Institute Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' University of Basel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Uni- versity of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Case Western Reserve University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Uni- versity of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Drexel University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Fermilab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' the Institute for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' the Japan Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Johns Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' the Joint Institute for Nuclear Astro- physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' the Kavli Institute for Particle Astrophysics and Cos- mology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' the Korean Scientist Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' the Chinese Academy of Sciences (LAMOST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Los Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Max Planck Institute for Astronomy (MPIA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' the Max Planck In- stitute for Astrophysics (MPA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' New Mexico State Univer- sity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' University of Pittsburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Uni- versity of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' the US Naval Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' and the University of Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' MNRAS 000, 1–14 (0000) 14 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' DATA AVAILABILITY The up-to-date MulGuisin algorithm can be downloaded at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='com/youngju20/Mulguisin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' REFERENCES Abazajian K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2009, ApJS, 182, 543 Alcock C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Paczynski B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1979, Nature, 281, 358 Appleby S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Hong S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2017, ApJ, 836, 45 Appleby S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Chingangbam P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Hong S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Gane- san V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2018, ApJ, 858, 87 Bae H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Woo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2014, ApJ, 795, 30 Ballinger W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Peacock J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Heavens A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1996, MNRAS, 282, 877 Barrow J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Bhavsar S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Sonoda D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1985, Monthly Notices of the Royal Astronomical Society, 216, 17 Berlind A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Weinberg D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2002, The Astrophysical Journal, 575, 587 Blanton M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2005, AJ, 129, 2562 Blumenthal G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Faber S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Primack J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Rees M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1984, Nature, 311, 517 Borůvka O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1926, Plàce mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' přírodově d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' spol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' v Brně III (3), pp 37–58 Bosman M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Corbal M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Costanzo D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Lami S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Paoletti R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Azuelos G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Strahl K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1998, ATLAS Software, 98, 038 Choi Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Han D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2010a, Journal of Korean As- tronomical Society, 43, 191 Choi Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Gott J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Richard I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Weinberg D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Vogeley M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', SDSS Collaboration 2010b, ApJS, 190, 181 Dall J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Christensen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2002, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' E, 66, 016121 Davis M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Efstathiou G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Frenk C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1985, ApJ, 292, 371 DeRose J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2019, ApJ, 875, 69 Diemer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2018, The Astrophysical Journal Supplement Series, 239, 35 Duarte M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Mamon G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2014, Monthly Notices of the Royal Astronomical Society, 440, 1763 Dunkley J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2009, ApJS, 180, 306 Ester M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kriegel H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Sander J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Xu X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1996, in kdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' pp 226–231 Fall S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Efstathiou G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1980, MNRAS, 193, 189 Farrens S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Abdalla F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Cypriano E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Sabiu C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Blake C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2011, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 417, 1402 Gifford D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Miller C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kern N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2013, ApJ, 773, 116 Gott J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Richard I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Melott A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Dickinson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1986, ApJ, 306, 341 Guo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Zehavi I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Zheng Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2012, ApJ, 756, 127 Hearin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2017, The Astronomical Journal, 154, 190 Hong S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2016, ApJ, 823, 103 Hoyle F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Vogeley M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Pan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2012, MNRAS, 426, 3041 Huchra J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Geller M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1982, The Astrophysical Journal, 257, 423 Hwang H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Elbaz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Choi Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2012, A&A, 538, A15 Jiang C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Jing Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Faltenbacher A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Lin W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Li C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2008, ApJ, 675, 1095 Kim J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', L’Huillier B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Hong S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2015, Journal of Korean Astronomical Society, 48, 213 Knebe A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2011, MNRAS, 415, 2293 Kravtsov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Berlind A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Wechsler R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Klypin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Got- tlober S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Allgood B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Primack J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2004, The Astrophysical Journal, 609, 35 Krzewina L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Saslaw W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1996, Monthly Notices of the Royal Astronomical Society, 278, 869 Lee G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Lee M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Choi Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2012a, ApJ, 745, 125 Lee G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Woo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Lee M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Hwang H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Lee J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Sohn J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Lee J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2012b, ApJ, 750, 141 Li X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Forero-Romero J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2014, ApJ, 796, 137 Limousin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Morandi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Sereno M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Meneghetti M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Ettori S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Bartelmann M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Verdugo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2013, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 177, 155 More S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kravtsov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Dalal N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Gottlöber S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2011, ApJS, 195, 4 Naidoo K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2019, Journal of Open Source Software, 4, 1721 Naidoo K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Whiteway L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Massara E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Gualdi D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Lahav O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Viel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Gil-Marún H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Font-Ribera A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2020, Monthly Notices of the Royal Astronomical Society, 491, 1709 Navarro J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Frenk C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1996, ApJ, 462, 563 Pan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Vogeley M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Hoyle F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Choi Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2012, MNRAS, 421, 926 Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Gott J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1991, ApJ, 378, 457 Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2010, ApJ, 715, L185 Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Choi Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Gott J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' Richard I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2012, ApJ, 759, L7 Park H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Sabiu C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Li X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Hong S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Tonegawa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Zheng Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2019, ApJ, 881, 146 Pedregosa F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2011, Journal of Machine Learning Research, 12, 2825 Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2016, A&A, 594, A13 Press W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Schechter P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1974, ApJ, 187, 425 Reid B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Seo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Leauthaud A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Tinker J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', White M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2014, MNRAS, 444, 476 Sander J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Ester M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2017, ACM Transactions on Database Systems, 42, 1 Serra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Diaferio A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2013, ApJ, 768, 116 Sheth R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1996, MNRAS, 279, 1310 Tago E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Einasto J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Saar E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Tempel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Einasto M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Vennik J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Müller V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2008, Astronomy & Astrophysics, 479, 927 Tempel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kipper R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Tamm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Gramann M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Einasto M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Sepp T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Tuvikene T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2016, Astronomy & Astrophysics, 588, A14 Tonegawa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Zheng Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Park H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Hong S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Hwang H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Kim J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2020, ApJ, 897, 17 White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', Rees M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 1978, MNRAS, 183, 341 Zehavi I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2002, ApJ, 571, 172 Zehavi I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=', 2011, ApJ, 736, 59 This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} +page_content=' MNRAS 000, 1–14 (0000)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQfkwTA/content/2301.03278v1.pdf'} diff --git a/19E2T4oBgHgl3EQfiwfE/vector_store/index.faiss b/19E2T4oBgHgl3EQfiwfE/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..084e488bf1e438518b4a8ee9daad390225b7c48a --- /dev/null +++ b/19E2T4oBgHgl3EQfiwfE/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8233042e62b8f96aed409f289f827d821b2b752a059bbc6e4533ac9767667c79 +size 7077933 diff --git a/1tAzT4oBgHgl3EQfDfp4/content/tmp_files/2301.00977v1.pdf.txt b/1tAzT4oBgHgl3EQfDfp4/content/tmp_files/2301.00977v1.pdf.txt new file mode 100644 index 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Ross,1 C. Reynolds,2 N. Seymour,1 J. R. Callingham,3,4 N. Hurley-Walker,1 and H. Bignall5,2 +1International Centre for Radio Astronomy Research, Curtin University, Bentley, WA 6102, Australia +2 CSIRO, Space and Astronomy, P.O. Box 1130, Bentley, WA 6102, Australia +3Leiden Observatory, Leiden University, PO Box 9513, Leiden, 2300 RA, The Netherlands +4ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, Dwingeloo, 7991 PD, The Netherlands +5Manly Astrophysics, 15/41-42 East Esplanade, Manly, NSW 2095, Australia +Author for correspondence: K. Ross, Email: kathryn.ross@icrar.org. +(Received 03 Aug 2022; revised 24 Nov 2022; accepted 31 Dec 2022; first published online XX) +Abstract +Spectral variability offers a new technique to identify small scale structures from scintillation, as well as determining the absorption mechanism +for peaked-spectrum (PS) radio sources. In this paper, we present very long baseline interferometry (VLBI) imaging using the Long Baseline +Array (LBA) of two PS sources, MRC 0225–065 and PMN J0322–4820, identified as spectrally variable from observations with the Murchison +Widefield Array (MWA). We compare expected milliarcsecond structures based on the detected spectral variability with direct LBA imaging. +We find MRC 0225–065 is resolved into three components, a bright core and two fainter lobes, roughly 430 pc projected separation. A +comprehensive analysis of the magnetic field, host galaxy properties, and spectral analysis implies that MRC 0225–065 is a young radio +source with recent jet activity over the last 102–103 years. We find PMN J0322–4820 is unresolved on milliarcsecond scales. We conclude +PMN J0322–4820 is a blazar with flaring activity detected in 2014 with the MWA. We use spectral variability to predict morphology and find +these predictions consistent with the structures revealed by our LBA images. +1. +Introduction +Peaked-spectrum (PS) sources, are a subset of active galac- +tic nuclei (AGN) that are identified by a peak in their radio +spectral energy distribution (O’Dea & Saikia, 2021), and are +also often associated with compact morphologies (≲ 20 kpc; +Phillips & Mutel, 1982; Tzioumis et al., 2010). PS sources +provide an interesting population of AGN as the evolutionary +pathway from PS source to extended (≳ 30 kpc) AGN is still +unclear. Two contending theories hypothesise the nature and +evolutionary pathway of PS sources: the youth scenario, where +the age of the PS source is ≤ 105 years and has not yet had +ample time to grow to the large-scale AGN (O’Dea & Baum, +1997; Owsianik & Conway, 1998; Tinti & de Zotti, 2006); +and the frustration scenario, when the PS source is confined +by a dense cloud of the interstellar medium (ISM) of the host +galaxy environment (van Breugel et al., 1984; Wilkinson et al., +1984; O’Dea et al., 1991). Furthermore, recent identifications +of embedded PS cores within remnant ageing lobes has been +attributed to restarted and episodic AGN activity (Hernández- +García et al., 2019), i.e. a cyclical evolution rather than linear +evolution. +Compact symmetric objects (CSOs) are a subset of PS +sources with similar morphologies to large scale AGN, namely +a central region (often quite faint, if detected) with emission +either side associated with hot spots and/or lobes. Unlike +typical AGN, CSOs show emission only on very compact scales, +typically ≤1 kpc, and thus require high-resolution imaging to +detect (Phillips & Mutel, 1982; Gugliucci et al., 2005). CSOs +are generally considered young AGN (< 104 yr; O’Dea & +Baum, 1997; Owsianik & Conway, 1998; Tinti & de Zotti, +2006), which may evolve into typical, radio-loud AGN. +Previous attempts to discriminate between youth and frus- +tration scenarios have relied on spectral modelling and high- +resolution imaging (e.g. Marr et al., 2014; Keim et al., 2019) +using very long baseline interferometry (VLBI). The cause of +absorption at low-frequencies, producing the spectral peak, +has typically been attributed to synchrotron-self absorption +(SSA) and/or free-free absorption (FFA) for the youth and +frustration scenarios respectively (Tingay & de Kool, 2003; +Callingham et al., 2015). Unfortunately, without sufficient +sampling below the spectral turnover, the cause of absorption +is often ambiguous (Callingham et al., 2017). In rare cases, a +SSA model can be ruled out if the optically thin spectral index +is sufficiently steep (α ≥ 2.5)a. +Many PS sources have been identified as a CSOs (e.g. +0108+388, 0710+439 and 2352+495; Readhead et al., 1996). +As CSOs are typically considered to be young AGN, identify- +ing PS sources that are also CSOs could help to differentiate +between the youth and frustrations scenarios. However, identi- +fying CSOs requires high resolution (mas) observations using +VLBI. Likewise, PS sources sometimes display extremely asym- +metrical mas structures, likely due to an inhomogeneous sur- +rounding environment influencing their growth (Orienti et al., +2006; Keim et al., 2019), compared with a fairly symmetrical +morphology associated with CSOs with minor asymmetries +likely coming from orientation effects (Orienti & Dallacasa, +2008). VLBI can also be used to measure proper motion of +aWe assume a power-law relation where Sν = S0να, thus the sign of α, +being negative or positive, also indicates either the optically thin or thick +spectral index respectively. +arXiv:2301.00977v1 [astro-ph.GA] 3 Jan 2023 + +2 +K. Ross et al. +hot-spots in lobes to estimate kinematic ages of ≤ 3×103years +(Polatidis & Conway, 2003; Gugliucci et al., 2005), consistent +with the theory that CSOs are young AGN. Indeed, Gugliucci +et al. (2005) find a majority of the CSOs with age estimates +were ≤ 500 yrs, suggesting CSOs may be short lived and few +would continue to grow to the scale of typical AGN, thereby +explaining the large fraction of CSO and PS sources thought +to be young relative to the number of large-scale radio galaxies +(O’Dea & Saikia, 2021). VLBI of PS sources can thus help to +identify populations of CSOs and elucidate the youth scenario +and AGN evolution. +Spectral variability at radio frequencies offers a new tech- +nique for identifying young or frustrated candidates. Many +variability surveys have identified PS sources that lost their PS +classification over time (Tinti et al., 2005; Torniainen et al., +2005; Ross et al., 2021, hereafter R21), or showed a signifi- +cant change in spectral shape likely due to a variable opacity +from the inhomogeneous surrounding ISM (Tingay et al., +2015; Ross et al., 2022, hereafter R22). Thus the population of +known PS sources, which is already biased from sparse spectral +coverage from a range of instruments and times, is likely con- +taminated by temporary PS sources. This is particularly true +at higher frequencies (∼GHz), which is sensitive to emission +from the core/jets. PS sources with a peak at lower frequen- +cies (∼MHz) appear to be less contaminated by sources only +showing a temporary peak (Callingham et al., 2017, R21). +Spectral variability offers the a new technique to find and +exclude contaminating “temporary” PS sources, as well as iden- +tify CSO candidates with a decreased risk of contaminating +sources. Variability of PS sources has been used to infer the +presence of compact (µas – mas) features based on scintillation +(Fanti et al., 1979; Chhetri et al., 2018, R21). Such compact +features are common for CSOs, but VLBI is required for con- +firmation of a CSO classification. Spectral variability has also +found PS sources that show changing spectral shape, inconsis- +tent with scintillation, which suggests that some PS sources +are frustrated or contaminating blazars (R22). +This paper aims to investigate the milliarcsecond scale struc- +tures of variable PS sources using VLBI to test predictions based +on spectral variability. In particular, we investigate PS sources +that have shown a consistent spectral shape with a variable +overall flux density, consistent with scintillation, suggesting a +compact feature on milliarcsecond scales (R21, R22), and use +VLBI to test a CSO classification. We also investigate variable +PS sources that R21 found as changing spectral shape. They +concluded the short timescale (∼1 year), and variable spectral +shape is inconsistent with interstellar scintillation and present +it as a blazar caught flaring. +In Section 2, we describe the three variable PS sources of +this study, in Section 3 we describe the observational strategy +and data reduction. Section 4 outlines the results of the LBA +imaging. We discuss the host galaxy properties including +their linear size compared to turnover in Section 5.1, the mid- +infrared (MIR) and optical emission in Section 5.2 and the radio +properties in Section 5.3. In Section 6 we present the likely +absorption mechanisms and source classification of our targets. +We adopt the standard Λ-cold dark matter cosmological model, +with ΩM = 0.286, ΩΛ = 0.714, and the Hubble constant +H0 = 69.6 km s–1 Mpc–1 (Wright, 2006; Hinshaw et al., 2013) +2. +Target Selection +Targets were selected for LBA imaging with the goal of com- +paring direct imaging of milliarcsecond structures with pre- +dicted morphologies based on their variability. Three targets +were selected based on the variability detected by R21 and +R22. MRC 0225–065 (GLEAM J022744-062106) was initially +identified as variable in R21 but further monitoring over a +year found no evidence of variability (R22). As such, it was +predicted MRC 0225–065 would have resolved structures on +milliarcsecond scales with a compact feature ≲ 25 mas, re- +sulting in variability from refractive interstellar scintillation +(RISS) on a longer timescale with a dampened modulation +index due to the extended structure. Conversely, PMN J0322– +4820 (GLEAM J032237–482010) was selected due to the vari- +able spectral shape identified in R21. To explain the variable +spectral shape, R21 concluded PMN J0322–4820 was likely a +blazar caught flaring in 2014. As such, it was predicted to show +a compact morphology even on milliarcsecond scales. Finally, +MRC 2236-454 (GLEAM J223933–451414) was identified by +R21 as the only PS source in their sample that showed sig- +nificant variability but maintained a constant peak frequency +below 231 MHz. A low peak frequency is typically associated +with PS sources that are of the order of tens of kilo-parsecs +across, but the RISS detected by R22 suggested MRC 2236-454 +is dominated by a compact feature, and showed variability due +to a surrounding inhomogeneous environment. As such, it was +predicted MRC 2236-454 may be resolved on milliarcsecond +scales and show an asymmetrical morphology, often associ- +ated with frustrated sources in an inhomogeneous surrounding +environment (Orienti et al., 2006). +3. +LBA Observations and Data Reduction +3.1 +Observations +LBA observations were taken on November 23, 2020 and +February 17, 2021 as part of project V600. The November +observation was centered at 2.4 GHz and the February obser- +vation was centered at 8.3 GHz and both utilised 128 MHz of +bandwidth in dual polarizations. Stations used in each obser- +vation and their diameter is listed in Table 1. Both observa- +tions cycled through phase calibrator scans and target scans +of lengths 2 min and 5 min, respectively. However, the spatial +separation of each target and their respective phase calibrator +meant each target had a different number of scans. A summary +of the targets, phase calibrators and number of scans each is +presented in Table 2. +Parkes at 2.4 GHz, and Katherine at both frequencies, ob- +served using their native linear feeds. These were converted +to a circular polarization basis post-correlation using the Pol- +Convert software (Martí-Vidal et al., 2016) +3.2 +Data Processing and Calibration +After correlation, data calibration and processing were done +using the NRAO’s Astronomical Imaging Processing System + +Publications of the Astronomical Society of Australia +3 +Table 1. LBA stations included in observations +Name +Code +Diameter (m) +Nov20 +Feb21 +ATCA, phased up +At +5×22 +Y +Y +Mopra +Mp +22 +Y +Y +Parkes +Pa +64 +Y +Y +Hobart +Ho +26 +Y +Y +Ceduna +Cd +30 +Y +Y +Yarragadee +Yg +12 +Y +Y +Warkworth +Ww +12 +Y +Y +Hartebeesthoek +Hh +26 +Y +Y +Katherine +Ke +12 +Y +Y +Tidbinbilla +Td +34 +Y +N +Table 2. Targets, associated calibrators and number of LBA scans for each +target source. +Source Name +Expected S5GHz (mJy) +Number of scans +MRC 0225–065 +0.238 +27 +PKS J0217+0144 (C) +0.666 +27 +PMN J0322–4820 +0.112 +40 +PMN J0335-4837 (C) +0.112 +40 +MRC 2236–454 +0.420 +48 +QSO B2227–445 (C) +0.386 +48 +(AIPS) (Wells, 1985). The calibration and flagging followed +the general procedure outlined in the AIPS cookbookb and +was implemented in a semi-automated script with the Parsel- +Tongue interface (Kettenis et al., 2006). Initial flagging of edge +channels and RFI was done using UVFLG. Auto-correlations +were scaled to unity across the band using ACCOR before +removing gross residual instrumental delays using FRING on +a short scan of a bright calibrator. Complex bandpass cor- +rections were derived using BPASS. The system temperature +and gain calibration were applied using APCAL. Delay, rate +and phase calibrations were determined from fringe fitting +using FRING from each target’s respective phase calibrator. +A phase referenced image was created for all targets except +for MRC 0225–065, as a first pass detection of the targets to +determine if a phase shift was needed. Lastly, UVFIX was used +to apply a phase shift to the data for any sources that were +∼arcsecond away from the phase centre used in correlation. +MRC 0225–065 had accurate VLBI coordinates and thus did +not require a phase shift. The calibrated and phase shifted data +were exported to be imaged using CASA. +3.3 +Imaging and Self-Calibration +Initial Stokes-I images were made with a quasi-natural weight- +ing with robust parameter set to +1 (Briggs, 1995) using the +tclean function in CASA (McMullin et al., 2007). Clean boxes +were used but were tightly restricted for the models used for +self-calibration to avoid inducing artificial structure from the +bThe AIPS cookbook can be found here http://www.aips.nrao.edu/cook. +html +complex point-spread-function. For each image, phase only +self calibration was performed and applied using the gaincal +and applycal functions respectively. Due to the sparse (u, v)- +coverage and low signal-to-noise (SNR), calibration solutions +were inspected and applied without flagging solutions that +had insufficient SNR. The slow rate of improvement necessi- +tated several (∼9) rounds of self-calibration. The SNR of the +main component and the root-mean-squared (rms) noise of +the image were inspected after each self calibration iteration +to ensure each round improved the overall image quality. For +each source the initial model assumed for the self-calibration +was an unresolved point source to avoid inducing any morpho- +logical features. Any resolved components were included in +subsequent rounds of imaging clean components and kept in +the model for self-calibration if this reduced the rms noise of +the image. The initial solution interval for the self calibration +was set to the scan length and decreased in further rounds of +self calibration. Phase only self calibration rounds were contin- +ued until the rms noise of the image increased. A final round of +both phase and amplitude self calibration was then performed +(provided it reduced the rms of the final image) with the so- +lution interval set to the scan length. For MRC 0225–065, an +amplitude self-calibration was applied to both frequencies, but +no amplitude self-calibration was applied to the 2.4 GHz image +of PMN J0322–4820. +4. +Results +Images of MRC 0225–065 at both 2.4 and 8.3 GHz are pre- +sented in Figure 1, and an image of PMN J0322–4820 at +2.4 GHz, presented in Figure 3. Unfortunately, due to large +phase errors from a pointing offset, we were unable to re- +cover images for MRC 2236–454 at either frequency, or for +PMN J0322–4820 at 8.3 GHz, this was because the source po- +sitions were beyond the observed correlated field of view for +recovery in each case. For MRC 2236–454, the pointing offset +was over 11 arcseconds for both the 2.4 GHz and 8.3 GHz ob- +servations, thus the phase errors from this pointing offset was +beyond recovery. PMN J0322–4820 also had a pointing offset +of ≈ 11.5 arcseconds, however, given it was bright (∼ 0.2 Jy), +there was sufficient sensitivity using a subset of antennas (flag- +ging the Hartebeesthoek antenna), and a phase shift combined +with self calibration to recover and image at 2.4 GHz. How- +ever, this method was not possible at 8.3 GHz due to the smaller +field-of-view and decreased sensitivity. Henceforth, we will +only discuss the results for MRC 0225–065 and PMN J0322– +4820. +Table 3. Properties for each LBA image: synthesised beam size and rms +background noise. +Source, ν (GHz) +rms (mJy/beam) +θbeam,maj +θbeam,min +PA +MRC 0225–065, 2.4 +2.7 +9.5 +3.2 +7.0 +MRC 0225–065, 8.3 +1.0 +4.4 +2.7 +83 +PMN J0322–4820, 2.4 +1.0 +30 +17 +-54 + +4 +K. Ross et al. +4.1 +MRC B0225–065 +MRC 0225–065 was resolved into three components morphol- +ogy at both 2.4 GHz and 8.3 GHz, as shown in Figure 1. The +final image was made with a robust parameter of -1 at 2.4 GHz +and -0.5 at 8.3 GHz (Briggs, 1995). MRC 0225–065 is resolved +into 3 regions: a bright, unresolved central component, with +an upper limit of source size of 2.5 × 4 mas assuming the beam +size at 8.3 GHz (labelled C in Figure 1), a fainter 16 × 11 mas +Western region (L1) and even fainter 14 × 10 mas Eastern +component (L2). The sizes of L1 and L2 are measured using +the contours in the 2.4 GHz image. The triple morphology is +roughly symmetrical with the distance between the C to L1 +and L2 being ∼ 40 mas each. Since it appears the components +of MRC 0225–065 may be resolved, we measured their flux +density over an irregular polygonc for each component. +We recovered all the flux density predictions from the +spectral fit to the R22 ATCA observations at 2.4 GHz, but +found that ∼ 35% of the flux density was lost at 8.3 GHz. +The flux densities for each component and their spectral index +are presented in Table 4. The irregular polygon was shaped +based on contour levels to ensure only real flux was included in +the final measurement. However, the missing flux density at +8.3 GHz may be due to extended structure being resolved out. +Consequently, the estimates for the spectral index presented +in Table 4 should be considered lower limits. +Table 4. Flux densities and two component spectral index for each compo- +nent of MRC 0225–065 found in the LBA images. The uncertainties for the +fluxdensitiesaremeasuredcalculatedusingthemeasureduncertaintyfrom +polygon flux and the rms noise of the image. The uncertainty for α is calcu- +lated using standard propagation of errors. The model prediction is calcu- +lated from the best spectral fit, a double SSA spectral model with an expo- +nential break. +Component +S2.4GHz (mJy) +S8.3GHz (mJy) +α +C +270±10 +78±7 +-0.95±0.08 +L1 +121±8 +30±5 +-1.1±0.2 +L2 +56±7 +18±4 +-0.9±0.2 +Integrated LBA +447±14 +126±10 +-0.97±0.07 +Model Prediction +400 +195 +N/A +The symmetrical triple morphology suggests MRC 0225– +065 is a CSO candidate with a core (C) and two lobes (L1 +and L2). +The spectral index of the central component is +αC = –0.95 ± 0.08, which is far steeper than expected for +a typical AGN “core", generally expected to have a α ≥ –0.5 +(Orienti et al., 2006; Hardcastle & Looney, 2008). However, +components have previously been identified as cores with spec- +tral indices as steep as –0.7 (Orienti et al., 2006). We present +the SED for MRC 0225–065 in Figure 2 including the MWA +flux densities from R22 as well as the flux densities and power- +law spectral model for each LBA component. The entire SED +is fit, using the most recent MWA epoch (2020-09), with a +double SSA model with an exponential break, which assumes +two synchrotron emitting regions that are self-absorbed and +cusing https://github.com/nhurleywalker/polygon-flux, (Hurley-Walker +et al., 2019) +ageing producing the exponential break, νb, separate from the +peak frequency. The break frequency is the frequency where +the spectrum begins to steepen as the electrons are ageing +and experiencing energy losses (Turner et al., 2018). We fit +the spectral model using the UltraNest packaged (Buchner, +2021), which uses a nested sampling Monte Carlo algorithm. +From the double SSA spectral model, we find the peak frequen- +cies for the two SSA components to be νp,1 =400±100 MHz +and νp,2=112±90 MHz, and find νb =14.3±2.7 GHz. +MRC 0225–065 has a spectroscopic redshift of 0.445 (Al- +bareti et al., 2017); thus, 1 mas corresponds to a linear scale of +5.25 pc. Using this redshift, we find the projected linear size +of MRC 0225–065 (from L1 to L2) to be ∼430 pc, the linear +distance from the core to either lobe to be ∼210 pc and place +an upper limit on the size of component C to be ≤26 pc. +4.2 +PMN J0322–4820 +Due to difficulties in the phase calibration, we were only able to +produce a high quality image of J0322–483 at 2.4 GHz, shown +in Figure 3. We do not resolve PMN J0322–4820 and it is +confined to the size of the beam: 56 × 40 mas. The final image +was made using a robust parameter of +0.5, and by flagging the +Hartebeesthoek antenna, thus the beam size for PMN 0322– +4820 compared to MRC 0225–065 for the same frequency is +much larger. Details of the image properties are presented in +Table 3. Compared to the spectral model fit to the ATCA and +2014 MWA observations, 18% of the flux density was missing. +We used a reported photometric redshift for PMN J0322–4820 +of 0.16 (Bilicki et al., 2014), thus 1 mas corresponds to a linear +size of 2.650 pc. We place an upper limit on the source size of +148 pc. +5. +Discussion +In this section, we will present a comprehensive analysis of both +MRC 0225–065 and PMN J0322–4820 to produce a unified +perspective of these two sources with the aim of concluding +whether they are young or frustrated PS sources. In Sec- +tion 5.1, we present our two sources in the linear size and +turnover relation, in Section 5.2, we discuss the host galaxy +properties according to mid-infrared, optical observations and +radio properties. +5.1 +Linear Size and Turnover Relation +PS sources follow an inverse relation between their linear size +and intrinsic turnover frequency, often referred to as the linear +size turnover relation, first presented by O’Dea (1998). This +relation is directly predicted from the youth scenario (O’Dea, +1998) where the peak frequency is due to SSA and thus the +linear size is directly related to the peak frequency (Keller- +mann & Pauliny-Toth, 1981). While modifications to models +in the frustration scenario can reproduce this relation (Bick- +nell et al., 2018), it is generally understood that PS sources +that fall below the linear size-turnover relation are likely com- +pact beyond what is expected for a young source and a thus +dhttps://johannesbuchner.github.io/UltraNest/ + +Publications of the Astronomical Society of Australia +5 +40 +20 +0 +-20 +-40 +40 +20 +0 +-20 +-40 +Relative R.A. (mas) +Relative Dec (mas) +C +L1 +L2 +MRC 0225-065 at 2.4GHz +52.5pc +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Intensity (Jy/beam) +40 +20 +0 +-20 +-40 +40 +20 +0 +-20 +-40 +Relative R.A. (mas) +Relative Dec (mas) +C +L1 +L2 +MRC 0225-065 at 8.3GHz +52.5pc +0.00 +0.01 +0.02 +0.03 +0.04 +Intensity (Jy/beam) +Figure 1. LBA images of MRC 0225–065 at 2.4 GHz (lef) and 8.3 GHz (right). Beam sizes are shown with a white ellipse in the bottom lef corner of each image +and dimensions are specified in Table 3. Contours are placed at (-3, 3, 4, 5, 6, 7, 10, 20, 50, 100, 200, 400, 800, 1600) times the rms noise of the image, also +specified in Table 3. Pixel brightness is plotted in a linear scale following the colour-bars to the right of each image. The resolved regions are labelled C, L1, +L2 and properties of each region are outlined in Table 4. Relative R.A and Dec are calculated from the position of the core (C) component with coordinates: +J2000 02h27m44.5s -06d21m06.7s. +0.1 +0.2 +0.5 +1.0 +2.0 +5.0 +10.0 +Frequency (GHz) +0.10 +0.20 +0.30 +0.40 +0.50 +0.60 +0.70 +0.80 +Flux Density (Jy) +MRC 0225-065 +2013 +2014 +2020-04 +2020-05 +2020-07 +2020-09 +ATCA 2020 +LBA int +C +L1 +L2 +40 +20 +0 +-20 +-40 +40 +20 +0 +-20 +-40 +Relative R.A. (mas) +Relative Dec (mas) +C +L1 +L2 +MRC 0225-065 Spectral Index Map +52.5pc +−1.8 +−1.6 +−1.4 +−1.2 +−1.0 +−0.8 +−0.6 +−0.4 +−0.2 +α +Figure 2. Spectral energy distribution (SED) for MRC 0225–065 (lef) and spectral index map (right). The spectral index map was created using by convolving +both the 8.3 GHz image and 2.4 GHz image to the same resolution. Data included in the SED are from R21 and R22 monitoring (circles) and coloured according +toepoch. LBAfluxdensitiesareplottedassquareswiththeintegratedfluxdensityofLBAplottedasblacksquares. ThespectralfittoeachLBApointisapower- +law with spectral index presented in Table 4. The grey spectral model to the entire SED is a double SSA model with an exponential break. Supplementary +data included: TIFR GMRT 150 MHz Sky Survey Alternative Data Release 1 (TGSS-ADR1; Intema, H. T. et al., 2017) (grey cross), Molonglo Reference Catalogue +(MRC; Large et al., 1981, 1991) (grey +), Rapid ASKAP Continuum Survey (RACS; McConnell et al., 2020; Hale et al., 2021) (grey ‘Y’), NRAO VLA Sky Survey (NVSS; +Condon et al., 1998), Australia Telescope 20 GHz (AT20G; Murphy et al., 2010) (grey right arrow). + +6 +K. Ross et al. +200 +100 +0 +-100 +-200 +200 +100 +0 +-100 +-200 +Relative R.A. (mas) +Relative Dec (mas) +C +L1 +L2 +PMN J0322-4820 at 2.4GHz +132.5pc +0.00 +0.02 +0.04 +0.06 +0.08 +Intensity (Jy/beam) +0.1 +0.2 +0.5 +1.0 +2.0 +5.0 +10.0 +Frequency (GHz) +0.10 +1.00 +0.50 +Flux Density (Jy) +PMN J0322-4820 +2013 +2014 +ATCA 2020 +LBA +Figure 3. LBA image for PMN J0322–4820 at 2.4 GHz (lef) and associated SED (right). The beam size is shown with a white ellipse in the bottom lef corner +and dimensions are specified in Table 3. Contours are placed at (-3, 3, 4, 5, 6, 7, 10, 20, 50, 100, 200, 400, 800, 1600) times the rms noise of the image, also +specified in Table 3. Pixel brightness is plotted in a linear scale following the colour-bars to the right of the image. Relative R.A and Dec are calculated from the +central coordinate: J2000 03h22m38.0s -48d20m16.2s. Data included in SED is from R21 and R22 (circles) and coloured according to epoch. LBA flux density +is plotted as a blue square. The grey spectral model to the entire SED is a single SSA model with an exponential break. Supplementary data included is: TIFR +GMRT 150 MHz Sky Survey Alternative Data Release 1 (TGSS-ADR1; Intema, H. T. et al., 2017) (grey cross), Sydney University Molonglo Sky Survey (SUMSS; +Mauch et al., 2003) (grey star), Rapid ASKAP Continuum Survey (RACS; McConnell et al., 2020; Hale et al., 2021) (grey ‘Y’). +assumed to be frustrated. We plot both MRC 0225–065 and +PMN J0322–4820 on the linear size-turnover relation in Fig- +ure 4, along with other known PS sources, details of which +are discussed by Keim et al. (2019). It is evident from Figure 4, +that MRC 0225–065 is entirely consistent with the relation +whereas PMN J0322–4820 sits somewhat below the relation, +particularly since the linear size is an upper limit. This would +suggest MRC 0225–065 is consistent with the youth scenario +whereas PMN J0322–4820 may be frustrated. However, it is +worth nothing, R21 identified PMN J0322–4820 as a variable +PS source with a changing spectral shape, and thus concluded +it was likely a blazar. Furthermore, R21 found the peak fre- +quency changed from ∼320 MHz in 2013 to ∼145 MHz in +2014. As the peak frequency is variable and PMN J0322–4820 +is known to exhibit a changing spectral shape, its position on +the linear size-turnover relation will also vary, shown by the +error bar in Figure 4 corresponding to the range of the peak +frequency from 2013 to 2014. Most likely, PMN J0322–4820 +is only a temporary PS source and thus should not be included +in this relation nor when considering the PS population at +large. +5.2 +Host Galaxy Properties +5.2.1 +WISE Colours +MIR colour selection techniques using the Wide-Field Infrared +Survey Explorer (Wright et al., 2010, WISE) are widely used to +efficiently distinguish between AGN and star-forming galax- +ies. +WISE is a MIR all sky survey covering four photometric +bands: 3.4, 4.6, 12, and 22 µm referred to as W1, W2, W3, and +W4 respectively. The MIR wavelengths are sensitive to the +emission from hot dust in the torus of the AGN, allowing for +the identification of AGN where X-ray and optical emission +10 +2 +10 +1 +100 +101 +102 +Linear Size (kpc) +102 +103 +104 +Rest-Frame Peak Frequency (MHz) +J0227-0621 +J0322-482 +Figure 4. Rest frame peak frequency versus linear size. Sources in black are +describedinKeimetal.(2019). Thedashedlineisthefittotherelationfound +by Orienti & Dallacasa (2014). Arrows indicate maximum linear sizes for un- +resolved sources. MRC 0225–065 (pink circle) and PMN J0322–4820 (purple +circle) are plotted with linear sizes calculated from LBA images. The error +bars for MRC 0225–065 represent the range for peak frequencies calculated +in R21. + +Publications of the Astronomical Society of Australia +7 +may be blocked by intervening gas and dust. This also makes +AGN stand out from star-bursting galaxies or stars due to their +extremely red MIR emission (Lonsdale et al., 2015). Obscured +AGN with red MIR emission have been identified by their MIR +colours, often by their place in a colour-colour diagram (Jarrett +et al., 2011; Lonsdale et al., 2015). The bulk of sources centred +around W1 – W2 = 1.2 and W2 – W3 = 3 correspond to the +region typically associated with quasars and AGN. MRC 0225– +065 is found in the region typically associated with emission +from star formation or stellar emission; i.e. there is no evidence +of hot AGN dust, however, there is evidence for moderate star +formation. As we know MRC 0225–065 is an AGN, it is likely +the emission at MIR is a combination of these two processes. +PMN J0322–4820 is well within the elliptical regime, thus +has low emission from star formation and no evidence of hot +AGN dust. Blazars are typically found to dominate the top +right region of the WISE colour-colour plot as the MIR emis- +sion is dominated by the emission of the blazar over the galaxy +(and associated stellar emission). A compact morphology and +variable spectral shape suggest PMN J0322–4820 is a blazar. +However, the WISE colours of PMN J0322–4820 suggest that +the host galaxy is an elliptical with predominantly red optical +emission. Therefore, the emission from the potential radio +blazar is not dominant in the MIR. While it is more common +to find blazars in the top right region of the WISE colour- +colour plot, the MIR colours, which suggest the host galaxy +for PMN J0322–4820 is an elliptical, are still consistent with a +blazar classification (Yang et al., 2015; D’Abrusco et al., 2019). +5.2.2 +Optical Spectra +MRC 0225–065 has an optical spectrum from the 13th data +release of the Sloan Digital Sky Survey (Albareti et al., 2017, +SDSS). From the fitted spectrum, Albareti et al. (2017) report +a spectroscopic redshift for MRC 0225–065 of z = 0.445 and +classify it as a broad-line, starburst quasar. The spectrum +additionally has low-ionisation nuclear emission-line region +(LINER) properties, evident from the strong NII, SiII and OI +lines. A LINER has a high energy radiation field. There is still +debate about whether this is AGN emission or star formation, +but likely the combination of the broad lines, strong OIII +emission and radio-loudness of MRC 0225–065 is evidence +of AGN. From the broad Hα, we can calculate the velocity +dispersion according to: +d(velocity) = cd(λ) +λ0 +, +(1) +where c is the speed of light, d(λ) is the wavelength dispersion +from the spectral fit, and λ0 is the rest-frame wavelength of +Hα. Using the reported fit to the broad Hα from SDSS where +λobserved = 9486 Å, we use the equivalent width, EW= 30±4 Å, +and find the velocity dispersion to be 900±100 km/s. This large +velocity dispersion may be from an extreme star formation +wind but it is also indicative of the broad-line regions from an +AGN, which is more consistent given our radio observations +identify MRC 0225–065 as an AGN. The broad Hα, and large +velocity dispersion, is consistent with an AGN that is quite +obscured, as reported by Albareti et al. (2017) who classify +it as a broad-line quasar. Perhaps of more interest are the +starburst properties of MRC 0225–065, namely OII and OIII +emission lines, identified by Albareti et al. (2017). Both OII and +OIII are forbidden lines with different origins: OII is mostly +due to star formation and thus is often used as an indicator +for star formation in galaxies; OIII is due to an AGN and +can be used as a proxy for the AGN bolometric luminosity. +This is also consistent with the WISE colours discussed in +Section 5.2.1, which find MRC 0225–065 consistent with a +galaxy with emission coming from both the AGN and star +formation. Combining the radio, MIR and optical properties +of MRC 0225–065, it is likely this galaxy has moderate star +formation with an obscured AGN. +5.3 +Radio Properties of MRC B0225–065 +Combining the spectral information and high resolution re- +solved structure of MRC 0225–065, we are able to determine +several intrinsic properties that can help differentiate between +SSA and FFA models. In this section, we estimate the magnetic +field strength and spectral ages to assess whether MRC 0225– +065 is consistent with the youth scenario. We do not consider +PMN J0322–4820 in this section due to its unresolved mor- +phology (even on mas scales) and since the radio variability +suggests it is a blazar with an added beaming effect producing +Doppler boosting and thus many of the assumptions required +for these calculations no longer hold. +5.3.1 +Magnetic Field +As a means of evaluating the validity of SSA compared to +an FFA, we can calculate the magnetic field estimates based +on a pure SSA model and on equipartition. Equipartition +assumes there is equal energy between the radiating particles +and the magnetic field. The comparison between magnetic +field estimates based on an SSA model and equipartition has +been used as evidence both for the SSA model (when the +estimates are in agreement; Orienti & Dallacasa, 2008) and +against (when there is a clear disparity; Keim et al., 2019). In +this section, we will first estimate the magnetic field assuming +a purely SSA model, then assuming equipartition and compare +these to determine whether SSA is a reasonable model for +MRC 0225–065. +We can estimate the magnetic field strength, in Gauss, +based on a purely SSA spectral model, BSSA, according to: +BSSA ≈ +(νpeak/f (αthin))5θsrc,min2θsrc,max2 +Speak +2(1 + z) +, +(2) +where νpeak is the observed peak frequency in GHz, Speak is the +flux density in Jy at the peak frequency for the source at redshift +z with angular minor and major component axis, θsrc,min and +θsrc,max, in mas (Kellermann & Pauliny-Toth, 1981). We note, +f (αthin) is as defined by Kellermann & Pauliny-Toth (1981), +where it is loosely related to αthin. We take f (αthin) = 8 based +on values from Marscher (1983); Orienti & Dallacasa (2008). + +8 +K. Ross et al. +Now, assuming equipartition, we calculate the magnetic +field strength, in Gauss, according to (Miley, 1980), as Bequi +by assuming the component has cylindrical symmetry such +that the width of the source on the sky is equivalent to the line +of sight path-length. +For both calculations, we calculate BSSA and Bequi for the +compact core region rather than the total source, to ensure we +are comparing a homogeneous region (Orienti & Dallacasa, +2008; Keim et al., 2019). For MRC 0225–065, using Equa- +tion 2, we estimate the magnetic field strength for a purely +SSA model to be BSSA ≈6±7 mG for the core region where +θsrc = 2.5 × 4 mas. To estimate Bequi, we assume a filling fac- +tor η = 1 and set k = 1e and find Bequi ≈6±2 mG. As BSSA is +within the uncertainties of Bequi, it suggests the core region of +MRC 0225–065 is in equipartition and consistent with a pure +SSA model. While this does not exclude the FFA model, it +does provide supportive evidence for the SSA model. Further- +more, it may not be a valid assumption that MRC 0225–065 +is in equipartition, thus the equation from Miley (1980) for +Bequi would not be a reasonable estimate of the magnetic field +strength. +We can also use the estimated magnetic field to calculate +the age of the electron population as a proxy for the age of the +jets/lobes. Calculating the spectral age of the electron popula- +tion requires an accurate estimate of the break frequency, νb. +We can thus calculate the spectral age, τspec, according to: +τspec = +aB1/2 +B2 + BiC2 +� +νb(1 + z) +�–1/2 +where +BiC = 0.318(1 + z)2 +a = +�243πme5c2 +4µ02e7 +�1/2 +(3) +where BiC is the magnitude of the microwave background +magnetic field in nT, B is the magnetic field of the source +in nT, νb is the break frequency in GHz, and the constants +me, c, µ0, and e are the mass of an electron, speed of light, +magnetic permeability of free space, and charge of an electron, +respectively. +It is possible the core is actually an unresolved double of +more recent AGN activity than the outer lobes, producing +the steep (α ≲ –1, see Table 4) spectral index. We assume a +constant expansion speed, v, and use the linear sizes to estimate +the dynamical age, τdyn, of the core and outer lobes. Using +the magnetic field calculated for the core region assuming +equipartition, i.e. setting B = Bequi = 6 ± 2 mG, and deter- +mining a break frequency, we can estimate the spectral age +of the core. Using a break frequency of νb = 14.3 ± 2.7 GHz, +calculated from the double SSA spectral model fit, we estimate +the spectral age of the core to be τspec ≈ 700 ± 100 years. +ek = 1 is equivalent to the minimum energy condition, however values for +k have ranged from 1 to 100, where k = 100 produces an order of magnitude +difference in Bequi (Pacholczyk & Roberts, 1971; Miley, 1980) +We then calculate an upper limit on the expected expansion +velocity of v ≤ 0.13 c (using simple speed = distance/time argu- +ments) for the core using the upper limit for the linear source +size of θsrc ≤ 26 pc, as outlined in Section 4.1. An expansion +velocity of v = 0.13 c is well within previous measurements of +the expansion speeds for compact AGN that have been found +to range from 0.1 c up to 0.7 c (Polatidis & Conway, 2003; +An & Baan, 2012; Orienti & Dallacasa, 2020). The range of +expansion velocities would correspond to a range in dynamical +ages for the core of 100 ≲ τdyn ≲ 900 years. If we assume the +expansion velocity of the core of “inner lobes" is roughly equal +to that of the outer lobes from a previous epoch of activity, we +can place an upper limit on the dynamical ages of the outer +lobes. We calculate the distance between the core and L1 as +∼ 210 pc, which corresponds to a dynamical age of 5000 years +for an expansion velocity of 0.13 c. For the range of dynamical +ages for typical PS sources, we expect the age of the outer lobes +to be 1000 ≲ τdyn ≲ 7000 years. Previous estimates for the +ages of PS sources using similar assumptions have estimated +ages from ∼ 101 to ∼ 105 years (Orienti et al., 2010), which +is entirely consistent with our age estimates for both the inner +core and outer lobes. +As the ages, expansion velocities, and magnetic fields that +we calculate are all consistent with the SSA model and a youth +scenario, it appears MRC 0225–065 is more consistent with +a young CSO rather than a frustrated compact AGN. How- +ever, there are several caveats and assumptions made in these +calculations. Thus, while these results are consistent with +the evolutionary scenario of MRC 0225–065 being the youth +model, it is not sufficient for excluding the frustration scenario +entirely. +6. +AUnifiedPerspectiveofMRCB0225–065andPMNJ0322– +4820 +Combining all the information we have obtained about MRC 0225– +065, we begin to create a unified perspective that suggests +MRC 0225–065 is a CSO with a peaked spectrum best ex- +plained by SSA and recent jet activity over the last 102–103 years. +A summary of the evidence in support of this conclusion are +as follows: +• Variability: R21 identified spectral variability of MRC 0225– +065 with a constant spectral shape, consistent with vari- +ability due to RISS. Further spectral variability monitor- +ing by R22 detected no further variability, suggesting a +resolved structure but consistent PS source classification. +This observation suggests it is unlikely MRC 0225–065 is +a contaminating blazar or source with only a temporary +PS source classification, such as frustrated sources with an +inhomogeneous surrounding medium. +• Radio morphology: Previously, it has been suggested +frustrated PS sources are more likely to show an asymmet- +rical morphology due to the asymmetrical environment +confining the growth of the lobes. Inversely, this suggests +young PS sources that are not frustrated may be more +likely to show a symmetrical morphology like that of a + +Publications of the Astronomical Society of Australia +9 +CSO. MRC 0225–065 has a very symmetrical morphology +according to our LBA images, suggesting it may not be +interacting with its surrounding environment. +• Linear size and turnover relation: We find MRC 0225– +065 is entirely consistent with the linear size turnover rela- +tion, a natural product of the youth scenario. Although, it +can be reproduced in certain frustration models. +• Host galaxy: Using the MIR colours reported in by WISE +and the optical spectrum from SDSS, we identify the MRC 0225– +065 as having an obscured AGN with moderate star forma- +tion. Since the AGN does not dominate the entire MIR and +optical emission, and there is still star formation present, it +is possible the AGN has only recently been switched on +and thus has not yet quenched all star formation in the +galaxy, which is not surprising given the compact size of +MRC 0225–065. +• Magnetic field: Estimating the magnetic field using a +purely SSA model and comparing it to the magnetic field +calculated assuming equipartition are entirely consistent, +suggesting the SSA model is a reasonable model for MRC 0225– +065 +• Spectral ages: Using spectral modelling of the break fre- +quency, we estimate the age of the radio emission (from +the core and lobes) to be roughly 700 years, consistent with +estimates of the age of PS sources in the youth scenario. +• Dynamical ages: Using the linear size from our LBA im- +ages and previous measurements of expansion velocity we +estimate MRC 0225–065 has two major epochs of activity, +one between 1000 to 7000 years ago and another more +recently from 100 to 900 years ago. This is also consistent +with previous estimates of the ages for young PS sources. +Furthermore, due to the missing flux density at 8.3 GHz, +this estimate should be considered an upper limit as the +spectral indices for each component may be artificially +steepened by the missing flux density. +We therefore conclude, MRC 0225–065 is likely a young AGN +and with the peak occurring due to SSA. +Likewise, combining all information of PMN J0322–4820, +we can also begin to create a unified picture that PMN J0322– +4820 is a blazar. A summary of the evidence for this conclusion +are: +• Spectral variability: R21 identified PMN J0322–4820 as +a variable source in and classified it as showing a changing +spectral shape. The dramatic change in spectral shape in the +megahertz regime on a timescale of ∼ 1 year is inconsistent +with evolutionary models for PS sources and predicted +variability due to RISS. The changing spectral shape is +most easily explained by the dynamical nature of blazars. +• Radio morphology: The high resolution image of PMN J0322– +4820 using the LBA found it was still compact on mas scales. +This is also entirely consistent with a blazar morphology, +which appears compact due to orientation effects. +• Linear size and turnover relation: PMN J0322–4820 sits +well below the linear size and turnover relation typically +associated with PS sources. This could either be because +it is a frustrated source and is thus more compact than +expected for it’s predicted age. However, more likely, is +that the temporary peak detected with the MWA in 2014 +was a result of the variability of a blazar with effects like +Doppler boosting influencing measurements and thus the +spectral peak is unrelated to the source age or absorption +mechanisms. +• WISE MIR Colours: PMN J0322–4820 has WISE colours +typically associated with elliptical galaxies and/or LERGs/BL +Lac blazars. +We therefore identify PMN J0322–4820 as a new blazar where +the jets are oriented along the line-of-sight. However, PMN J0322– +4820 was not in the ROMA-bzcat catalogue of γ-ray emitting +blazars. This is potentially due to the steep spectrum at fre- +quencies over 1 GHz where PMN J0322–4820 is too faint to be +detected by traditional blazar searches. We suggest further ob- +servations using higher frequency observations in the X-ray or +γ regimes to search for any high frequency counterpart (Mas- +saro et al., 2009, 2015). We conclude PMN J0322–4820 should +not be included in any future population studies of PS sources +as it is a contaminating blazar and not a genuine PS source. +Furthermore, this highlights the possibility of a population +of blazars with steep spectra at high frequencies (ν ≥ 1 GHz) +that aren’t detected in traditional blazar searches and thus may +be contaminating populations of PS sources. Low-frequency +spectral variability thus presents as a new method for identify- +ing blazar candidates. +7. +Conclusion +We have sought to compare detections of spectral variabil- +ity for two PS sources with small scale (∼mas) morphology +and structures. The images produced using observations with +the LBA have identified one resolved and one unresolved PS +source. We have also combined our observations with archival +observations of the host galaxies of our sources to provide +evidence for either the youth or frustration scenario. +We find PMN J0322–4820 is unresolved with the LBA at +2.4 GHz, and pace an upper limit of the source size to be 148 pc, +using a photometric redshift of 0.16. In R21, PMN J0322–4820 +was found to show a changing spectral shape and was presented +as a blazar candidate. Comparing our compact morphology +with the spectral variability of R21, we find PMN J0322–4820 +is consistent with a blazar classification, and suggest high fre- +quency (X-ray or Gamma) to confirm. +We resolve MRC 0225–065 into three components at both +2.4 GHz and 8.3 GHz: a bright central region containing +∼50% of the total flux density, and two fainter regions roughly +equal distance from the central region. In R21 and R22, +MRC 0225–065 was found to show low levels of variability +with a constant spectral shape, and presented as showing vari- +ability due to ISS from a compact morphology with resolved +structure on mas scales. We find the projected linear size to +be 430 pc, using a spectroscopic redshift of 0.445. Using spec- +tral modelling, we calculate the magnetic field assuming a +purely SSA model, and find it is in agreement with the mag- +netic field calculated assuming equipartition. We therefore +conclude MRC 0225–065 is a young CSO, with a PS classifi- + +10 +K. Ross et al. +cation due to SSA. We found the core to have a spectral age of +τspec = 700 ± 100 years, which is consistent with previous age +estimates of young CSO sources of 101 – 105 years (Orienti +et al., 2010; Orienti & Dallacasa, 2020). Furthermore, we use +the spectral age of the core and the upper limit of core size to +calculate and expected expansion velocity (assuming the simple +relation speed = distance/time), and place an upper limit on +the expansion velocity of the lobes to be v = 0.13c, well within +previous measurements of expansion velocities for PS sources +of 0.1c ≲ v ≲ 0.7c (Orienti & Dallacasa, 2020). Lastly, we +use this to estimate the dynamical age of the outer lobes and +estimate their age to be τdyn ≈ 5000 years, again, well within +previous estimates of ages for young PS sources. +Our findings highlight the advantage of spectral variability +in identifying different milliarcsecond structures in PS sources +traditionally acquired using VLBI. Furthermore, we have con- +firmed the use of identifying contaminating sources displaying +only a temporary spectral peak and present spectral variability +as a new method for identifying steep spectrum blazars. We +also suggest future observations of MRC 0225–065 to search +for direct observations of expansion to better constraining the +expansion velocity and age. We recommend observations of +MRC 0225–065 with the VLBA for improved sensitivity and +more u, v-coverage on short baselines to recover more flux +density from extended structures. Likewise, with improved ac- +curacy of the position for MRC 2236-454, we suggest another +VLBI observation. +Acknowledgement +We thank the referees for their comments that improved the +overall quality of this work. KR acknowledges a Doctoral +Scholarship and an Australian Government Research Training +Programme scholarship administered through Curtin Univer- +sity of Western Australia. JRC thanks the Nederlandse Organ- +isatie voor Wetenschappelijk Onderzoek (NWO) for support +via the Talent Programme Veni grant. NHW is supported +by an Australian Research Council Future Fellowship (project +number FT190100231) funded by the Australian Government. +The Long Baseline Array is part of the Australia Telescope +National Facility https://ror.org/05qajvd42 which is funded by +the Australian Government for operation as a National Facility +managed by CSIRO. This work was supported by resources +provided by the Pawsey Supercomputing Centre with funding +from the Australian Government and the Government of West- +ern Australia. LBA data was correlated at the Pawsey Super- +computer Centre using the DiFX software (Deller et al., 2011). +This scientific work uses data obtained from Inyarrimanha +Ilgari Bundara/the Murchison Radio-astronomy Observatory. +We acknowledge the Wajarri Yamaji People as the Traditional +Owners and native title holders of the Observatory site. The +Australian SKA Pathfinder is part of the Australia Telescope +National Facility https://ror.org/05qajvd42 which is managed +by CSIRO. Operation of ASKAP is funded by the Australian +Government with support from the National Collaborative +Research Infrastructure Strategy. ASKAP uses the resources of +the Pawsey Supercomputing Centre. Establishment of ASKAP, +the Murchison Radio-astronomy Observatory and the Pawsey +Supercomputing Centre are initiatives of the Australian Gov- +ernment, with support from the Government of Western Aus- +tralia and the Science and Industry Endowment Fund. This +paper includes archived data obtained through the CSIRO +ASKAP Science Data Archive, CASDA (https://data.csiro.au). +This research made use of NASA’s Astrophysics Data System, +the VizieR catalog access tool, CDS, Strasbourg, France. We +also make use of the IPYTHON package (Pérez & Granger, +2007); SciPy (Virtanen et al., 2020); MATPLOTLIB, a PYTHON +library for publication quality graphics (Hunter, 2007); AS- +TROPY, a community-developed core PYTHON package for +astronomy (Astropy Collaboration et al., 2013; Price-Whelan +et al., 2018); PANDAS, a data analysis and manipulation PYTHON +module (pandas development team, 2020; Wes McKinney, +2010); and NUMPY (van der Walt et al., 2011). We also made +extensive use of the visualisation and analysis packages DS9f +and Topcat (Taylor, 2005). This work was compiled in the +useful online LATEX editor Overleaf. +References +Albareti, F. D., Allende Prieto, C., Almeida, A., et al. 2017, ApJS, 233, 25 +An, T., & Baan, W. A. 2012, ApJ, 760, 77 +Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, +A33 +Bicknell, G. V., Mukherjee, D., Wagner, A. Y., Sutherland, R. S., & Nesvadba, +N. P. H. 2018, MNRAS, 475, 3493 +Bilicki, M., Jarrett, T. H., Peacock, J. A., Cluver, M. E., & Steward, L. 2014, +ApJS, 210, 9 +Briggs, D. S. 1995, PhD thesis, New Mexico Institute of Mining and Tech- +nology +Buchner, J. 2021, The Journal of Open Source Software, 6, 3001 +Callingham, J. R., Gaensler, B. M., Ekers, R. D., et al. 2015, ApJ, 809, 168 +Callingham, J. R., Ekers, R. D., Gaensler, B. M., et al. 2017, ApJ, 836, 174 +Chhetri, R., Morgan, J., Ekers, R. D., et al. 2018, MNRAS, 474, 4937 +Condon, J. J., Cotton, W. D., Greisen, E. W., et al. 1998, AJ, 115, 1693 +D’Abrusco, R., Álvarez Crespo, N., Massaro, F., et al. 2019, ApJS, 242, 4 +Deller, A. T., Brisken, W. F., Phillips, C. J., et al. 2011, PASP, 123, 275 +Fanti, R., Ficarra, A., Mantovani, F., Padrielli, L., & Weiler, K. 1979, A&AS, +36, 359 +Gugliucci, N. E., Taylor, G. B., Peck, A. B., & Giroletti, M. 2005, ApJ, 622, +136 +Hale, C. L., McConnell, D., Thomson, A. J. M., et al. 2021, PASA, 38, e058 +Hardcastle, M. J., & Looney, L. W. 2008, Monthly Notices of the Royal +Astronomical Society, 388, 176 +Hernández-García, L., Panessa, F., Bassani, L., et al. 2019, MNRAS, 489, 4049 +Hinshaw, G., Larson, D., Komatsu, E., et al. 2013, ApJS, 208, 19 +Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90 +Hurley-Walker, N., Gaensler, B. M., Leahy, D. A., et al. 2019, PASA, 36, +e048 +Intema, H. T., Jagannathan, P., Mooley, K. P., & Frail, D. A. 2017, A&A, 598, +A78 +Jarrett, T. H., Cohen, M., Masci, F., et al. 2011, ApJ, 735, 112 +Keim, M. A., Callingham, J. R., & Röttgering, H. J. A. 2019, A&A, 628, A56 +Kellermann, K. I., & Pauliny-Toth, I. I. K. 1981, ARA&A, 19, 373 +Kettenis, M., van Langevelde, H. J., Reynolds, C., & Cotton, B. 2006, in +Astronomical Society of the Pacific Conference Series, Vol. 351, Astronom- +ical Data Analysis Software and Systems XV, ed. C. Gabriel, C. Arviset, +D. Ponz, & S. Enrique, 497 +fhttp://ds9.si.edu/site/Home.html + +Publications of the Astronomical Society of Australia +11 +Large, M. I., Cram, L. E., & Burgess, A. M. 1991, The Observatory, 111, 72 +Large, M. I., Mills, B. Y., Little, A. G., Crawford, D. F., & Sutton, J. M. 1981, +MNRAS, 194, 693 +Lonsdale, C. J., Lacy, M., Kimball, A. E., et al. 2015, ApJ, 813, 45 +Marr, J. M., Perry, T. M., Read, J., Taylor, G. B., & Morris, A. O. 2014, ApJ, +780, 178 +Marscher, A. P. 1983, ApJ, 264, 296 +Martí-Vidal, I., Roy, A., Conway, J., & Zensus, A. J. 2016, A&A, 587, A143 +Massaro, E., Giommi, P., Leto, C., et al. 2009, A&A, 495, 691 +Massaro, E., Maselli, A., Leto, C., et al. 2015, Ap&SS, 357, 75 +Mauch, T., Murphy, T., Buttery, H. J., et al. 2003, MNRAS, 342, 1117 +McConnell, D., Hale, C. L., Lenc, E., et al. 2020, PASA, 37, e048 +McMullin, J. P., Waters, B., Schiebel, D., Young, W., & Golap, K. 2007, in +Astronomical Society of the Pacific Conference Series, Vol. 376, Astronom- +ical Data Analysis Software and Systems XVI, ed. R. A. Shaw, F. Hill, & +D. J. Bell, 127 +Miley, G. 1980, ARA&A, 18, 165 +Murphy, T., Sadler, E. M., Ekers, R. D., et al. 2010, MNRAS, 402, 2403 +O’Dea, C. P. 1998, The Publications of the Astronomical Society of the Pacific, +110, 493 +O’Dea, C. P., & Baum, S. A. 1997, AJ, 113, 148 +O’Dea, C. P., Baum, S. A., & Stanghellini, C. 1991, ApJ, 380, 66 +O’Dea, C. P., & Saikia, D. J. 2021, The Astronomy and Astrophysics Review, +29, 3 +Orienti, M., & Dallacasa, D. 2008, A&A, 487, 885 +—. 2014, MNRAS, 438, 463 +—. 2020, MNRAS, 499, 1340 +Orienti, M., Dallacasa, D., Tinti, S., & Stanghellini, C. 2006, A&A, 450, 959 +Orienti, M., Murgia, M., & Dallacasa, D. 2010, MNRAS, 402, 1892 +Owsianik, I., & Conway, J. E. 1998, A&A, 337, 69 +Pacholczyk, A. G., & Roberts, J. 1971, Physics Today, 24, 57 +pandas development team, T. 2020, pandas-dev/pandas: Pandas, doi:10.5281/ +zenodo.3509134 +Pérez, F., & Granger, B. E. 2007, Computing in Science and Engineering, 9, +21 +Phillips, R. B., & Mutel, R. L. 1982, A&A, 106, 21 +Polatidis, A. G., & Conway, J. E. 2003, PASA, 20, 69 +Price-Whelan, A. M., Sipőcz, B. M., Günther, H. M., et al. 2018, AJ, 156, 123 +Readhead, A. C. S., Taylor, G. B., Pearson, T. J., & Wilkinson, P. N. 1996, +ApJ, 460, 634 +Ross, K., Hurley-Walker, N., Seymour, N., et al. 2022, MNRAS, 512, 5358 +Ross, K., Callingham, J. R., Hurley-Walker, N., et al. 2021, MNRAS, 501, +6139 +Taylor, M. B. 2005, Astronomical Society of the Pacific Conference Series, Vol. +347, TOPCAT & STIL: Starlink Table/VOTable Processing Software +(Shopbell, P. and Britton, M. and Ebert, R.), 29 +Tingay, S. J., & de Kool, M. 2003, AJ, 126, 723 +Tingay, S. J., Macquart, J. P., Collier, J. D., et al. 2015, AJ, 149, 74 +Tinti, S., Dallacasa, D., de Zotti, G., Celotti, A., & Stanghellini, C. 2005, +A&A, 432, 31 +Tinti, S., & de Zotti, G. 2006, A&A, 445, 889 +Torniainen, I., Tornikoski, M., Teräsranta, H., Aller, M. F., & Aller, H. D. +2005, A&A, 435, 839 +Turner, R. J., Shabala, S. S., & Krause, M. G. H. 2018, MNRAS, 474, 3361 +Tzioumis, A. K., Tingay, S. J., Stansby, B., et al. 2010, AJ, 140, 1506 +van Breugel, W., Miley, G., & Heckman, T. 1984, AJ, 89, 5 +van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science +Engineering, 13, 22 +Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, +261 +Wells, D. C. 1985, Nrao’S Astronomical Image Processing System (AIPS), ed. +V. D. Gesù, L. Scarsi, P. Crane, J. H. Friedman, & S. Levialdi (Boston, MA: +Springer US), 195–209 +Wes McKinney. 2010, in Proceedings of the 9th Python in Science Confer- +ence, ed. Stéfan van der Walt & Jarrod Millman, 56 – 61 +Wilkinson, P. N., Booth, R. S., Cornwell, T. J., & Clark, R. R. 1984, Nature, +308, 619 +Wright, E. L. 2006, The Publications of the Astronomical Society of the +Pacific, 118, 1711 +Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868 +Yang, X.-h., Chen, P.-s., & Huang, Y. 2015, MNRAS, 449, 3191 + diff --git a/1tAzT4oBgHgl3EQfDfp4/content/tmp_files/load_file.txt b/1tAzT4oBgHgl3EQfDfp4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..95ee197be2268d7e4833c72dfe35d8b40946ea18 --- /dev/null +++ b/1tAzT4oBgHgl3EQfDfp4/content/tmp_files/load_file.txt @@ -0,0 +1,1058 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf,len=1057 +page_content='Publications of the Astronomical Society of Australia (), 1–11 doi: ARTICLE Milliarcsecond Structures of Variable Peaked-Spectrum Sources K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Ross,1 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Reynolds,2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Seymour,1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Callingham,3,4 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Hurley-Walker,1 and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Bignall5,2 1International Centre for Radio Astronomy Research, Curtin University, Bentley, WA 6102, Australia 2 CSIRO, Space and Astronomy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Box 1130, Bentley, WA 6102, Australia 3Leiden Observatory, Leiden University, PO Box 9513, Leiden, 2300 RA, The Netherlands 4ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, Dwingeloo, 7991 PD, The Netherlands 5Manly Astrophysics, 15/41-42 East Esplanade, Manly, NSW 2095, Australia Author for correspondence: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Ross, Email: kathryn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='ross@icrar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' (Received 03 Aug 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' revised 24 Nov 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' accepted 31 Dec 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' first published online XX) Abstract Spectral variability offers a new technique to identify small scale structures from scintillation, as well as determining the absorption mechanism for peaked-spectrum (PS) radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' In this paper, we present very long baseline interferometry (VLBI) imaging using the Long Baseline Array (LBA) of two PS sources, MRC 0225–065 and PMN J0322–4820, identified as spectrally variable from observations with the Murchison Widefield Array (MWA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We compare expected milliarcsecond structures based on the detected spectral variability with direct LBA imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We find MRC 0225–065 is resolved into three components, a bright core and two fainter lobes, roughly 430 pc projected separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A comprehensive analysis of the magnetic field, host galaxy properties, and spectral analysis implies that MRC 0225–065 is a young radio source with recent jet activity over the last 102–103 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We find PMN J0322–4820 is unresolved on milliarcsecond scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We conclude PMN J0322–4820 is a blazar with flaring activity detected in 2014 with the MWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We use spectral variability to predict morphology and find these predictions consistent with the structures revealed by our LBA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Introduction Peaked-spectrum (PS) sources, are a subset of active galac- tic nuclei (AGN) that are identified by a peak in their radio spectral energy distribution (O’Dea & Saikia, 2021), and are also often associated with compact morphologies (≲ 20 kpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Phillips & Mutel, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Tzioumis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' PS sources provide an interesting population of AGN as the evolutionary pathway from PS source to extended (≳ 30 kpc) AGN is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Two contending theories hypothesise the nature and evolutionary pathway of PS sources: the youth scenario, where the age of the PS source is ≤ 105 years and has not yet had ample time to grow to the large-scale AGN (O’Dea & Baum, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Owsianik & Conway, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Tinti & de Zotti, 2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' and the frustration scenario, when the PS source is confined by a dense cloud of the interstellar medium (ISM) of the host galaxy environment (van Breugel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Wilkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' O’Dea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Furthermore, recent identifications of embedded PS cores within remnant ageing lobes has been attributed to restarted and episodic AGN activity (Hernández- García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2019), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' a cyclical evolution rather than linear evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Compact symmetric objects (CSOs) are a subset of PS sources with similar morphologies to large scale AGN, namely a central region (often quite faint, if detected) with emission either side associated with hot spots and/or lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Unlike typical AGN, CSOs show emission only on very compact scales, typically ≤1 kpc, and thus require high-resolution imaging to detect (Phillips & Mutel, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Gugliucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' CSOs are generally considered young AGN (< 104 yr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' O’Dea & Baum, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Owsianik & Conway, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Tinti & de Zotti, 2006), which may evolve into typical, radio-loud AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Previous attempts to discriminate between youth and frus- tration scenarios have relied on spectral modelling and high- resolution imaging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Marr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Keim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2019) using very long baseline interferometry (VLBI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The cause of absorption at low-frequencies, producing the spectral peak, has typically been attributed to synchrotron-self absorption (SSA) and/or free-free absorption (FFA) for the youth and frustration scenarios respectively (Tingay & de Kool, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Callingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Unfortunately, without sufficient sampling below the spectral turnover, the cause of absorption is often ambiguous (Callingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' In rare cases, a SSA model can be ruled out if the optically thin spectral index is sufficiently steep (α ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Many PS sources have been identified as a CSOs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 0108+388, 0710+439 and 2352+495;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Readhead et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' As CSOs are typically considered to be young AGN, identify- ing PS sources that are also CSOs could help to differentiate between the youth and frustrations scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' However, identi- fying CSOs requires high resolution (mas) observations using VLBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Likewise, PS sources sometimes display extremely asym- metrical mas structures, likely due to an inhomogeneous sur- rounding environment influencing their growth (Orienti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Keim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2019), compared with a fairly symmetrical morphology associated with CSOs with minor asymmetries likely coming from orientation effects (Orienti & Dallacasa, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' VLBI can also be used to measure proper motion of aWe assume a power-law relation where Sν = S0να, thus the sign of α, being negative or positive, also indicates either the optically thin or thick spectral index respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='00977v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='GA] 3 Jan 2023 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' hot-spots in lobes to estimate kinematic ages of ≤ 3×103years (Polatidis & Conway, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Gugliucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2005), consistent with the theory that CSOs are young AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Indeed, Gugliucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' (2005) find a majority of the CSOs with age estimates were ≤ 500 yrs, suggesting CSOs may be short lived and few would continue to grow to the scale of typical AGN, thereby explaining the large fraction of CSO and PS sources thought to be young relative to the number of large-scale radio galaxies (O’Dea & Saikia, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' VLBI of PS sources can thus help to identify populations of CSOs and elucidate the youth scenario and AGN evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Spectral variability at radio frequencies offers a new tech- nique for identifying young or frustrated candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Many variability surveys have identified PS sources that lost their PS classification over time (Tinti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Torniainen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2021, hereafter R21), or showed a signifi- cant change in spectral shape likely due to a variable opacity from the inhomogeneous surrounding ISM (Tingay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2022, hereafter R22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Thus the population of known PS sources, which is already biased from sparse spectral coverage from a range of instruments and times, is likely con- taminated by temporary PS sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This is particularly true at higher frequencies (∼GHz), which is sensitive to emission from the core/jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' PS sources with a peak at lower frequen- cies (∼MHz) appear to be less contaminated by sources only showing a temporary peak (Callingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2017, R21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Spectral variability offers the a new technique to find and exclude contaminating “temporary” PS sources, as well as iden- tify CSO candidates with a decreased risk of contaminating sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Variability of PS sources has been used to infer the presence of compact (µas – mas) features based on scintillation (Fanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Chhetri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2018, R21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Such compact features are common for CSOs, but VLBI is required for con- firmation of a CSO classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Spectral variability has also found PS sources that show changing spectral shape, inconsis- tent with scintillation, which suggests that some PS sources are frustrated or contaminating blazars (R22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This paper aims to investigate the milliarcsecond scale struc- tures of variable PS sources using VLBI to test predictions based on spectral variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' In particular, we investigate PS sources that have shown a consistent spectral shape with a variable overall flux density, consistent with scintillation, suggesting a compact feature on milliarcsecond scales (R21, R22), and use VLBI to test a CSO classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We also investigate variable PS sources that R21 found as changing spectral shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' They concluded the short timescale (∼1 year), and variable spectral shape is inconsistent with interstellar scintillation and present it as a blazar caught flaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' In Section 2, we describe the three variable PS sources of this study, in Section 3 we describe the observational strategy and data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Section 4 outlines the results of the LBA imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We discuss the host galaxy properties including their linear size compared to turnover in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1, the mid- infrared (MIR) and optical emission in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 and the radio properties in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' In Section 6 we present the likely absorption mechanisms and source classification of our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We adopt the standard Λ-cold dark matter cosmological model, with ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='286, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='714, and the Hubble constant H0 = 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='6 km s–1 Mpc–1 (Wright, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Hinshaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2013) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Target Selection Targets were selected for LBA imaging with the goal of com- paring direct imaging of milliarcsecond structures with pre- dicted morphologies based on their variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Three targets were selected based on the variability detected by R21 and R22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' MRC 0225–065 (GLEAM J022744-062106) was initially identified as variable in R21 but further monitoring over a year found no evidence of variability (R22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' As such, it was predicted MRC 0225–065 would have resolved structures on milliarcsecond scales with a compact feature ≲ 25 mas, re- sulting in variability from refractive interstellar scintillation (RISS) on a longer timescale with a dampened modulation index due to the extended structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Conversely, PMN J0322– 4820 (GLEAM J032237–482010) was selected due to the vari- able spectral shape identified in R21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' To explain the variable spectral shape, R21 concluded PMN J0322–4820 was likely a blazar caught flaring in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' As such, it was predicted to show a compact morphology even on milliarcsecond scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Finally, MRC 2236-454 (GLEAM J223933–451414) was identified by R21 as the only PS source in their sample that showed sig- nificant variability but maintained a constant peak frequency below 231 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A low peak frequency is typically associated with PS sources that are of the order of tens of kilo-parsecs across, but the RISS detected by R22 suggested MRC 2236-454 is dominated by a compact feature, and showed variability due to a surrounding inhomogeneous environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' As such, it was predicted MRC 2236-454 may be resolved on milliarcsecond scales and show an asymmetrical morphology, often associ- ated with frustrated sources in an inhomogeneous surrounding environment (Orienti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' LBA Observations and Data Reduction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1 Observations LBA observations were taken on November 23, 2020 and February 17, 2021 as part of project V600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The November observation was centered at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz and the February obser- vation was centered at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz and both utilised 128 MHz of bandwidth in dual polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Stations used in each obser- vation and their diameter is listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Both observa- tions cycled through phase calibrator scans and target scans of lengths 2 min and 5 min, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' However, the spatial separation of each target and their respective phase calibrator meant each target had a different number of scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A summary of the targets, phase calibrators and number of scans each is presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Parkes at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz, and Katherine at both frequencies, ob- served using their native linear feeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' These were converted to a circular polarization basis post-correlation using the Pol- Convert software (Martí-Vidal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2016) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 Data Processing and Calibration After correlation, data calibration and processing were done using the NRAO’s Astronomical Imaging Processing System Publications of the Astronomical Society of Australia 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' LBA stations included in observations Name Code Diameter (m) Nov20 Feb21 ATCA, phased up At 5×22 Y Y Mopra Mp 22 Y Y Parkes Pa 64 Y Y Hobart Ho 26 Y Y Ceduna Cd 30 Y Y Yarragadee Yg 12 Y Y Warkworth Ww 12 Y Y Hartebeesthoek Hh 26 Y Y Katherine Ke 12 Y Y Tidbinbilla Td 34 Y N Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Targets, associated calibrators and number of LBA scans for each target source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Source Name Expected S5GHz (mJy) Number of scans MRC 0225–065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='238 27 PKS J0217+0144 (C) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='666 27 PMN J0322–4820 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='112 40 PMN J0335-4837 (C) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='112 40 MRC 2236–454 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='420 48 QSO B2227–445 (C) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='386 48 (AIPS) (Wells, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The calibration and flagging followed the general procedure outlined in the AIPS cookbookb and was implemented in a semi-automated script with the Parsel- Tongue interface (Kettenis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Initial flagging of edge channels and RFI was done using UVFLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Auto-correlations were scaled to unity across the band using ACCOR before removing gross residual instrumental delays using FRING on a short scan of a bright calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Complex bandpass cor- rections were derived using BPASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The system temperature and gain calibration were applied using APCAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Delay, rate and phase calibrations were determined from fringe fitting using FRING from each target’s respective phase calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A phase referenced image was created for all targets except for MRC 0225–065, as a first pass detection of the targets to determine if a phase shift was needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Lastly, UVFIX was used to apply a phase shift to the data for any sources that were ∼arcsecond away from the phase centre used in correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' MRC 0225–065 had accurate VLBI coordinates and thus did not require a phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The calibrated and phase shifted data were exported to be imaged using CASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 Imaging and Self-Calibration Initial Stokes-I images were made with a quasi-natural weight- ing with robust parameter set to +1 (Briggs, 1995) using the tclean function in CASA (McMullin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Clean boxes were used but were tightly restricted for the models used for self-calibration to avoid inducing artificial structure from the bThe AIPS cookbook can be found here http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='aips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='nrao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='edu/cook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' html complex point-spread-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' For each image, phase only self calibration was performed and applied using the gaincal and applycal functions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Due to the sparse (u, v)- coverage and low signal-to-noise (SNR), calibration solutions were inspected and applied without flagging solutions that had insufficient SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The slow rate of improvement necessi- tated several (∼9) rounds of self-calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The SNR of the main component and the root-mean-squared (rms) noise of the image were inspected after each self calibration iteration to ensure each round improved the overall image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' For each source the initial model assumed for the self-calibration was an unresolved point source to avoid inducing any morpho- logical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Any resolved components were included in subsequent rounds of imaging clean components and kept in the model for self-calibration if this reduced the rms noise of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The initial solution interval for the self calibration was set to the scan length and decreased in further rounds of self calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Phase only self calibration rounds were contin- ued until the rms noise of the image increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A final round of both phase and amplitude self calibration was then performed (provided it reduced the rms of the final image) with the so- lution interval set to the scan length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' For MRC 0225–065, an amplitude self-calibration was applied to both frequencies, but no amplitude self-calibration was applied to the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz image of PMN J0322–4820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Results Images of MRC 0225–065 at both 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz are pre- sented in Figure 1, and an image of PMN J0322–4820 at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz, presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Unfortunately, due to large phase errors from a pointing offset, we were unable to re- cover images for MRC 2236–454 at either frequency, or for PMN J0322–4820 at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz, this was because the source po- sitions were beyond the observed correlated field of view for recovery in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' For MRC 2236–454, the pointing offset was over 11 arcseconds for both the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz ob- servations, thus the phase errors from this pointing offset was beyond recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' PMN J0322–4820 also had a pointing offset of ≈ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5 arcseconds, however, given it was bright (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 Jy), there was sufficient sensitivity using a subset of antennas (flag- ging the Hartebeesthoek antenna), and a phase shift combined with self calibration to recover and image at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' How- ever, this method was not possible at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz due to the smaller field-of-view and decreased sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Henceforth, we will only discuss the results for MRC 0225–065 and PMN J0322– 4820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Properties for each LBA image: synthesised beam size and rms background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Source, ν (GHz) rms (mJy/beam) θbeam,maj θbeam,min PA MRC 0225–065, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0 MRC 0225–065, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='7 83 PMN J0322–4820, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0 30 17 54 4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1 MRC B0225–065 MRC 0225–065 was resolved into three components morphol- ogy at both 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The final image was made with a robust parameter of -1 at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5 at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz (Briggs, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' MRC 0225–065 is resolved into 3 regions: a bright, unresolved central component, with an upper limit of source size of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5 × 4 mas assuming the beam size at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz (labelled C in Figure 1), a fainter 16 × 11 mas Western region (L1) and even fainter 14 × 10 mas Eastern component (L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The sizes of L1 and L2 are measured using the contours in the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The triple morphology is roughly symmetrical with the distance between the C to L1 and L2 being ∼ 40 mas each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Since it appears the components of MRC 0225–065 may be resolved, we measured their flux density over an irregular polygonc for each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We recovered all the flux density predictions from the spectral fit to the R22 ATCA observations at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz, but found that ∼ 35% of the flux density was lost at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The flux densities for each component and their spectral index are presented in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The irregular polygon was shaped based on contour levels to ensure only real flux was included in the final measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' However, the missing flux density at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz may be due to extended structure being resolved out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Consequently, the estimates for the spectral index presented in Table 4 should be considered lower limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Flux densities and two component spectral index for each compo- nent of MRC 0225–065 found in the LBA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The uncertainties for the fluxdensitiesaremeasuredcalculatedusingthemeasureduncertaintyfrom polygon flux and the rms noise of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The uncertainty for α is calcu- lated using standard propagation of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The model prediction is calcu- lated from the best spectral fit, a double SSA spectral model with an expo- nential break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Component S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4GHz (mJy) S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3GHz (mJy) α C 270±10 78±7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='08 L1 121±8 30±5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 L2 56±7 18±4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 Integrated LBA 447±14 126±10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='97±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='07 Model Prediction 400 195 N/A The symmetrical triple morphology suggests MRC 0225– 065 is a CSO candidate with a core (C) and two lobes (L1 and L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The spectral index of the central component is αC = –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='08, which is far steeper than expected for a typical AGN “core", generally expected to have a α ≥ –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5 (Orienti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Hardcastle & Looney, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' However, components have previously been identified as cores with spec- tral indices as steep as –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='7 (Orienti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We present the SED for MRC 0225–065 in Figure 2 including the MWA flux densities from R22 as well as the flux densities and power- law spectral model for each LBA component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The entire SED is fit, using the most recent MWA epoch (2020-09), with a double SSA model with an exponential break, which assumes two synchrotron emitting regions that are self-absorbed and cusing https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='com/nhurleywalker/polygon-flux, (Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2019) ageing producing the exponential break, νb, separate from the peak frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The break frequency is the frequency where the spectrum begins to steepen as the electrons are ageing and experiencing energy losses (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We fit the spectral model using the UltraNest packaged (Buchner, 2021), which uses a nested sampling Monte Carlo algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' From the double SSA spectral model, we find the peak frequen- cies for the two SSA components to be νp,1 =400±100 MHz and νp,2=112±90 MHz, and find νb =14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='7 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' MRC 0225–065 has a spectroscopic redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='445 (Al- bareti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' thus, 1 mas corresponds to a linear scale of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='25 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Using this redshift, we find the projected linear size of MRC 0225–065 (from L1 to L2) to be ∼430 pc, the linear distance from the core to either lobe to be ∼210 pc and place an upper limit on the size of component C to be ≤26 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 PMN J0322–4820 Due to difficulties in the phase calibration, we were only able to produce a high quality image of J0322–483 at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz, shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We do not resolve PMN J0322–4820 and it is confined to the size of the beam: 56 × 40 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The final image was made using a robust parameter of +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5, and by flagging the Hartebeesthoek antenna, thus the beam size for PMN 0322– 4820 compared to MRC 0225–065 for the same frequency is much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Details of the image properties are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Compared to the spectral model fit to the ATCA and 2014 MWA observations, 18% of the flux density was missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We used a reported photometric redshift for PMN J0322–4820 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='16 (Bilicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2014), thus 1 mas corresponds to a linear size of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='650 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We place an upper limit on the source size of 148 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Discussion In this section, we will present a comprehensive analysis of both MRC 0225–065 and PMN J0322–4820 to produce a unified perspective of these two sources with the aim of concluding whether they are young or frustrated PS sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' In Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1, we present our two sources in the linear size and turnover relation, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2, we discuss the host galaxy properties according to mid-infrared, optical observations and radio properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1 Linear Size and Turnover Relation PS sources follow an inverse relation between their linear size and intrinsic turnover frequency, often referred to as the linear size turnover relation, first presented by O’Dea (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This relation is directly predicted from the youth scenario (O’Dea, 1998) where the peak frequency is due to SSA and thus the linear size is directly related to the peak frequency (Keller- mann & Pauliny-Toth, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' While modifications to models in the frustration scenario can reproduce this relation (Bick- nell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2018), it is generally understood that PS sources that fall below the linear size-turnover relation are likely com- pact beyond what is expected for a young source and a thus dhttps://johannesbuchner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='io/UltraNest/ Publications of the Astronomical Society of Australia 5 40 20 0 20 40 40 20 0 20 40 Relative R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' (mas) Relative Dec (mas) C L1 L2 MRC 0225-065 at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4GHz 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5pc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='10 Intensity (Jy/beam) 40 20 0 20 40 40 20 0 20 40 Relative R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' (mas) Relative Dec (mas) C L1 L2 MRC 0225-065 at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3GHz 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5pc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='04 Intensity (Jy/beam) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' LBA images of MRC 0225–065 at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz (lef) and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Beam sizes are shown with a white ellipse in the bottom lef corner of each image and dimensions are specified in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Contours are placed at (-3, 3, 4, 5, 6, 7, 10, 20, 50, 100, 200, 400, 800, 1600) times the rms noise of the image, also specified in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Pixel brightness is plotted in a linear scale following the colour-bars to the right of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The resolved regions are labelled C, L1, L2 and properties of each region are outlined in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Relative R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='A and Dec are calculated from the position of the core (C) component with coordinates: J2000 02h27m44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5s -06d21m06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='7s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='80 Flux Density (Jy) MRC 0225-065 2013 2014 2020-04 2020-05 2020-07 2020-09 ATCA 2020 LBA int C L1 L2 40 20 0 20 40 40 20 0 20 40 Relative R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' (mas) Relative Dec (mas) C L1 L2 MRC 0225-065 Spectral Index Map 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5pc −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 α Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Spectral energy distribution (SED) for MRC 0225–065 (lef) and spectral index map (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The spectral index map was created using by convolving both the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz image and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz image to the same resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Data included in the SED are from R21 and R22 monitoring (circles) and coloured according toepoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' LBAfluxdensitiesareplottedassquareswiththeintegratedfluxdensityofLBAplottedasblacksquares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' ThespectralfittoeachLBApointisapower- law with spectral index presented in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The grey spectral model to the entire SED is a double SSA model with an exponential break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Supplementary data included: TIFR GMRT 150 MHz Sky Survey Alternative Data Release 1 (TGSS-ADR1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Intema, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2017) (grey cross), Molonglo Reference Catalogue (MRC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Large et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 1981, 1991) (grey +), Rapid ASKAP Continuum Survey (RACS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' McConnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Hale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2021) (grey ‘Y’), NRAO VLA Sky Survey (NVSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 1998), Australia Telescope 20 GHz (AT20G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2010) (grey right arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 200 100 0 100 200 200 100 0 100 200 Relative R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' (mas) Relative Dec (mas) C L1 L2 PMN J0322-4820 at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4GHz 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5pc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='08 Intensity (Jy/beam) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='50 Flux Density (Jy) PMN J0322-4820 2013 2014 ATCA 2020 LBA Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' LBA image for PMN J0322–4820 at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz (lef) and associated SED (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The beam size is shown with a white ellipse in the bottom lef corner and dimensions are specified in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Contours are placed at (-3, 3, 4, 5, 6, 7, 10, 20, 50, 100, 200, 400, 800, 1600) times the rms noise of the image, also specified in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Pixel brightness is plotted in a linear scale following the colour-bars to the right of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Relative R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='A and Dec are calculated from the central coordinate: J2000 03h22m38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='0s -48d20m16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Data included in SED is from R21 and R22 (circles) and coloured according to epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' LBA flux density is plotted as a blue square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The grey spectral model to the entire SED is a single SSA model with an exponential break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Supplementary data included is: TIFR GMRT 150 MHz Sky Survey Alternative Data Release 1 (TGSS-ADR1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Intema, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2017) (grey cross), Sydney University Molonglo Sky Survey (SUMSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Mauch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2003) (grey star), Rapid ASKAP Continuum Survey (RACS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' McConnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Hale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2021) (grey ‘Y’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' assumed to be frustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We plot both MRC 0225–065 and PMN J0322–4820 on the linear size-turnover relation in Fig- ure 4, along with other known PS sources, details of which are discussed by Keim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' It is evident from Figure 4, that MRC 0225–065 is entirely consistent with the relation whereas PMN J0322–4820 sits somewhat below the relation, particularly since the linear size is an upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This would suggest MRC 0225–065 is consistent with the youth scenario whereas PMN J0322–4820 may be frustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' However, it is worth nothing, R21 identified PMN J0322–4820 as a variable PS source with a changing spectral shape, and thus concluded it was likely a blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Furthermore, R21 found the peak fre- quency changed from ∼320 MHz in 2013 to ∼145 MHz in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' As the peak frequency is variable and PMN J0322–4820 is known to exhibit a changing spectral shape, its position on the linear size-turnover relation will also vary, shown by the error bar in Figure 4 corresponding to the range of the peak frequency from 2013 to 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Most likely, PMN J0322–4820 is only a temporary PS source and thus should not be included in this relation nor when considering the PS population at large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 Host Galaxy Properties 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1 WISE Colours MIR colour selection techniques using the Wide-Field Infrared Survey Explorer (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2010, WISE) are widely used to efficiently distinguish between AGN and star-forming galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' WISE is a MIR all sky survey covering four photometric bands: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='6, 12, and 22 µm referred to as W1, W2, W3, and W4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The MIR wavelengths are sensitive to the emission from hot dust in the torus of the AGN, allowing for the identification of AGN where X-ray and optical emission 10 2 10 1 100 101 102 Linear Size (kpc) 102 103 104 Rest-Frame Peak Frequency (MHz) J0227-0621 J0322-482 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Rest frame peak frequency versus linear size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Sources in black are describedinKeimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='(2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Thedashedlineisthefittotherelationfound by Orienti & Dallacasa (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Arrows indicate maximum linear sizes for un- resolved sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' MRC 0225–065 (pink circle) and PMN J0322–4820 (purple circle) are plotted with linear sizes calculated from LBA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The error bars for MRC 0225–065 represent the range for peak frequencies calculated in R21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Publications of the Astronomical Society of Australia 7 may be blocked by intervening gas and dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This also makes AGN stand out from star-bursting galaxies or stars due to their extremely red MIR emission (Lonsdale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Obscured AGN with red MIR emission have been identified by their MIR colours, often by their place in a colour-colour diagram (Jarrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Lonsdale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The bulk of sources centred around W1 – W2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 and W2 – W3 = 3 correspond to the region typically associated with quasars and AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' MRC 0225– 065 is found in the region typically associated with emission from star formation or stellar emission;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' there is no evidence of hot AGN dust, however, there is evidence for moderate star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' As we know MRC 0225–065 is an AGN, it is likely the emission at MIR is a combination of these two processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' PMN J0322–4820 is well within the elliptical regime, thus has low emission from star formation and no evidence of hot AGN dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Blazars are typically found to dominate the top right region of the WISE colour-colour plot as the MIR emis- sion is dominated by the emission of the blazar over the galaxy (and associated stellar emission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A compact morphology and variable spectral shape suggest PMN J0322–4820 is a blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' However, the WISE colours of PMN J0322–4820 suggest that the host galaxy is an elliptical with predominantly red optical emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Therefore, the emission from the potential radio blazar is not dominant in the MIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' While it is more common to find blazars in the top right region of the WISE colour- colour plot, the MIR colours, which suggest the host galaxy for PMN J0322–4820 is an elliptical, are still consistent with a blazar classification (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' D’Abrusco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2 Optical Spectra MRC 0225–065 has an optical spectrum from the 13th data release of the Sloan Digital Sky Survey (Albareti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2017, SDSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' From the fitted spectrum, Albareti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' (2017) report a spectroscopic redshift for MRC 0225–065 of z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='445 and classify it as a broad-line, starburst quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The spectrum additionally has low-ionisation nuclear emission-line region (LINER) properties, evident from the strong NII, SiII and OI lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A LINER has a high energy radiation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' There is still debate about whether this is AGN emission or star formation, but likely the combination of the broad lines, strong OIII emission and radio-loudness of MRC 0225–065 is evidence of AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' From the broad Hα, we can calculate the velocity dispersion according to: d(velocity) = cd(λ) λ0 , (1) where c is the speed of light, d(λ) is the wavelength dispersion from the spectral fit, and λ0 is the rest-frame wavelength of Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Using the reported fit to the broad Hα from SDSS where λobserved = 9486 Å, we use the equivalent width, EW= 30±4 Å, and find the velocity dispersion to be 900±100 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This large velocity dispersion may be from an extreme star formation wind but it is also indicative of the broad-line regions from an AGN, which is more consistent given our radio observations identify MRC 0225–065 as an AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The broad Hα, and large velocity dispersion, is consistent with an AGN that is quite obscured, as reported by Albareti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' (2017) who classify it as a broad-line quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Perhaps of more interest are the starburst properties of MRC 0225–065, namely OII and OIII emission lines, identified by Albareti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Both OII and OIII are forbidden lines with different origins: OII is mostly due to star formation and thus is often used as an indicator for star formation in galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' OIII is due to an AGN and can be used as a proxy for the AGN bolometric luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This is also consistent with the WISE colours discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1, which find MRC 0225–065 consistent with a galaxy with emission coming from both the AGN and star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Combining the radio, MIR and optical properties of MRC 0225–065, it is likely this galaxy has moderate star formation with an obscured AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 Radio Properties of MRC B0225–065 Combining the spectral information and high resolution re- solved structure of MRC 0225–065, we are able to determine several intrinsic properties that can help differentiate between SSA and FFA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' In this section, we estimate the magnetic field strength and spectral ages to assess whether MRC 0225– 065 is consistent with the youth scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We do not consider PMN J0322–4820 in this section due to its unresolved mor- phology (even on mas scales) and since the radio variability suggests it is a blazar with an added beaming effect producing Doppler boosting and thus many of the assumptions required for these calculations no longer hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1 Magnetic Field As a means of evaluating the validity of SSA compared to an FFA, we can calculate the magnetic field estimates based on a pure SSA model and on equipartition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Equipartition assumes there is equal energy between the radiating particles and the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The comparison between magnetic field estimates based on an SSA model and equipartition has been used as evidence both for the SSA model (when the estimates are in agreement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Orienti & Dallacasa, 2008) and against (when there is a clear disparity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Keim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' In this section, we will first estimate the magnetic field assuming a purely SSA model, then assuming equipartition and compare these to determine whether SSA is a reasonable model for MRC 0225–065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We can estimate the magnetic field strength, in Gauss, based on a purely SSA spectral model, BSSA, according to: BSSA ≈ (νpeak/f (αthin))5θsrc,min2θsrc,max2 Speak 2(1 + z) , (2) where νpeak is the observed peak frequency in GHz, Speak is the flux density in Jy at the peak frequency for the source at redshift z with angular minor and major component axis, θsrc,min and θsrc,max, in mas (Kellermann & Pauliny-Toth, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We note, f (αthin) is as defined by Kellermann & Pauliny-Toth (1981), where it is loosely related to αthin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We take f (αthin) = 8 based on values from Marscher (1983);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Orienti & Dallacasa (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Now, assuming equipartition, we calculate the magnetic field strength, in Gauss, according to (Miley, 1980), as Bequi by assuming the component has cylindrical symmetry such that the width of the source on the sky is equivalent to the line of sight path-length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' For both calculations, we calculate BSSA and Bequi for the compact core region rather than the total source, to ensure we are comparing a homogeneous region (Orienti & Dallacasa, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Keim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' For MRC 0225–065, using Equa- tion 2, we estimate the magnetic field strength for a purely SSA model to be BSSA ≈6±7 mG for the core region where θsrc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5 × 4 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' To estimate Bequi, we assume a filling fac- tor η = 1 and set k = 1e and find Bequi ≈6±2 mG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' As BSSA is within the uncertainties of Bequi, it suggests the core region of MRC 0225–065 is in equipartition and consistent with a pure SSA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' While this does not exclude the FFA model, it does provide supportive evidence for the SSA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Further- more, it may not be a valid assumption that MRC 0225–065 is in equipartition, thus the equation from Miley (1980) for Bequi would not be a reasonable estimate of the magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We can also use the estimated magnetic field to calculate the age of the electron population as a proxy for the age of the jets/lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Calculating the spectral age of the electron popula- tion requires an accurate estimate of the break frequency, νb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We can thus calculate the spectral age, τspec, according to: τspec = aB1/2 B2 + BiC2 � νb(1 + z) �–1/2 where BiC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='318(1 + z)2 a = �243πme5c2 4µ02e7 �1/2 (3) where BiC is the magnitude of the microwave background magnetic field in nT, B is the magnetic field of the source in nT, νb is the break frequency in GHz, and the constants me, c, µ0, and e are the mass of an electron, speed of light, magnetic permeability of free space, and charge of an electron, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' It is possible the core is actually an unresolved double of more recent AGN activity than the outer lobes, producing the steep (α ≲ –1, see Table 4) spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We assume a constant expansion speed, v, and use the linear sizes to estimate the dynamical age, τdyn, of the core and outer lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Using the magnetic field calculated for the core region assuming equipartition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' setting B = Bequi = 6 ± 2 mG, and deter- mining a break frequency, we can estimate the spectral age of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Using a break frequency of νb = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='7 GHz, calculated from the double SSA spectral model fit, we estimate the spectral age of the core to be τspec ≈ 700 ± 100 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' ek = 1 is equivalent to the minimum energy condition, however values for k have ranged from 1 to 100, where k = 100 produces an order of magnitude difference in Bequi (Pacholczyk & Roberts, 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Miley, 1980) We then calculate an upper limit on the expected expansion velocity of v ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='13 c (using simple speed = distance/time argu- ments) for the core using the upper limit for the linear source size of θsrc ≤ 26 pc, as outlined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' An expansion velocity of v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='13 c is well within previous measurements of the expansion speeds for compact AGN that have been found to range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1 c up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='7 c (Polatidis & Conway, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' An & Baan, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Orienti & Dallacasa, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The range of expansion velocities would correspond to a range in dynamical ages for the core of 100 ≲ τdyn ≲ 900 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' If we assume the expansion velocity of the core of “inner lobes" is roughly equal to that of the outer lobes from a previous epoch of activity, we can place an upper limit on the dynamical ages of the outer lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We calculate the distance between the core and L1 as ∼ 210 pc, which corresponds to a dynamical age of 5000 years for an expansion velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='13 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' For the range of dynamical ages for typical PS sources, we expect the age of the outer lobes to be 1000 ≲ τdyn ≲ 7000 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Previous estimates for the ages of PS sources using similar assumptions have estimated ages from ∼ 101 to ∼ 105 years (Orienti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2010), which is entirely consistent with our age estimates for both the inner core and outer lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' As the ages, expansion velocities, and magnetic fields that we calculate are all consistent with the SSA model and a youth scenario, it appears MRC 0225–065 is more consistent with a young CSO rather than a frustrated compact AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' How- ever, there are several caveats and assumptions made in these calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Thus, while these results are consistent with the evolutionary scenario of MRC 0225–065 being the youth model, it is not sufficient for excluding the frustration scenario entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' AUnifiedPerspectiveofMRCB0225–065andPMNJ0322– 4820 Combining all the information we have obtained about MRC 0225– 065, we begin to create a unified perspective that suggests MRC 0225–065 is a CSO with a peaked spectrum best ex- plained by SSA and recent jet activity over the last 102–103 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A summary of the evidence in support of this conclusion are as follows: Variability: R21 identified spectral variability of MRC 0225– 065 with a constant spectral shape, consistent with vari- ability due to RISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Further spectral variability monitor- ing by R22 detected no further variability, suggesting a resolved structure but consistent PS source classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This observation suggests it is unlikely MRC 0225–065 is a contaminating blazar or source with only a temporary PS source classification, such as frustrated sources with an inhomogeneous surrounding medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Radio morphology: Previously, it has been suggested frustrated PS sources are more likely to show an asymmet- rical morphology due to the asymmetrical environment confining the growth of the lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Inversely, this suggests young PS sources that are not frustrated may be more likely to show a symmetrical morphology like that of a Publications of the Astronomical Society of Australia 9 CSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' MRC 0225–065 has a very symmetrical morphology according to our LBA images, suggesting it may not be interacting with its surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Linear size and turnover relation: We find MRC 0225– 065 is entirely consistent with the linear size turnover rela- tion, a natural product of the youth scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Although, it can be reproduced in certain frustration models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Host galaxy: Using the MIR colours reported in by WISE and the optical spectrum from SDSS, we identify the MRC 0225– 065 as having an obscured AGN with moderate star forma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Since the AGN does not dominate the entire MIR and optical emission, and there is still star formation present, it is possible the AGN has only recently been switched on and thus has not yet quenched all star formation in the galaxy, which is not surprising given the compact size of MRC 0225–065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Magnetic field: Estimating the magnetic field using a purely SSA model and comparing it to the magnetic field calculated assuming equipartition are entirely consistent, suggesting the SSA model is a reasonable model for MRC 0225– 065 Spectral ages: Using spectral modelling of the break fre- quency, we estimate the age of the radio emission (from the core and lobes) to be roughly 700 years, consistent with estimates of the age of PS sources in the youth scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Dynamical ages: Using the linear size from our LBA im- ages and previous measurements of expansion velocity we estimate MRC 0225–065 has two major epochs of activity, one between 1000 to 7000 years ago and another more recently from 100 to 900 years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This is also consistent with previous estimates of the ages for young PS sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Furthermore, due to the missing flux density at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz, this estimate should be considered an upper limit as the spectral indices for each component may be artificially steepened by the missing flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We therefore conclude, MRC 0225–065 is likely a young AGN and with the peak occurring due to SSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Likewise, combining all information of PMN J0322–4820, we can also begin to create a unified picture that PMN J0322– 4820 is a blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A summary of the evidence for this conclusion are: Spectral variability: R21 identified PMN J0322–4820 as a variable source in and classified it as showing a changing spectral shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The dramatic change in spectral shape in the megahertz regime on a timescale of ∼ 1 year is inconsistent with evolutionary models for PS sources and predicted variability due to RISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The changing spectral shape is most easily explained by the dynamical nature of blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Radio morphology: The high resolution image of PMN J0322– 4820 using the LBA found it was still compact on mas scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This is also entirely consistent with a blazar morphology, which appears compact due to orientation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Linear size and turnover relation: PMN J0322–4820 sits well below the linear size and turnover relation typically associated with PS sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This could either be because it is a frustrated source and is thus more compact than expected for it’s predicted age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' However, more likely, is that the temporary peak detected with the MWA in 2014 was a result of the variability of a blazar with effects like Doppler boosting influencing measurements and thus the spectral peak is unrelated to the source age or absorption mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' WISE MIR Colours: PMN J0322–4820 has WISE colours typically associated with elliptical galaxies and/or LERGs/BL Lac blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We therefore identify PMN J0322–4820 as a new blazar where the jets are oriented along the line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' However, PMN J0322– 4820 was not in the ROMA-bzcat catalogue of γ-ray emitting blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This is potentially due to the steep spectrum at fre- quencies over 1 GHz where PMN J0322–4820 is too faint to be detected by traditional blazar searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We suggest further ob- servations using higher frequency observations in the X-ray or γ regimes to search for any high frequency counterpart (Mas- saro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2009, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We conclude PMN J0322–4820 should not be included in any future population studies of PS sources as it is a contaminating blazar and not a genuine PS source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Furthermore, this highlights the possibility of a population of blazars with steep spectra at high frequencies (ν ≥ 1 GHz) that aren’t detected in traditional blazar searches and thus may be contaminating populations of PS sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Low-frequency spectral variability thus presents as a new method for identify- ing blazar candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Conclusion We have sought to compare detections of spectral variabil- ity for two PS sources with small scale (∼mas) morphology and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The images produced using observations with the LBA have identified one resolved and one unresolved PS source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We have also combined our observations with archival observations of the host galaxies of our sources to provide evidence for either the youth or frustration scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We find PMN J0322–4820 is unresolved with the LBA at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz, and pace an upper limit of the source size to be 148 pc, using a photometric redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' In R21, PMN J0322–4820 was found to show a changing spectral shape and was presented as a blazar candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Comparing our compact morphology with the spectral variability of R21, we find PMN J0322–4820 is consistent with a blazar classification, and suggest high fre- quency (X-ray or Gamma) to confirm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We resolve MRC 0225–065 into three components at both 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='4 GHz and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3 GHz: a bright central region containing ∼50% of the total flux density, and two fainter regions roughly equal distance from the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' In R21 and R22, MRC 0225–065 was found to show low levels of variability with a constant spectral shape, and presented as showing vari- ability due to ISS from a compact morphology with resolved structure on mas scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We find the projected linear size to be 430 pc, using a spectroscopic redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Using spec- tral modelling, we calculate the magnetic field assuming a purely SSA model, and find it is in agreement with the mag- netic field calculated assuming equipartition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We therefore conclude MRC 0225–065 is a young CSO, with a PS classifi- 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' cation due to SSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We found the core to have a spectral age of τspec = 700 ± 100 years, which is consistent with previous age estimates of young CSO sources of 101 – 105 years (Orienti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Orienti & Dallacasa, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Furthermore, we use the spectral age of the core and the upper limit of core size to calculate and expected expansion velocity (assuming the simple relation speed = distance/time), and place an upper limit on the expansion velocity of the lobes to be v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='13c, well within previous measurements of expansion velocities for PS sources of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='1c ≲ v ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='7c (Orienti & Dallacasa, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Lastly, we use this to estimate the dynamical age of the outer lobes and estimate their age to be τdyn ≈ 5000 years, again, well within previous estimates of ages for young PS sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Our findings highlight the advantage of spectral variability in identifying different milliarcsecond structures in PS sources traditionally acquired using VLBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Furthermore, we have con- firmed the use of identifying contaminating sources displaying only a temporary spectral peak and present spectral variability as a new method for identifying steep spectrum blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We also suggest future observations of MRC 0225–065 to search for direct observations of expansion to better constraining the expansion velocity and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We recommend observations of MRC 0225–065 with the VLBA for improved sensitivity and more u, v-coverage on short baselines to recover more flux density from extended structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Likewise, with improved ac- curacy of the position for MRC 2236-454, we suggest another VLBI observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Acknowledgement We thank the referees for their comments that improved the overall quality of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' KR acknowledges a Doctoral Scholarship and an Australian Government Research Training Programme scholarship administered through Curtin Univer- sity of Western Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' JRC thanks the Nederlandse Organ- isatie voor Wetenschappelijk Onderzoek (NWO) for support via the Talent Programme Veni grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' NHW is supported by an Australian Research Council Future Fellowship (project number FT190100231) funded by the Australian Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The Long Baseline Array is part of the Australia Telescope National Facility https://ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='org/05qajvd42 which is funded by the Australian Government for operation as a National Facility managed by CSIRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This work was supported by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of West- ern Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' LBA data was correlated at the Pawsey Super- computer Centre using the DiFX software (Deller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This scientific work uses data obtained from Inyarrimanha Ilgari Bundara/the Murchison Radio-astronomy Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We acknowledge the Wajarri Yamaji People as the Traditional Owners and native title holders of the Observatory site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' The Australian SKA Pathfinder is part of the Australia Telescope National Facility https://ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='org/05qajvd42 which is managed by CSIRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Operation of ASKAP is funded by the Australian Government with support from the National Collaborative Research Infrastructure Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' ASKAP uses the resources of the Pawsey Supercomputing Centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Establishment of ASKAP, the Murchison Radio-astronomy Observatory and the Pawsey Supercomputing Centre are initiatives of the Australian Gov- ernment, with support from the Government of Western Aus- tralia and the Science and Industry Endowment Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This paper includes archived data obtained through the CSIRO ASKAP Science Data Archive, CASDA (https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This research made use of NASA’s Astrophysics Data System, the VizieR catalog access tool, CDS, Strasbourg, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We also make use of the IPYTHON package (Pérez & Granger, 2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' MATPLOTLIB, a PYTHON library for publication quality graphics (Hunter, 2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' AS- TROPY, a community-developed core PYTHON package for astronomy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Price-Whelan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' PANDAS, a data analysis and manipulation PYTHON module (pandas development team, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Wes McKinney, 2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' and NUMPY (van der Walt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' We also made extensive use of the visualisation and analysis packages DS9f and Topcat (Taylor, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' This work was compiled in the useful online LATEX editor Overleaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' References Albareti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Allende Prieto, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Almeida, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2017, ApJS, 233, 25 An, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Baan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2012, ApJ, 760, 77 Astropy Collaboration, Robitaille, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Tollerud, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2013, A&A, 558, A33 Bicknell, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Mukherjee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Wagner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Sutherland, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Nesvadba, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2018, MNRAS, 475, 3493 Bilicki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Jarrett, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Peacock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Cluver, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Steward, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2014, ApJS, 210, 9 Briggs, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1995, PhD thesis, New Mexico Institute of Mining and Tech- nology Buchner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2021, The Journal of Open Source Software, 6, 3001 Callingham, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Gaensler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Ekers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2015, ApJ, 809, 168 Callingham, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Ekers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Gaensler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2017, ApJ, 836, 174 Chhetri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Morgan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Ekers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2018, MNRAS, 474, 4937 Condon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Cotton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Greisen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1998, AJ, 115, 1693 D’Abrusco, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Álvarez Crespo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Massaro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2019, ApJS, 242, 4 Deller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Brisken, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Phillips, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2011, PASP, 123, 275 Fanti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Ficarra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Mantovani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Padrielli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Weiler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1979, A&AS, 36, 359 Gugliucci, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Taylor, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Peck, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Giroletti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2005, ApJ, 622, 136 Hale, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', McConnell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Thomson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2021, PASA, 38, e058 Hardcastle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Looney, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2008, Monthly Notices of the Royal Astronomical Society, 388, 176 Hernández-García, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Panessa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Bassani, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2019, MNRAS, 489, 4049 Hinshaw, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Larson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Komatsu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2013, ApJS, 208, 19 Hunter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2007, Computing in Science & Engineering, 9, 90 Hurley-Walker, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Gaensler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Leahy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2019, PASA, 36, e048 Intema, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Jagannathan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Mooley, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Frail, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2017, A&A, 598, A78 Jarrett, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Cohen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Masci, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2011, ApJ, 735, 112 Keim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Callingham, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Röttgering, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2019, A&A, 628, A56 Kellermann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Pauliny-Toth, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1981, ARA&A, 19, 373 Kettenis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', van Langevelde, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Reynolds, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Cotton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2006, in Astronomical Society of the Pacific Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 351, Astronom- ical Data Analysis Software and Systems XV, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Gabriel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Arviset, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Ponz, & S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Enrique, 497 fhttp://ds9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='edu/site/Home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='html Publications of the Astronomical Society of Australia 11 Large, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Cram, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Burgess, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1991, The Observatory, 111, 72 Large, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Mills, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Little, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Crawford, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Sutton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1981, MNRAS, 194, 693 Lonsdale, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Lacy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Kimball, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2015, ApJ, 813, 45 Marr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Perry, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Read, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Taylor, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Morris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2014, ApJ, 780, 178 Marscher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1983, ApJ, 264, 296 Martí-Vidal, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Roy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Conway, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Zensus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2016, A&A, 587, A143 Massaro, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Giommi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Leto, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2009, A&A, 495, 691 Massaro, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Maselli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Leto, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2015, Ap&SS, 357, 75 Mauch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Murphy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Buttery, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2003, MNRAS, 342, 1117 McConnell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Hale, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Lenc, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2020, PASA, 37, e048 McMullin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Waters, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Schiebel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Young, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Golap, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2007, in Astronomical Society of the Pacific Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 376, Astronom- ical Data Analysis Software and Systems XVI, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Shaw, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Hill, & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Bell, 127 Miley, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1980, ARA&A, 18, 165 Murphy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Sadler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Ekers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2010, MNRAS, 402, 2403 O’Dea, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1998, The Publications of the Astronomical Society of the Pacific, 110, 493 O’Dea, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Baum, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1997, AJ, 113, 148 O’Dea, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Baum, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Stanghellini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1991, ApJ, 380, 66 O’Dea, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Saikia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2021, The Astronomy and Astrophysics Review, 29, 3 Orienti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Dallacasa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2008, A&A, 487, 885 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2014, MNRAS, 438, 463 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2020, MNRAS, 499, 1340 Orienti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Dallacasa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Tinti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Stanghellini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2006, A&A, 450, 959 Orienti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Murgia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Dallacasa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2010, MNRAS, 402, 1892 Owsianik, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Conway, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1998, A&A, 337, 69 Pacholczyk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Roberts, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1971, Physics Today, 24, 57 pandas development team, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2020, pandas-dev/pandas: Pandas, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='5281/ zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='3509134 Pérez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Granger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2007, Computing in Science and Engineering, 9, 21 Phillips, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Mutel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1982, A&A, 106, 21 Polatidis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Conway, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2003, PASA, 20, 69 Price-Whelan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Sipőcz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Günther, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2018, AJ, 156, 123 Readhead, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Taylor, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Pearson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Wilkinson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1996, ApJ, 460, 634 Ross, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Hurley-Walker, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Seymour, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2022, MNRAS, 512, 5358 Ross, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Callingham, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Hurley-Walker, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2021, MNRAS, 501, 6139 Taylor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2005, Astronomical Society of the Pacific Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 347, TOPCAT &' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' STIL: Starlink Table/VOTable Processing Software (Shopbell, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' and Britton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' and Ebert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='), 29 Tingay, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & de Kool, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2003, AJ, 126, 723 Tingay, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Macquart, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Collier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2015, AJ, 149, 74 Tinti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Dallacasa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', de Zotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Celotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Stanghellini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2005, A&A, 432, 31 Tinti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & de Zotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2006, A&A, 445, 889 Torniainen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Tornikoski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Teräsranta, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Aller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Aller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2005, A&A, 435, 839 Turner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Shabala, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Krause, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2018, MNRAS, 474, 3361 Tzioumis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Tingay, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Stansby, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2010, AJ, 140, 1506 van Breugel, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Miley, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Heckman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1984, AJ, 89, 5 van der Walt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Colbert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Varoquaux, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2011, Computing in Science Engineering, 13, 22 Virtanen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Gommers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Oliphant, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2020, Nature Methods, 17, 261 Wells, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1985, Nrao’S Astronomical Image Processing System (AIPS), ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Gesù, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Scarsi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Crane, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Friedman, & S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Levialdi (Boston, MA: Springer US), 195–209 Wes McKinney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2010, in Proceedings of the 9th Python in Science Confer- ence, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' Stéfan van der Walt & Jarrod Millman, 56 – 61 Wilkinson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Booth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Cornwell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Clark, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 1984, Nature, 308, 619 Wright, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2006, The Publications of the Astronomical Society of the Pacific, 118, 1711 Wright, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Eisenhardt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Mainzer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2010, AJ, 140, 1868 Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content='-s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=', & Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} +page_content=' 2015, MNRAS, 449, 3191' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfDfp4/content/2301.00977v1.pdf'} diff --git a/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf b/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..45c3faa33e38c99d0491815286b55307739275a4 --- /dev/null +++ b/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c3b7734f65acee4200a3a8b2fce7e83f7c5e9cc3b7fd6b83f6ae3d401870fd9b +size 266979 diff --git a/3NAyT4oBgHgl3EQfb_eW/content/tmp_files/2301.00274v1.pdf.txt b/3NAyT4oBgHgl3EQfb_eW/content/tmp_files/2301.00274v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e3ac610f2d31d95beb59eee47cd9168c1d0d8d5 --- /dev/null +++ b/3NAyT4oBgHgl3EQfb_eW/content/tmp_files/2301.00274v1.pdf.txt @@ -0,0 +1,4259 @@ +CONVERGENCE OF INDUCTIVE SEQUENCES OF SPECTRAL TRIPLES FOR THE +SPECTRAL PROPINQUITY +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +ABSTRACT. In the context of metric geometry, we introduce a new necessary and suf- +ficient condition for the convergence of an inductive sequence of quantum compact +metric spaces for the Gromov-Hausdorff propinquity, which is a noncommutative ana- +logue of the Gromov-Hausdorff distance for compact metric spaces. This condition is +easy to verify in many examples, such as quantum compact metric spaces associated +to AF algebras or certain twisted convolution C*-algebras of discrete inductive limit +groups. Our condition also implies the convergence of an inductive sequence of spectral +triples in the sense of the spectral propinquity, a generalization of the Gromov-Hausdorff +propinquity on quantum compact metric spaces to the space of metric spectral triples. +In particular we show the convergence of the state spaces of the underlying C*-algebras +as quantum compact metric spaces, and also the convergence of the quantum dynamics +induced by the Dirac operators in the spectral triples. We apply these results to new +classes of inductive limit of even spectral triples on noncommutative solenoids and +Bunce-Deddens C*-algebras. Our construction, which involves length functions with +bounded doubling, adds geometric information and highlights the structure of these +twisted C*-algebras as inductive limits. +CONTENTS +1. +Introduction +2 +2. +A Characterization of Convergence in the Propinquity for Inductive Sequences +6 +2.1. +Preliminaries: the Gromov-Hausdorff Propinquity +7 +2.2. +Main result +10 +3. +Convergence of Inductive Sequences of Metric Spectral Triples for the Spectral +Propinquity +21 +3.1. +Preliminaries: The Spectral Propinquity +21 +3.2. +Preliminaries: Inductive Limits of Spectral Triples +25 +3.3. +Main result +26 +4. +Even Spectral Triples on Twisted Group C ∗-algebras +31 +4.1. +Discrete Groups, Proper Length Functions, 2-Cocycles, and Classical +Spectral Triples. +31 +4.2. +The Spectral Triples +32 +4.3. +Main result +39 +References +52 +Date: January 3, 2023. +2000 Mathematics Subject Classification. Primary: 46L89, 46L30, 58B34. +Key words and phrases. Spectral triples, Noncommutative metric geometry, quantum Gromov-Hausdorff +distance, Monge-Kantorovich distance, Quantum Metric Spaces, Quantum Tori, Noncommutative solenoids, +Bunce-Deddens algebras. +1 +arXiv:2301.00274v1 [math.OA] 31 Dec 2022 + +2 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +1. INTRODUCTION +Spectral triples, introduced by Connes in 1985 as a noncommutative generalization of +Dirac operators acting on bundles over manifolds [11, 12], have emerged as a powerful +means to encode geometric information over noncommutative operator algebras. Mo- +tivated in part by ideas from mathematical physics, and by the recurrent usefulness of +various notions of limits of C*-algebras, the second author introduced in [47] a distance +on metric spectral triples, up to an obvious notion of unitary equivalence, thus enabling +the discussion of approximations of certain spectral triples by others, in a geometric +sense. This distance is named the spectral propinquity, and is built from a noncommuta- +tive analogue of the Gromov-Hausdorff distance for noncommutative geometry, called +the Gromov-Hausdorff propinquity [35, 38, 39, 40]. Thus, convergence of spectral triples +is defined as part of a larger framework for convergence of quantum compact metric +spaces, which are noncommutative analogues of algebras of Lipschitz functions over +compact metric spaces. Within this framework, the propinquity was extended to certain +modules over quantum compact metric spaces [48], and even C*-correspondences [46] +with additional metric data inspired by metric connections. The propinquity also was +extended to various dynamical systems [41, 44]. These extensions have been used by the +second author to define the spectral propinquity over metric spectral triples. +The spectral propinquity Λspec has been applied to approximations of spectral triples +on fractals [29] and on quantum tori [45], with the latter example rooted in matrix mod- +els in physics and the problem of their convergence. Indeed, the spectral propinquity +endows the space of all metric spectral triples with its own geometry, and it allows to cap- +ture some geometric intuition within the well understood framework of a topology. For +instance, while quantum tori are not inductive limits of finite dimensional C*-algebras, +spectral triples over quantum tori can now be approximated by spectral triples over full +matrix algebras to arbitrary precision using the spectral propinquity — a common heuris- +tics in mathematical physics, now formalized. Convergence for the spectral propinquity +implies convergence of the state spaces of the underlying algebras for a form of Gromov- +Hausdorff distance, convergence of the quantum dynamics obtained by exponentiating +the Dirac operators, and implies convergence of the spectra and the bounded continuous +functional calculus for the Dirac operators, with implications for the convergence of +physically important quantities such as the spectral actions [31]. +In this paper, we consider the question of when an inductive sequence of metric spectral +triples [20] converges, in the sense of the spectral propinquity, to its inductive limit. To +illustrate the power of our result, besides the class of AF algebras, we construct even +metric spectral triples on noncommutative solenoids [49] and on some Bunce-Deddens +algebras [8, 14] and show that they are limits of metric spectral triples on, respectively, +quantum tori and bundles of full matrix algebras over the circle, in the sense of the +spectral propinquity Λspec. In this way, we provide a noncommutative geometric version +of the fact that solenoid groups can be seen as metric limits of tori, and Bunce-Deddens +algebras are metric limits of algebras of matrix valued functions over the circle. +A spectral triple (A,H , /D) is given by a unital C*-algebra A acting on a Hilbert space +H and a (usually unbounded) self-adjoint operator /D on H , which has bounded com- +mutator with the elements of a dense ∗-subalgebra of A, and has compact resolvent (see +Definition (3.1)). Spectral triples contain much geometric information, including metric +data. Indeed, Connes noted in [12] that spectral triples define a canonical extended +pseudo-distance on the state space of their underlying C*-algebras, which, in particular, + +3 +recovers the geodesic distance when working with the usual spectral triple given by the +Dirac operator acting on the square integrable sections of the spinor bundle of a compact +connected Riemannian spin manifold without boundary. +Rieffel in [55, 56] then cast this metric aspect of noncommutative geometry under +a new light, starting from the observation that Connes’ distance induced by a spectral +triple is a noncommutative analogue of the Monge-Kantorovich metric [27, 28]; it was +thus natural to define a quantum compact metric space as an ordered pair (A,L) of a +unital C*-algebra A and a noncommutative analogue of a Lipschitz seminorm L such +that, in particular, if we set, for any two states ϕ,ψ of A, +mkL(ϕ,ψ) := sup +� +|ϕ(a)−ψ(a)| : L(a) � 1 +� +then mkL is a distance inducing the weak-∗ topology on the state space of A. The exact +list of requirements on the seminorm L have evolved as the study of noncommutative +metric geometry matured, and we will use the definition of a quantum compact metric +space given in [38, 39] and recalled in Definition (2.3). Indeed, a spectral triple whose +Connes’ metric induces the weak-∗ topology on the state space of its underlying C*- +algebra then automatically gives a quantum compact metric space; such a spectral triple +is called a metric spectral triple. +Metric spectral triples may thus be studied within the context of noncommutative +metric geometry. As a result, the second author introduced a distance on the space +of metric spectral triples. The first step in defining this distance, called the spectral +propinquity, is the construction of a noncommutative geometric analogue of the Gromov- +Hausdorff distance [17, 22, 23] between quantum compact metric spaces, which we +will recall in subsection (2.1). The first such analogue was introduced by Rieffel [57], +motivated by the possibility of formalizing certain convergence results found in the +mathematical physics literature. While several such analogues have been offered, we +will work with the Gromov-Hausdorff propinquity Λ∗, introduced by the second author +in [35, 38, 39, 40] precisely to be well adapted to C*-algebras theory and the type of +seminorms given by spectral triples. The propinquity in general is designed precisely +to enable distance computations between quantum compact metric spaces defined on +unrelated C*-algebras, such as between matrix algebra and quantum tori. However, in +this work, we investigate what additional properties of the propinquity we can derive +when we work with inductive limits of C*-algebras. +We begin this work by establishing a characterization of convergence of inductive +limits of quantum compact metric spaces to their inductive limit, in terms of bridge +builders, a type of ∗-automorphism with a natural relation to quantum metrics. +Definition (Definition (2.20)). For each n ∈ N ∪ {∞}, let (An,Ln) be a quantum com- +pact metric space, such that A∞ = cl(� +n∈NAn), where (An)n∈N is an increasing (for ⊆) +sequence of C*-subalgebras of A∞, with the unit of A∞ in A0. +A ∗-automorphism π : A∞ → A∞ is a bridge builder for ((An,Ln)n∈N,(A∞,L∞)) when, +for all ε > 0, there exists N ∈ N such that if n � N, then +∀a ∈ dom(L∞) +∃b ∈ dom(Ln) : +Ln(b) � L∞(a) and ∥π(a)−b∥A∞ < εL∞(a) +and +∀b ∈ dom(Ln) +∃a ∈ dom(L∞) : +L∞(a) � Ln(b) and ∥π(a)−b∥A∞ < εLn(b), +where ∥·∥A∞ is the C*-norm on A∞. + +4 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +Bridge builders are powerful means to prove metric convergence for the propinquity +and notable because it is usually very difficult to find necessary conditions for metric +convergence in the sense of the propinquity (besides the trivial convergence for the +diameters). Thus, this theorem is of independent interest from our study of spectral +triples, and addresses the relationship between inductive limits and limits in a metric +sense as in [47, 35]. Our first main result is therefore the following theorem about +convergence for the propinquity Λ∗ of certain inductive sequences. +Theorem (Theorem (2.22)). For each n ∈ N ∪{∞}, let (An,Ln) be a quantum compact +metric space, where (An)n∈N is an increasing (for ⊆) sequence of C*-subalgebras of A∞ +such that A∞ = cl(� +n∈NAn), with the unit of A∞ in A0. We assume that there exists +∃M > 0 such that for all n ∈ N: +1 +M Ln � L∞ � M ·Ln on dom(Ln). +Then +lim +n→∞Λ∗ ((An,Ln),(A∞,L∞)) = 0, +if, and only if, for any subsequence (Ag(n),Lg(n))n∈N of (An,Ln)n∈N, there exists a strictly +increasing function f : N → N and a bridge builder π for ((Ag◦f (n),Lg◦f (n))n∈N,(A∞,L∞)). +The second step in the construction of the spectral propinquity Λspec on the space of +metric spectral triples is the extension of the Gromov-Hausdorff propinquity to a distance +on the class of C*-correspondences over quantum compact metric spaces endowed with +a form of quantum metric, and with a compatible action of some monoid. The C*- +correspondence associated with a metric spectral triple (A,H , /D) is the Hilbert space +H , seen as a A-C-C*-correspondence, with the quantum metric given by the graph norm +of /D, and with the action of [0,∞) on H given by t ∈ [0,∞) �→ exp(it /D). Convergence for +the spectral propinquity, by design, implies the convergence of the underlying quantum +compact metric spaces, but the converse does not hold in general. These matters will be +recalled in detail in Subsection (3.1). +We then turn to the more specific context of inductive sequences of metric spectral +triples. Inductive sequences of spectral triples were introduced in [20], and are a natural +source of spectral triples; our interest is in the convergence of such sequences for the +spectral propinquity, i.e. in the sense of an actual metric. We establish in the present +work, as our second main result, that an inductive sequence of metric spectral triples +converges for the spectral propinquity when there exists a fully quantum isometric bridge +builder for the underlying sequence of quantum compact metric spaces. Again, it is a +surprising result that a mild strengthening of convergence for the Gromov-Hausdorff +propinquity implies the much stronger convergence for the spectral propinquity, a fact +which does not hold for arbitrary sequences of metric spectral triples, but holds thanks +to the structure of inductive limits. Our second main theorem is given as follows. +Theorem (Theorem (3.17)). Let (A∞,H∞, /D∞) be a metric spectral triple which is the +inductive limit of a sequence (An,Hn, /Dn)n∈N of metric spectral triples, in the sense of +Definition (3.15). For each n ∈ N∪{∞}, let +dom(Ln) := +� +a ∈ An : a = a∗,a dom( /Dn) ⊆ dom( /Dn) and [ /Dn,a] is bounded +� +, +and, for all a ∈ dom(Ln), let Ln(a) be the operator norm of [ /Dn,a]. + +5 +If there exists a bridge builder π : (A∞,L∞) → (A∞,L∞) for ((An,Ln)n∈N,(A∞,L∞)) +which is a full quantum isometry of (A∞,L∞), i.e. such that π(dom(L∞)) ⊆ dom(L∞) and +L∞ ◦π = L∞ on dom(L∞), then +lim +n→∞Λspec((An,Hn, /Dn),(A∞,H∞, /D∞)) = 0. +We conclude our paper with the construction of new even spectral triples on certain +twisted group C*-algebras C ∗(G,σ) where the discrete group G = � +n∈NGn is the union +of a strictly increasing sequence of subgroups Gn of G. These examples include noncom- +mutative solenoids [49] and certain Bunce-Deddens algebras [8]. Our construction is +motivated by the desire to see our new spectral triples over C ∗(G,σ) as limits, for the +spectral propinquity, of an inductive sequence of metric spectral triples constructed over +the inductive sequence (C ∗(Gn,σ))n∈N. This metric aspect distinguishes our spectral +triples from other spectral triples on noncommutative solenoids [1, 2] or Bunce-Deddens +algebras [24], and is applicable, in principle, to many other examples. Moreover, non- +commutative solenoids were shown in [48] to be limits, for the propinquity, of quantum +tori, for a different family of quantum metrics which did not come from a spectral triple. +In general, it is difficult to prove that a given spectral triple is metric. Examples of +metric spectral triples can be found over certain manifolds, quantum tori [12, 15, 16, 34, +45], or more generally, over unital C*-algebras endowed with ergodic actions of compact +Lie groups [21, 55], over certain C*-crossed-products [24], over quantum groups [13], +over Podle´s spheres [3], over AF algebras [7], over certain fractals [10, 30], and more. We +note that there are known examples of spectral triples which are not metric [26]. +It is therefore quite interesting to obtain new examples of metric spectral triples, and +moreover, to prove that they are interesting limits of spectral triples for the spectral +propinquity. We thus establish the following third main result of this paper, which draws +on the first two in its proof. +Theorem (Simplified form of Theorem (4.16)). Let G = � +n∈NGn be an Abelian discrete +group, with (Gn)n∈N a strictly increasing sequence of subgroups of G. Let σ be a 2-cocycle +of G, with values in T := {z ∈ C : |z| = 1}. +Let LH be a length function over G whose restriction to Gn is proper for all n ∈ N, such +that the sequence (Gn)n∈N converges to G for the Hausdorff distance induced on the closed +subsets of G by LH. Let +F : g ∈ G �−→ scale(min{n ∈ N : g ∈ Gn}), +where scale : N → [0,∞) is a strictly increasing function. +If the proper length function L := max{LH,F} satisfies that, for some θ > 1, there exists +c > 0 such that for all r � 1: +��� +g ∈ G : L(g) � θ ·r +��� � c +��� +g ∈ G : L(g) � r +���, +then +lim +n→∞Λspec((C ∗(G,σ),ℓ2(G)⊗C2, /D),(C ∗(Gn,σ),ℓ2(Gn)⊗C2, /Dn)) = 0, +where for all n ∈ N∪{∞} and for all (ξ1,ξ2) in +� +ξ ∈ ℓ2(Gn)⊗C2 : +� +g∈Gn +(LH(g)2 +F(g)2) +��ξ(g) +��2 +C2 < ∞ +� +, +we set +/Dξ : g ∈ G �−→ +�F(g)ξ2(g)+LH(g)ξ1(g) +F(g)ξ2(g)−LH(g)ξ1(g) +� +. + +6 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +In the above spectral triples, C ∗(G,σ) and C ∗(Gn,σ) act via their left regular σ-projective +representations. +We then apply this theorem to construct metric spectral triples on noncommutative +solenoids, i.e. the twisted group C*-algebras C ∗ +�� +Z +� +1 +p +��2 +,σ +� +where +Z +� 1 +p +� +:= +� k +pn : k ∈ Z,n ∈ N +� +, +with p a prime natural number, and where σ is a 2-cocycle of +� +Z +� +1 +p +��2 +. In this case, +using the notation of the above theorem, we choose LH to be the restriction to +� +Z +� +1 +p +��2 +of any norm on R2, while F can be chosen by setting F(g) := p +min +� +n∈N:g∈ +� +1 +pn Z +�2� +for all +g = (g1,g2) ∈ +� +Z +� +1 +p +��2 +. Alternatively, following the ideas of [19], which motivated the +present work, we can choose F(g1,g2) := max{|g1|p,|g2|p} for all g1,g2 ∈ Z +� +1 +p +� +, where +|·|p is the p-adic absolute value. +Similarly, we can apply [52] to see that the Bunce-Deddens algebras are given as the +twisted group C*-algebra C ∗ (Z(α)×Z,σ) for an appropriate choice of a 2-cocycle σ and +a sequence α = (αn)n∈N of nonzero natural numbers such that αn+1 +αn +is a prime number +for all n ∈ N, where the group Z(α) is the subgroup of the circle group T given by all roots +of unity of order αn for n ranging over N. We endow Z(α) with the discrete topology. +The supernatural number number describing the ∗-isomorphism class of the Bunce- +Deddens algebra thus obtained is +� +p|{n∈N: αn+1 +αn =p}|� +p prime . For our purpose, we will work +with sequences α for which +� +αn+1 +αn +� +n∈N is bounded. In this case, we will choose LH to be +the sum or the max (or one of many other choices) of the restriction of a length function +over T to Z(α), and the absolute value on Z. Observing that +Z(α) = +� +n∈N +� +Z⧸αn, +where � +Z⧸m is the group of all m-th roots of unity, we then set F(ζ,z) := min{αn : ζ ∈ +� +Z⧸αn} for all (ζ,z) ∈ Z(α)×Z. This provides a new way to look at Bunce-Deddens algebras +as limits of algebras of continuous sections of bundles of matrix algebras over circles +in a geometric sense, as an echo of the topological fact that they are AT algebras. This +work thus provides an approach to endowing Bunce-Deddens algebras with a different +quantum metric from [29], with the advantage that our quantum metrics are induced by +spectral triples — solving the main difficulty in [29], at least for these Bunce-Deddens +algebras to which our present work applies. +Acknowledgements. This work was partially supported by the Simons Foundation (Si- +mons Foundation collaboration grant #523991 [C. Farsi] and # 31698 [J. Packer].) +2. A CHARACTERIZATION OF CONVERGENCE IN THE PROPINQUITY FOR INDUCTIVE +SEQUENCES +We introduce in this section the notion of bridge builders associated with inductive +sequences of quantum compact metric spaces, which can be used to characterize the + +7 +convergence of such sequences to their inductive limits in the sense of the Gromov- +Hausdorff propinquity. We begin with a review of the notions of quantum compact +metric spaces and propinquity, and then we prove our main theorem, which underlies +all the rest of our work. +2.1. Preliminaries: the Gromov-Hausdorff Propinquity. Our work is concerned with +quantum compact metric spaces, which are noncommutative analogues of the algebras +of Lipschitz functions over a compact metric space. Our definition is the result of a +natural evolution from the notion of compact quantum metric spaces introduced in [55] +by Rieffel, designed as the natural context for the construction of the propinquity. This +subsection will also set some of the basic notation which we will use throughout this +paper. +Notation 2.1. By default, we denote the norm of a normed vector space E by ∥·∥E , and +for us, the set N of natural numbers always contains zero. +Notation 2.2. If A is a unital C*-algebra, then the unit of A will simply be denoted by +1. The state space of the C*-algebra A is denoted by S (A). For any a ∈ A, we write +ℜa = a+a∗ +2 +and ℑa = a−a∗ +2i +. The space {a ∈ A : a = a∗} is denoted by sa(A) and is closed +under the Jordan product a,b ∈ sa(A) �→ ℜ(ab) and the Lie product a,b ∈ sa(A) �→ ℑ(ab), +making sa(A) a Jordan-Lie algebra. +Definition 2.3 ([11, 38, 39, 55, 57, 58]). Fix Ω � 1 and Ω′ � 0. An (Ω,Ω′)-quantum com- +pact metric space (A,L) is given by a unital C*-algebra A and a seminorm L defined on a +dense Jordan-Lie subalgebra dom(L) of sa(A) such that: +(1) {a ∈ dom(L) : L(a) = 0} = R1, +(2) the Monge-Kantorovich metric mkL, defined on the state space S (A) of A, by, +for all ϕ,ψ ∈ S (A): +mkL(ϕ,ψ) := sup +� +|ϕ(a)−ψ(a)| : a ∈ dom(L),L(a) � 1 +� +is a metric which induces the weak-∗ topology on S (A), +(3) for all a,b ∈ sa(A), +max{L(ℜ(ab)),L(ℑ(ab))} � Ω(∥a∥AL(b)+L(a)∥b∥A)+Ω′L(a)L(b); +this inequality being referred to as the (Ω,Ω′)-Leibniz inequality, +(4) the set {a ∈ dom(L) : L(a) � 1} is closed in A. +Any such a seminorm L is called a Lipschitz seminorm on A. +Convention 2.4. By convention, if L is a Lipschitz seminorm on some unital C*-algebra +A, we will write L(a) = ∞ whenever a ∉ dom(L), with the convention that 0∞ = 0 and +∞+x = x+∞ = ∞ for all x ∈ [0,∞]. With this convention, L is lower semicontinuous over +sa(A) as a [0,∞]-valued function (not just on dom(L) but on the entire space sa(A)). +Convention 2.5. Throughout this paper, we fix Ω � 1 and Ω′ � 0. These parameters will +be implicit in our notation; when working with spectral triples, one may always assume +Ω = 1 and Ω′ = 0. +Remark 2.6. If (A,L) is a quantum compact metric space, then we record the following +fact which we shall use repeatedly: if a ∈ dom(L), then L(a +t1) = L(a) for all t ∈ R, since +L(a) = L(a+t1−t1) � L(a+t1)+L(t1) = L(a+t1)+t L(1) +=0 += L(a+t1) � L(a)+tL(1) = L(a). + +8 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +Since the state space of a quantum compact metric space is a compact metric space +for the Monge-Kantorovich metric, it has bounded diameter. Moreover, its diameter can +used to obtain a natural bound on the norm of some self-adjoint elements, which is a +simple but very useful result, which we now recall. +Notation 2.7. The diameter of a metric space (E,d) is denoted by diam(E,d). If (A,L) is +a quantum compact metric space, then we will write qdiam(A,L) for diam +�S (A),mkL +� +. +If E is actually a normed vector space, then we simply write diam(A,E) for the diameter +of any subset A of E for the norm ∥·∥E of E. +We recall the following fact, which we will use repeatedly. +Theorem 2.8 ([55, Propostion 1.6]). If (A,L) is a quantum compact metric space, and if +µ ∈ S (A), then +��a −µ(a)1 +��A � L(a) qdiam(A,L). +Proof. For all ϕ ∈ S (A), we note that |ϕ(a −µ(a)1)| = |ϕ(a)−µ(a)| � L(a)qdiam(A,L). +Since a −µ(a)1 is self-adjoint, we conclude that +��a −µ(a)1 +��A � L(a)qdiam(A,L). +□ +The property difficult to establish when working with quantum compact metric spaces +is, of course, that the Monge-Kantorovich metric induces the weak-∗ topology. Rieffel +provided various characterizations; we will find the following helpful in this paper: +Theorem 2.9 ([51]). Let L be a seminorm defined on some dense subspace dom(L) of +sa(A) for some unital C*-algebra A such that {a ∈ dom(L) : L(a) = 0} = R1. If we set +mkL(ϕ,ψ) = sup +� +|ϕ(a)−ψ(a)| : a ∈ dom(L),L(a) � 1 +� +, for all ϕ,ψ ∈ S (A), then the fol- +lowing assertions are equivalent: +• mkL is a metric on the state space S (A) of A inducing the weak-∗ topology, +• there exists a state µ ∈ S (A) such that {a ∈ dom(L) : L(a) � 1,µ(a) = 0} is totally +bounded in sa(A), +• for all states µ ∈ S (A), the set {a ∈ dom(L) : L(a) � 1,µ(a) = 0} is totally bounded +in sa(A). +We record the following helpful result, which we will also use often. +Corollary 2.10 ([55]). If (A,L) is a quantum compact metric space, µ ∈ S (A), and if K > 0, +then the set +� +a ∈ dom(L) : L(a) � 1,|µ(a)| � K +� +is compact in A. +Proof. We first note that the set +� +a ∈ dom(L) : L(a) � 1,|µ(a)| � K +� +is closed since L is +lower semicontinuous and µ is continuous. +Let (an)n∈N be a sequence in dom(L) such that L(an) � 1 and |µ(an)| � K for all n ∈ N. +Since (|µ(an)|)n∈N is bounded in R, it has a convergent subsequence (|µ(a f (n))|)n∈N. +On the other hand, (a f (n) −µ(a f (n))1)n∈N has a convergent subsequence (a f (g(n)) − +µ(a f (g(n))))n∈N by Theorem (2.9). It now follows that (a f (g(n)))n∈N is a convergent se- +quence. +□ +Quantum compact metric spaces are the points of a (pseudo-)metric space, where +the metric is the Gromov-Hausdorff propinquity, an analogue of the Gromov-Hausdorff +distance in noncommutative geometry. The construction of the propinquity thus relies +on an appropriate notion of quantum isometries. + +9 +Definition 2.11. Let (A1,L1) and (A2,L2) be two quantum compact metric spaces. A Lips- +chitz morphism π : (A1,L1) → (A2,L2) from (A1,L1) to (A2,L2) is a surjective ∗-morphism +π from A1 to A2 such that π(dom(L1)) ⊆ dom(L2). Moreover, if, for all b ∈ dom(L2): +L2(b) = inf{L1(a) : π(a) = b}, +then π is called a quantum isometry. If π is a quantum isometry and a bijection whose +inverse is also a quantum isometry, then π is called a full quantum isometry; in this case +π is a ∗-isomorphism such that for all a ∈ sa(A1): +L2 ◦π(a) = L1(a). +The propinquity is a metric computed by isometrically “embedding” two quantum +compact metric spaces into an arbitrary third one, which in the contravariant picture of +noncommutative geometry, leads us to the following definition for a tunnel. Crucially, a +non-negative number can be associated to a tunnel using the Hausdorff distance. +Notation 2.12. The Hausdorff distance induced by the distance function of a metric +space (X ,d) on the hyperspace of closed subsets of X is denoted by Haus[d]. If N is a +norm on a vector space, we denote by Haus[N] the Hausdorff distance induced by the +metric given by the norm N. By default, if E is a normed vector space, we simplify our +notation and simply write Haus[E] for the Hausdorff distance induced by the distance +defined by the norm ∥·∥E of E. +Notation 2.13. If π : A → B is a unital ∗-morphism, then we define +π∗ : ϕ ∈ S (B) �−→ ϕ◦π ∈ S (A). +Definition 2.14 ([35, Definition 3.1],[40, Definition 2.11,Definition 3.6]). Let (A1,L1) and +(A2,L2) be two quantum compact metric spaces. A tunnel τ = (D,LD,π1,π2) is given by +a quantum compact metric space (D,LD) and two quantum isometries π1 : (D,LD) → +(A1,L1) and π2 : (D,LD) → (A2,L2). The domain dom(τ) of τ is (A1,L1) and the codomain +codom(τ) of τ is (A2,L2). +The extent χ(τ) of τ is the non-negative number: +χ(τ) := max +j∈{1,2}Haus +�mkLD +� � +π∗ +j (S (Aj )),S (D) +� +. +Remark 2.15. We emphasize that all quantum compact metric spaces involved in our +tunnels in this paper must satisfy the same (Ω,Ω′)-Leibniz inequality for our fixed Ω,Ω′. +There always exists a tunnel between any two quantum compact metric spaces, and +the extent of a tunnel is always finite. We thus define: +Definition 2.16. The (dual) Gromov-Hausdorff propinquity Λ∗((A,LA),(B,LB)) be- +tween any two quantum compact metric spaces (A,LA) to (B,LB) is defined by: +Λ∗((A,LA),(B,LB)) := inf +� +χ(τ) : τ tunnel from (A,LA) to (B,LB) +� +. +The (dual) propinquity is well-behaved, as summarized in the following theorem: +Theorem 2.17 ([38, 35]). The dual propinquity is a complete metric up to full quantum +isometry. Moreover, if (Xn,dn)n∈N is a sequence of compact metric spaces, then (Xn,dn)n∈N +converges to a compact metric space (X ,d) for the Gromov-Hausdorff distance if, and +only if limn→∞ Λ∗((C(Xn),Ldn),(C(X ),Ld)) = 0, where Ld denotes the Lipschitz seminorm +induced by any metric d. + +10 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +There are several interesting known examples of convergence for the propinquity, in- +cluding approximations of quantum tori by fuzzy tori [33], approximations of spheres by +matrix algebras [9], continuity of quantum tori in their cocycle parameter [33], continuity +of UHF algebras with respect to the Baire space seen as their natural parameter space, +continuity of the Effros-Shen algebras in their irrational parameters [5], and more. +2.2. Main result. We begin with a simple sufficient condition to ensure that a seminorm +is indeed a Lipschitz seminorm on an inductive limit of unital C*-algebras, when each +of the C*-subalgebra in the inductive sequence is already equipped with a Lipschitz +seminorm. This condition is quite natural and generalizes, for instance, the idea behind +the construction of Lipschitz seminorms on AF algebras in [5]. +Proposition 2.18. Let A∞ be a unital C*-algebra. For each n ∈ N, let (An,Ln) be a quan- +tum compact metric space, where (An)n∈N is an increasing sequence of C*-subalgebras +of A∞ with the unit of A∞ in A0. Assume moreover that A∞ = cl(� +n∈NAn). Let L∞ be a +seminorm defined on a dense Jordan-Lie subalgebra dom(L∞) of sa(A∞), such that: +(1) {a ∈ dom(L∞) : L∞(a) = 0} = R1, +(2) the unit ball of L∞ is closed in A∞, +(3) L∞ is (Ω,Ω′)-Leibniz. +If there exists a unital isometric positive linear map π : A∞ → A∞ such that, for all +ε > 0, there exists N ∈ N with the property that: +∀a ∈ dom(L∞) +∃b ∈ dom(LN) : +LN(b) � L∞(a) and ∥π(a)−b∥A∞ < εL∞(a), +then (A∞,L∞) is a quantum compact metric space. +Proof. Let µ ∈ S (A∞). By assumption, µ ∈ S (An) for all n ∈ N — where we use the same +symbol µ to denote the restriction of µ to An. Let +B∞ := +� +a ∈ dom(L∞) : µ◦π(a) = 0,L∞(a) � 1 +� +. +Now, let ε > 0 and let n ∈ N. We set +Bn := +� +a ∈ dom(Ln) : |µ(a)| < ε +4,Ln(a) � 1 +� +. +Let a ∈ Bn, and let ϕ ∈ S (An). By Theorem (2.8), we have the following inclusion: +Bn ⊆ +� +a ∈ dom(Ln) : Ln(a) � 1,∥a∥An � qdiam(An,Ln)+ ε +4 +� +and the latter set is compact since Ln is a Lipschitz seminorm, by Corollary (2.10). So Bn +is totally bounded. In fact, since Ln is lower semicontinuous and µ is continuous, the set +Bn is also closed in the complete space A∞, so Bn is compact. +By assumption on π, there exists N ∈ N such that +∀a ∈ B∞ +∃b ∈ dom(LN) : +LN(b) � 1 and ∥π(a)−b∥A∞ < ε +4. +In particular, if a ∈ B∞ and b ∈ dom(LN) with LN(b) � 1 and ∥π(a)−b∥A∞ < ε +4, then +|µ(b)| � ∥b −π(a)∥A∞ +|µ(π(a))| < ε +4, so b ∈ BN. +Since BN is compact in sa(AN) by Corollary (2.10), there exists a ε +4-dense subset +F ⊆ BN of BN. So +Haus[A∞](π(B∞),F) � Haus[A∞](π(B∞),BN)+Haus[A∞](BN,F) < ε +2. + +11 +The domain dom(L∞) is dense in sa(A), so it is not empty and thus {a ∈ dom(L∞) : +L∞(a) � 1} is not empty, since L is a seminorm. Thus, by Remark (2.6), the set B∞ is not +empty as well. We thus obtain: +� ̸= B∞ = +� +b∈F +� +a ∈ B∞ : ∥π(a)−b∥A∞ < ε +2 +� +. +Therefore, if we define +G := +� +b ∈ F : +� +a ∈ B∞ : ∥π(a)−b∥A∞ < ε +2 +� +̸= � +� +, +then G ̸= � and B∞ = � +b∈G +� +a ∈ B∞ : ∥π(a)−b∥A∞ < ε +2 +� +. For each b ∈ G, we pick t(b) ∈ +B∞ such that ∥π(t(b))−b∥A∞ < ε +2. Let now a ∈ B∞. There exists b ∈ G such that +∥π(a)−b∥A∞ < ε +2. Then +∥a − t(b)∥A∞ = ∥π(a − t(b))∥A∞ +� ∥π(a)−b∥A∞ +∥b −π(t(b))∥A∞ +< ε +2 + ε +2 = ε. +Thus, t(G) is a ε-dense subset of B∞. So B∞ is totally bounded in A∞. Therefore, noting +that µ◦π is a state of A∞, we conclude by Theorem (2.9) that mkL∞ induces the weak-∗ +topology on S (A∞). Since all other required properties are assumed, L∞ is indeed a +Lipschitz seminorm. +□ +The next natural question is to find a sufficient condition to strengthen Proposition +(2.18) and obtain convergence of the sequence (An,Ln)n∈N to (A∞,L∞) in the sense of +the propinquity. To this end, we introduce the notion of a bridge builder — a map which, +among other things, satisfy the condition in Proposition (2.18). In fact, we basically “sym- +metrize” the condition in Proposition (2.18) and require that we work with ∗-morphism +(which will allow us to construct seminorms with the Leibniz property), rather than just +positive linear maps. +Notation 2.19. We will write N := N∪{∞} for the one point compactification of N. +Definition 2.20. For each n ∈ N∪{∞}, let (An,Ln) be a quantum compact metric space, +where (An)n∈N is an increasing (for ⊆) sequence of C*-subalgebras of A∞ such that +A∞ = cl(� +n∈NAn) and the unit of A∞ is in A0. +A ∗-automorphism π : A∞ → A∞ is a bridge builder for ((An,Ln)n∈N,(A∞,L∞)) when, +for all ε > 0, there exists N ∈ N such that if n � N, then +∀a ∈ dom(L∞) +∃b ∈ dom(Ln) : +Ln(b) � L∞(a) and ∥π(a)−b∥A∞ < εL∞(a) +and +∀b ∈ dom(Ln) +∃a ∈ dom(L∞) : +L∞(a) � Ln(b) and ∥π(a)−b∥A∞ < εLn(b). +Proposition 2.21. For each n ∈ N∪{∞}, let (An,Ln) be a quantum compact metric space, +where (An)n∈N is an increasing (for ⊆) sequence of C*-subalgebras of A∞ such that A∞ = +cl(� +n∈NAn) and the unit of A∞ is in A0. +If there exists a bridge builder for ((An,Ln)n∈N,(A∞,L∞)), then +lim +n→∞Λ∗((An,Ln),(A∞,L∞)) = 0. +Proof. Let π : A∞ → A∞ be the given bridge builder. Let ε > 0. There exists N ∈ N such +that if n � N, then +• ∀a ∈ dom(L∞) +∃b ∈ dom(Ln) : +Ln(b) � L∞(a)∧∥π(a)−b∥A∞ < εL∞(a), + +12 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +• ∀b ∈ dom(Ln) +∃a ∈ dom(L∞) : +L∞(a) � Ln(b)∧∥π(a)−b∥A∞ < εLn(b). +Fix n � N. We define, for all a ∈ dom(L∞) and b ∈ dom(Ln): +Tn(a,b) := max +� +L∞(a),Ln(b), 1 +ε ∥π(a)−b∥A∞ +� +. +It is a standard argument that (A∞ ⊕An,Tn) is a quantum compact metric space: +(1) the domain dom(Tn) = dom(L∞)⊕dom(Ln) of Tn is dense in sa(A∞ ⊕An) since +dom(L∞) is dense in sa(A∞) and dom(Ln) is dense in sa(An), +(2) if Tn(a,b) = 0 for some (a,b) ∈ dom(Tn), then L∞(a) = 0 so a = t1 for some t ∈ R, +and Ln(b) = 0 so b = s1 for some s ∈ R (it matters here that the unit is the same +in A∞ and An), and 0 = ∥π(a)−b∥A∞ = |t − s| so (a,b) = t(1,1); +(3) Tn is the maximum of two lower semicontinuous functions and one continuous +function, so it is lower semicontinuous over sa(A∞ ⊕An); +(4) a direct computation shows that Tn is (Ω,Ω′)-Leibniz since L∞ and Ln both are, +and π is a ∗-morphism; +(5) fixing any state µ of A∞ and setting ϕ : (a,b) ∈ A∞⊕An �→ µ(a), then ϕ ∈ S (A∞⊕ +An), and +� +(a,b) ∈ dom(Tn) : Tn(a,b) � 1,ϕ(a,b) = 0 +� +⊆ +� +a ∈ dom(L∞) : L∞(a),µ(a) = 0 +� +× +� +b ∈ dom(Ln) : Ln(b) � 1,|µ◦π−1(b)| � ε +� +and, as seen in the proof of Proposition (2.18), the set on the right hand side is +a product of two compact set, and thus compact; thus the set on the left hand +side is compact (closed in a compact set) and thus, Tn is indeed a Lipschitz +seminorm, invoking Theorem (2.9). +We now check that τn := (A∞ ⊕An,Tn,ψn,θn), with ψn : (a,b) ∈ A∞ ⊕An �→ a ∈ A∞ +and θn : (a,b) ∈ A∞ ⊕An �→ b ∈ An, is a tunnel, in the sense of Definition (2.14). +Let a ∈ dom(L∞). By assumption, there exists b ∈ dom(Ln) with Ln(b) � L∞(a) and +∥π(a)−b∥A∞ < εL∞(a). Therefore, Tn(a,b) = L∞(a). Since by construction, Tn(a,c) � +L∞(a) for all a ∈ dom(L∞) and c ∈ dom(Ln), we have shown that ψn is a quantum +isometry by Definition (2.11). +Let now b ∈ dom(Ln). Again by assumption on π, there exists a ∈ dom(L∞) such +that ∥π(a)−b∥A∞ < εLn(b) and L∞(a) � Ln(b). Thus Tn(a,b) = Ln(b). Once again, +Tn(c,b) � Ln(b) by construction for all c ∈ dom(L∞), so θn is indeed a quantum isometry, +so τn is a tunnel. +We now compute the extent of τn, in the sense of Definition (2.14). Let ϕ ∈ S (A∞ ⊕ +An). Using Hahn-Banach theorem, we extend ϕ to a state ϕ′ of A∞ ⊕A∞. Let µ : a ∈ +A∞ �→ ϕ′(a,π(a)); since π is a unital ∗-morphism, µ is a state of A∞. By construction, if +Tn(a,b) � 1 then ∥π(a)−b∥A∞ � ε and thus +|ϕ(a,b)−µ◦ψn(a,b)| = |ϕ′(a,b)−ϕ′(a,π(a))| +� |ϕ′(0,b −π(a))| +� ∥b −π(a)∥A∞ � ε. +Thus Haus +�mkTn +� +(ψ∗ +n(S (A∞)),S (A∞ ⊕An)) � ε. + +13 +Let now µ′ : b ∈ An �→ ϕ(π−1(b),b). Since π is a ∗-automorphism of A∞, the map µ′ is +a state of An. Moreover: +|ϕ(a,b)−µ′ ◦θn(a,b)| = |ϕ(a,b)−ϕ(π−1(b),b)| += |ϕ(a −π−1(b),0)| +� +��a −π−1(b) +��A∞ += ∥π(a)−b∥A∞ � ε. +Thus Haus +�mkTn +� +(θ∗ +n(S (An)),S (A∞)) � ε. +Hence, the extent χ(τn) of τn is at most ε. By Definition (2.16), we thus have shown +that for all n � N, +(2.1) +Λ∗((An,Ln),(A∞,L∞)) � ε, +which concludes our proof. +□ +Our main result in this section is the following theorem, which shows that the natural +sufficient condition in Definition (2.20) and Proposition (2.21) is, in fact, very close to +necessary, under a mild and natural condition. This is notable because in general, it +is difficult to exhibit nontrivial necessary conditions for convergence in the sense of +the propinquity (besides, say, the fact that diameters must converge). It also shows +that the existence of bridge builders is the natural setup for establishing convergence +of inductive limits in the sense of the propinquity, thus providing a complete answer +for the relationship between convergence of inductive sequences of quantum compact +metric spaces in the categorical sense and the propinquity sense, under a commonly +met condition. +Theorem 2.22. For each n ∈ N∪{∞}, let (An,Ln) be a quantum compact metric space, +where (An)n∈N is an increasing (for ⊆) sequence of C*-subalgebras of A∞ such that A∞ = +cl(� +n∈NAn) and the unit of A∞ is in A0. We assume that there exists M > 0 such that for +all n ∈ N: +1 +M Ln � L∞ � M ·Ln on dom(Ln). +Then +lim +n→∞Λ∗ ((An,Ln),(A∞,L∞)) = 0, +if, and only if, for any subsequence (Ag(n),Lg(n))n∈N of (An,Ln)n∈N, there exists a strictly +increasing function f : N → N and a bridge builder π for ((Ag◦f (n),Lg◦f (n))n∈N,(A∞,L∞)). +Proof. First, assume that for any subsequence (Ag(n),Lg(n))n∈N, there exists a strictly +increasing function f : N → N and a bridge builder π for ((Ag◦f (n),Lg◦f (n))n∈N,(A∞,L∞)). +By Proposition (2.21), we conclude that every subsequence of (An,Ln)n∈N has a subse- +quence converging to (A∞,L∞). Therefore, (An,Ln)n∈N converges to (A∞,L∞) since the +propinquity is, indeed, a metric (up to full quantum isometry). +Let us now assume that (An,Ln)n∈N converges to (A∞,L∞) for the propinquity. Since +any subsequence will converge as well, it is sufficient to prove our statement for g being +the identity, and this will simplify our notation. +Since (An,Ln)n∈N converges to (A∞,L∞), there exists a sequence +(τn)n∈N := (Dn,Tn,ψn,θn)n∈N +of tunnels, as in Definition (2.14), with limn→∞ χ(τn) = 0, while, for each n ∈ N, we +have dom(τn) = (A∞,L∞) and codom(τn) = (An,Ln). To ease notation, the target set + +14 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +of a ∈ dom(L∞) with l � L∞(a) defined by τn will be denoted by tn (a|l), rather than +tτn (a|l); we recall from [35, 38] that: +tn (a|l) = +� +θn(d) : d ∈ ψ−1 +n ({a}),Tn(d) � l +� +. +This proof heavily relies on the properties of target sets, as discussed in [35, 38, 39, 40]. In +[35], various estimates which we will refer to in this proof are expressed using the length +λ(τ) of a tunnel τ, rather than the extent χ(τ); however as seen in [40, Proposition 2.12], +for any tunnel τ, we have λ(τ) � χ(τ) � 2λ(τ). We will use this inequality without further +mention to express all our results here in terms of extents. +Claim 2.23. For all a ∈ dom(L∞), there exists a strictly increasing function f : N → N +and an element π(a) ∈ dom(L∞) such that, for all l � L∞(a), +lim +n→∞Haus[A∞] +�tf (n) (a|l),{π(a)} +� += 0, +and ∥π(a)∥A∞ = ∥a∥A∞. +Proof of Claim (2.23). First, since the sequence (χ(τn))n∈N converges (to 0), it is bounded; +let K ′ > 0 such that χ(τn) � K ′ for all n ∈ N. +Let a ∈ dom(L∞). Let l = L∞(a). For any K > 0, let +A∞[K ] := +� +b ∈ dom(L∞) : L∞(b) � K ,∥b∥A∞ � ∥a∥A∞ +K K ′� +. +The set A∞[K ] is compact in sa(A∞) by Corollary (2.10). By [35, Corollary 4.5] and since +L∞ � MLn on dom(Ln), the sequence (tn (a|l))n∈N is a sequence of compact subsets of +A∞[Ml], and +lim +n→∞diam(tn (a|l),A∞) = 0. +Since A∞[Ml] is compact in A∞, the Hausdorff distance Haus[A∞] induced on the set +of closed subsets of A∞[Ml] by the norm ∥·∥A∞ of A∞ gives a compact topology as well. +Therefore, there exists a subsequence (tf (n) (a|l))n∈N of (tn (a|l))n∈N which converges, +for Haus[A∞], to a singleton {π(a)} of A∞[Ml]. In particular, L∞(π(a)) � Ml = ML∞(a). +Let now L � l. By definition, tf (n) (a|l) ⊆ tf (n) (a|L) for all n ∈ N and +lim +n→∞diam +�tf (n) (a|L),A∞ +� += 0, +so we conclude easily as well that +lim +n→∞Haus[A∞] +�tf (n) (a|L),{π(a)} +� += 0. +By [35, Proposition 4.4], we also note that if bn ∈ tf (n) (a|l) for each n ∈ N, then +∥π(a)∥A∞ = lim +n→∞∥bn∥A∞ � limsup +n→∞ +� +∥a∥A∞ +χ +� +τf (n) +� +l +� += ∥a∥A∞ . +Similarly, since a ∈ tτ−1 +f (n) (bn|l), we also have +∥a∥A∞ � limsup +n→∞ +� +∥bn∥A∞ +lχ +� +τf (n) +�� += ∥π(a)∥A∞ . +So indeed, ∥π(a)∥A∞ = ∥a∥A∞. This proves our claim. +Q.E.D. +Claim 2.24. There exists a unital ∗-endomorphism π of A∞ such that π(dom(L∞)) ⊆ +dom(L∞), and a strictly increasing function f : N → N such that, for all a ∈ dom(L∞), +and for all l � L∞(a), +lim +n→∞Haus[A∞] +�tf (n) (a|l),{π(a)} +� += 0. + +15 +Proof of Claim (2.24). Since A∞ is separable, there exists a countable dense subset S∞ +of sa(A∞) with S∞ ⊆ dom(L∞). Using Claim (2.23), a diagonal argument shows that +there exists a strictly increasing sequence f : N → N such that, for all a ∈ S∞ and for all +l � L∞(a), we have limn→∞Haus[A∞] +�tf (n) (a|l),{π(a)} +� += 0. +Let now a ∈ dom(L∞), and let l � L∞(a). Let ε > 0. Since S∞ is dense in dom(L∞), +there exists aε ∈ dom(L∞) such that ∥a − aε∥A∞ < ε +5. Note that L∞(aε) < ∞ but in general, +there is no relation between L∞(aε) and L∞(a). +Let l = max{L∞(a),L∞(aε)}. Since it is convergent for the Hausdorff distance Haus[A∞], +the sequence +�tf (n) (aε|l) +� +n∈N is Cauchy for Haus[A∞]. +Therefore, there exists N ∈ N such that, for all p,q � N, we have +Haus[A∞] +�tf (p) (aε|l),tf (q) (aε|l) +� +< ε +5. +Since limn→∞ χ +� +τf (n) +� += 0, there exists N ′ ∈ N such that if n � N ′ then χ +� +τf (n) +� +< +ε +5(l+1). Therefore, if n � N ′, then by [35, Corollary 4.5], +Haus[A∞] +�tf (n) (a|l),tf (n) (aε|l) +� +� ∥a − aε∥A∞ +lχ +� +τf (n) +� +< 2ε +5 . +Let now p,q � max{N,N ′}. We compute: +Haus[A∞] +�tf (p) (a|l),tf (q) (a|l) +� +� Haus[A∞] +�tf (p) (a|l),tf (p) (aε|l) +� ++Haus[A∞] +�tf (p) (aε|l),tf (q) (aε|l) +� ++Haus[A∞] +�tf (q) (aε|l),tf (q) (a|l) +� +< 2ε +5 + ε +5 + 2ε +5 = ε. +Thus, +�tf (n) (a|l) +� +n∈N is Cauchy for Haus[A∞]. Since sa(A∞) is complete, so is the +set of all closed subsets of sa(A∞) with the Hausdorff distance Haus[A∞]. Therefore, +�tf (n) (a|l) +� +n∈N converges to some compact subset in sa(A∞). In fact, since +lim +n→∞diam +�tf (n) (a|l),A∞ +� += 0 +by [35, Corollary 4.5], the sequence +�tf (n) (a|l) +� +n∈N converges to some singleton. As +observed in Claim (2.23), this limit does not depend on l; we denote it by {π(a)}. Again +using the same argument, we also note that ∥π(a)∥A∞ = ∥a∥A∞. +Since L∞ is lower semicontinuous over A∞, and since by construction, π(a) is the limit +in A∞ of any sequence (bn)n∈N with bn ∈ tf (n) (a|L∞(a)) for all n ∈ N, we also conclude +that +L∞(π(a)) � liminf +n→∞ L∞(bn) +by lower semicontinuity of L∞, +� liminf +n→∞ M ·Ln(b) +since L∞ � M ·Ln for all n ∈ N, +� M ·L∞(a) +since Ln(b) � L∞(a), as b ∈ tf (n) (a|L∞(a)) . +Let a,a′ ∈ dom(L∞). Let t ∈ R. Since tf (n) (a|l)+t ·tf (n) +� +a′��l +� +⊆ tf (n) +� +a + ta′��(1+|t|)l +� +for all n ∈ N by [35, Corollary 4.5], we immediately conclude that {π(a)} + t · {π(a′)} ⊆ +{π(a + ta′)}, i.e. π is linear. A similar argument shows that π is a Jordan-Lie morphism +over dom(L∞), using [35, Proposition 4.8]. +As a linear map π with ∥π(a)∥A∞ = ∥a∥A∞ for all a ∈ dom(L∞), we can uniquely +extend π to sa(A∞) as a uniformly continuous map over sa(A∞); this map is of course +again a Jordan-Lie morphism from sa(A∞) to sa(A∞) and an isometry. + +16 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +A straightforward argument shows that we can uniquely extent π to a continuous +Jordan-Lie algebra endomorphism of A∞, and thus π thus extended is a unital ∗-endo- +morphism with L∞ ◦π � L∞ over dom(L∞). +We already know that π is an isometry on sa(A∞) and a ∗-morphism, so it is injective +on A∞: if π(a) = 0 then π(ℜa) = 0 so ℜa = 0, and π(ℑa) = 0 so ℑa = 0, and thus a = 0. In +particular, since π is now an injective ∗-morphism, it is an isometry on A∞ (rather than +just sa(A∞)). This proves our claim. +Q.E.D. +Claim 2.25. For all ε > 0, there exists N ∈ N such that for all n � N, and for all a ∈ +dom(L∞) with L∞(a) � 1, we have +Haus[A∞] +� +{π(a)},tf (n) (a|1) +� +< ε. +Proof of Claim (2.25). Let ε > 0. Fix µ ∈ S (A∞). The set +B := +� +a ∈ dom(L∞) : L∞(a) � 1,µ(a) = 0 +� +is compact in sa(A∞) by Theorem (2.9). Therefore, there exists a finite subset F ⊆ B such +that Haus[A∞](F,B) < ε +4. Since F is finite, by Claim (2.24), there exists N ∈ N such that, +for all a ∈ F and for all n � N, we have Haus[A∞] +� +{π(a)},tf (n) (a|L∞(a)) +� +< ε +4. Moreover, +there exists N ′ ∈ N such that, if n � N ′, then χ(τn) < ε +4. +Now, let n � max{N,N ′}, a ∈ B and b ∈ tf (n) (a|1). There exists a′ ∈ F such that +��a − a′��A∞ < ε +4. Let b′ ∈ tf (n) +� +a′��1 +� +. By [35, Corollary 4.5], we compute the following +expression: +∥π(a)−b∥A∞ � +��π(a)−π(a′) +��A∞ + +��π(a′)−b′��A∞ + +��b′ −b +��A∞ +� +��π(a − a′) +��A∞ +π is linear ++ +ε +4 +by choice of N ++ +��a − a′��A∞ +χ(τn) +by [35] +� 2 +��a − a′��A∞ + ε +4 + ε +4 +� ε +2 + ε +4 + ε +4 = ε. +We thus have proven our uniform convergence claim over B. Let now a ∈ dom(L∞). +Then of course, a − µ(a)1 ∈ B, since L∞(a − µ(a)1) � L∞(a) + L∞(µ(a)1) = L∞(a) � 1 +(in fact, L∞(a) = L∞(a − µ(a)1)). If b ∈ tf (n) (a|1) then b − µ(a)1 ∈ tf (n) +� +a −µ(a)1 +��1 +� +by +construction, and thus ∥π(a)−b∥A∞ = +��π(a −µ(a)1)−(b −µ(a)1) +��A∞ < ε. +Thus, as claimed, Haus[A∞] +� +{π(a)},tf (n) (a|1) +� +< ε for all n � max{N,N ′} and for all +a ∈ B. This proves our claim. +Q.E.D. +Claim 2.26. For all ε > 0, there exists N ∈ N such that, if n � N, then +• ∀a ∈ dom(L∞) +∃b ∈ dom(Ln) : +Ln(b) � L∞(a) and ∥π(a)−b∥A∞ < εL∞(a), +• ∀b ∈ dom(Ln) +∃a ∈ dom(L∞) : +L∞(a) � Ln(b) and ∥π(a)−b∥A∞ < εLn(b). +Proof of Claim (2.26). Let ε > 0. Let N ∈ N be chosen as in Claim (2.25), so that for all +a ∈ dom(L∞) with L∞(a) � 1, and for all n � N, we have Haus[A∞]({π(a)},tf (n) (a|1)) < ε. +Let now n � N. +If a ∈ dom(L∞)\R1A∞, and if b ∈ tf (n) (a|L∞(a)), then L∞(a) > 0, Ln(b) � L∞(a), and +b +L∞(a) ∈ tf (n) +� +a +L∞(a) +���1 +� +and thus +���π +� +a +L∞(a) +� +− +b +L∞(a) +���A∞ < ε. So ∥π(a)−b∥A∞ < εL∞(a), as +needed. +Now, let b ∈ dom(Ln) \ R1A. Let b′ = +b +Ln(b), so Ln(b′) = 1. Let a′ ∈ tτ−1 +f (n) +� +b′��1 +� +, so +in particular L∞(a′) � 1. By symmetry, b′ ∈ tf (n) +� +a′��1 +� +. Therefore, +��π(a′)−b′��A∞ < ε. + +17 +Hence, letting a = Ln(b)a′, we conclude that ∥π(a)−b∥A∞ � Ln(b)ε and L∞(a) � Ln(b), +as desired. +Last, it is immediate that since π(1) = 1, our claim holds whenever L∞(a) = 0 or +Ln(b) = 0, i.e., for any a,b ∈ R1. This proves our claim. +Q.E.D. +Claim 2.27. The map π constructed in Claim (2.24) is a ∗-automorphism. +Proof of Claim (2.27). The map isometry of A∞, hence it is a ∗-monomorphism of A∞, +via Claim (2.24), +Now, let b ∈ � +n∈N dom(Ln), so b ∈ dom(Lm) for some m ∈ N. Thus b ∈ dom(L∞) by +assumption. Let l = Lm(b). By assumption, L∞(b) � MLm(b) = Ml. Let ε > 0 and let +N ∈ N given by Claim (2.26). Since Ln(b) � ML∞(b) � M2l, for all n � max{N,m}, and +there exists an ∈ A∞ with ∥π(an)−b∥A∞ < εM2l (and L∞(a) � Ln(b), which we do not +need for this claim). As ε > 0 was arbitrary, the element b lies in the closure of the range of +π. Since A∞ is complete and π is an isometry, the range of π is closed, and we now have +shown that the range of π is a closed set containing the total subspace � +n∈N dom(Ln) of +A∞; consequently, π is a surjection as well. Thus as claimed, π is a ∗-automorphism of +A∞. +Moreover, by construction, for all a ∈ dom(L∞), as noted in Claim (2.23), we have +L∞(π(a)) � ML∞(a) — in particular, π(a) ∈ dom(L∞). So π(dom(L∞)) ⊆ dom(L∞) and +thus π is a Lipschitz morphism. This proves our claim. +Q.E.D. +This concludes the proof of our theorem. +□ +Remark 2.28. Limits, for the propinquity, are unique up to full quantum isometry. There- +fore, the appearance of some map π in Theorem (2.22) is to be expected. However, the +map π in Theorem (2.22) is quite a bit more general than a full quantum isometry — in +fact, it need not be Lipschitz for us to use Proposition (2.21) — even though Theorem +(2.22) shows that it can always be chosen to be so. The map π is really used here as a +tool to construct a special kind of bridge. In general, the function π is not expected to +be unique: if Ln is just the restriction to An of L∞ for all n ∈ N, and if θ is a full quantum +isometry of (A∞,L∞), then π ◦ θ can be used in place of π, of course. The situation is +more delicate when Ln varies, but there will usually be many maps π if there is one. +Theorem (2.22) characterizes the convergence of inductive sequences in the sense of +the propinquity, under the condition of uniform equivalence of the Lipschitz seminorms +on the sequence. The condition of uniform equivalence of Lipschitz seminorms is in +essence our compatibility condition between the Lipschitz seminorms and the inductive +limit structure in Theorem (2.22): using the notation of Theorem (2.22), as seen in [37], +under the hypothesis that dom(Ln) = An ∩dom(L∞), the Lipschitz seminorms Ln and +L∞ are equivalent for each n ∈ N, and we require, in the assumptions of Theorem (2.22), +that we want this equivalence be uniform. This leads us to several natural questions: +does convergence of (An,Ln)n∈N imply some uniform equivalence of the Lipschitz semi- +norms Ln (n ∈ N) (i.e. is our assumption redundant)? Does the existence of a bridge +builder imply uniform equivalence of the Lipschitz seminorms? Does convergence of +an inductive limit for the propinquity imply the existence of a bridge builder without +the assumption of uniform equivalence of the Lipschitz seminorms? Moreover, does the +convergence of (An,Ln)n∈N to (A∞,L∞) for the propinquity imply the convergence of +(An,Lk)k�n to (An,L∞) for a fixed n ∈ N, for the propinquity? +We now will show with two examples that all of the above questions have negative +answers, so there is no obvious generalization of Theorem (2.22). First, we see that it is +possible to have convergence for the propinquity of an inductive sequence of quantum + +18 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +0 +dist +1 +n ·dist +1−n−2 +0 +dist +1 +(n +1)·dist +1−(n +1)−2 +··· +n → ∞ +0 +dist +1 +X with dist : x, y ∈ X �→ |x − y| +FIGURE 1. Approximating [0,1] with itself by modifying the metric on a +small interval at the end (red) +compact metric spaces, using the identity as a bridge builder, and yet, not have uniform +equivalence of the Lipschitz seminorms. +Example 2.29. Let X = [0,1] with its usual metric. If Y ⊆ X with at least two points, then +we set LY (f ) = sup +� |f (x)−f (y)| +|x−y| +: x ̸= y,x, y ∈ Y +� +for all f ∈ C(X ), allowing for ∞. For each +n ∈ N, and for all f ∈ C(X ), we set: +Ln(f ) = L� +0,1− 1 +n2 +�(f )+ 1 +n L� +1− 1 +n2 ,1 +�(f ), +allowing again for ∞. +Let +fn : x ∈ [0,1] �−→ +� +0 +if x � 1− 1 +n2 , +x −(1− 1 +n2 ) +otherwise. +By construction, L[0,1](fn) = 1 for all n ∈ N. On the other hand, Ln(f ) = 0+ 1 +n ·1 = 1 +n . +So there does not exists M > 0 such that L[0,1] � MLn on the common domain of these +Lipschitz seminorms (the algebra of Lipschitz functions for the usual metric). +We now prove that (C(X ),Ln)n∈N converges for the propinquity to (C(X ),L[0,1]) — this +could be done here just as easily by proving the convergence for the Gromov-Hausdorff +distance of X with a sequence of distances which agree with the usual distance on +[0,1− 1 +n2 ] and is a dilation by a factor n of the usual distance on [1− 1 +n2 ,1], but we will +keep with our functional analytic perspective here. +We thus define, for all n ∈ N, and for all f ,g Lipschitz functions over [0,1] with its +usual metric: +Tn(f ,g) := max +� +L[0,1](f ),Ln(g),(n +1) +��f − g +�� +C(X ) +� +. +Let f ∈ C(X ) with L[0,1](f ) = 1. Then Ln(f ) � 1+ 1 +n . From this, we see that +Tn +� +f , +1 +1+ 1 +n +f − +1 +n +1 f (0) +� +� max +� +1, n +1 +n +1 +��f − f (0)1 +�� +C(X ) +� +� 1. +Let now g ∈ C(X ) with Ln(g) = 1. Thus L� +0,1− 1 +n2 +�(g) � 1 and L� +1− 1 +n2 ,1 +�(g) � n. In particular, +for all x ∈ [1− 1 +n2 ,1], we have +���g(x)− g +� +1− 1 +n2 +���� < n|x −1+ 1 +n2 | � 1 +n . +Let h ∈ C(X ) defined by h(x) = g(x) if x ∈ +� +0,1− 1 +n2 +� +, and h(x) = g +� +1− 1 +n2 +� +otherwise. +By construction, L[0,1](h) � 1 and +��g −h +�� +C(X ) < 1 +n . Thus Tn(h,g) = 1 = Ln(g). + +19 +1 +1 +2 +2 +1 +3 +3 +1 +4 +4 +1 +5 +5 +1 +6 +0 +... +1 +n+1 +1 +2 +3 +4 +5 +0 +... +1 +n+2 +1 +2 +3 +4 +5 +0 +... +1 +n+3 +··· +n → ∞ +0 +1 +2 +3 +4 +... +FIGURE 2. Approximating N by itself, by merging the first two points +at ∞ +Therefore, (C(X )⊕C(X ),Tn,p1,p2), with p1 : (f ,g) ∈ C(X )⊕C(X ) �→ f and p2 : (f ,g) ∈ +C(X )⊕C(X ) �→ g, is easily seen to be a tunnel whose extent is at most 1 +n (the method is +analogous to Proposition (2.21)). +Hence (C(X ),Ln)n∈N converges to (C(X ),L[0,1])n∈N for the propinquity. Moreover, +the identity map satisfies Condition (2) of Theorem (2.22). Nonetheless, there is no +M > 0 such that ∀n ∈ N +L[0,1] � MLn on the common domain of these seminorms. So +convergence in the propinquity does not imply uniform equivalence of the Lipschitz +seminorms, even when working with a fixed, Abelian C*-algebra. +Now, we can also ask whether convergence for the propinquity of an inductive se- +quence, implies the existence of a bridge builder, and as we shall see in the next example, +this is not the case: once again, convergence occurs without uniform equivalence of +Lipschitz seminorms (and we prove that we have neither uniform dominance or uniform +domination using both examples). Moreover, we see that (An,Lm)m�n does not converge +to (An,L∞) in this case, for any n ∈ N. +Example 2.30. Let A∞ be the C*-algebra of convergent sequences with values in C. +For each n ∈ N, let An = {(xk)k∈N : (xk)k�n is constant }, so An is a C*-subalgebra of A∞ +sharing the unit (1)n∈N of A∞. +For all n ∈ N, and for all (xk)k∈N ∈ An, we set +Ln((xk)k∈N) := sup +� +|xp − xq| +|ϕn(p)−ϕn(q)| : p,q ∈ N,p ̸= q +� +where: +ϕn : m ∈ N �→ +� 1 +m if m > 0, +1+ 1 +n if m = 0. +Of course, Ln is indeed a seminorm on the finite dimensional C*-subalgebra An of A∞. +We also set L∞((xk)k∈N) = sup +� +|xp−xq| +��� +1 +p+1 − +1 +q+1 +��� : p,q ∈ N,p ̸= q +� +for all (xk)k∈N ∈ A∞, al- +lowing for the value ∞. Of course, � +n∈NAn ⊆ dom(L∞). +Now, let +x : n ∈ N �→ +� +1 if n = 0, +0 otherwise. + +20 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +By construction, L∞(x) = 2, yet Ln(x) = n. So there is no M > 0 such that, for all n ∈ N, +the inequality MLn � L∞ on dom(Ln) holds. +On the other hand, limn→∞ Λ∗((An,Ln),(A∞,L∞)) = 0. Indeed, let π : (xk)k∈N �→ +(x0,x0,x1,x2,x3,...) ∈ A∞, B = π(A∞), and let θ : (xk)k∈N ∈ B �→ (xk+1)k∈N ∈ A∞ — of +course, θ is a ∗-isomorphism from B onto A∞ such that π = θ−1. We define LB(π(x)) = +L∞(x) for all x ∈ dom(L∞). This way, π is easily checked to be a full quantum isometry +from (A∞,L∞) to (B,LB). +Let ε > 0 and let N ∈ N be such that if n � N, then +1 +n+1 < ε +2. If x = (xk)k∈N with +L∞(x) � 1, and if l = lims→∞ xs, then by construction, +|xk −l| +1 +k+1 += lim +s→∞ +|xk − xs| +1 +k+1 − +1 +s+1 +� 1 +so |xk −l| � +1 +k+1 < ε +2 for all k � N. Therefore, if k � N then |xk − xN| < ε. +Now, let n � N. Let Dn = An ⊕B, and for all (a,b) ∈ dom(An)⊕dom(B), we set: +Tn(a,b) := max +� +Ln(a),LB(b), 1 +ε ∥π(a)−b∥B +� +. +We also set pn : (a,b) ∈ Dn �→ a ∈ An and qn : (a,b) ∈ Dn �→ θ(b) ∈ A∞. We are now +going to prove that τn := (Dn,TNn,pn,qn) is indeed a tunnel from (An,Ln) to (A∞,L∞). +Let a := (xk)k∈N ∈ dom(L∞) with L∞(a) = 1, and let +a′ := (x0,x0,x1,x2,...,xN−1,xN,xN,xN ...) ∈ An. +By construction, Ln(a′) � 1 and +��π(a)− a′��A∞ < ε by our choice of N. Also by construc- +tion, LB(π(a)) = L∞(a) = 1. Thus Tn(a′,π(a)) � L∞(a) = 1. So, we have shown that, for +any a ∈ dom(L∞) with L∞(a) = 1, there exists an element d := (a′,π(a)) ∈ Dn such that +TNn(d) = 1 = L∞(a) and qn(d) = θ(π(a)) = a. Therefore, the map qn is indeed a quantum +isometry from (Dn,Tn) to (A∞,L∞). +Let now a = (xk)k∈N ∈ dom(Ln) with Ln(a) = 1. By definition, |x1 − x0| � 1 +n < ε. Let +b = (x1,x1,x2,x3,x4,...). +By construction, b ∈ dom(LB) with LB(b) � Ln(b), and ∥a −b∥A∞ = |x1 − x0| < ε. Thus +again Tn(a,b) = Ln(b). So pn : (a,b) ∈ Dn �→ a ∈ An is a quantum isometry. Therefore, +(Dn,Tn,pn,qn) is indeed a tunnel from (An,Ln) to (A∞,L∞). We now compute an upper +bound on its extent. +Let ϕ ∈ S (Dn) be a state of Dn. If we set µ : a ∈ An �→ ϕ(a,π(a)), then µ ∈ S (An) is +again a state of An. If (a,b) ∈ dom(Tn) with Tn(a,b) � 1, then +|ϕ(a,b)−µ◦ pn(a,b)| = |ϕ(a,b)−ϕ(a,π(a))| += |ϕ(0,b −π(a))| +� ∥b −π(a)∥A∞ < ε, +so indeed Haus +�mkTn +� +(S (Dn),p∗ +nS (An)) < ε. +On the other hand, let ν : a ∈ A∞ �→ ϕ′(a,π(a)) where ϕ′ is an extension of ϕ to a state +of A∞ ⊕B by the Hahn-Banach theorem. Once again, it is immediate that mkTn(ϕ,ν◦ +qn) < ε. So Haus +�mkTn +� +(S (D),q∗ +nS (A∞)) < ε. +Thus, for all n � N, the extend of χ(τn) is at most ε. We conclude: +lim +n→∞Λ∗((An,Ln),(A∞,L∞)) = 0. +However, for any fixed p ∈ N, it is easy to check, by a similar method, that +lim +n→∞Λ∗((Ap,Ln),(Ap−1,L∞)) = 0, + +21 +and since dimAp−1 < dimAp, the sequence (Ap,Ln)n�p does not converge to (Ap,L∞). +The map π we have used here is not surjective. In fact, there is no bridge builder in +this case. Indeed, assume that we have a unital ∗-morphism π : A∞ → A∞ such that for +all ε > 0, there exists Nπ(ε) ∈ N with the property that if n � Nπ(ε), and if a ∈ dom(L∞), +then there exists b ∈ dom(Ln) with Ln(b) � L∞(a) and ∥π(a)−b∥A∞ < ε +2L∞(a). +Fix a ∈ dom(L∞) with L∞(a) = 1. Let ε > 0 and let n � Nπ(ε) such that 1 +n < ε +2. De- +fine (yk)k∈N := π(a). Then there exists b := (bk)k∈N ∈ dom(Ln) such that Ln(b) � 1 and +∥π(a)−b∥A∞ < ε +2. By definition of Ln, we thus conclude that |b1 − b0| � 1 +n < ε +2. Thus, +|y1−y0| < ε. As ε > 0 is arbitrary, we conclude that y1 = y0. Thus π can never be surjective +— in fact, it is valued in B. So no bridge builder exists for this example. +As seen in Example (2.30), convergence of (An,Ln)n∈N to (A∞,L∞) for the propinquity +does not imply the convergence of (An,Lp)p∈N to (An,L∞). We have the following +immediate consequence of our work: +Corollary 2.31. Let A∞ be a unital separable C*-algebra, such that A∞ = cl(� +n∈NAn), +where (An)n∈N is an increasing (for ⊆) sequence of C*-subalgebras of A∞, with the unit of +A∞ in A0. For each n ∈ N, let Ln be a Lipschitz seminorm on An. If there exists a bridge +builder π : A∞ → A∞ for ((An,Ln)n∈N,(A∞,L∞)) such that π(An) ⊆ An for each n ∈ N, +then for all n ∈ N, +lim +p→∞ +p�n +Λ∗((An,Lp),(An,L∞)) = 0, +and limn→∞ Λspec((An,Ln),(A∞,L∞)) = 0. +Proof. This follows by observing that the restriction of π to An is a bridge builder for +((An,Lp)p�n,(An,L∞)). Our result then follows from Proposition (2.21). +□ +3. CONVERGENCE OF INDUCTIVE SEQUENCES OF METRIC SPECTRAL TRIPLES FOR THE +SPECTRAL PROPINQUITY +We now study the convergence of certain families of metric spectral triples for the +spectral propinquity [47], whose construction we will recall below. We thus begin this +section with the definition of a spectral triple, due to Connes, and the foundational +concept for noncommutative Riemannian geometry. +Definition 3.1 ([12, 11]). A spectral triple (A,H , /D) is given by a unital C*-algebra A of +bounded linear operators on a Hilbert space H , and a self-adjoint operator /D defined +on some dense subspace dom( /D) of H , such that: +(1) {a ∈ A : a ·dom( /D) ⊆ dom( /D),[ /D,a] is bounded } is a dense ∗-algebra in A, +(2) +/D has compact resolvent. +The operator /D is referred to as the Dirac operator of the spectral triple. +3.1. Preliminaries: The Spectral Propinquity. The spectral propinquity is a distance, +up to unitary equivalence, on the class of metric spectral triples. +Notation 3.2. If T : D ⊆ E → F is a linear operator defined from a dense subspace D of a +normed vector space E to a normed vector space F, then we write: +|||T |||F +E := sup +� +∥T ξ∥F : ξ ∈ D,∥ξ∥E � 1 +� +allowing for the value ∞. If F = E, then |||T |||F +E is simply denoted by |||T |||E. + +22 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +Definition 3.3. A spectral triple (A,H , /D) is metric if the Connes extended pseudo- +distance, defined on the state space S (A) of A by: +mk /D : ϕ,ψ ∈ S (A) �→ sup +� +|ϕ(a)−ψ(a)| : a dom( /D) ⊆ dom( /D) and |||[ /D,a]|||H � 1 +� +is in fact a metric on S (A), which induces the weak-∗ topology on S (A). +As soon as a spectral triple is metric, it induces a structure of quantum compact metric +space on its underlying C*-algebra in a natural manner. +Proposition 3.4 ([47, Proposition 1.10]). Let (A,H , /D) be a spectral triple. We set: +dom(L /D) := {a ∈ sa(A) : a dom( /D) ⊆ dom( /D) and [ /D,a] is bounded } +and for all a ∈ dom(L /D): +L /D(a) := |||[ /D,a]|||H . +The spectral triple (A,H , /D) is metric if, and only if, (A,L /D) is a quantum compact +metric space. +The construction of the spectral propinquity begins with the following observation. +Recall from [47] that if (A,H , /D) is a metric spectral triple, and if we set +• for all ξ ∈ dom( /D): +(3.1) +DN /D(ξ) := ∥ξ∥H +∥ /Dξ∥H , +• dom(L /D) := {a ∈ sa(A) : a dom( /D) ⊆ dom( /D), [ /D,a] is bounded } +• for all a ∈ dom(L /D): +L /D(a) := |||[ /D,a]|||H , +then +metCor(A,H , /D) := (H ,DN /D,A,L /D,C,0) +is an example of a metrical C*-correspondence, in the following sense: +Definition 3.5. An A-B-C ∗-correspondence (M ,A,B), for two C*-algebras A and B, is +a right Hilbert module M over B (whose B-valued inner product is denoted by 〈·,·〉M ), +together with a unital ∗-morphism from A to the C*-algebra of adjoinable B-linear +operators over M . +Definition 3.6 ([47, Definition 2.2]). An (Ω,Ω′,Ωmod,Ωinner)-metrical C*-correspondence +(M ,DN,A,L,B,S), where Ω,Ωinner � 1, Ωmod � 2, and Ω′ � 0, is given by two (Ω,Ω′)- +quantum compact metric spaces (A,L) and (B,S), an A-B C*-correspondence (M ,A,B), +and a norm DN defined on a dense C-subspace dom(TN) of M , such that +(1) ∀ω ∈ dom(DN) +DN(ω) � ∥ω∥M := +���〈ω,ω〉M +��B, +(2) {ω ∈ dom(DN) : DN(ω) � 1} is compact in (M ,∥·∥M ), +(3) for all a ∈ dom(L) and ω ∈ dom(TN), +DN(aω) � Ωmod(∥a∥A +L(a))DN(ω), +(4) for all ω,η ∈ dom(DN), +max{S(ℜ〈ω,η〉M ),S(ℑ〈ω,η〉M )} � ΩinnerDN(ω)DN(η). +In particular, the norm DN is called a D-norm. +Convention 3.7. In this work, we fix Ωmod � 2 and Ωinner � 1 all throughout the paper. +All quantum compact metric spaces will be assumed to be in the class of (Ω,Ω′)-quantum +compact metric spaces and all metrical C*-correspondences will be assume to be in the +class of (Ω,Ω′,Ωmod,Ωinner)-metrical C*-correspondences, unless otherwise specified. + +23 +Note that the compactness condition in Definition (3.6) borrows and extends on +Theorem (2.9). +The importance of Definition (3.6) is that one can extend the propinquity to metrical +C*-correspondences as follows. First, we employ a natural notion of morphism between +metrical C*-correspondences. +Definition 3.8 ([47, Definition 2.13]). For each j ∈ {1,2}, let +Mj = +�Mj ,DNj ,Aj ,Lj ,Bj ,Sj +� +be a metrical C*-correspondence. +A metrical quantum isometry (Π,π,θ) from M1 to M2 is a given by: +(1) a continuous, surjective C-linear map Π : M1 → M2, +(2) a quantum isometry π : (A1,L1) → (A2,L2), +(3) a quantum isometry θ : (B1,S1) → (B2,S2), +such that +(1) ∀a ∈ A +∀ω ∈ M1 +Π(aω) = π(a)Π(ω), +(2) ∀b ∈ B +∀ω ∈ M2 +Π(ω·b) = Π(ω)θ(b), +(3) ∀ω,η ∈ M1 +θ(〈ω,η〉M1) = 〈Π(ω),Π(η)〉M2, +(4) Π(dom(DN1)) ⊆ dom(DN2) and, for all ω ∈ dom(DN2), the equality DN2(ω) = +inf +�DN1(η) : η ∈ dom(DN1),Π(η) = ω +� +. +The definition of a distance between metrical C*-correspondences, called the metrical +propinquity, relies on a notion of isometric embedding called a tunnel, and is defined as +follows. +Definition 3.9 ([47, Definition 2.19]). Let M1 and M2 be two metrical C*-correspondences. +A (metrical) tunnel τ = (J,Π1,Π2) from M1 to M2 is a triple given by a metrical C*- +correspondence J, and for each j ∈ {1,2}, a metrical quantum isometry Πj : J �→ Mj . +Remark 3.10. It is important to note that our tunnels involve (Ω,Ω′,Ωmod,Ωinner)-C*- +metrical correspondences only (as per Convention (3.7)). We will dispense calling our tun- +nels (Ω,Ω′,Ωmod,Ωinner)-tunnels, to keep our notation simple, but it should be stressed +that fixing (Ω,Ω′,Ωmod,Ωinner) and staying within the class of (Ω,Ω′,Ωmod,Ωinner)-C*- +metrical correspondences is crucial to obtain a metric from tunnels. +We now proceed by defining the extent of a metrical tunnel; remarkably this only +involves our previous notion of extent of a tunnel between quantum compact metric +spaces. +Definition 3.11 ([47, Definition 2.21]). Let Mj = (Mj ,DNj ,Aj ,Lj ,Bj ,Sj ) be a metrical +C*-correspondence, for each j ∈ {1,2}. Let τ = (P,(Π1,π1,θ1),(Π2,π2,θ2)) be a metrical +tunnel from M1 to M2, with P = (P,TN,D,LD,E,LE). +The extent χ(τ) of a metrical tunnel τ is +χ(τ) := max +� +χ(D,LD,π1,π2),χ(E,TE,θ1,θ2) +� +. +Given two metric spectral triples, we can thus either take the Gromov-Hausdorff +distance between their underlying quantum compact metric spaces, or take the metri- +cal propinquity [42, 46] between the metrical C*-correspondence they define, which is +defined as the infimum of the extent of every possible metrical tunnels between them. +However, the spectral propinquity involves our work on the geometry of quantum dy- +namics [43, 44, 47] as well. We recall the construction of the spectral propinquity; the new + +24 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +quantity called the ε-magnitude was introduced in [47, Definition 3.31], but is simpler to +express for spectral triples, based on [31]. +Definition 3.12 ([31, Theorem 3.6]). Let (A1,H1, /D1) and (A2,H2, /D2) be two metric +spectral triples. Let +τ := +� +� (P,TN,D,LD,E,S) +metrical C*-correspondence +, +(Π1,π1,θ1) +metrical quantum isometry +, +(Π2,π2,θ2) +metrical quantum isometry +� +� +be a metrical tunnel from metCor(A1,H1, /D1) to metCor(A2,H2, /D2), We define the +ε-magnitude µ(τ|ε) of τ as the maximum of the extent χ(τ) of τ, and the ε-reach of τ, +which is the number: +(3.2) +sup +ξ∈dom +� +/D j +� +DNj (ξ)�1 +inf +η∈dom( /Dk) +DNk(η)�1 +sup +ω∈dom(TN) +TN(ω)�1 +0�t� 1 +ε +���〈exp(it /D j )ξ,Πj (ω)〉Hj −〈exp(it /Dk)η,Πk(ω)〉Hk +���, +for {j,k} = {1,2}. +Definition 3.13 ([47, Definition 4.2]). The spectral propinquity between two metric +spectral triples (A1,H1, /D1) and (A2,H2, /D2) is +Λspec((A1,H1, /D1),(A2,H2, /D2)) := +inf +�� +2 +2 ,ε > 0 : µ(τ|ε) < ε for τ a tunnel +from metCor(A1,H1, /D1) to metCor(A2,H2, /D2) +� +. +The key property of the spectral propinquity is that, for any two metric spectral triples +(A1,H1, /D1) and (A2,H2, /D2), we have the following equivalence: +Λspec((A1,H1, /D1),(A2,H2, /D2)) = 0 +if, and only if, there exists a unitary U : H1 → H2 such that +• Udom( /D1) = dom( /D2), +• U /D1 = /D2U on dom( /D1), +• a ∈ A1 �→UaU ∗ is a ∗-isomorphism from A1 onto A2. +A nontrivial example of convergence in the sense of the spectral propinquity is pro- +vided in [45] with the approximation of spectral triples on quantum tori by spectral +triples of certain matrix algebras known as fuzzy tori. These examples include many +examples of previously informally stated convergences in mathematical physics, dealing +with matrix models and their limits as the dimension of the algebra grows to infinity. +Such examples are a major motivation for the construction of the spectral propinquity. +Another example on fractals is presented in [29]. Moreover, convergence for the spec- +tral propinquity implies convergence of the spectra of the Dirac operators and, in an +appropriate sense, the convergence of the bounded functional calculi of these operators, +among other properties. Of course, convergence for the spectral propinquity implies +convergence of the underlying quantum compact metric spaces for the propinquity. +In this paper, we will construct new examples of convergence for new spectral triples +defined over noncommutative solenoids and over Bunce-Deddens algebras, seen as +limits of spectral triples. + +25 +3.2. Preliminaries: Inductive Limits of Spectral Triples. While the spectral propinquity +allows the discussion of convergence of spectral triples defined on vastly different C*- +algebras, there are certain more restricted situations where the C*-algebras of a sequence +of spectral triples may be related in a manner compatible with the spectral triples. In +[20], a simple notion of inductive limit for spectral triples is introduced, based on the +following encoding of such a compatibility via a natural, and rigid, notion of morphism +between spectral triples. +Definition 3.14 ([20]). An isometric morphism (π,S) from (A1,H1, /D1) to (A2,H2, /D2) is +given by a unital ∗-morphism π : A1 → A2 and a linear isometry S : H1 → H2 such that: +(1) π(dom(L1)) ⊆ dom(L2), +(2) Sdom( /D1) ⊆ dom( /D2) and S /D1 = /D2S on dom( /D1), +(3) ∀a ∈ A1 +Sa = π(a)S. +Since S is a linear isometry, H1 can be identified with the closed subspace SH1 of H2 +via S at no cost in our definition. In that case, /D1 is only defined on H1 ⊆ H2, and we +simply require that /D1 is the restriction of /D2 to dom( /D1). +We also note that if π(a) = 0 for some a ∈ A1, then π(a)S = Sa = 0. Since S is an +isometry, a = 0. So π is actually automatically a ∗-monomorphism, and we thus can +also identify A1 with the C*-subalgebra π(A1) of A2, since Definition (3.14) ensures that +aH1 ⊆ H1 and [ /D1,a] is identified with P[ /D2,π(a)]P = P[ /D2,π(a)] = [ /D2,π(a)]P where +P is the orthogonal projection of H2 onto H1. Furthermore, since π is unital, the unit of +A2 is contained in A1 with this identification. +An inductive sequence of spectral triples, as defined in [20], with a somewhat more +involved notation, is simply a sequence of the form ((An,Hn, /Dn),(πn,Sn))n∈N where +(An,Hn, /Dn) is a spectral triple and (πn,Sn) is an isometric morphism from (An,Hn, /Dn) +to (An+1,Hn+1, /Dn+1), for each n ∈ N. As we have seen above, we can identify such a +sequence with one of the following type, which we will take as our notion of inductive +limit of spectral triples. +Definition 3.15. Let A∞ = cl(� +n∈NAn) be a C*-algebra which is the closure of an increas- +ing sequence of C*-subalgebras (An)n∈N in A∞, with the unit of A∞ in A0. A spectral +triple (A∞,H∞, /D∞) is the inductive limit of a sequence (An,Hn, /Dn)n∈N of spectral +triples when: +(1) H∞ = cl(� +n∈N)Hn, where each Hn is a Hilbert subspace of H∞, +(2) for each n ∈ N, the restriction of /D∞ to dom( /Dn) is /Dn, +(3) for each n ∈ N, the subspace Hn is reducing for An, which is equivalent to +AnHn ⊆ Hn. +We note, using the notation of Definition (3.15), that the operator which, to any +ξ ∈ � +n∈N dom( /Dn), associates /Dnξ whenever ξ ∈ dom( /Dn) for any n ∈ N, is indeed well- +defined, and shown in [20] to be essentially self-adjoint, so /D∞ is the closure of this +operator. +For our purpose, the following result from [20] will play an important role. +Theorem 3.16 ([20, Theorem 3.1, partial]). If (An,Hn, /Dn)n∈N is an inductive sequence +of spectral triples converging to a spectral triple (A∞,H∞, /D∞), then for any C-valued +continuous function f ∈ C0(R) which vanishes at infinity, the sequence (Pn f ( /Dn)Pn)n∈N +converges to f ( /D∞) in norm. +This section is concerned with the question: if a spectral triple is an inductive limit +of spectral triples, then what additional assumptions should be made to get a more + +26 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +geometric convergence, specifically in the sense of the spectral propinquity? In order to +make sense of this question, we will work with metric spectral triples, which give rise to +quantum compact metric spaces, and lie within the realm of noncommutative metric +geometry. +3.3. Main result. The notion of inductive limit of spectral triples is simpler to define +than the spectral propinquity but only applies to rather narrow examples — it is not +applicable to fuzzy and quantum tori [45] or the fractals in [29]. It is certainly interesting +to wonder how much metric information from the spectral triples are continuous with +respect to the inductive limit process. In this section, we establish a sufficient condition +for the convergence, in the sense of the spectral propinquity, of a sequence of metric +spectral triples which already converges to a metric spectral triple in the categorical sense. +This sufficient condition is simply the existence of an appropriate bridge builder which +is also a full quantum isometry. Thus, the main difficulty in establishing convergence for +the spectral propinquity, in this context, reduces to proving metric convergence for the +propinquity using adequate tunnels. +Theorem 3.17. Let (A∞,H∞, /D∞) be a metric spectral triple which is the inductive limit +of a sequence of metric spectral triples (An,Hn, /Dn), in the sense of Definition (3.15). For +each n ∈ N, let +dom(Ln) := {a ∈ sa(An) : a dom( /Dn) ⊆ dom( /Dn) and [ /Dn,a] is bounded}, +and for all a ∈ dom(Ln), define +Ln(a) := |||[ /Dn,a]|||Hn. +If there exists a full quantum isometry π : (A∞,L∞) → (A∞,L∞) which is also a bridge +builder for ((An,Ln)n∈N,(A∞,L∞)), then +lim +n→∞Λspec((An,Hn, /Dn),(A∞,H∞, /D∞)) = 0. +Proof. Fix ε > 0. By Proposition (2.21), the sequence (An,Ln)n∈N converges to (A∞,L∞) +for the propinquity. More specifically, set, for convenience, ˜ε = ε +2 > 0. Let Nπ ∈ N be given +so that, for all n � Nπ, we have: +• ∀a ∈ dom(L∞) +∃b ∈ dom(Ln) : +Ln(b) � L∞(a) and ∥π(a)−b∥A∞ < ˜εL∞(a), +• ∀b ∈ dom(Ln) +∃a ∈ dom(L∞) : +L∞(a) � Ln(b) and ∥π(a)−b∥A∞ < ˜εLn(b). +For each n ∈ N, we constructed in Proposition (2.21) a tunnel τn = (Dn,Tn,ψn,θn) +with Dn = A∞ ⊕An, and for all (a,b) ∈ dom(L∞)⊕dom(Ln), +Tn(a,b) := max +� +L∞(a),Ln(b), 1 +˜ε ∥π(a)−b∥A∞ +� +, +while ψn : (a,b) ∈ Dn �→ a, θn : (a,b) ∈ Dn �→ b. We proved that χ(τn) < ˜ε. It is immediate, +since π is a full quantum isometry, that τ′ +n := (Dn,Tn,π◦ψn,θn) is also a tunnel with the +same extent as τn. +For each n ∈ N and for all ξ ∈ dom( /Dn), we define +DNn(ξ) := ∥ξ∥Hn +∥ /Dnξ∥Hn , +following Expression (3.1). +Now, since DN∞ is a D-norm, the set X∞ = {ξ ∈ dom( /D∞) : DN∞(ξ) � 1} is compact +in H∞. Thus, there exists a finite subset F ⊆ X∞ of X∞ such that Haus[H∞](X∞,F) < ˜ε +3. +As /D∞ is the closure of an operator on � +n∈NHn by [20], for any ξ ∈ F, there exists +a sequence (ξn)n∈N, with ξn ∈ � +j∈NHj for all n ∈ N, such that limn→∞ ξn = ξ, and + +27 +limn→∞ /D∞ξn = /D∞ξ. Since F is finite, there exists NF ∈ N such that if n � NF and ξ ∈ F, +then ∥ξ−ξn∥H∞ < ˜ε +3 and ∥ /D∞ξ− /D∞ξn∥H∞ < ˜ε +3. Again by Definition (3.15), we also +have /D∞ξn = /Dnξn. +Fix n ∈ N,n � N := max{Nπ,NF }. Let Mn := H∞ ⊕ Hn, seen as a Dn-(C ⊕ C) C*- +correspondence, with the C*-correspondence structure: +∀(a,b) ∈ Dn +∀(ξ,η) ∈ Mn +(a,b)◁(ξ,η) := (π(a)ξ,bη), +and +∀(ξ,η),(ξ′,η′) ∈ Mn +〈(ξ,ξ′),(η,η′)〉n := +� +〈ξ,ξ′〉H∞,〈η,η′〉Hn +� +∈ C⊕C, +while +∀(t,s) ∈ C⊕C +∀(ξ,η) ∈ Mn +(ξ,η)·(t,s) := (tξ,sη). +We note that here, C2 is the C*-algebra of C-valued functions over a two points set, and +in particular, the norm of (z,w) ∈ C2 is max{|z|,|w|}. +We then define, for all (ξ,η) ∈ dom( /D∞)⊕dom( /Dn): +TNn(ξ,η) := max +� +DN∞(ξ),DNn(η), 1 +˜ε +��ξ−η +��H∞ +� +, +while we also set +Q : (z,w) ∈ C⊕C �→ 1 +˜ε|z − w|. +It is immediate to see that Q is a Lipschitz seminorm on C ⊕ C (it is, in fact, the +Lipschitz seminorm for the metric on the two point set which places these two points +exactly ˜ε apart). +Now, we check that TNn is a D-norm. Of course, for all (ξ,η) ∈ Mn: +TNn(ξ,η) � max{DN∞(ξ),DNn(η)} � max +� +∥ξ∥H∞ , +��η +��Hn +� += +��(ξ,η) +��Mn . +We observe that +{(ξ,η) ∈ Mn : TNn(ξ,η) � 1} ⊆ +{ξ ∈ dom( /D∞) :DN∞(ξ) � 1}×{η ∈ dom( /Dn) : DNn(η) � 1}, +the latter set being compact as a product of two compact sets – since DNn and DN∞ +are indeed D-norms. Since in addition, TNn is lower semicontinuous over Mn as the +maximum of three lower semicontinuous functions over this space, the unit ball of TNn +is indeed closed, hence compact, in Mn (which is complete). We now check the Leibniz +inequalities. If (a,b) ∈ dom(Tn) and (ξ,η) ∈ dom(TNn), then we compute: +��(a,b)◁(ξ,η) +��H∞ = +��π(a)ξ−bη +��H∞ +� ∥π(a)−b∥A∞ ∥ξ∥H∞ +∥b∥A∞ +��ξ−η +��H∞ +� ˜εTn(a,b)DNn(ξ)+∥(a,b)∥Dn ˜εTNn(ξ,η) +� ˜ε +�Tn(a,b)+∥(a,b)∥Dn +�TNn(ξ,η). +From this, it follows that for all (a,b) ∈ dom(Tn) and for all (ξ,η) ∈ dom(TNn), +TNn((a,b)◁(ξ,η)) � +�Tn(a,b)+∥(a,b)∥Dn +�TNn(ξ,η). + +28 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +On the other hand, if (ξ,η),(ξ′,η′) ∈ dom(TNn), we have: +Q(〈(ξ,η),(ξ′,η′)〉Mn) = 1 +˜ε +��〈ξ,ξ′〉H∞ −〈η,η′〉H∞ +�� +� 1 +˜ε +���〈ξ−η,ξ′〉H∞ +��+ +��〈η,ξ′ −η′〉H∞ +��� +� 1 +˜ε +���ξ−η +��H∞ +��ξ′��H∞ + +��η +��H∞ +��ξ′ −η′��H∞ +� +� TNn(ξ,η) +��ξ′��H∞ + +��η +��H∞ TNn(ξ′,η′) +� 2TNn(ξ,η)TNn(ξ′,η′). +We now define the maps: +Πn : (ξ,η) ∈ Mn �→ ξ ∈ H∞, and Θn : (ξ,η) ∈ Mn �→ η ∈ Hn. +Our goal is to show that +Υn := +�Mn,(Πn,π◦ψn),(Θn,θn) +� +where Mn := (Mn,TNn,Dn,Tn,C⊕C,Q) +is a metrical tunnel, using Definition (3.9). +By construction, Πn(a ·ξ,b ·η) = π(a)ξ = π◦ψn(a,b)Πn(ξ,η) and Θn(a ·ξ,b ·η) = bη = +θn(a,b)Θn(ξ,η), for all (a,b) ∈ Dn and (ξ,η) ∈ Mn. +Now, let ξ ∈ H∞ with DN∞(ξ) = 1. By construction of F, there exists ξ′ ∈ F such that +��ξ−ξ′��H∞ < ˜ε +3. By our choice of N, there exists η(= ξ′ +n) ∈ Hn such that DNn(η) � 1+ ˜ε +3 +and +��ξ′ −η +��H∞ < ˜ε +3. Let χ = +1 +1+ ˜ε +3 +η ∈ Hn, so that DNn(χ) � 1. Moreover, +��ξ−χ +��H∞ � +��ξ−η +��H∞ + +˜ε +3 +1+ ˜ε +3 +��η +��H∞ +� +��ξ−η +��H∞ + +˜ε +3 +1+ ˜ε +3 +DNn(η) +� +��ξ−ξ′��H∞ + +��ξ′ −η +��H∞ + ˜ε +3 +< ˜ε. +Thus TNn(ξ,χ) = 1, and therefore, (Πn,π◦ψn) is indeed a metrical quantum isometry. +Now, let η ∈ Hn. By construction, /D∞η = /Dnη, so DN∞(η) = DNn(η). Therefore, +TNn(η,η) = DNn(η). Again, we conclude that (Θn,θn) is a metrical quantum isometry as +well. +Therefore, Υn is a metrical tunnel. It is immediate, of course, that the canonical +surjections from C⊕C to C are quantum isometries — the only Lipschitz seminorm on +C being the 0 function. So Υn is a metrical tunnel. +We now compute the extent of Υn. It is, by Definition (3.11), the maximum of the +extent of the tunnel τ′ +n, which is at most ˜ε, and the extent of the tunnel (C,0) ←− (C ⊕ +C,Q) −→ (C,0), which is immediately computed to be ˜ε. So the extent of Υn is ˜ε as well. +Therefore, for all n � N, we have +Λ∗met((Hn,DNn,An,Ln,C,0),(H∞,DN∞,A∞,L∞,C,0)) � χ(Υn) = ˜ε < ε, +and therefore, +lim +n→∞Λ∗met((Hn,DNn,An,Ln,C,0),(H∞,DN∞.A∞,L∞,C,0)) = 0. + +29 +It remains to compute an upper bound for the ε-reach of our tunnels Υn (see Defini- +tion 3.12). We will once again use our finite set F with Haus[H∞](F,X∞) < ˜ε +3 where X∞ +is the closed unit ball of DN∞. +Now, let (vk)k∈N be a sequence of continuous functions on R vanishing at ∞, valued +in [0,1], and converging pointwise to 1 over R. Therefore, (vk( /D∞))k∈N converges to /D∞ +in the strong operator topology. Since F is finite, there exists k ∈ N such that, for all ξ ∈ F +(3.3) +∥vk( /D∞)ξ−ξ∥H∞ < ˜ε +12. +We identify, from now on, /Dn with the linear operator on H∞ whose restriction to Hn +is /Dn, and whose restriction to H ⊥ +n is 0; thus dom( /Dn) is replaced with dom( /Dn)⊕H ⊥ +n . +We denote by Pn the orthogonal projection of H∞ onto Hn, so that Pn /Dn = /DnPn = /Dn +on dom( /Dn). +For each t ∈ [0,∞), let ut : s ∈ R �→ exp(its), and for each n ∈ N, we denote ut( /Dn) by +U t +n. +Fix t ∈ R. The function ut vk is continuous over R and vanishes at infinity. By Theorem +(3.16), since (A∞,H∞, /D∞) is a spectral triple, and the inductive limit of the sequence +(An,Hn, /Dn)n∈N of spectral triples, the sequence of operators (Pnut vk( /Dn)Pn)n∈N con- +verges in norm to ut vk( /D∞). Moreover, ut vk( /Dn)Pn = Pnut vk( /Dn)Pn for all n ∈ N by +construction. +Let F ′ be a finite subset of the compact set +� +0, 1 +ε +� +such that Haus[R](F ′, +� +0, 1 +ε +� +) < +˜ε +12. +Since F ′ is finite, there exists Nν ∈ N such that if n � Nν, then for all t ∈ F ′: +(3.4) +������U t +n(vk( /Dn))Pn −U t +∞(vk( /D∞)) +������H∞ < ˜ε +12. +Let n ∈ N. Now, we note that if ξ ∈ dom(DNn) with DNn(ξ) � 1, then for all s < t ∈ R: +��U t +nξ−U s +nξ +��Hn � +�t +s +���� +d +dr U r +nξ +����Hn +dr +� +�t +s +��U r +n /Dnξ +��Hn dr +� |s − t|. +Thus, for all s,t ∈ R and ξ ∈ dom(DNn) with DNn(ξ) � 1, we have +��U t +nξ−U s +nξ +��Hn � |s − t|. +Now, let n � N ′ := max{Nν,NF }. Since /Dn and Pn commute, if ξ ∈ X∞, then DNn(ξ) � +DN∞(ξ) and: +DNn(νk( /Dn)Pnξ) = ∥vk( /Dn)Pnξ∥H∞ +∥ /Dnvk( /Dn)Pnξ∥H∞ +� ∥vk∥C0(R) |||Pn|||H∞DNn(ξ) � 1. + +30 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +For all ξ ∈ X∞ and t ∈ +� +0, 1 +˜ε +� +, let s ∈ F ′ and ξ′ ∈ F such that |s − t| < ˜ε +12, +��ξ−ξ′��H∞ < ˜ε +3, +then: +��U t +nνk( /Dn)Pnξ−U t +∞ξ +��H∞ � +��U t +nνk( /Dn)Pnξ−U s +nνk( /Dn)Pnξ +��H∞ +�|t−s|< ˜ε +12 since DNn(νk( /Dn)Pnξ)�1 ++ +��U s +nνk( /Dn)Pnξ−U s +∞νk( /D∞)ξ +��H∞ +�|||U snνk( /Dn)Pn−U s∞νk( /D∞)|||H∞< ˜ε +12 by Eq. (3.4) ++ +��U s +∞νk( /D∞)(ξ−ξ′) +��H∞ +�∥ξ−ξ′∥H∞< ˜ε +3 by Eq. (3.3) ++ +��U s +∞νk( /D∞)ξ′ −U s +∞ξ′��H∞ +�∥νk( /D∞)ξ′−ξ′∥H∞< ˜ε +12 ++ +��U s +∞ξ′ −U t +∞ξ′��H∞ +�|s−t|< ˜ε +12 ++ +��U t +∞ξ′ −U t +∞ξ +��H∞ +�∥ξ−ξ′∥H∞< ˜ε +3 +< ˜ε. +Let ξ ∈ dom( /D∞) with DN∞(ξ) � 1. Let n � N ′, and set η = νk( /D∞)Pnξ. For all +t ∈ +� +0, 1 +˜ε +� +, we have, η ∈ dom( /Dn) and DNn(η) � 1. Therefore, +inf +η∈dom(DNn) +DNn(η)�1 +sup +ω∈dom(TNn) +TNn(ω)�1 +���〈U t +nη,Θn(ω)〉Hn −〈U t +∞ξ,Πn(ω)〉H∞ +��� +� +sup +ω∈dom(TNn) +TNn(ω)�1 +���〈U t +nνk( /Dn)Pnξ,Θn(ω)〉Hn −〈U t +∞ξ,Πn(ω)〉H∞ +��� +� +sup +ω∈dom(TNn) +TNn(ω)�1 +� +�� +��U t +nνk( /Dn)ξ−U t +∞ξ +��H∞ ∥ω∥Mn + +��U t +∞ξ +��H∞ ∥Θn(ω)−Πn(ω)∥H∞ +<˜ε since TNn(ω)�1 +� +�� +< ˜ε+ ˜ε = ε. +Now, take ξ ∈ Hn, with DNn(ξ) � 1. By construction, ∥ /D∞ξ∥H∞ = ∥ /Dnξ∥Hn, and +U t +nξ =U t +∞ξ. So for all ξ ∈ dom(DNn) with DNn(ξ) � 1, we have, for all t ∈ R: +inf +η∈dom( /D∞) +DN∞(η)�1 +sup +ω∈dom(TNn) +TNn(ω)�1 +���〈U t +∞η,Πn(ω)〉H∞ −〈U t +nξ,Θn(ω)〉Hn +��� +� +���〈U t +∞ξ,Πn(ω)〉H∞ −〈U t +nξ,Θn(ω)〉Hn +��� = 0 < ε. +Therefore, for all n � max{N,N ′}, the ε-reach of Υn is no more than ε, and thus the +ε-magnitude µ(Υn|ε) of Υn is no more than ε (by Definition (3.12)). Therefore for all +n � N: +Λspec((An,Hn, /Dn),(A∞,H∞, /D∞)) � µ(Υn|ε) < ε, +and thus +lim +n→∞Λspec((An,Hn, /Dn),(A∞,H∞, /D∞)) = 0, +as claimed. +□ + +31 +Remark 3.18. A corollary of Theorem (3.17) is that we obtain convergence for the +bounded continuous functional calculus for the Dirac operators from the work in [31], +which extends Theorem (3.16). +4. EVEN SPECTRAL TRIPLES ON TWISTED GROUP C ∗-ALGEBRAS +We now apply our results of the previous sections to the construction of inductive +limits of spectral triples for the spectral propinquity on twisted C*-algebras of discrete +groups endowed with length functions. In particular we will prove in this section our +third main theorem, Theorem (4.11). Our approach introduces new metric spectral +triples on certain twisted group C*-algebras which generalize the related, though distinct, +past constructions using length functions over discrete groups such as the ones in [19]. +Our main applications would be the construction of new spectral triples over noncom- +mutative solenoids and some Bunce-Deddens algebras. In particular, we shall prove that +the noncommutative solenoids spectral triples are limits of spectral triples over quantum +2-tori for the spectral propinquity. We will start with detailing in the next two subsections +some background material that will be used to state and prove our main result. +4.1. Discrete Groups, Proper Length Functions, 2-Cocycles, and Classical Spectral +Triples. Let G∞ be a discrete group, and let σ be a 2-cocycle over G∞. Let λ be the +left regular σ-projective representation of G∞ on ℓ2(G∞), defined by, for all g ∈ G∞ and +for all ξ ∈ ℓ2(G∞): +λ(g)ξ : h ∈ G∞ �−→ σ(g,g −1h)ξ(g −1h). +Of course, each operator λ(g) is unitary for each g ∈ G∞. Let C ∗ +red(G∞,σ) be the re- +duced C*-algebra of G∞ twisted by σ, i.e. the C*-algebra of operators on ℓ2(G∞) gener- +ated by +� +λ(g) : g ∈ G∞ +� +. For any f ∈ ℓ1(G∞), the operator λ(f ) on ℓ2(G∞) is defined as +� +g∈G∞ f (g)λ(g) — it is easily checked that +������λ(f ) +������ +ℓ2(G∞) � +��f +�� +ℓ1(f ). The reduced group +C*-algebra C ∗ +red(G) is, in particular, C ∗ +red(G∞,1). +In [11], Connes introduced spectral triples (C ∗ +red(G∞),ℓ2(G∞),ML) using any proper +length function L overG∞, where ML is the operator of multiplication by L, defined on its +natural domain in the Hilbert space ℓ2(G∞). Connes proved that +������[ML,λ(g)] +������ +ℓ2(G∞) = +L(g) — which immediately follows from the triangle inequality and the fact that [ML,λ(g)]δe += L(g)σ(g,1)δg , where, for all g ∈ G∞: +δg : h ∈ G∞ �→ +� +1 if g = h, +0 otherwise. +It then follows that for the ∗-algebra Cc(G∞) of C-valued functions with finite support, +we obtain the inequality, for all f ∈ Cc(G∞): +(4.1) +������� +ML,λ(g) +������� +ℓ2(G∞) � +� +g∈G∞ +|f (g)|L(g), +since λ(g) is unitary for all g ∈ G∞. +Note that by construction, for the multiplication operator by L to have compact +resolvent, the spectral projection of this operator on any compact interval must have +finite rank. Thus, in particular, the set {δh ∈ ℓ2(G∞) : L(h) � r} must be finite for all r � 0. +In other words, all closed balls in G∞ for L must be finite, i.e., L must indeed be proper. +However, natural length functions on G∞ may not be proper, or even give the discrete +topology. An example of this situation is given when G∞ is the additive group +� +Z +� +1 +p +��2 + +32 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +where: +Z +� 1 +p +� +:= +� a +pn : n ∈ N,a ∈ Z +� +, +and where p ∈ N is prime. It is natural to regard Z +� +1 +p +� +as a subgroup of Q, and thus equip +it with the induced length function from the usual absolute value on Q (see Figure (3)). +However, this length function is not proper — and induces a non-discrete topology. We +moreover note that Z +� +1 +p +� += � +n∈N +1 +pn Z, and we would like to capture this inductive limit +structure metrically; while the sequence +� +1 +pn Z +� +n∈N converges to Z +� +1 +p +� +for the Hausdorff +distance induced by | · |, we can not apply this observation directly to the associated +twisted C*-algebras since |·| does not define a spectral triple using Connes’ methods. +Let us discuss this situation by returning to a general discrete group G∞ and some +2-cocycle σ on G∞. We now assume that we are given a strictly increasing sequence +(Gn)n∈N of subgroups of G∞ such that G∞ = � +n∈NGn — in fancier terms, G∞ is the +inductive limit of the sequence of groups (Gn)n∈N, which we identify with a sequence of +subgroups of G∞ from now on. We also identify σ with its restriction to Gn for all n ∈ N. +We now have a conundrum. If we choose a proper length function L on G∞, then, +since G∞ = � +n∈NGn with (Gn)n∈N increasing, any finite subset of G∞ is contained in +some GN (and thus in all Gn with n � N). This implies that (Gn)n∈N converges to G∞ for +the pointed Gromov-Hausdorff distance for proper metric space, where we always use +1 as our base point, and the metrics are induced by L (see [22]). On the other hand, as +soon as G∞ is infinite — which is the only interesting case to consider when G∞ is the +union of countably many groups, otherwise of course G∞ is just Gn for n large enough +— not only the diameter of G∞ is infinite — it can not be a closed ball as these are finite +— but the subgroups Gn are not close to G∞ for the Hausdorff distance induced by L in +general. So, we can define the spectral triples (C ∗ +red(Gn,σ),ℓ2(Gn),ML) as before since L +is proper, but in general, there is no apparent reason why |||[ML,a]|||ℓ2(G∞) is particularly +close to |||[ML,a]|||ℓ2(Gn) for a ∈ C ∗ +red(Gn,σ). +On the other hand, there may be length functions on G∞ for which (Gn)n∈N does +converge in the Hausdorff distance for these length functions, but these length functions +are not proper whenever G∞ is infinite. We are thus led to build a new type of spectral +triples which combine these two apparently opposite situations: one where we do not +know how to build a spectral triple using a non-proper length with otherwise good metric +properties for our purpose, and one with a proper length function which has bad metric +property. The following construction is inspired, but different from [19], where a proper +length function is constructed as a sum of a non-proper length function with a p-norm. +4.2. The Spectral Triples. We now define our new spectral triples on a particular type of +twisted group C*-algebras, which are the subject of our main third theorem, Theorem +(4.11), and its corollaries. +From now on we assume that G∞ is a discrete group endowed with a 2-cocycle σ with +values in T := {z ∈ C : |z| = 1}, and that G∞ is the union of a strictly increasing sequence +for inclusion, (Gn), of subgroups of G∞ such that G∞ = � +n∈NGn. +We also assume that we are given a length function LH on G∞, whose restriction to +each Gn is proper for each n ∈ N, and with the property that +(4.2) +lim +n→∞Haus[LH](G∞,Gn) = 0. + +33 +Z +� 1 +2 +� +⊆ Q +-3 +-2 +-1 +0 +1 +2 +3 +(A) The geometry of Z +� +1 +2 +� +for |·| in Q +LH +log2 ◦F +0 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3 +(B) The geometry of Z +� +1 +2 +� +using F +FIGURE 3. The geometry of Z +� 1 +2 +� +In addition we require that we are given a strictly increasing unbounded function scale : +N → [0,∞), together with F : G∞ → [0,∞) such that for all g ∉ G0: +F(g) = scale(min{n ∈ N : g ∈ Gn}), +while F restricted to G0 satisfies: +• ∀g ∈ G0 +F(g) = F(g −1), +• ∀g,h ∈ G0 +F(gh) � max{F(g),F(h)}, +• ∀g ∈ G0 +F(g) ∈ [0,scale(0)], +• F(1) = 0. +Clearly, the above assumptions provide us with many length functions on G∞ and Gn; +we will use them in our spectral triples constructions. +One of our main examples for this section will be the noncommutative solenoids, +whose fundamental components are described below. We will give more details on this +example later in this work. +Example 4.1. Let d � 2 and p a prime number. Let G∞ = +� +Z +� +1 +p +��d +, and let Gn = +� +1 +pn Z +�d +for all n ∈ N. We note that G∞ = � +n∈NGn. We can then choose LH to be the restriction +of any norm on Rd, and scale : n ∈ N → pn ∈ [0,∞), so that: +F : g ∈ G∞ �→ scale +� +min +� +n ∈ N : g ∈ +� 1 +pn Z +�d�� +. +Now, for any function f : Gn → C, we denote by M f the operator of multiplication by +f on the subspace: +dom +� +M f +� +:= +� +ξ ∈ ℓ2(Gn) : (h ∈ Gn �→ f (h)ξ(h)) ∈ ℓ2(Gn) +� +of ℓ2(Gn). Of course, M f is bounded by +��f +�� +C(Gn) if f is bounded, and unbounded +otherwise; nonetheless dom +� +M f +� +always contains Cc(Gn) and thus is always dense in +ℓ2(Gn). +Let E be a finite dimensional Hilbert space with inner product 〈·,·〉E and dimE ∈ +2N\{0}, and let c be a ∗-representation of the Clifford algebra of C2 on E. Let γ1 = c +��1 +0 +�� + +34 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +and γ2 = c +��0 +1 +�� +. For our purpose, we record that for all j,k ∈ {1,2}: +γj γk +γkγj = +� +2 if j = k, +0 otherwise. +. +Remark 4.2. There is no particular reason to restrict ourselves to E = C2, though it is the +natural choice. In this case, we can choose the usual Weyl matrices: +γ1 = +�1 +0 +0 +−1 +� +and γ2 = +�0 +1 +1 +0 +� +as the most natural choice for our construction. +For each n ∈ N := N∪{∞}, we identify the Hilbert space ℓ2(Gn,E) of E-valued func- +tions over Gn (with inner product 〈ξ,η〉ℓ2(Gn,E) := � +g∈Gn 〈ξ(g),η(g)〉E) with ℓ2(Gn)⊗E. We +then let +dom( /Dn) := +� +ξ ∈ ℓ2(Gn,E) : (LH(g)γ1ξ(g)+F(g)γ2ξ(g))g∈Gn ∈ ℓ2(Gn,E) +� +and on dom( /Dn), we define the Dirac operator: +(4.3) +/Dn := MLH ⊗γ1 + MF ⊗γ2. +We now prove that (C ∗(Gn,σ),ℓ2(Gn)⊗E, /Dn), as defined above, are indeed spectral +triples, for all n ∈ N. A first step is the computation of the domain of our Dirac operators +of Equation (4.3). To do so, we will need the following lemma. Recall that a norm ∥·∥R2 +on R2 is monotone when it is increasing with respect to the product order on R2; the +most important such norm for our purpose will be the max norm x, y ∈ R2 �→ +��(x, y) +�� +∞ = +max{|x|,|y|}; we also note that we will often write elements of Rd as simple d-tuples. +Lemma 4.3. With the notation and assumptions of this section, the following identities +hold. +(1) For all g ∈ G∞: +F(g −1) = F(g); +(2) For all g,h ∈ G∞: +F(gh) � max +�F(g),F(h) +� +� F(g)+F(h). +Moreover, if ∥·∥R2 is any monotone norm on R2, then g ∈ G∞ �→ +��(LH(g),F(g)) +��R2 is a +proper, unbounded length function over G∞. +Proof. Let g ∈ G∞, and let n ∈ N be the unique natural number such that F(g) = scale(n), +or n = 0 if F(g) < scale(0). If n = 0 then F(g) = F(g −1) by assumption. If n > 0, then +g ∈ Gn and g ∉ Gp for p < n; therefore, g −1 ∈ Gn and g −1 ∉ Gp if p < n; hence, F(g −1) = +scale(n) = F(g). +Now, take h ∈ G∞. Again, let m ∈ N be uniquely defined by F(h) = scale(m) or m = 0 +otherwise. Let k = max{m,n}. Thus g,h ∈ Gk and therefore, gh ∈ Gk. First, if g,h,gh ∈ G0, +then F(gh) � max{F(g),F(h)} by assumption on F. Otherwise, k > 0, and we simply +observe that either gh ∈ G0 and then F(gh) � scale(0) < scale(k), or gh ∉ G0, and again +F(gh) � scale(k); either way we observe: +F(gh) � scale(k) = scale(max{n,m}) = max{scale(n),scale(m)} = max{F(g),F(h)}. +Fix a monotone norm ∥·∥R2 on R2 and let +L : g ∈ G∞ �−→ +��(LH(g),F(g)) +��R2 . + +35 +It is then immediate to check that if g,h ∈ G∞, then, since ∥·∥R2 is monotone: +��(LH(gh),F(gh)) +��R2 � +��(LH(g)+LH(h),F(g)+F(h)) +��R2 +� +��(LH(g),F(g)) +��R2 +∥(LH(h),F(h))∥R2 . +Moreover +��(LH(g −1),F(g −1)) +��R2 = +��(LH(g),F(g)) +��R2 for all g ∈ G∞. +Finally, if +��(LH(g),F(g)) +��R2 = 0, then LH(g) = 0, which in turns implies g = 1. On the +other hand, F(1) = 0 and LH(1) = 0, so L(1) = 0. Thus as claimed, L is a length function +on G∞. +Now, let be more specific in our choice of ∥·∥R2, and fix it to be the usual max norm +∥·∥∞; we then rename our length L∞; so +L∞(g) := max +�LH(g),F(g) +� +. +Fix n ∈ N. By definition, the following equality between closed balls hold: +� +g ∈ G∞ : L(g) � scale(n) +� += +� +g ∈ Gn : LH(g) � scale(n) +� +. +Since LH is proper on Gn, this set is finite. So L is indeed proper on G∞. +By assumption, the function scale is unbounded on N and, for all n ∈ N, there exists +g ∈ G∞ \Gn (since (Gn)n∈N is assumed to be strictly increasing), i.e. F(g) � scale(n), so +L is unbounded. +We now return to a general monotone norm ∥·∥R2 on R2. Since all norms on R2 are +equivalent, there exists c > 0 such that 1 +c ∥·∥∞ � ∥·∥R2 � c ∥·∥∞. Therefore, +1 +c L∞ � L � cL∞. +It is now easy to check that L is again proper and unbounded on G∞. This concludes our +proof. +□ +Remark 4.4. It is quite natural to simply set F(g) = scale(0) for all g ∈ G0 \ {1}. The +difference between such a choice of F, vs any other F′, which meets our assumptions over +G0, is a bounded perturbation. We refer to [36] for a discussion on bounded perturbations +of spectral triples from the metric perspective. +As seen in the above discussion, the above length function LH will not be proper, so it +won’t define a spectral triple by itself, however L is proper, and thus can be used to define +a spectral triple on C ∗(G∞,σ). However, we take a slightly different route by working with +what we shall prove is an even spectral triple, replacing the linear geometry of G∞ with a +sort of “two-dimensional” geometry (see Figure (3) for the noncommutative solenoid +case). +We now prove that in the above hypotheses we can indeed define spectral triples. We +begin with a computation of the domain of the proposed Dirac operators defined in +Equation (4.3). +Lemma 4.5. With the notation and assumptions of this section, the following assertion +holds; for all ξ ∈ E and for all a,b ∈ R: +��(aγ1 +bγ2)ξ +��2 +E = +� +a2 +b2� +∥ξ∥2 +E . +In particular, for all n ∈ N, the domain dom( /Dn) of the Dirac operator /Dn is given by +� +ξ ∈ ℓ2(Gn,E) : +� +g∈Gn +(LH(g)2 +F(g)2) +��ξ(g) +��2 +E < ∞ +� +. + +36 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +Proof. Let ξ ∈ E. The following identity holds for all a,b ∈ R: +��aγ1ξ+bγ2ξ +��2 +E = a2〈γ1ξ,γ1ξ〉E + ab〈γ1ξ,γ2ξ〉E + ab〈γ2ξ,γ1ξ〉E +b2〈γ2ξ,γ2ξ〉E += a2〈γ2 +1ξ,ξ〉E + ab〈(γ1γ2 +γ2γ1)ξ,ξ〉E +b2〈γ2 +2ξ,ξ〉E += (a2 +b2)∥ξ∥2 +E . +as claimed. +The computation of the dom( /Dn), for all n ∈ N, follows immediately. +□ +We now prove that our Dirac operators are indeed self-adjoint with compact resolvent, +and that they can be used to define spectral triples. We also establish some useful +estimates which will later allow us to prove that our construction gives metric spectral +triples over noncommutative solenoids. +Definition 4.6. If a is a bounded operator on ℓ2(G∞), we denote by a◦ the operator a⊗1E +acting on ℓ2(G∞,E). We also define the representation λ of C ∗(G∞,σ) on ℓ2(G∞,E) by +setting λE := λ⊗1E, so for all f ∈ Cc(G∞), we have λ(f )◦ := λE(f ). Moreover +(1) For each n ∈ N, define: +dom(Ln) := +� +a ∈ sa +� +C ∗ +red(Gn,σ) +� +: a◦dom( /Dn) ⊆ dom( /Dn) and [ /Dn,a◦] is bounded +� +. +(2) For all a ∈ dom(Ln) define: +Ln(a) := +������[ /Dn,a◦] +������ +ℓ2(Gn,E). +We conclude this subsection by proving that we indeed defined even spectral triples, +and lay the groundwork for our third main theorem in the next subsection. Recall that, +by Lemma (4.3), LH +F is proper and unbounded on G∞. +Lemma 4.7. With the notation and assumptions of this section, for any fixed n ∈ N, the +ordered triple +(C ∗ +red(Gn,σ),ℓ2(Gn,E), /Dn) +is an even spectral triple, where the grading on ℓ2(Gn,E) is given by 1ℓ2(Gn) ⊗iγ1γ2. More- +over, for all a ∈ dom(Ln): +������[MLH ,a] +������ +ℓ2(Gn) � +������[ /Dn,a◦] +������ +ℓ2(Gn,E), +together with: +|||[MF,a]|||ℓ2(Gn) � +������[ /Dn,a◦] +������ +ℓ2(Gn,E). +In particular, if we define L := LH +F, then for all n ∈ N and all a ∈ dom(Ln): +|||[ML,a]|||ℓ2(Gn) � 2 +������[ /Dn,a◦] +������ +ℓ2(Gn,E). +If, for any n ∈ N, the spectral triple (C ∗ +red(Gn,σ),ℓ2(Gn), /DL) is metric, then so is (C ∗ +red(Gn,σ), +ℓ2(Gn,E), /Dn). +Proof. We will start by showing that, for any fixed n ∈ N, /Dn is self-adjoint with compact +resolvent. Fixed any n ∈ N, note that the domain of /Dn contains all finitely supported +functions in ℓ2(Gn,E) and is therefore dense. Moreover, since γ1 and γ2 are self-adjoint, + +37 +if ξ,η ∈ dom( /Dn), it follows that: +〈 /Dnξ,η〉ℓ2(Gn,E) = +� +g∈Gn +〈 +�LH(g)γ1 +F(g)γ2 +� +ξ,η〉E += +� +g∈Gn +〈ξ, +�LH(g)γ1 +F(g)γ2 +� +η〉E += 〈ξ, /Dnη〉ℓ2(Gn,E), +so /Dn is also a symmetric operator. By using Lemma (4.5), we now note that: +dom +� +/D2 +n +� += +� +ξ ∈ ℓ2(Gn,E) : +� +g∈Gn +�LH(g)2 +F(g)2�2 ��ξ(g) +��2 +E < ∞ +� +and, over dom +� +/D2 +n +� +, the Clifford algebra relations imply: +/D2 +n +1 = +� +M2 +LH + M2 +F +1 +� +⊗1E. +Now define an operator K on ℓ2(Gn,E) by setting, for all ξ ∈ ℓ2(Gn,E): +K ξ : g ∈ Gn �→ +1 +� +LH(g)2 +F(g)2 +1 +ξ(g). +By construction, K is positive. Moreover, if n ∈ N, then LH restricted to Gn is proper and +F is bounded over Gn by our hypotheses, so K is compact. If n = ∞, by our hypotheses, +for all r � 0, the set {g ∈ G∞ : F(g) � r} is a subset of Gk for some k ∈ N. Since LH is +proper on Gk, the set {g ∈ G∞ : L2 +H(g)+F2(g) � r} is finite. Thus, the eigenspaces of K +are all finite dimensional. It follows easily that K is compact, as well. +In any case, i.e., for all n ∈ N, ( /D2 +n +1)K 2ξ = ξ for all ξ ∈ ℓ2(Gn,E), while K 2( /D2 +n +1)ξ = +ξ for all ξ ∈ dom +� +/D2 +n +� +, as seen by a direct computation; in particular, we note that +K ℓ2(Gn,E) = dom( /Dn) by construction. +By Lemma (4.5), for all ξ ∈ ℓ2(Gn,E), we obtain: +� +g∈Gn +�� /DnK ξ(g) +��2 +E += +� +g∈Gn +����� +LH(g) +� +LH(g)2 +F(g)2 +1 +(γ1ξ(g))+ +F(g) +� +LH(g)2 +F(g)2 +1 +(γ2ξ(g)) +����� +2 +E += +� +g∈Gn +LH(g)2 +F(g)2 +LH(g)2 +F(g)2 +1 +��ξ(g) +��2 +E +� ∥ξ∥2 +ℓ2(Gn,E) . +Thus, /DnK is bounded, of norm at most 1. Consequently, ( /Dn ± i)K is also bounded +on ℓ2(Gn,E). Therefore, ( /D ±i)K 2 is compact. It follows that /D ±i both have compact +inverse ( /D ∓i)K 2. Specifically for our purpose, if ξ ∈ ℓ2(Gn,E), then: +( /Dn +i) +� +( /Dn −i)K 2� +ξ = ( /D2 +n +1)K 2ξ = ξ. +Therefore, the range of /Dn +i is ℓ2(Gn,E). Similarly, the range of /Dn −i is also ℓ2(Gn,E). +As /Dn is also a symmetric operator defined on a dense domain, we conclude by [53, Sec. +VIII.2] and [45, Lemma 2.48] that /Dn is indeed self-adjoint, with compact resolvent (since +the inverse of /Dn +i is the compact ( /Dn −i)K 2). +We will now verify the commutator spectral triples condition. Note that if g ∈ Gn, then +������[ /Dn,λE(g)] +������ +ℓ2(Gn,E) � LH(g)+F(g) = L(g). + +38 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +Therefore, if f ∈ Cc(Gn), then the operator [ /Dn,λE(f )] is bounded, and in fact, +������[ /Dn,λE(f )] +������ +ℓ2(Gn,E) � +� +g∈Gn +|f (g)|L(g). +We conclude that (C ∗ +red(G∞,σ),ℓ2(G∞), /D) is a spectral triple for all n ∈ N. +We will now prove that our spectral triple is metric. Let a ∈ dom(Ln) for some n ∈ N. +We then note that, +(1⊗γ1)[ /Dn,a◦]+[ /Dn,a◦](1⊗γ1) = [MLH ,a]⊗2, +which implies: +������[MLH ,a] +������ +ℓ2(Gn) � 1 +2 +������(1⊗γ1)[ /Dn,a◦]+[ /Dn,a◦](1⊗γ1) +������ +ℓ2(Gn,E) +� 1 +2 +�������(1⊗γ1)[ /Dn,a◦] +������ +ℓ2(Gn,E) + +������[ /Dn,a◦](1⊗γ1) +������ +ℓ2(Gn,E) +� +� 1 +2 +�������[ /Dn,a◦] +������ +ℓ2(Gn,E) + +������[ /Dn,a◦] +������ +ℓ2(Gn,E) +� += +������[ /Dn,a◦] +������ +ℓ2(Gn,E). +The same reasoning, with 1⊗γ2 in place of 1⊗γ1, leads to +|||[MF,a]|||ℓ2(Gn) � +������[ /Dn,a◦] +������ +ℓ2(Gn,E). +Therefore, for all a ∈ dom(Ln), we obtain: +|||[ML,a]|||ℓ2(Gn) � +������[MLF ,a] +������ +ℓ2(Gn) +|||[MF,a]|||ℓ2(Gn) � 2 +������[ /Dn,a◦] +������ +ℓ2(Gn,E). +In particular, if (C ∗ +red(Gn,σ),ℓ2(Gn), /DL) is a metric spectral triple, then, by [55, Lemma +1.10], so is (C ∗ +red(Gn,σ),ℓ2(Gn), /Dn). +Finally, we will show that our spectral triples are in fact even, with grading given by +1ℓ2(Gn) ⊗ γ where γ := iγ1γ2. By construction, γ2 is the identity, and γ∗ = γ, so γ is a +self-adjoint unitary; therefore so is 1ℓ2(Gn) ⊗γ, which splits ℓ2(Gn,E) in its two spectral +subspaces for 1 and −1, in such a way that λE commutes with 1⊗γ, while /Dn(1⊗γ) = +−(1⊗γ) /Dn. So (C ∗ +red(Gn,σ),ℓ2(Gn,E), /Dn) is an even spectral triple. +□ +Remark 4.8. With the notation of Lemma (4.7), we note that for each finite n ∈ N, the +spectral triple (C ∗ +red(Gn),ℓ2(Gn,E), /Dn) is, in some sense, a bounded perturbation of the +odd spectral triple (C ∗ +red(Gn),ℓ2(Gn),ML), since F is bounded on Gn. The situation is +quite different when n = ∞, of course. +Remark 4.9. Suppose ρ is some other 2-cocycle of G∞, which is equivalent to σ, i.e., for +some function f : G∞ → T, the following holds: +∀g,h ∈ G∞ +ρ(g,h) = f (g)f (h)f (gh)σ(g,h). +The operator M f is then a unitary which intertwines the left regular σ and ρ projective +representation of G∞. Thus, (AdM f )◦ implements a *-isomorphism from λE(C ∗(G∞,ρ)) +onto λE(C ∗(G∞,σ)). Furthermore, M◦ +f commutes with /D∞. Therefore, the spectral +triples (C ∗(G∞,σ),ℓ2(G∞,E), /D∞) and (C ∗(G∞,ρ),ℓ2(G∞,E), /D∞) are unitarily equiva- +lent. In particular, whenever one is metric, so is the other, and then they are at distance +zero from each others for the spectral propinquity. + +39 +4.3. Main result. We begin this section by making some basic identifications that will +be used throughout the rest of the paper. We will use the notation introduced in the +above sections. Fixed n ∈ N, the C*-algebra C ∗ +red(Gn,σ) is technically the closure, in +the operator norm, of the linear span of the operators λn(g) defined on ℓ2(Gn) by +λn(g)ξ : h ∈ ℓ2(Gn) �→ σ(g,g −1h)ξ(g −1h). On the other hand, since Gn ⊆ G∞, we obtain a +different unitary σ-projective representation of Gn, via the restriction of the σ-projective +representation λ of G∞ to Gn on ℓ2(G∞), giving us an alternative C*-algebra generated +by {λ(h) : h ∈ Gn}. If S ⊆ G∞ is any nonempty subset of G∞, we identify the space ℓ2(S) +with +{ξ ∈ ℓ2(G∞) : ∀g ∈ G∞ \S +ξ(g) = 0}. +Let Qn ⊆ G∞ be a subset of G∞ such that every right coset of Gn in G∞ is of the form Gnk +for a unique k ∈ Qn. Of course, +ℓ2(G∞) = +� +k∈Qnℓ2(Gnk), +where ⊕ is the Hilbert sum, i.e. the closure of the direct sum. +Now, we set, for all k ∈ G∞ and ξ ∈ ℓ2(G∞): +(4.4) +ρ(k)ξ : h ∈ ℓ2(G∞) �→ σ(hk,k−1)ξ(hk). +Thus defined, ρ is the right regular ˘σ-projective representation of G∞ on ℓ2(G∞), where +˘σ : g,h ∈ G∞ �→ σ(h−1,g −1) is indeed a 2-cocycle of G∞. +Remark 4.10. If the 2-cocycle σ is normalized, i.e. σ(g,g −1) = 1 for all g ∈ G∞, then ˘σ = σ; +we will however not need to work with normalized cocycles here. +Since σ is a 2-cocycle, we obtain, for all g,h,k ∈ G∞ and ξ ∈ ℓ2(G∞): +λ(g)ρ(k)ξ(h) = σ(g,g −1h)ρ(k)ξ(g −1h) += σ(g,g −1h)σ(g −1hk,k−1)ξ(g −1hk) += σ( g +=:x +,g −1hk +=:y +k−1 +=:z +)σ(g −1hk +=y +,k−1 +=z +)ξ(g −1hk) += σ( g +=x +,g −1hk +=y +)σ(hk +=xy +,k−1 +=z +)ξ(g −1hk) += σ(hk,k−1)(λ(g)ξ)(hk) += ρ(k)λ(g)ξ(h). +Therefore, λ(g) and ρ(k) commute, for all g,k ∈ G∞. It is moreover immediate that ρ(k−1) +maps ℓ2(Gnk) onto ℓ2(Gn). +We now define the unitary V from ℓ2(G∞) = � +k∈Qnℓ2(Gnk) to � +k∈Qnℓ2(Gn) by setting, +for all ξ = (ξk)k∈Qn ∈ � +k∈Qnℓ2(Gnk): +V ξ = +� +ρ(k−1)ξk +� +k∈Qn ∈ +� +k∈Qnℓ2(Gn). +By construction, V is unitary, and moreover, for any g ∈ Gn: +V λ(g)V ∗(ξk)k∈Qn = (λn(g)ξk)k∈Qn. +Thus, AdV is a ∗-isomorphism from the C*-subalgebra generated by {λ(g) : g ∈ Gn} and +the C*-algebra C ∗ +red(Gn,σ) which maps λ(g) to λn(g) for all g ∈ Gn. +From now on, we thus identify C ∗ +red(Gn,σ) with the C*-algebra generated by {λ(g) : g ∈ +Gn} in C ∗ +red(G∞,σ) and work exclusively in the latter. We will also identify ℓ2(Gn,E) with +� +ξ ∈ ℓ2(G∞,E) : ∀g ∈ G∞ \Gn +ξ(g) = 0 +� +. + +40 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +To complete our picture, we also identify /Dn with the operator defined for ξ = ξn + +ξ⊥ +n , with ξn ∈ ℓ2(Gn,E)2 and ξ⊥ +n ∈ ℓ2(Gn.E) = � +k∈Qn\Gnℓ2(Gnk,E), by /Dnξ = /Dnξn ∈ +ℓ2(Gn,E). We then observe that if Pn is the orthogonal projection from ℓ2(G∞) onto +ℓ2(Gn), we have, for all ξ ∈ dom( /Dn) and for all g ∈ G∞: +P◦ +n /Dnξ(g) = +� +(LH(g)⊗γ1 +F(g)⊗γ2)ξ(g) if g ∈ Gn, +0 otherwise, += /D∞P◦ +nξ(g). +We thus have shown that P◦ +ndom( /Dn) ⊆ dom( /D∞) and P◦ +n /Dn = /D∞P◦ +n. Moreover, P◦ +n /D∞P◦ +n = +/Dn and thus, for all a ∈ dom(Ln) we compute the following expression, using the fact +that [Pn,a] = 0,: +P◦ +n[ /D∞,a◦]P◦ +n = P◦ +n /D∞a◦P◦ +n −P◦ +na◦ /D∞P◦ +n += P◦ +n /D∞P◦ +na◦ − a◦P◦ +n /D∞P◦ +n += /Dna◦ − a◦ /Dn. +So we have, for all a ∈ dom(L∞): +(4.5) +Ln(a) = +������[ /Dn,a◦] +������ +ℓ2(Gn,E) = +������P◦ +n[ /D∞,a◦]P◦ +n +������ +ℓ2(G∞,E) � L∞(a). +With all of the above identifications, we thus have a natural unital ∗-morphism from +C ∗ +red(Gn,σ) into C ∗ +red(G∞,σ) which becomes just the natural inclusion, and +λ(g)ℓ2(Gnk) ⊆ ℓ2(Gnk) +for each g ∈ Gn and k ∈ G∞. By linearity and continuity, we conclude that if a ∈ +C ∗ +red(Gn,σ), then aℓ2(Gnk) ⊆ ℓ2(Gnk) for all k ∈ G∞. We also note that [ /D∞,a◦]ℓ2(Gnk,E) ⊆ +ℓ2(Gnk,E) for all k ∈ G∞ and a ∈ dom(Ln). +We will work for the rest of this section with the above identifications and their basic +properties without further mention. +Our main theorem in this section involves, in particular, a strong result about the +convergence of some of the quantum compact metric spaces induced by our spectral +triples: namely, we obtain some convergence in the sense of the Lipschitz distance. +The Lipschitz distance LipD, extended to noncommutative metric geometry in [37], is +defined between any two quantum compact metric spaces (A,LA) and (B,LB), by +LipD((A,LA),(B,LB)) := +inf +� +ln(k) : ∃π : (A,LA) → (B,LB) Lipschitz *-isomorphism with 1 +k LA � LB ◦π � kLA +� +, +with the convention that inf� = ∞. Thus LipD is finite only between quantum compact +metric spaces built over isomorphic C*-algebras. As shown in [37], the Lipschitz distance +dominates the Gromov-Hausdorff propinquity; in fact, closed balls for the Lipschitz +distance are compact in the propinquity. +In particular, if A is a unital C*-algebra, and if L1 and L2 are two Lipschitz seminorms +over A with the same domain, then the identity is bi-Lipschitz, and we do obtain, by +definition: +LipD((A,L1),(A,L2)) � ln(C) if 1 +C L1 � L2 � CL1. +We now prove our main result about inductive limits of discrete groups and the +convergence, for the spectral propinquity, of their spectral triples. Note that below we +use the notation established in Definition (4.6). + +41 +Theorem 4.11. With the notation and assumptions of Subsection 4.2, if +(C ∗ +red(Gn,σ),ℓ2(Gn,E), /Dn) +is a metric spectral triple for all n ∈ N, and if +{a ∈ dom(Ln) : Ln(a) � 1} = cl({a ∈ Cc(Gn) : Ln(a) � 1}), +then +lim +n→∞Λspec � +(C ∗ +red(Gn,σ),ℓ2(Gn,E), /Dn),(C ∗ +red(G∞,σ),ℓ2(G∞,E), /D∞) +� += 0. +Moreover, for any fixed k ∈ N, the sequence (C ∗ +red(Gk,σ),Ln)n�k converges in the Lips- +chitz distance LipD to the quantum compact metric space (C ∗ +red(Gk,σ),L∞). +Proof. We shall check that the identity automorphism of C ∗ +red(G∞,σ) satisfies the hypoth- +esis of Theorem (3.17). +Obviously, the identity is a full quantum isometry of (C ∗ +red(G∞,σ),L∞). +Let C = 2qdiam(C ∗(G∞,σ),L∞) — note that since G∞ ̸= {1}, we have C > 0. Let tr : a ∈ +C ∗ +red(G∞,σ) �→ 〈aδ1,δ1〉ℓ2(G∞); tr is a tracial state of C ∗(G∞,σ) which maps a ∈ Cc(G∞) to +a(1). +Fix ε ∈ +� +0, C +2 +� +. Since (C ∗ +red(G∞,σ),L∞) is a quantum compact metric space by assump- +tion, the set X∞ := {a ∈ dom(L∞) : L∞(a) � 1,tr(a) = 0} is compact. Thus, there exists +a finite ε-dense subset X ε +∞ ⊆ X∞. Since X∞ = cl({a ∈ Cc(G∞) : L∞(a) � 1,tr(a) = 0}), we +can moreover assume that X ε +∞ ⊆ Cc(G∞) as well. +Since X ε +∞ is finite and each of its element has finite support, there exists a finite subset +S ⊆ G∞ which contains the support of all the elements in X ε +∞. Since G∞ = � +n∈NGn and +(Gn)n∈N is increasing, there exists N1 ∈ N such that, for all n � N1, we have S ⊆ Gn. Thus +X ε +∞ ⊆ Cc(Gn). Moreover, by Expression (4.5), we also obtain Ln(a) � L∞(a) for all a ∈ X ε +∞. +In summary, +∀a ∈ X∞ +∃b ∈ X ε +∞ ⊆ Cc(Gn) ⊆ C ∗ +red(Gn,σ) : +∥a −b∥C∗ +red(G∞,σ) < ε and Ln(a) � L∞(a) � 1. +If a ∈ dom(L∞), then there exists b ∈ X ε +∞ such that ∥a −tr(a)−b∥C∗ +red(G∞,σ) < ε. Of course, +b +tr(a) ∈ C ∗ +red(Gn,σ) and Ln(b +tr(a)) = Ln(b) � 1. By homogeneity, it follows that for all +a ∈ dom(L∞), and for all n � N1, there exists b ∈ dom(Ln) such that ∥a −b∥C∗ +red(G∞,σ) < +εL∞(a) and Ln(b) � L∞(a). +Now, using our assumption of Equation (4.2), there exists N2 ∈ N, with N2 � N1, such +that +Haus[LH](G∞,Gn) < ε +C 2 . +For each right coset c of Gn in G∞, let k ∈ c. Since the distance for LH from k ∈ G∞ to +Gn is strictly less than +ε +C2 , there exists g ∈ Gn such that LH(g −1k) < ε +C2 . Setting kc = g −1k, +we have by definition of right cosets that c = Gnkc. Therefore, there exists a subset +Qn ⊆ G∞ of G∞ such that, if k ∈ Qn then LH(k) < ε +C2 , and if c is a right coset of Gn in G∞, +then there exists a unique k ∈ Qn such that c = Gnk. +Let n � N2 and let b ∈ Cc(Gn) ⊆ C ∗ +red(Gn,σ) with b(1) = tr(b) = 0. Note that b ∈ +dom(L∞)∩dom(Ln) so, in particular, both Ln(b) and L∞(b) are finite. +We thus have ℓ2(G∞) = ⊕k∈Qnℓ2(Gnk), where ⊕ is the Hilbert sum (the closure of +the sum). If h ∈ Gn, then, by definition of a right coset, λ(h)ℓ2(Gnk) ⊆ ℓ2(Gnk) for +all k ∈ Qn. As /D∞ (dom( /Dn)) ⊆ ℓ2(Gnk,E) as well for all k ∈ Qn, we conclude that + +42 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +[ /D∞,b◦] +� +ℓ2(Gnk,E) +� +⊆ ℓ2(Gnk,E) — i.e. b, /D∞ and its commutators are all block di- +agonal in this decomposition of ℓ2(G∞). It follows that +(4.6) +������[ /D∞,b◦] +������ +ℓ2(G∞,E) = sup +k∈Qn +������[ /Dn,b◦] +������ +ℓ2(Gnk,E), +allowing for any of the above norm to be infinite. +Now, the restriction of /D∞ to dom( /Dn) is exactly /Dn, so: +������[ /D∞,b◦] +������ +ℓ2(Gn,E) = +������[ /Dn,b◦] +������ +ℓ2(Gn,E) = Ln(b). +Now, fix k ∈ Qn and k ∉ Gn. By assumption, and using repeatedly that (Gp)p∈N +is increasing, we observe that F(gk) = F(k) for all g ∈ Gn: Lemma (4.3) implies that +F(gk) � F(k) since F(g) � n < F(k); on the other hand, if p ∈ {0,...,m −1} where F(k) = +scale(m), noting that m > 0 since k ∉ G0, then gk ∈ Gp implies k = g −1gk ∈ Gn, which is +a contradiction; hence F(kg) = scale(m), as claimed. +Therefore, the operator MF is constant on ℓ2(Gnk), and thus [MF,b] = 0 on ℓ2(Gnk). +So, +������[ /D∞,b◦] +������ +ℓ2(Gnk,E) = +������[MLH ,b] +������ +ℓ2(Gnk). +We will use the ˘σ-projective right representation of G∞ on ℓ2(G∞), as defined in +Expression (4.4). By construction, the restriction of ρ(k) to ℓ2(Gn) (which we will keep +denoting by ρ(k)) is a unitary onto ℓ2(Gnk) (with inverse the restriction to ℓ2(Gnk) of its +adjoint ρ(k)∗ = ρ(k−1)). Therefore, +(4.7) +������[MLH ,b] +������ +ℓ2(Gnk) = +������[MLH ,b]ρ(k) +������ℓ2(Gnk) +ℓ2(Gn) . +Next, a simple computation shows (like with λ) that the unitary ρ(k) maps dom +� +MLH +� +to itself, and for all ξ ∈ ℓ2(G∞) and h ∈ G∞: +[MLH ,ρ(k)]ξ(h) = (LH(h)−LH(hk))σ(hk,k−1)ξ(hk) +so +������[MLH ,ρ(k)]ξ +������ +ℓ2(Gn) � suph∈Gn |LH(h)−LH(hk)|∥ξ∥ℓ2(Gn) � LH(k)∥ξ∥ℓ2(Gn). Choos- +ing ξ = δ1, we obtain +(4.8) +������[MLH ,ρ(k)] +������ +ℓ2(Gn) = LH(k). +Now, since ρ(k) commutes with λ(g) for all g ∈ G∞, we conclude [b,ρ(k)] = 0, and +thus, on dom +� +MLH +� +[MLH ,b]ρ(k) = MLH bρ(k)−bMLH ρ(k) +(4.9) += MLH ρ(k)b +[b,ρ(k)]=0 +−bρ(k)MLH −b[MLH ,ρ(k)] += [MLH ,ρ(k)]b +ρ(k)MLH b − ρ(k)b +[ρ(k),b]=0 +MLH −b[MLH ,ρ(k)] += [MLH ,ρ(k)]b +ρ(k)[MLH ,b]−b[MLH ,ρ(k)]. + +43 +Therefore, by Equation (4.7), +������[ /D∞,b◦] +������ +ℓ2(Gnk,E) += +������[MLH ,b]ρ(k) +������ℓ2(Gnk) +ℓ2(Gn) +� +������[MLH ,ρ(k)]b +������ℓ2(Gnk) +ℓ2(Gn) + +������ρ(k)[MLH ,b] +������ℓ2(Gnk) +ℓ2(Gn) + +������b[MLH ,ρ(k)] +������ℓ2(Gnk) +ℓ2(Gn) +by Eq. (4.9) +� +������[MLH ,ρ(k)] +������ℓ2(Gnk) +ℓ2(Gn) +�LH (k) by Eq. (4.8) +∥b∥C∗(Gn,σ) + +������ρ(k) +������ℓ2(Gnk) +ℓ2(Gn) +=1 as ρ(k) is unitary +������[MLH ,b] +������ +ℓ2(Gn) +�Ln(b) by Lemma (4.7) ++∥b∥C∗(Gn,σ) +������[MLH ,ρ(k)] +������ℓ2(Gnk) +ℓ2(Gn) +�LH (k) by Eq. (4.8) +� LH(k)∥b∥C∗ +red(Gn,σ) +� C +2 L∞(b) ++Ln(b)+∥b∥C∗ +red(Gn,σ)LH(k) +� ε +C 2 +C +2 L∞(b)+Ln(b)+ C +2 L∞(b) ε +C 2 +� Ln(b)+ ε +C L∞(b). +By Expression (4.6), we thus get +L∞(b) � Ln(b)+ ε +C L∞(b). +Therefore, we have shown that since ε ∈ +� +0, C +2 +� +, +(4.10) +∀b ∈ Cc(Gn) +tr(b) = 0 =⇒ L∞(b) � +1 +1− ε +C +Ln(b). +Now, let b ∈ Cc(Gn). We then easily compute: +L∞(b) = L∞(b −tr(b)1) � +1 +1− ε +C +Ln(b −tr(b)1) = +1 +1− ε +C +Ln(b). +Now, let a ∈ dom(Ln) with Ln(a) � 1. By assumption, there exists a sequence (ak)k∈N +converging in C ∗ +red(Gn,σ) to a such that Ln(ak) � 1 and ak ∈ Cc(Gn) for all k ∈ N. We thus +have, by lower semicontinuity of Ln, and Expression (4.10): +L∞(a) � liminf +k→∞ L∞(ak) � +1 +1− ε +C +liminf +k→∞ Ln(ak) � +1 +1− ε +C +. +Thus, we have shown that, for all n � N, if a ∈ dom(Ln), then a ∈ dom(L∞), and more- +over, +∀a ∈ dom(Ln) +L∞(a) � +1 +1− ε +C +Ln(a). +It is immediate by construction that Ln � L∞ on dom(Ln). Thus we have proven that +for all n � N and k � n, we have Lk � L∞ � +1 +1− ε +C Lk(a). As a byproduct of this, we have +shown that limk→∞LipD((C ∗(Gn,σ),Lk),(C ∗(Gn,σ),L∞) = 0. +We now pause to note that, thanks to our identifications discussed prior to this theo- +rem, and the observation that dom(Ln) ⊆ dom(L∞) which we have just now established, +(C ∗ +red(Gn,σ),ℓ2(Gn)⊗E, /Dn)n∈N is an inductive sequence of spectral triples in the sense +of [20], where the ∗-morphisms from C ∗ +red(Gn,σ) to C ∗ +red(Gn+1,σ) and the linear isometry + +44 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +from ℓ2(Gn) to ℓ2(Gn+1) are just the inclusion maps. Moreover (C ∗ +red(G∞,σ),ℓ2(G∞,E), /D∞) +is indeed the inductive limit of this system. +We now note that since L∞ � +� +1 +1− ε +C +� +Ln and ε ∈ +� +0, C +2 +� +, we have +qdiam +� +C ∗ +red(Gn,σ),Ln +� +� +� +1 +1− ε +C +� +qdiam +� +C ∗ +red(G∞,σ),L∞ +� += +C 2 +2(C −ε) � C. +Let b ∈ dom(Ln), and let a = +� +1− ε +C +� +b ∈ dom(L∞). We then compute: +∥b − a∥C∗ +red(G∞,σ) = +���b − +� +1− ε +C +� +b +��� +C∗ +red(G∞,σ) +� ε +C ∥b∥C∗ +red(G∞,σ) +� ε +C qdiam +� +C ∗ +red(Gn,σ),Ln +�Ln(b) +� ε +C C Ln(b) = εLn(b), +while +L∞(a) = L∞ +�� +1− ε +C +� +b +� +� +1 +1− ε +C +Ln +�� +1− ε +C +� +b +� += Ln(b). +Hence, if n � N2, then: +• ∀a ∈ dom(L∞) +∃b ∈ dom(Ln) : +Ln(b) � L∞(a) and ∥b − a∥C∗ +red(G∞,σ) < εL∞(a), +• ∀b ∈ dom(Ln) +∃a ∈ dom(L∞) : +L∞(a) � Ln(b) and ∥a −b∥C∗ +red(G∞,σ) < εLn(b). +Therefore, by Theorem (3.17), we conclude that +lim +n→∞Λspec((C ∗ +red(Gn,σ),ℓ2(Gn,E), /Dn),(C ∗ +red(G∞,σ),ℓ2(G∞,E), /D∞)) = 0, +as claimed. +□ +We now wish to apply Theorem (4.11) to the family in Example (4.1), as well as to the +Bunce-Deddens algebras. Thus, we shall now focus on Abelian groups. +So from now on we assume that G∞ is Abelian. Therefore we will employ the additive +notation for the groups Gn (n ∈ N). Since Abelian groups are amenable, we will also from +now on identify C ∗ +red(Gn,σ) with C ∗(Gn,σ) for all n ∈ N. +A key condition for Theorem (4.11) is always met when working with Abelian groups, +as seen in the following lemma. +Lemma 4.12. With the assumptions and notation of Subsection (4.2), for any n ∈ N, if Gn +is Abelian, then we have that +{a ∈ dom(Ln) : Ln(a) � 1} = cl({a ∈ Cc(Gn) : Ln(a) � 1}). +Proof. Fix n ∈ N. Since Ln is lower semicontinuous, we get +cl({a ∈ dom(Ln)∩Cc(Gn) : Ln(a) � 1}) ⊆ {a ∈ dom(Ln) : Ln(a) � 1}. +We now prove that when Gn is Abelian, the converse inclusion holds. +Let � +Gn be the Pontryagin dual of Gn (we will use the multiplicative notation for � +Gn). +The dual action β of � +Gn on C ∗(Gn,σ) is unitarily implemented by defining, for each +z ∈ � +Gn, the unitary vz of ℓ2(Gn,E) which is given by, for all ξ ∈ ℓ2(Gn)⊗E: +vzξ : g ∈ Gn �−→ z(g)ξ(g)(= z(−g)ξ(g)). + +45 +It is easily checked that z ∈ � +Gn �→ vz is a unitary representation of � +Gn. We then note +that: +∀z ∈ � +Gn +vzλE(g)(vz)∗ = βzλE(g). +By construction, /Dn commutes with vz for all z ∈ � +Gn, so β acts by full quantum +isometries on (C ∗(Gn,σ),Ln). +Let µ be the Haar probability measure on � +Gn. As seen in [32, Lemma 3.1],[57, Theorem +8.2], there exists a sequence (ϕk)k∈N of non-negative functions over � +Gn, each obtained as +a linear combination of characters of � +Gn (i.e. of the form z ∈ � +Gn �→ z(g) for some g ∈ Gn, +by Pontryagin duality), such that +� +� +Gn ϕk dµ = 1 for all k ∈ N, and (ϕk)k∈N converges, in +the sense of distributions, to the Dirac measure at 1 ∈ � +Gn, i.e., for all f ∈ C(� +Gn), +lim +k→∞ +� +� +Gn +f (z)ϕk(z)dµ(z) = f (1). +We define, for each k ∈ N, the continuous linear endomorphism: +βϕk : a ∈ C ∗(Gn,σ) �→ +� +� +Gn +βz(a)ϕk(z)dµ(z), +acting on C ∗(Gn,σ). Since the dual action is strongly continuous, we conclude that, for +all a ∈ C ∗(Gn,σ): +lim +k→∞ +��βϕk (a)− a +�� +C∗(Gn) = 0. +Since Ln is lower semicontinuous, ϕk � 0 and +� +� +G∞ ϕk dµ = 1 for all k ∈ N, and β acts +by quantum isometries, we also get, for all a ∈ dom(Ln), +Ln +� +βϕk (a) +� +� +� +� +Gn +ϕ(z)Ln(a)dµ(z) = Ln(a). +As a quick digression, lower semicontinuity also implies that Ln(a) � liminfk→∞Ln(βϕk(a)), +so altogether we have shown that Ln(a) = liminfk→∞Ln(βϕk (a))). +For each k ∈ N, as ϕk is a linear combination of characters of � +Gn, there exists a +finite subset F ⊆ Gn and a function t : F → C such that ϕk : z ∈ � +Gn �→ � +g∈F t(g)z(g); +the range of βϕk is then the finite dimensional subspace of Cc(Gn) consisting of the +functions supported on F. For our purpose, the main observations here are that, given +a ∈ dom(Ln), and ε > 0, there exists K ∈ N such that if k � K , then +��a −βϕk (a) +�� +C∗(Gn,σ) < +ε and Ln(βϕk (a)) � Ln(a). In particular, again since Ln is lower semi-continuous, it +follows that: +(4.11) +{a ∈ dom(L /D) : L /D(a) � 1} = cl({a ∈ dom(L /D)∩Cc(Gn) : L /D(a) � 1}), +as claimed. +□ +Remark 4.13. With the notation of the proof of Lemma (4.12), fix ϕ ∈ S (C ∗(Gn,σ)). Since, +for all k ∈ N, we have +� +� +Gn ϕk dµ = 1, we conclude that βϕk is a unital map, and thus +sup +���a −βϕk (a) +�� +C∗(Gn) : a ∈ dom(Ln),Ln(a) � 1 +� += sup +���a −βϕk (a) +�� +C∗(Gn) : a ∈ dom(Ln),Ln(a) � 1,µ(a) = 0 +� +where the second supremum is indeed finite since X = {a ∈ dom(Ln) : Ln(a) � 1,µ(a) = 0} +is compact and we take the supremum of a continuous function over this set. In fact, +Arzelà-Ascoli theorem can be applied here to prove that the convergence of (βϕk )k∈N to + +46 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +the identity on X is uniform, though we here offer a simple ε +3-type argument. First, note +that for all a,b ∈ C ∗(G∞), and for all k ∈ N, +��βϕk (a)−βϕk (b) +�� +C∗(G∞) � +� +� +G∞ +∥a −b∥C∗(G∞) ϕk(z)dµ(z) = ∥a −b∥C∗(G∞) . +Moreover, for all ε > 0, there exists a finite ε +3-dense subset Xε of X ; as Xε is finite, there +exists K ∈ N such that, for all k � K and for all a ∈ Xε, then +��a −βϕk (a) +�� +C∗(G∞) < ε +3, as +seen above; therefore for all k � K , we have +��a −βϕk (a) +�� +C∗(G∞) � +��a − a′�� +C∗(G∞) + +��a′ −βϕk (a′) +�� +C∗(G∞) + +��βϕk (a′ − a) +�� +C∗(G∞) +< ε +3 + ε +3 + ε +3 = ε. +This proves that indeed, (βϕk )k∈N converges uniformly to the identity over X . +We will prove that some of the spectral triples introduced in Subsection (4.2) are +metric by invoking a property central to the work in [9, 50], called bounded doubling, +which we now recall in the formulation of [50]. +Definition 4.14 ([9, 50]). A proper length function L on a discrete group G satisfies the +bounded doubling property when there exists θ > 1 and c > 0 such that, for all r � 1: +��� +g ∈ G : L(g) � θ ·r +��� � c +��� +g ∈ G : L(g) � r +���. +The bounded doubling property indeed ensures the following result. +Lemma 4.15. The spectral triples constructed in Subsection (4.2) are metric if the proper +length function L := max{LH,F} has the bounded doubling property. +Proof. We note that Lemma (4.3) proves that L is a proper unbounded length function. +By [9, 50], since all our groups are Abelian hence nilpotent, for any µ ∈ S (C ∗(Gn),σ), +the set +� +a ∈ Cc(Gn) : |||[ML,a]|||ℓ2(Gn) � 1,µ(a) = 0 +� +is totally bounded. Since |||[ML,·]|||ℓ2(Gn) � Ln on Cc(Gn), we thus conclude that +� +a ∈ Cc(Gn) : Ln(a) � 1,µ(a) = 0 +� +⊆ +� +a ∈ Cc(Gn) : |||[ML,a]|||ℓ2(Gn) � 1,µ(a) = 0 +� +and thus +� +a ∈ Cc(Gn) : Ln(a) � 1,µ(a) = 0 +� +is also totally bounded. By Lemma (4.12), we +also have: +{a ∈ dom(Ln) : Ln(a) � 1,µ(a) = 0} = cl +�� +a ∈ Cc(Gn) : Ln(a) � 1,µ(a) = 0 +�� +so {a ∈ dom(Ln) : Ln(a) � 1,µ(a) = 0} is compact. Thus by Theorem (2.9), Ln is a Lipschitz +seminorm, i.e. our spectral triples are metric. +□ +We are now ready to establish the following theorem. +Theorem 4.16. Let G = � +n∈NGn be an Abelian discrete group, arising as the union of a +strictly increasing sequence (Gn)n∈N of subgroups of G. Let σ be a 2-cocycle of G and LH a +length function on G such that +lim +n→∞Haus[LH](Gn,G) = 0, +and whose restriction to Gn is proper for all n ∈ N. Assume scale : N → [0,∞) is a strictly +increasing, unbounded function such that, if we set +F : g ∈ G �−→ scale(min{n ∈ N : g ∈ Gn}) + +47 +then the proper length function L := max{LH,F} has the bounded doubling property. +Then, for any Hermitian space E, +lim +n→∞Λspec((C ∗(G,σ),ℓ2(G)⊗E, /D),(C ∗(Gn,σ),ℓ2(Gn)⊗E, /Dn)) = 0, +where +• +/D = MLH ⊗γ1 +MF ⊗γ2 on +� +ξ ∈ ℓ2(G)⊗E : � +g∈G(LH(g)2 +F(g)2) +��ξ(g) +��2 +E < ∞ +� +, +with γ1,γ2 unitaries of E such that, for all j,k ∈ {1,2}: +γj γk +γkγj = +� +2 if j = k, +0 otherwise. +• ℓ2(Gn)⊗E is identified with the subspace of Gn-supported vectors in ℓ2(G)⊗E, +• +/Dn is the restriction of /D to dom( /D)∩ +� +ℓ2(Gn)⊗E +� +, +• C ∗(G,σ) and C ∗(Gn,σ) act via their left regular σ-projective representations. +Proof. Our theorem follows from Theorem (4.11). We first note that Lemma (4.15) proves +that all our spectral triples are metric. By Lemma (4.12), since G∞ is Abelian, we conclude +that, for all n ∈ N, +{a ∈ dom(Ln) : Ln(a) � 1} = cl({a ∈ Cc(Gn) : Ln(a) � 1}). +Since all hypotheses of Theorem (4.11) are met, the result follows. +□ +In particular, for the noncommutative solenoids of Example (4.1), we obtain the +following. +Corollary 4.17. Fix a prime number p ∈ N and d ∈ N\{0,1}. For each n ∈ N, let +Gn := +� 1 +pn Z +�d +and +G∞ := +� +Z +� 1 +p +��d +. +Fix a 2-cocycle σ on G∞ such that ∀g ∈ G∞ +σ(g,−g) = 1. +Let LH be the restriction to G∞ of some norm on R2. We define F by setting, for all +g ∈ G∞: +F(g) := min +� +pn : g ∈ +� 1 +pn +�d� +. +Let E be an even dimensional hermitian space, with γ1,γ2 be two unitaries on E such +that, for all j,k ∈ {1,2}: +γj γk +γkγj = +� +2 if j = k, +0 otherwise. +If we define, for all n ∈ N, the operator +/Dn := MLH ⊗γ1 + MF ⊗γ2 on dom( /Dn) +on the domain +dom( /Dn) := +� +ξ ∈ ℓ2(Gn,E) : +� +g∈Gn +(LH(g)2 +F(g)2) +��ξ(g) +��2 +E < ∞ +� +, +then, for all n ∈ N, the triple (C ∗(Gn,σ),ℓ2(Gn,E), /Dn) is a metric spectral triple, and: +lim +n→∞Λspec((C ∗(Gn,σ),ℓ2(Gn,E), /Dn),(C ∗(G∞,σ),ℓ2(G∞,E), /D∞)) = 0. + +48 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +Moreover, for each n ∈ N, the sequence (C ∗(Gn,σ),Lk)k�n of quantum compact metric +spaces converge to (C ∗(Gn,σ),L∞) in the Lipschitz distance. +Proof. We first establish the bounded doubling property of certain related length func- +tions. +Fix a prime number p and d � 2. For all g ∈ G∞, let +L′(g) = max +� +��g +��Rd ,p +min +� +n∈N:g∈ +� +1 +pn Z +�d �� +, +where the norm we choose on Rd for this proof is the max norm. By Lemma (4.3), the +function L′ is an unbounded proper length function. By Lemma (4.7), we have that +|||[ML′,·]|||ℓ2(Gn) � Ln on C(Gn) for all n ∈ N. By [11], the triple (C ∗(Gn,σ),ℓ2(Gn),ML′) is +a spectral triple. +Assume L′(g) � pn. Since g ∈ +� +1 +pn Z +�d +, we can write g = +� aj +pn +� +1�j�d for a1,...,ad ∈ Z. +Since +��g +��Rd � pn, we also have a1,...,ad ∈ [−p2n,p2n]. Conversely, if g = +� aj +pn +� +1�j�d +with −p2n � a j � p2n for all j ∈ {1,...,d}, then L′(g) � pn by definition. Hence, the closed +ball of center (0,0) and radius pn has cardinal (2p2n +1)d. +Consequently: +��� +g ∈ G∞ : L′(g) � pn+1��� = (2p2n+2 +1)d +� (2p2n+2 + p2)d += p2d(2p2n +1)d +� p2d ��� +g ∈ G∞ : L′(g) � pn���. +Therefore, L′ is a proper unbounded length with the bounded doubling property. +Let LH be any norm on Rd. Since all the norms on Rd are equivalent, there exists +C > 0 such that 1 +C LH � ∥·∥Rd � CLH. Then +1 +C (max{LH,F}) � L′ � C max{LH,F}. +Therefore, +��� +g ∈ G∞ : max +�LH(g),F(g) +� +� pn+1��� � C 2p2d ��� +g ∈ G∞ : max +�LH(g),F(g) +� +� pn���. +Write L := max{LH,F} on Cc(Gn). We thus have shown that L, which is unbounded +and proper by Lemma (4.3), also has the bounded doubling property. +By Lemma (4.12), since G∞ is Abelian, we conclude that +∀n ∈ N +{a ∈ dom(Ln) : Ln(a) � 1} = cl({a ∈ Cc(Gn) : Ln(a) � 1}). +Thus, our corollary follows from Theorem (4.11). +□ +We can choose somewhat different length functions over +� +Z +� +1 +p +��d +, by varying not +only LH, but also F. For instance, Corollary (4.17) remains valid if we replace F by +F′ : (g1,...,gd) ∈ G∞ �→ maxd +j=1 |g j |p, where |·|p is now the p-norm. The resulting length +function max{LH,F′} has the bounded doubling property, as seen by applying [19, Propo- +sition 3.17] up to an equivalence of metrics. We also note that for this construction to +give us something different from Corollary (4.17), we require that LH(g) < F′(g) for at +least one g ∈ Zd \{0}. In general, the difference is only up to a bounded perturbation of +the underlying Dirac operator. + +49 +Another interesting family of C*-algebras to which our work applies are certain Bunce- +Deddens algebras. +Notation 4.18. Let P be the set of all sequences (αn)n∈N of nonzero natural numbers +such that αn+1 +αn +is a prime number for all n ∈ N. +Notation 4.19. For any integer m ∈ Z, we denote the quotient group Z⧸mZ simply by +Z⧸m. +Notation 4.20. Let α := (αn)n∈N ∈ P. If n ∈ N, then αn divides αn+1, and thus the map +ρn : (m +mod αn+1) ∈ Z⧸αn+1 → (m +mod αn) ∈ Z⧸αn +where x mod y is the equivalence class of x ∈ Z modulo y ∈ Z \ {0}, is a well-defined +surjective group morphism. The projective limit of the projective sequence +Z⧸α0 +ρ0 +←−−− +Z⧸α1 +ρ1 +←−−− +Z⧸α2 +ρ2 +←−−− +Z⧸α3 +ρ4 +←−−− +··· +is denoted by Z⧸α. By construction, we observe that: +Z⧸α = +� +(zn)n∈N ∈ +∞ +� +j=0 +Z⧸αn : ρn(zn+1) = zn +� +. +We endow Z⧸α with its topology as a projective space of compact spaces, i.e. with the +topology induced by the product topology on �∞ +j=0Z⧸αn, which is compact by Tychonoff +theorem. +We identify, for any m ∈ N\{0}, the Pontryagin dual � +Z⧸m of Z⧸m with the subgroup +of T of m-th roots of unity in the obvious manner — while of course, Z⧸m is self-dual, +this identification will be helpful to our presentation. The Pontryagin dual Z(α) := � +Z⧸α +of Z⧸α is thus, by contravariant functoriality, the limit of the inductive sequence: +� +Z⧸α0 +j0 +−−−→ +� +Z⧸α1 +j1 +−−−→ � +Z⧸α2 +j2 +−−−→ +� +Z⧸α3 +j3 +−−−→ +··· +where j1,j2,..., are simply the injection maps. Of course, by construction: +Z(α) = +� +ζ ∈ T : ∃n ∈ N +ζαn = 1 +� +, +where T = {u ∈ C : |u| = 1} is the circle group; moreover Z(α) is a discrete group as the +dual of a compact group (i.e. we do not endow it with the topology inherited as a subset +of T). +The Pontryagin duality pairing between Z(α) and its dual Z⧸α is given for all ζ ∈ Z(α) +and for all z := (zn)n∈N ∈ Z⧸α by ζz := limn→∞ ζzn, noting that the sequence (ζzn)n∈N is +eventually constant, by construction. +In the special case when α = (p,p2,p3,...), the group Z(α) is the Prüfer group Z(p∞) +and the group Z⧸α is the group Zp of p-adic integers. +Lemma 4.21. Let α := (αn)n∈N ∈ P. Let LH be a length function over the circle group T +restricted to Z(α) such that limn→∞Haus[LH] +� +� +Z⧸αn ,Z(α) +� += 0. +For all ζ ∈ Z(α), we define +F(ζ) := min +� +p ∈ N : ζp = 1 +� +. +Let ∥·∥R2 be any monotone norm on R2. The function +L : ζ ∈ Z(α) �→ ∥(LH(ζ),F(ζ))∥R2 + +50 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +is a proper unbounded length function over Z(α). +Moreover, L has the bounded doubling property if, and only if, the sequence +� +αn+1 +αn +� +n∈N +is bounded. +Proof. First, it is easy to see that, for all ζ ∈ Z(α)\{1}, +F(ζ) = p +min +� +n∈N:ζ∈� +Z⧸αn +� +, +while F(1) = 0. Therefore, by Lemma (4.3), we already know that L is a proper unbounded +length function on Z(α). +For now, let us assume ∥·∥R2 is the max norm. +For any ρ > 0, we write B[ρ] the cardinality of the closed ball centered at (1,0) ∈ +Z(α)×Z of radius ρ. For any d ∈ N, we compute the following expression: +B [αd] = |{ζ ∈ Z(α) : L(ζ) � αd}| = +���� +� +ζ ∈ � +Z⧸αd +����� = αd. +Now, let R � 1. Then, there exists d ∈ N such that αd � R � αd+1. We note that since +B[R] � B[αd+1] < ∞, our length function L is indeed proper; we also note that since +B[R] � B[αd] = αd � 2d, the length function L is also unbounded. +Now, assume that M := supn∈N +αn+1 +αn < ∞. We then compute: +B[2R] � B[2αd+1] � B[αd+2] = αd+2 += αd+2 +αd+1 +αd+1 +αd +αd � M2αd = M2B[αd] � M2B[R]. +Therefore, our length L has the bounding doubling property. Now, if we allow for a +different choice of monotone norm for ∥·∥R2, then, as all norms on R2 are equivalent, +the resulting length function still has the property of bounded doubling. +Now, assume instead that supn∈N +αn+1 +αn = ∞. Let n ∈ N, and let rn = αn+1/2. We then +note, using our above computation, that +B[2rn] = αn+1 = αn+1 +αn +·B[rn], +and thus αn+1 +αn += B[2rn] +B[rn] for all n ∈ N; therefore, our length L does not actually have the +bounded doubling property. +□ +Corollary 4.22. Let α = (αn)n∈N be a sequence of nonzero natural numbers such that +� +αn+1 +αn +� +n∈N is a bounded sequence of prime numbers, and let +Z(α) := +� +ζ ∈ C : ∃n ∈ N +ζαn = 1 +� +. +Define: +G∞ := Z(α)×Z and ∀n ∈ N +Gn := � +Z⧸αn ×Z, +i.e. Gn = {(ζ,z) ∈ G∞ : z ∈ Z,ζαn = 1}. Let σ be a 2-cocycle of G∞. +Let LZ be the restriction of any continuous length function on T to Z(α), and define +LH : (u,z) ∈ G∞ �→ LZ (u)+|z|. +For all ζ ∈ Z(α), set: +F(ζ) := min{n ∈ N : un = 1}. +Let E be a Hermitian vector space, and let γ1,γ2 be unitaries such that γ1γ2 = −γ2γ1 +and γ2 +1 = γ2 +2 = 1E. +If we set, for all n ∈ N, +/Dn := MLH ⊗γ1 + MF ⊗γ2, + +51 +then for all n ∈ N, the spectral triple (C ∗(Gn,σ),ℓ2(Gn)⊗E, /Dn) is metric, and +lim +n→∞Λspec �� +C ∗(Gn,σ),ℓ2(Gn)⊗E, /Dn +� +, +� +C ∗(Z(α)×Z,σ),ℓ2(Z(α)×Z)⊗E, /D∞ +�� += 0. +Proof. A straightforward computation shows that |·| is proper with the bounded doubling +property. +By [19, Proposition 3.7] applied to the proper unbounded lengths | · | and LZ , we +conclude that L := (ζ,z) ∈ G∞ �→ LZ (ζ)+F(ζ)+|m| has the bounded doubling property. +Since LZ is continuous on T, it induces the usual topology on T (as a subset of C). +Therefore, the topology of the Hausdorff distance Haus[LH] is the Vietoris topology for +the usual topology of T, and thus the same as the topology induced by Haus[T], when T +is endowed with the restriction of the usual metric on C. It then follows that: +lim +n→∞Haus[LH] +� +� +Z⧸αn,Z(α) +� += 0. +As all the other assumptions are now met, we conclude that our corollary holds, by +Theorem (4.16). +□ +The map +ϖ : z ∈ Z �→ (z +mod αn)n∈N ∈ Z⧸α +is an injective *-morphism of group with dense range. Now, we define the following +automorphism of Z(α): +τ : u ∈ Z(α) �→ u +ϖ(1). +The C*-crossed-product C(Z(α))⋊τ Z is the Bunce-Deddens algebra associated to the +“supernatural” number +n := +� +p|{n∈N: αn+1 +αn =p}|� +p prime . +It is also *-isomorphic to C ∗(Z(α) × Z,σ), as defined above, when σ is the 2-cocycle +defined by setting, for all (ζ,z),(η, y) ∈ G∞: +σ((ζ,z),(η, y)) := ηz. +Indeed, this isomorphism can be obtained by using [52]. We begin with the observa- +tion that Bunce-Deddens algebras [8] are C*-crossed products [54, 18]. Now, let us briefly +explain the construction of this isomorphism. Since the natural inclusion j : Z(α) → T +is a character of Z(α), it is given by the pairing with an element in Z⧸α; this element is +precisely our ϖ(1) defined above. In our case, we note that λ(1,1)λζ,0λ∗ +(1,1) = ζ−1λζ,0 for all +ζ ∈ Z(α). If f ∈ Cc +�Z⧸α +� +, we denote its Fourier transform by �f ; specifically +�f : ζ ∈ Z(α) �→ +� +z∈Z⧸α +f (z)ζ−z. +A straightforward computation shows that � +τ(f )(ζ) = ζ−1 �f (ζ). Thus, we conclude that +λ(1,1)λ( �f )λ∗ +(1,1) = λ +� +� +τ(f ) +� +. A similar computation invoking the inverse Fourier transform +can be done by using the canonical generators of the C*-crossed product C +�Z⧸α +� +⋊τ +Z. By universality of the C*-crossed-product and the twisted group C*-algebra (here, +since our groups are Abelian, these algebras agree with their image by their left regular +representations), we conclude the description of our isomorphism. +Thus, we have constructed metric spectral triples over Bunce-Deddens algebra for +bounded supernatural numbers, and these triples are limits of sequences of metric +spectral triples for the spectral propinquity. + +52 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +In particular, C ∗(Z(α)×Z,σ) is seen to be the inductive limit and the limit for the +propinquity, with the quantum metrics described here, of the C*-algebras C ∗ +� +� +Z⧸αn +×Z,σ +� +as n ∈ N approaches ∞. Notably, C ∗ +� +� +Z⧸αn ×Z,σ +� +is actually *-isomorphic to +the C*-algebra of continuous sections of a vector bundle over the circle T with fibers +the algebras of square αn-matrices. This situation is of course reminiscent of the fact +that in particular, Bunce-Deddens algebras are AT algebras. However, starting from the +usual description of Bunce-Deddens algebras as AT algebras led to difficulties in [6], +where the quantum metrics on the Bunce-Deddens algebra do not arise from a spectral +triple, and the convergence is only proven in the sense of Rieffel’s quantum Gromov- +Hausdorff distance. Thus, for Bunce-Deddens algebras associated with supernatural +numbers consisting of only finitely many prime numbers, we have now constructed +metric spectral triples which actually capture their inductive limit structure within our +geometric framework. We hope that Theorems (4.11) and (4.16) will prove useful in +constructing other examples of metric spectral triples over twisted group C*-algebras for +interesting inductive limits of groups. +REFERENCES +[1] V. Aiello, D. Guido, T. Isola, Spectral triples for noncommutative solenoidal spaces from self-coverings, 448 +(2017), 1378–1412. +[2] A. Austad and F. Luef, Modulation spaces as a smooth structure in noncommutative geometry, Banach J. +Math. Anal. 15 (2021) 15–35. +[3] K. Aguilar and J. Kaad, The Podle´s sphere as a spectral metric space., J. Geom. Phys. 133 (2018), 260–278. +[4] K. Aguilar, F. Latrèmolière, and T. Rainone, Bunce-Deddens algebras as quantum Gromov-Hausdorff +distance limits of circle algebras, Integral Equations Operator Theory 94 (2022), no. 1, Paper No. 2, 42 pp. +[5] K. Aguilar and F. Latrémolière, Quantum ultrametrics on AF algebras and the Gromov-Hausdorff propin- +quity, Studia Math. 231 (2015), no. 2, 149–194, ArXiv: 1511.07114. +[6] K. Aguilar, F. Latrémolière, and T. Rainone, Bunce-Deddens algebras as quantum Gromov-Hausorff distance +limits of circle algebras, Integral Equations Operator Theory 94 (2022), no. 1, Paper no. 2, 42pp, ArXiv: +2008.07676. +[7] C. Antonescu and E. Christensen, Spectral triples for af C∗-algebras and metrics on the Cantor set, J. Oper. +Theory 56 (2006), no. 1, 17–46, arXiv: 0309044. +[8] J. W. Bunce and J. Deddens, A family of simple C∗-algebras related to weighted shift operators, J. Functional +Analysis 19 (1975), 13–24. +[9] M. Christ and M. A. Rieffel, Nilpotent group C∗-algebras-algebras as compact quantum metric spaces, +Canad. Math. Bull. 60 (2017), no. 1, 77–94, arXiv: 1508.00980. +[10] E. Christensen, C. Ivan, and M. Lapidus, Dirac operators and spectral triples for some fractal sets built on +curves, Adv. Math. 217 (2008), no. 1, 42–78. +[11] A. Connes, Compact metric spaces, Fredholm modules and hyperfiniteness, Ergodic Theory Dynam. Systems +9 (1989), no. 2, 207–220. +[12] +, Noncommutative geometry, Academic Press, San Diego, 1994. +[13] L. Dabrowski, G. Landi, A. Sitarz, W. van Suijlekom, and J. C. Varilly, The Dirac operator on SUq(2), Comm. +Math. Phys. 259 (2005), 729–759. +[14] K. R. Davidson, C∗−Algebras by example, Fields Institute Monographs, American Mathematical Society, +1996. +[15] L. Da¸browski and A. Sitarz, Curved noncommutative torus and Gauß-Bonnet, J. Math. Phys. 013518 (2013). +[16] +, An asymmetric noncommutative torus, SIGMA 11 (2015), 11 pages, arXiv: 1406.4645. +[17] David A. Edwards, The structure of superspace, Studies in topology (Proc. Conf., Univ. North Carolina, +Charlotte, N. C., 1974; dedicated to Math. Sect. Polish Acad. Sci.), 1975, p. 121–133. MR 0401069 +[18] R. Exel, The Bunce-Deddens algebras as crossed products by partial automorphisms, Bol. Soc. Brasil. Mat. +(N.S.) 25 (1994), 173–179. +[19] C. Farsi, T. Basa Landry, N. S. Larsen, and J. Packer, Spectral triples for noncommutative solenoids and a +Wiener’s lemma, (2022), ArXiv: 2212.07470. + +53 +[20] R. Floricel and A. Ghorbanpour, On inductive limit spectral triples, Proc. Amer. Math. Soc. 147 (2019), no. 8, +3611–3619, ArXiv: 1712.09621. +[21] O. Gabriel and M. Grensing, Ergodic actions and spectral triples, J. Oper. Theory 76 (2016), no. 2, 307–334. +[22] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. Inst. Hautes Études Sci. 53 +(1981), 53–78. +[23] +, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, +BirkhÀuser, 1999. +[24] A. Hawkins, A. Skalski, S. White, and J. Zacharias, On spectral triples on crossed products arising from +equicontinuous actions, Math. Scand. 113 (2013), no. 2, 262–291. +[25] D. A. Herron, Gromov-Hausdorff distance for pointed metric spaces, J. Anal. 24 (2016), no. 1, 1–38. +[26] Antoine Julien and I. Putnam, Spectral triples for subshifts, J. Funct. Anal. 270 (2016), no. 3, 1031–1063, +arXiv: 1411.6800. +[27] L. V. Kantorovich, On one effective method of solving certain classes of extremal problems, Dokl. Akad. Nauk. +USSR 28 (1940), 212–215. +[28] L. V. Kantorovich and G. Sh. Rubinstein, On the space of completely additive functions, Vestnik Leningrad +Univ., Ser. Mat. Mekh. i Astron. 13 (1958), no. 7, 52–59, In Russian. +[29] T. Landry, M. Lapidus, and F. Latrémolière, Metric approximations of the spectral triple on the Sierpinki +gasket and other fractals, Adv. Math. 385 (2021), Paper No. 107771, 43 pp. +[30] M. Lapidus and J. Sarhad, Dirac operators and geodesic metric on the harmonic Sierpi´nskigasket and other +fractal sets, J. Noncommut. Geom. 8 (2014), no. 4, 947–985. +[31] F. Latrémolière, Continuity of the spectrum of Dirac operators of spectral triples for the spectral propinquity, +Submitted (2022), 49 pages. +[32] F. Latrémolière, Approximation of the quantum tori by finite quantum tori for the quantum Gromov- +Hausdorff distance, J. Funct. Anal. 223 (2005), 365–395, math.OA/0310214. +[33] +, Convergence of fuzzy tori and quantum tori for the quantum Gromov-Hausdorff propinquity: an +explicit approach., Münster J. Math. 8 (2015), no. 1, 57–98, arXiv: math/1312.0069. +[34] +, Curved noncommutative tori as Leibniz compact quantum metric spaces, J. Math. Phys. 56 (2015), +no. 12, 123503, 16 pages, arXiv: 1507.08771. +[35] +, The dual Gromov-Hausdorff propinquity, J. Math. Pures Appl. 103 (2015), no. 2, 303–351, arXiv: +1311.0104. +[36] +, Quantum metric spaces and the Gromov-Hausdorff propinquity, Noncommutative geometry and +optimal transport, Contemp. Math., no. 676, Amer. Math. Soc., 2015, arXiv: 150604341, p. 47–133. +[37] +, Equivalence of quantum metrics with a common domain, J. Math. Anal. Appl. 443 (2016), 1179– +1195, arXiv: 1604.00755. +[38] +, The quantum Gromov-Hausdorff propinquity, Trans. Amer. Math. Soc. 368 (2016), no. 1, 365–411. +[39] +, A compactness theorem for the dual Gromov-Hausdorff propinquity, Indiana Univ. Math. J. 66 +(2017), no. 5, 1707–1753, arXiv: 1501.06121. +[40] +, The triangle inequality and the dual Gromov-Hausdorff propinquity, Indiana Univ. Math. J. 66 +(2017), no. 1, 297–313, arXiv: 1404.6633. +[41] +, Convergence of Cauchy sequences for the covariant Gromov-Hausdorff propinquity, J. Math. Anal. +Appl. 469 (2019), no. 1, 378–404, arXiv: 1806.04721. +[42] +, The modular Gromov-Hausdorff propinquity, Dissertationes Math. 544 (2019), 1–70, arXiv: +1608.04881. +[43] +, Convergence of Heisenberg modules for the modular Gromov-Hausdorff propinquity, J. Operator +Theory 84 (2020), no. 1, 211–237. +[44] +, The covariant Gromov-Hausdorff propinquity, Studia Math. 251 (2020), no. 2, 135–169, arXiv: +1805.11229. +[45] +, Convergence of spectral triples on fuzzy tori to spectral triples on quantum tori, Comm. Math. Phys. +388 (2021), no. 2, 1049–1128, arXiv: 2102.03729. +[46] +, The dual-modular Gromov-Hausdorff propinquity and completeness, J. Noncomm. Geometry 115 +(2021), no. 1, 347–398. +[47] +, The Gromov-Hausdorff propinquity for metric spectral triples, Adv. Math. 404 (2022), Paper No. +108393, 56. +[48] F. Latrémolière and J. Packer, Noncommutative solenoids and the Gromov-Hausdorff propinquity, Proc. +Amer. Math. Soc. 145 (2017), no. 5, 1179–1195, arXiv: 1601.02707. +[49] +, Noncommutative solenoids, New York J. Math. 24A (2018), 155–191, arXiv: 1110.6227. +[50] B. Long and W. Wu, Twisted bounded-dilation group C∗-algebras as C∗-metric algebras, Sci. China Math. +64 (2021), no. 3, 547–572. + +54 +CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER +[51] N. Ozawa and M. A. Rieffel, Hyperbolic group C∗-algebras and free product C∗-algebras as compact +quantum metric spaces, Canad. J. of Math. 57 (2005), 1056–1079, arXiv: math/0302310. +[52] J. Packer, Twisted group C∗-algebras corresponding to nilpotent discrete groups, Mathematica Scandinavica +64 (1989), no. 1, 109–122. +[53] M. Reed and B. Simon, Functional analysis, Methods of modern Mathematical Physics, Academic Press, +San Diego, 1980. +[54] N. Riedel, Classification of the C∗-algebras associated with minimal rotations, Pacific J. Math. 101 (1982), +153–161. +[55] M. A. Rieffel, Metrics on states from actions of compact groups, Doc. Math. 3 (1998), 215–229, +math.OA/9807084. +[56] +, Metrics on state spaces, Doc. Math. 4 (1999), 559–600, math.OA/9906151. +[57] +, Gromov-Hausdorff distance for quantum metric spaces, Mem. Amer. Math. Soc. 168 (2004), no. 796, +1–65, math.OA/0011063. +[58] +, Leibniz seminorms for "matrix algebras converge to the sphere", Clay Mathematics Proceedings 11 +(2010), 543–578, arXiv: 0707.3229. +Email address: carla.farsi@colorado.edu +DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COLORADO AT BOULDER, BOULDER CO 80309-0395 +Email address: frederic@math.du.edu +URL: http://www.math.du.edu/~frederic +DEPARTMENT OF MATHEMATICS, UNIVERSITY OF DENVER, DENVER CO 80208 +Email address: judith.jesudason@colorado.edu +DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COLORADO AT BOULDER, BOULDER CO 80309-0395 + diff --git a/3NAyT4oBgHgl3EQfb_eW/content/tmp_files/load_file.txt b/3NAyT4oBgHgl3EQfb_eW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..437e1194e1baca0a5b238e4a64096707062a304d --- /dev/null +++ b/3NAyT4oBgHgl3EQfb_eW/content/tmp_files/load_file.txt @@ -0,0 +1,1903 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf,len=1902 +page_content='CONVERGENCE OF INDUCTIVE SEQUENCES OF SPECTRAL TRIPLES FOR THE SPECTRAL PROPINQUITY CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In the context of metric geometry, we introduce a new necessary and suf- ficient condition for the convergence of an inductive sequence of quantum compact metric spaces for the Gromov-Hausdorff propinquity, which is a noncommutative ana- logue of the Gromov-Hausdorff distance for compact metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This condition is easy to verify in many examples, such as quantum compact metric spaces associated to AF algebras or certain twisted convolution C*-algebras of discrete inductive limit groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our condition also implies the convergence of an inductive sequence of spectral triples in the sense of the spectral propinquity, a generalization of the Gromov-Hausdorff propinquity on quantum compact metric spaces to the space of metric spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular we show the convergence of the state spaces of the underlying C*-algebras as quantum compact metric spaces, and also the convergence of the quantum dynamics induced by the Dirac operators in the spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We apply these results to new classes of inductive limit of even spectral triples on noncommutative solenoids and Bunce-Deddens C*-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our construction, which involves length functions with bounded doubling, adds geometric information and highlights the structure of these twisted C*-algebras as inductive limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' CONTENTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A Characterization of Convergence in the Propinquity for Inductive Sequences 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Preliminaries: the Gromov-Hausdorff Propinquity 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Main result 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Convergence of Inductive Sequences of Metric Spectral Triples for the Spectral Propinquity 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Preliminaries: The Spectral Propinquity 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Preliminaries: Inductive Limits of Spectral Triples 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Main result 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Even Spectral Triples on Twisted Group C ∗-algebras 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Discrete Groups, Proper Length Functions, 2-Cocycles, and Classical Spectral Triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The Spectral Triples 32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Main result 39 References 52 Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Primary: 46L89, 46L30, 58B34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Spectral triples, Noncommutative metric geometry, quantum Gromov-Hausdorff distance, Monge-Kantorovich distance, Quantum Metric Spaces, Quantum Tori, Noncommutative solenoids, Bunce-Deddens algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='00274v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='OA] 31 Dec 2022 2 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' INTRODUCTION Spectral triples, introduced by Connes in 1985 as a noncommutative generalization of Dirac operators acting on bundles over manifolds [11, 12], have emerged as a powerful means to encode geometric information over noncommutative operator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Mo- tivated in part by ideas from mathematical physics, and by the recurrent usefulness of various notions of limits of C*-algebras, the second author introduced in [47] a distance on metric spectral triples, up to an obvious notion of unitary equivalence, thus enabling the discussion of approximations of certain spectral triples by others, in a geometric sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This distance is named the spectral propinquity, and is built from a noncommuta- tive analogue of the Gromov-Hausdorff distance for noncommutative geometry, called the Gromov-Hausdorff propinquity [35, 38, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, convergence of spectral triples is defined as part of a larger framework for convergence of quantum compact metric spaces, which are noncommutative analogues of algebras of Lipschitz functions over compact metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Within this framework, the propinquity was extended to certain modules over quantum compact metric spaces [48], and even C*-correspondences [46] with additional metric data inspired by metric connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The propinquity also was extended to various dynamical systems [41, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' These extensions have been used by the second author to define the spectral propinquity over metric spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The spectral propinquity Λspec has been applied to approximations of spectral triples on fractals [29] and on quantum tori [45], with the latter example rooted in matrix mod- els in physics and the problem of their convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Indeed, the spectral propinquity endows the space of all metric spectral triples with its own geometry, and it allows to cap- ture some geometric intuition within the well understood framework of a topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For instance, while quantum tori are not inductive limits of finite dimensional C*-algebras, spectral triples over quantum tori can now be approximated by spectral triples over full matrix algebras to arbitrary precision using the spectral propinquity — a common heuris- tics in mathematical physics, now formalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Convergence for the spectral propinquity implies convergence of the state spaces of the underlying algebras for a form of Gromov- Hausdorff distance, convergence of the quantum dynamics obtained by exponentiating the Dirac operators, and implies convergence of the spectra and the bounded continuous functional calculus for the Dirac operators, with implications for the convergence of physically important quantities such as the spectral actions [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In this paper, we consider the question of when an inductive sequence of metric spectral triples [20] converges, in the sense of the spectral propinquity, to its inductive limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' To illustrate the power of our result, besides the class of AF algebras, we construct even metric spectral triples on noncommutative solenoids [49] and on some Bunce-Deddens algebras [8, 14] and show that they are limits of metric spectral triples on, respectively, quantum tori and bundles of full matrix algebras over the circle, in the sense of the spectral propinquity Λspec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In this way, we provide a noncommutative geometric version of the fact that solenoid groups can be seen as metric limits of tori, and Bunce-Deddens algebras are metric limits of algebras of matrix valued functions over the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A spectral triple (A,H , /D) is given by a unital C*-algebra A acting on a Hilbert space H and a (usually unbounded) self-adjoint operator /D on H , which has bounded com- mutator with the elements of a dense ∗-subalgebra of A, and has compact resolvent (see Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Spectral triples contain much geometric information, including metric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Indeed, Connes noted in [12] that spectral triples define a canonical extended pseudo-distance on the state space of their underlying C*-algebras, which, in particular, 3 recovers the geodesic distance when working with the usual spectral triple given by the Dirac operator acting on the square integrable sections of the spinor bundle of a compact connected Riemannian spin manifold without boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Rieffel in [55, 56] then cast this metric aspect of noncommutative geometry under a new light, starting from the observation that Connes’ distance induced by a spectral triple is a noncommutative analogue of the Monge-Kantorovich metric [27, 28];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' it was thus natural to define a quantum compact metric space as an ordered pair (A,L) of a unital C*-algebra A and a noncommutative analogue of a Lipschitz seminorm L such that, in particular, if we set, for any two states ϕ,ψ of A, mkL(ϕ,ψ) := sup � |ϕ(a)−ψ(a)| : L(a) � 1 � then mkL is a distance inducing the weak-∗ topology on the state space of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The exact list of requirements on the seminorm L have evolved as the study of noncommutative metric geometry matured, and we will use the definition of a quantum compact metric space given in [38, 39] and recalled in Definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Indeed, a spectral triple whose Connes’ metric induces the weak-∗ topology on the state space of its underlying C*- algebra then automatically gives a quantum compact metric space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' such a spectral triple is called a metric spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Metric spectral triples may thus be studied within the context of noncommutative metric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As a result, the second author introduced a distance on the space of metric spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The first step in defining this distance, called the spectral propinquity, is the construction of a noncommutative geometric analogue of the Gromov- Hausdorff distance [17, 22, 23] between quantum compact metric spaces, which we will recall in subsection (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The first such analogue was introduced by Rieffel [57], motivated by the possibility of formalizing certain convergence results found in the mathematical physics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' While several such analogues have been offered, we will work with the Gromov-Hausdorff propinquity Λ∗, introduced by the second author in [35, 38, 39, 40] precisely to be well adapted to C*-algebras theory and the type of seminorms given by spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The propinquity in general is designed precisely to enable distance computations between quantum compact metric spaces defined on unrelated C*-algebras, such as between matrix algebra and quantum tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' However, in this work, we investigate what additional properties of the propinquity we can derive when we work with inductive limits of C*-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We begin this work by establishing a characterization of convergence of inductive limits of quantum compact metric spaces to their inductive limit, in terms of bridge builders, a type of ∗-automorphism with a natural relation to quantum metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition (Definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N ∪ {∞}, let (An,Ln) be a quantum com- pact metric space, such that A∞ = cl(� n∈NAn), where (An)n∈N is an increasing (for ⊆) sequence of C*-subalgebras of A∞, with the unit of A∞ in A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A ∗-automorphism π : A∞ → A∞ is a bridge builder for ((An,Ln)n∈N,(A∞,L∞)) when, for all ε > 0, there exists N ∈ N such that if n � N, then ∀a ∈ dom(L∞) ∃b ∈ dom(Ln) : Ln(b) � L∞(a) and ∥π(a)−b∥A∞ < εL∞(a) and ∀b ∈ dom(Ln) ∃a ∈ dom(L∞) : L∞(a) � Ln(b) and ∥π(a)−b∥A∞ < εLn(b), where ∥·∥A∞ is the C*-norm on A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 4 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER Bridge builders are powerful means to prove metric convergence for the propinquity and notable because it is usually very difficult to find necessary conditions for metric convergence in the sense of the propinquity (besides the trivial convergence for the diameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, this theorem is of independent interest from our study of spectral triples, and addresses the relationship between inductive limits and limits in a metric sense as in [47, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our first main result is therefore the following theorem about convergence for the propinquity Λ∗ of certain inductive sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Theorem (Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='22)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N ∪{∞}, let (An,Ln) be a quantum compact metric space, where (An)n∈N is an increasing (for ⊆) sequence of C*-subalgebras of A∞ such that A∞ = cl(� n∈NAn), with the unit of A∞ in A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We assume that there exists ∃M > 0 such that for all n ∈ N: 1 M Ln � L∞ � M ·Ln on dom(Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Then lim n→∞Λ∗ ((An,Ln),(A∞,L∞)) = 0, if, and only if, for any subsequence (Ag(n),Lg(n))n∈N of (An,Ln)n∈N, there exists a strictly increasing function f : N → N and a bridge builder π for ((Ag◦f (n),Lg◦f (n))n∈N,(A∞,L∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The second step in the construction of the spectral propinquity Λspec on the space of metric spectral triples is the extension of the Gromov-Hausdorff propinquity to a distance on the class of C*-correspondences over quantum compact metric spaces endowed with a form of quantum metric, and with a compatible action of some monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The C*- correspondence associated with a metric spectral triple (A,H , /D) is the Hilbert space H , seen as a A-C-C*-correspondence, with the quantum metric given by the graph norm of /D, and with the action of [0,∞) on H given by t ∈ [0,∞) �→ exp(it /D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Convergence for the spectral propinquity, by design, implies the convergence of the underlying quantum compact metric spaces, but the converse does not hold in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' These matters will be recalled in detail in Subsection (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We then turn to the more specific context of inductive sequences of metric spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Inductive sequences of spectral triples were introduced in [20], and are a natural source of spectral triples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' our interest is in the convergence of such sequences for the spectral propinquity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' in the sense of an actual metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We establish in the present work, as our second main result, that an inductive sequence of metric spectral triples converges for the spectral propinquity when there exists a fully quantum isometric bridge builder for the underlying sequence of quantum compact metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Again, it is a surprising result that a mild strengthening of convergence for the Gromov-Hausdorff propinquity implies the much stronger convergence for the spectral propinquity, a fact which does not hold for arbitrary sequences of metric spectral triples, but holds thanks to the structure of inductive limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our second main theorem is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Theorem (Theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let (A∞,H∞, /D∞) be a metric spectral triple which is the inductive limit of a sequence (An,Hn, /Dn)n∈N of metric spectral triples, in the sense of Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N∪{∞}, let dom(Ln) := � a ∈ An : a = a∗,a dom( /Dn) ⊆ dom( /Dn) and [ /Dn,a] is bounded � , and, for all a ∈ dom(Ln), let Ln(a) be the operator norm of [ /Dn,a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 5 If there exists a bridge builder π : (A∞,L∞) → (A∞,L∞) for ((An,Ln)n∈N,(A∞,L∞)) which is a full quantum isometry of (A∞,L∞), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' such that π(dom(L∞)) ⊆ dom(L∞) and L∞ ◦π = L∞ on dom(L∞), then lim n→∞Λspec((An,Hn, /Dn),(A∞,H∞, /D∞)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We conclude our paper with the construction of new even spectral triples on certain twisted group C*-algebras C ∗(G,σ) where the discrete group G = � n∈NGn is the union of a strictly increasing sequence of subgroups Gn of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' These examples include noncom- mutative solenoids [49] and certain Bunce-Deddens algebras [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our construction is motivated by the desire to see our new spectral triples over C ∗(G,σ) as limits, for the spectral propinquity, of an inductive sequence of metric spectral triples constructed over the inductive sequence (C ∗(Gn,σ))n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This metric aspect distinguishes our spectral triples from other spectral triples on noncommutative solenoids [1, 2] or Bunce-Deddens algebras [24], and is applicable, in principle, to many other examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, non- commutative solenoids were shown in [48] to be limits, for the propinquity, of quantum tori, for a different family of quantum metrics which did not come from a spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In general, it is difficult to prove that a given spectral triple is metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Examples of metric spectral triples can be found over certain manifolds, quantum tori [12, 15, 16, 34, 45], or more generally, over unital C*-algebras endowed with ergodic actions of compact Lie groups [21, 55], over certain C*-crossed-products [24], over quantum groups [13], over Podle´s spheres [3], over AF algebras [7], over certain fractals [10, 30], and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We note that there are known examples of spectral triples which are not metric [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is therefore quite interesting to obtain new examples of metric spectral triples, and moreover, to prove that they are interesting limits of spectral triples for the spectral propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We thus establish the following third main result of this paper, which draws on the first two in its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Theorem (Simplified form of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let G = � n∈NGn be an Abelian discrete group, with (Gn)n∈N a strictly increasing sequence of subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let σ be a 2-cocycle of G, with values in T := {z ∈ C : |z| = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let LH be a length function over G whose restriction to Gn is proper for all n ∈ N, such that the sequence (Gn)n∈N converges to G for the Hausdorff distance induced on the closed subsets of G by LH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let F : g ∈ G �−→ scale(min{n ∈ N : g ∈ Gn}), where scale : N → [0,∞) is a strictly increasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If the proper length function L := max{LH,F} satisfies that, for some θ > 1, there exists c > 0 such that for all r � 1: ��� g ∈ G : L(g) � θ ·r ��� � c ��� g ∈ G : L(g) � r ���, then lim n→∞Λspec((C ∗(G,σ),ℓ2(G)⊗C2, /D),(C ∗(Gn,σ),ℓ2(Gn)⊗C2, /Dn)) = 0, where for all n ∈ N∪{∞} and for all (ξ1,ξ2) in � ξ ∈ ℓ2(Gn)⊗C2 : � g∈Gn (LH(g)2 +F(g)2) ��ξ(g) ��2 C2 < ∞ � , we set /Dξ : g ∈ G �−→ �F(g)ξ2(g)+LH(g)ξ1(g) F(g)ξ2(g)−LH(g)ξ1(g) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 6 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER In the above spectral triples, C ∗(G,σ) and C ∗(Gn,σ) act via their left regular σ-projective representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We then apply this theorem to construct metric spectral triples on noncommutative solenoids, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' the twisted group C*-algebras C ∗ �� Z � 1 p ��2 ,σ � where Z � 1 p � := � k pn : k ∈ Z,n ∈ N � , with p a prime natural number, and where σ is a 2-cocycle of � Z � 1 p ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In this case, using the notation of the above theorem, we choose LH to be the restriction to � Z � 1 p ��2 of any norm on R2, while F can be chosen by setting F(g) := p min � n∈N:g∈ � 1 pn Z �2� for all g = (g1,g2) ∈ � Z � 1 p ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Alternatively, following the ideas of [19], which motivated the present work, we can choose F(g1,g2) := max{|g1|p,|g2|p} for all g1,g2 ∈ Z � 1 p � , where |·|p is the p-adic absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Similarly, we can apply [52] to see that the Bunce-Deddens algebras are given as the twisted group C*-algebra C ∗ (Z(α)×Z,σ) for an appropriate choice of a 2-cocycle σ and a sequence α = (αn)n∈N of nonzero natural numbers such that αn+1 αn is a prime number for all n ∈ N, where the group Z(α) is the subgroup of the circle group T given by all roots of unity of order αn for n ranging over N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We endow Z(α) with the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The supernatural number number describing the ∗-isomorphism class of the Bunce- Deddens algebra thus obtained is � p|{n∈N: αn+1 αn =p}|� p prime .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For our purpose, we will work with sequences α for which � αn+1 αn � n∈N is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In this case, we will choose LH to be the sum or the max (or one of many other choices) of the restriction of a length function over T to Z(α), and the absolute value on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Observing that Z(α) = � n∈N � Z⧸αn, where � Z⧸m is the group of all m-th roots of unity, we then set F(ζ,z) := min{αn : ζ ∈ � Z⧸αn} for all (ζ,z) ∈ Z(α)×Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This provides a new way to look at Bunce-Deddens algebras as limits of algebras of continuous sections of bundles of matrix algebras over circles in a geometric sense, as an echo of the topological fact that they are AT algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This work thus provides an approach to endowing Bunce-Deddens algebras with a different quantum metric from [29], with the advantage that our quantum metrics are induced by spectral triples — solving the main difficulty in [29], at least for these Bunce-Deddens algebras to which our present work applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This work was partially supported by the Simons Foundation (Si- mons Foundation collaboration grant #523991 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Farsi] and # 31698 [J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Packer].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A CHARACTERIZATION OF CONVERGENCE IN THE PROPINQUITY FOR INDUCTIVE SEQUENCES We introduce in this section the notion of bridge builders associated with inductive sequences of quantum compact metric spaces, which can be used to characterize the 7 convergence of such sequences to their inductive limits in the sense of the Gromov- Hausdorff propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We begin with a review of the notions of quantum compact metric spaces and propinquity, and then we prove our main theorem, which underlies all the rest of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Preliminaries: the Gromov-Hausdorff Propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our work is concerned with quantum compact metric spaces, which are noncommutative analogues of the algebras of Lipschitz functions over a compact metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our definition is the result of a natural evolution from the notion of compact quantum metric spaces introduced in [55] by Rieffel, designed as the natural context for the construction of the propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This subsection will also set some of the basic notation which we will use throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By default, we denote the norm of a normed vector space E by ∥·∥E , and for us, the set N of natural numbers always contains zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If A is a unital C*-algebra, then the unit of A will simply be denoted by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The state space of the C*-algebra A is denoted by S (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For any a ∈ A, we write ℜa = a+a∗ 2 and ℑa = a−a∗ 2i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The space {a ∈ A : a = a∗} is denoted by sa(A) and is closed under the Jordan product a,b ∈ sa(A) �→ ℜ(ab) and the Lie product a,b ∈ sa(A) �→ ℑ(ab), making sa(A) a Jordan-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3 ([11, 38, 39, 55, 57, 58]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix Ω � 1 and Ω′ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' An (Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='Ω′)-quantum com- pact metric space (A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='L) is given by a unital C*-algebra A and a seminorm L defined on a dense Jordan-Lie subalgebra dom(L) of sa(A) such that: (1) {a ∈ dom(L) : L(a) = 0} = R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (2) the Monge-Kantorovich metric mkL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' defined on the state space S (A) of A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' for all ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='ψ ∈ S (A): mkL(ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='ψ) := sup � |ϕ(a)−ψ(a)| : a ∈ dom(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='L(a) � 1 � is a metric which induces the weak-∗ topology on S (A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (3) for all a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='b ∈ sa(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' max{L(ℜ(ab)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='L(ℑ(ab))} � Ω(∥a∥AL(b)+L(a)∥b∥A)+Ω′L(a)L(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' this inequality being referred to as the (Ω,Ω′)-Leibniz inequality, (4) the set {a ∈ dom(L) : L(a) � 1} is closed in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Any such a seminorm L is called a Lipschitz seminorm on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Convention 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By convention, if L is a Lipschitz seminorm on some unital C*-algebra A, we will write L(a) = ∞ whenever a ∉ dom(L), with the convention that 0∞ = 0 and ∞+x = x+∞ = ∞ for all x ∈ [0,∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' With this convention, L is lower semicontinuous over sa(A) as a [0,∞]-valued function (not just on dom(L) but on the entire space sa(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Convention 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Throughout this paper, we fix Ω � 1 and Ω′ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' These parameters will be implicit in our notation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' when working with spectral triples, one may always assume Ω = 1 and Ω′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If (A,L) is a quantum compact metric space, then we record the following fact which we shall use repeatedly: if a ∈ dom(L), then L(a +t1) = L(a) for all t ∈ R, since L(a) = L(a+t1−t1) � L(a+t1)+L(t1) = L(a+t1)+t L(1) =0 = L(a+t1) � L(a)+tL(1) = L(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 8 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER Since the state space of a quantum compact metric space is a compact metric space for the Monge-Kantorovich metric, it has bounded diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, its diameter can used to obtain a natural bound on the norm of some self-adjoint elements, which is a simple but very useful result, which we now recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The diameter of a metric space (E,d) is denoted by diam(E,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If (A,L) is a quantum compact metric space, then we will write qdiam(A,L) for diam �S (A),mkL � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If E is actually a normed vector space, then we simply write diam(A,E) for the diameter of any subset A of E for the norm ∥·∥E of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We recall the following fact, which we will use repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='8 ([55, Propostion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If (A,L) is a quantum compact metric space, and if µ ∈ S (A), then ��a −µ(a)1 ��A � L(a) qdiam(A,L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For all ϕ ∈ S (A), we note that |ϕ(a −µ(a)1)| = |ϕ(a)−µ(a)| � L(a)qdiam(A,L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since a −µ(a)1 is self-adjoint, we conclude that ��a −µ(a)1 ��A � L(a)qdiam(A,L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ The property difficult to establish when working with quantum compact metric spaces is, of course, that the Monge-Kantorovich metric induces the weak-∗ topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Rieffel provided various characterizations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' we will find the following helpful in this paper: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='9 ([51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let L be a seminorm defined on some dense subspace dom(L) of sa(A) for some unital C*-algebra A such that {a ∈ dom(L) : L(a) = 0} = R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If we set mkL(ϕ,ψ) = sup � |ϕ(a)−ψ(a)| : a ∈ dom(L),L(a) � 1 � , for all ϕ,ψ ∈ S (A), then the fol- lowing assertions are equivalent: mkL is a metric on the state space S (A) of A inducing the weak-∗ topology, there exists a state µ ∈ S (A) such that {a ∈ dom(L) : L(a) � 1,µ(a) = 0} is totally bounded in sa(A), for all states µ ∈ S (A), the set {a ∈ dom(L) : L(a) � 1,µ(a) = 0} is totally bounded in sa(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We record the following helpful result, which we will also use often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='10 ([55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If (A,L) is a quantum compact metric space, µ ∈ S (A), and if K > 0, then the set � a ∈ dom(L) : L(a) � 1,|µ(a)| � K � is compact in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We first note that the set � a ∈ dom(L) : L(a) � 1,|µ(a)| � K � is closed since L is lower semicontinuous and µ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let (an)n∈N be a sequence in dom(L) such that L(an) � 1 and |µ(an)| � K for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since (|µ(an)|)n∈N is bounded in R, it has a convergent subsequence (|µ(a f (n))|)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' On the other hand, (a f (n) −µ(a f (n))1)n∈N has a convergent subsequence (a f (g(n)) − µ(a f (g(n))))n∈N by Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It now follows that (a f (g(n)))n∈N is a convergent se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ Quantum compact metric spaces are the points of a (pseudo-)metric space, where the metric is the Gromov-Hausdorff propinquity, an analogue of the Gromov-Hausdorff distance in noncommutative geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The construction of the propinquity thus relies on an appropriate notion of quantum isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 9 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let (A1,L1) and (A2,L2) be two quantum compact metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A Lips- chitz morphism π : (A1,L1) → (A2,L2) from (A1,L1) to (A2,L2) is a surjective ∗-morphism π from A1 to A2 such that π(dom(L1)) ⊆ dom(L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, if, for all b ∈ dom(L2): L2(b) = inf{L1(a) : π(a) = b}, then π is called a quantum isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If π is a quantum isometry and a bijection whose inverse is also a quantum isometry, then π is called a full quantum isometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' in this case π is a ∗-isomorphism such that for all a ∈ sa(A1): L2 ◦π(a) = L1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The propinquity is a metric computed by isometrically “embedding” two quantum compact metric spaces into an arbitrary third one, which in the contravariant picture of noncommutative geometry, leads us to the following definition for a tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Crucially, a non-negative number can be associated to a tunnel using the Hausdorff distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The Hausdorff distance induced by the distance function of a metric space (X ,d) on the hyperspace of closed subsets of X is denoted by Haus[d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If N is a norm on a vector space, we denote by Haus[N] the Hausdorff distance induced by the metric given by the norm N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By default, if E is a normed vector space, we simplify our notation and simply write Haus[E] for the Hausdorff distance induced by the distance defined by the norm ∥·∥E of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If π : A → B is a unital ∗-morphism, then we define π∗ : ϕ ∈ S (B) �−→ ϕ◦π ∈ S (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='14 ([35, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1],[40, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11,Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let (A1,L1) and (A2,L2) be two quantum compact metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A tunnel τ = (D,LD,π1,π2) is given by a quantum compact metric space (D,LD) and two quantum isometries π1 : (D,LD) → (A1,L1) and π2 : (D,LD) → (A2,L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The domain dom(τ) of τ is (A1,L1) and the codomain codom(τ) of τ is (A2,L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The extent χ(τ) of τ is the non-negative number: χ(τ) := max j∈{1,2}Haus �mkLD � � π∗ j (S (Aj )),S (D) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We emphasize that all quantum compact metric spaces involved in our tunnels in this paper must satisfy the same (Ω,Ω′)-Leibniz inequality for our fixed Ω,Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' There always exists a tunnel between any two quantum compact metric spaces, and the extent of a tunnel is always finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We thus define: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The (dual) Gromov-Hausdorff propinquity Λ∗((A,LA),(B,LB)) be- tween any two quantum compact metric spaces (A,LA) to (B,LB) is defined by: Λ∗((A,LA),(B,LB)) := inf � χ(τ) : τ tunnel from (A,LA) to (B,LB) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The (dual) propinquity is well-behaved, as summarized in the following theorem: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='17 ([38, 35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The dual propinquity is a complete metric up to full quantum isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, if (Xn,dn)n∈N is a sequence of compact metric spaces, then (Xn,dn)n∈N converges to a compact metric space (X ,d) for the Gromov-Hausdorff distance if, and only if limn→∞ Λ∗((C(Xn),Ldn),(C(X ),Ld)) = 0, where Ld denotes the Lipschitz seminorm induced by any metric d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 10 CARLA FARSI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' FRÉDÉRIC LATRÉMOLIÈRE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' AND JUDITH PACKER There are several interesting known examples of convergence for the propinquity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' in- cluding approximations of quantum tori by fuzzy tori [33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' approximations of spheres by matrix algebras [9],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' continuity of quantum tori in their cocycle parameter [33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' continuity of UHF algebras with respect to the Baire space seen as their natural parameter space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' continuity of the Effros-Shen algebras in their irrational parameters [5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We begin with a simple sufficient condition to ensure that a seminorm is indeed a Lipschitz seminorm on an inductive limit of unital C*-algebras, when each of the C*-subalgebra in the inductive sequence is already equipped with a Lipschitz seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This condition is quite natural and generalizes, for instance, the idea behind the construction of Lipschitz seminorms on AF algebras in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let A∞ be a unital C*-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N, let (An,Ln) be a quan- tum compact metric space, where (An)n∈N is an increasing sequence of C*-subalgebras of A∞ with the unit of A∞ in A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Assume moreover that A∞ = cl(� n∈NAn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let L∞ be a seminorm defined on a dense Jordan-Lie subalgebra dom(L∞) of sa(A∞), such that: (1) {a ∈ dom(L∞) : L∞(a) = 0} = R1, (2) the unit ball of L∞ is closed in A∞, (3) L∞ is (Ω,Ω′)-Leibniz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If there exists a unital isometric positive linear map π : A∞ → A∞ such that, for all ε > 0, there exists N ∈ N with the property that: ∀a ∈ dom(L∞) ∃b ∈ dom(LN) : LN(b) � L∞(a) and ∥π(a)−b∥A∞ < εL∞(a), then (A∞,L∞) is a quantum compact metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let µ ∈ S (A∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By assumption, µ ∈ S (An) for all n ∈ N — where we use the same symbol µ to denote the restriction of µ to An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let B∞ := � a ∈ dom(L∞) : µ◦π(a) = 0,L∞(a) � 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let ε > 0 and let n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We set Bn := � a ∈ dom(Ln) : |µ(a)| < ε 4,Ln(a) � 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let a ∈ Bn, and let ϕ ∈ S (An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='8), we have the following inclusion: Bn ⊆ � a ∈ dom(Ln) : Ln(a) � 1,∥a∥An � qdiam(An,Ln)+ ε 4 � and the latter set is compact since Ln is a Lipschitz seminorm, by Corollary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So Bn is totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In fact, since Ln is lower semicontinuous and µ is continuous, the set Bn is also closed in the complete space A∞, so Bn is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By assumption on π, there exists N ∈ N such that ∀a ∈ B∞ ∃b ∈ dom(LN) : LN(b) � 1 and ∥π(a)−b∥A∞ < ε 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular, if a ∈ B∞ and b ∈ dom(LN) with LN(b) � 1 and ∥π(a)−b∥A∞ < ε 4, then |µ(b)| � ∥b −π(a)∥A∞ +|µ(π(a))| < ε 4, so b ∈ BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since BN is compact in sa(AN) by Corollary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='10), there exists a ε 4-dense subset F ⊆ BN of BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So Haus[A∞](π(B∞),F) � Haus[A∞](π(B∞),BN)+Haus[A∞](BN,F) < ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 11 The domain dom(L∞) is dense in sa(A), so it is not empty and thus {a ∈ dom(L∞) : L∞(a) � 1} is not empty, since L is a seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, by Remark (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6), the set B∞ is not empty as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We thus obtain: � ̸= B∞ = � b∈F � a ∈ B∞ : ∥π(a)−b∥A∞ < ε 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, if we define G := � b ∈ F : � a ∈ B∞ : ∥π(a)−b∥A∞ < ε 2 � ̸= � � , then G ̸= � and B∞ = � b∈G � a ∈ B∞ : ∥π(a)−b∥A∞ < ε 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each b ∈ G, we pick t(b) ∈ B∞ such that ∥π(t(b))−b∥A∞ < ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let now a ∈ B∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' There exists b ∈ G such that ∥π(a)−b∥A∞ < ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Then ∥a − t(b)∥A∞ = ∥π(a − t(b))∥A∞ � ∥π(a)−b∥A∞ +∥b −π(t(b))∥A∞ < ε 2 + ε 2 = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, t(G) is a ε-dense subset of B∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So B∞ is totally bounded in A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, noting that µ◦π is a state of A∞, we conclude by Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='9) that mkL∞ induces the weak-∗ topology on S (A∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since all other required properties are assumed, L∞ is indeed a Lipschitz seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ The next natural question is to find a sufficient condition to strengthen Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='18) and obtain convergence of the sequence (An,Ln)n∈N to (A∞,L∞) in the sense of the propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' To this end, we introduce the notion of a bridge builder — a map which, among other things, satisfy the condition in Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In fact, we basically “sym- metrize” the condition in Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='18) and require that we work with ∗-morphism (which will allow us to construct seminorms with the Leibniz property), rather than just positive linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will write N := N∪{∞} for the one point compactification of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N∪{∞}, let (An,Ln) be a quantum compact metric space, where (An)n∈N is an increasing (for ⊆) sequence of C*-subalgebras of A∞ such that A∞ = cl(� n∈NAn) and the unit of A∞ is in A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A ∗-automorphism π : A∞ → A∞ is a bridge builder for ((An,Ln)n∈N,(A∞,L∞)) when, for all ε > 0, there exists N ∈ N such that if n � N, then ∀a ∈ dom(L∞) ∃b ∈ dom(Ln) : Ln(b) � L∞(a) and ∥π(a)−b∥A∞ < εL∞(a) and ∀b ∈ dom(Ln) ∃a ∈ dom(L∞) : L∞(a) � Ln(b) and ∥π(a)−b∥A∞ < εLn(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N∪{∞}, let (An,Ln) be a quantum compact metric space, where (An)n∈N is an increasing (for ⊆) sequence of C*-subalgebras of A∞ such that A∞ = cl(� n∈NAn) and the unit of A∞ is in A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If there exists a bridge builder for ((An,Ln)n∈N,(A∞,L∞)), then lim n→∞Λ∗((An,Ln),(A∞,L∞)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let π : A∞ → A∞ be the given bridge builder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' There exists N ∈ N such that if n � N, then ∀a ∈ dom(L∞) ∃b ∈ dom(Ln) : Ln(b) � L∞(a)∧∥π(a)−b∥A∞ < εL∞(a), 12 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER ∀b ∈ dom(Ln) ∃a ∈ dom(L∞) : L∞(a) � Ln(b)∧∥π(a)−b∥A∞ < εLn(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix n � N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We define, for all a ∈ dom(L∞) and b ∈ dom(Ln): Tn(a,b) := max � L∞(a),Ln(b), 1 ε ∥π(a)−b∥A∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is a standard argument that (A∞ ⊕An,Tn) is a quantum compact metric space: (1) the domain dom(Tn) = dom(L∞)⊕dom(Ln) of Tn is dense in sa(A∞ ⊕An) since dom(L∞) is dense in sa(A∞) and dom(Ln) is dense in sa(An), (2) if Tn(a,b) = 0 for some (a,b) ∈ dom(Tn), then L∞(a) = 0 so a = t1 for some t ∈ R, and Ln(b) = 0 so b = s1 for some s ∈ R (it matters here that the unit is the same in A∞ and An), and 0 = ∥π(a)−b∥A∞ = |t − s| so (a,b) = t(1,1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (3) Tn is the maximum of two lower semicontinuous functions and one continuous function, so it is lower semicontinuous over sa(A∞ ⊕An);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (4) a direct computation shows that Tn is (Ω,Ω′)-Leibniz since L∞ and Ln both are, and π is a ∗-morphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (5) fixing any state µ of A∞ and setting ϕ : (a,b) ∈ A∞⊕An �→ µ(a), then ϕ ∈ S (A∞⊕ An), and � (a,b) ∈ dom(Tn) : Tn(a,b) � 1,ϕ(a,b) = 0 � ⊆ � a ∈ dom(L∞) : L∞(a),µ(a) = 0 � × � b ∈ dom(Ln) : Ln(b) � 1,|µ◦π−1(b)| � ε � and, as seen in the proof of Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='18), the set on the right hand side is a product of two compact set, and thus compact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' thus the set on the left hand side is compact (closed in a compact set) and thus, Tn is indeed a Lipschitz seminorm, invoking Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now check that τn := (A∞ ⊕An,Tn,ψn,θn), with ψn : (a,b) ∈ A∞ ⊕An �→ a ∈ A∞ and θn : (a,b) ∈ A∞ ⊕An �→ b ∈ An, is a tunnel, in the sense of Definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let a ∈ dom(L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By assumption, there exists b ∈ dom(Ln) with Ln(b) � L∞(a) and ∥π(a)−b∥A∞ < εL∞(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, Tn(a,b) = L∞(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since by construction, Tn(a,c) � L∞(a) for all a ∈ dom(L∞) and c ∈ dom(Ln), we have shown that ψn is a quantum isometry by Definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let now b ∈ dom(Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Again by assumption on π, there exists a ∈ dom(L∞) such that ∥π(a)−b∥A∞ < εLn(b) and L∞(a) � Ln(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus Tn(a,b) = Ln(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Once again, Tn(c,b) � Ln(b) by construction for all c ∈ dom(L∞), so θn is indeed a quantum isometry, so τn is a tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now compute the extent of τn, in the sense of Definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let ϕ ∈ S (A∞ ⊕ An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Using Hahn-Banach theorem, we extend ϕ to a state ϕ′ of A∞ ⊕A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let µ : a ∈ A∞ �→ ϕ′(a,π(a));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' since π is a unital ∗-morphism, µ is a state of A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, if Tn(a,b) � 1 then ∥π(a)−b∥A∞ � ε and thus |ϕ(a,b)−µ◦ψn(a,b)| = |ϕ′(a,b)−ϕ′(a,π(a))| � |ϕ′(0,b −π(a))| � ∥b −π(a)∥A∞ � ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus Haus �mkTn � (ψ∗ n(S (A∞)),S (A∞ ⊕An)) � ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 13 Let now µ′ : b ∈ An �→ ϕ(π−1(b),b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since π is a ∗-automorphism of A∞, the map µ′ is a state of An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover: |ϕ(a,b)−µ′ ◦θn(a,b)| = |ϕ(a,b)−ϕ(π−1(b),b)| = |ϕ(a −π−1(b),0)| � ��a −π−1(b) ��A∞ = ∥π(a)−b∥A∞ � ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus Haus �mkTn � (θ∗ n(S (An)),S (A∞)) � ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Hence, the extent χ(τn) of τn is at most ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By Definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='16), we thus have shown that for all n � N, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1) Λ∗((An,Ln),(A∞,L∞)) � ε, which concludes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ Our main result in this section is the following theorem, which shows that the natural sufficient condition in Definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='20) and Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='21) is, in fact, very close to necessary, under a mild and natural condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This is notable because in general, it is difficult to exhibit nontrivial necessary conditions for convergence in the sense of the propinquity (besides, say, the fact that diameters must converge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It also shows that the existence of bridge builders is the natural setup for establishing convergence of inductive limits in the sense of the propinquity, thus providing a complete answer for the relationship between convergence of inductive sequences of quantum compact metric spaces in the categorical sense and the propinquity sense, under a commonly met condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N∪{∞}, let (An,Ln) be a quantum compact metric space, where (An)n∈N is an increasing (for ⊆) sequence of C*-subalgebras of A∞ such that A∞ = cl(� n∈NAn) and the unit of A∞ is in A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We assume that there exists M > 0 such that for all n ∈ N: 1 M Ln � L∞ � M ·Ln on dom(Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Then lim n→∞Λ∗ ((An,Ln),(A∞,L∞)) = 0, if, and only if, for any subsequence (Ag(n),Lg(n))n∈N of (An,Ln)n∈N, there exists a strictly increasing function f : N → N and a bridge builder π for ((Ag◦f (n),Lg◦f (n))n∈N,(A∞,L∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' First, assume that for any subsequence (Ag(n),Lg(n))n∈N, there exists a strictly increasing function f : N → N and a bridge builder π for ((Ag◦f (n),Lg◦f (n))n∈N,(A∞,L∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='21), we conclude that every subsequence of (An,Ln)n∈N has a subse- quence converging to (A∞,L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, (An,Ln)n∈N converges to (A∞,L∞) since the propinquity is, indeed, a metric (up to full quantum isometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let us now assume that (An,Ln)n∈N converges to (A∞,L∞) for the propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since any subsequence will converge as well, it is sufficient to prove our statement for g being the identity, and this will simplify our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since (An,Ln)n∈N converges to (A∞,L∞), there exists a sequence (τn)n∈N := (Dn,Tn,ψn,θn)n∈N of tunnels, as in Definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='14), with limn→∞ χ(τn) = 0, while, for each n ∈ N, we have dom(τn) = (A∞,L∞) and codom(τn) = (An,Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' To ease notation, the target set 14 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER of a ∈ dom(L∞) with l � L∞(a) defined by τn will be denoted by tn (a|l), rather than tτn (a|l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' we recall from [35, 38] that: tn (a|l) = � θn(d) : d ∈ ψ−1 n ({a}),Tn(d) � l � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This proof heavily relies on the properties of target sets, as discussed in [35, 38, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In [35], various estimates which we will refer to in this proof are expressed using the length λ(τ) of a tunnel τ, rather than the extent χ(τ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' however as seen in [40, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='12], for any tunnel τ, we have λ(τ) � χ(τ) � 2λ(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will use this inequality without further mention to express all our results here in terms of extents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For all a ∈ dom(L∞), there exists a strictly increasing function f : N → N and an element π(a) ∈ dom(L∞) such that, for all l � L∞(a), lim n→∞Haus[A∞] �tf (n) (a|l),{π(a)} � = 0, and ∥π(a)∥A∞ = ∥a∥A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof of Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' First, since the sequence (χ(τn))n∈N converges (to 0), it is bounded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' let K ′ > 0 such that χ(τn) � K ′ for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let a ∈ dom(L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let l = L∞(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For any K > 0, let A∞[K ] := � b ∈ dom(L∞) : L∞(b) � K ,∥b∥A∞ � ∥a∥A∞ +K K ′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The set A∞[K ] is compact in sa(A∞) by Corollary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By [35, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='5] and since L∞ � MLn on dom(Ln), the sequence (tn (a|l))n∈N is a sequence of compact subsets of A∞[Ml], and lim n→∞diam(tn (a|l),A∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since A∞[Ml] is compact in A∞, the Hausdorff distance Haus[A∞] induced on the set of closed subsets of A∞[Ml] by the norm ∥·∥A∞ of A∞ gives a compact topology as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, there exists a subsequence (tf (n) (a|l))n∈N of (tn (a|l))n∈N which converges, for Haus[A∞], to a singleton {π(a)} of A∞[Ml].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular, L∞(π(a)) � Ml = ML∞(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let now L � l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By definition, tf (n) (a|l) ⊆ tf (n) (a|L) for all n ∈ N and lim n→∞diam �tf (n) (a|L),A∞ � = 0, so we conclude easily as well that lim n→∞Haus[A∞] �tf (n) (a|L),{π(a)} � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By [35, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='4], we also note that if bn ∈ tf (n) (a|l) for each n ∈ N, then ∥π(a)∥A∞ = lim n→∞∥bn∥A∞ � limsup n→∞ � ∥a∥A∞ +χ � τf (n) � l � = ∥a∥A∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Similarly, since a ∈ tτ−1 f (n) (bn|l), we also have ∥a∥A∞ � limsup n→∞ � ∥bn∥A∞ +lχ � τf (n) �� = ∥π(a)∥A∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So indeed, ∥π(a)∥A∞ = ∥a∥A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' There exists a unital ∗-endomorphism π of A∞ such that π(dom(L∞)) ⊆ dom(L∞), and a strictly increasing function f : N → N such that, for all a ∈ dom(L∞), and for all l � L∞(a), lim n→∞Haus[A∞] �tf (n) (a|l),{π(a)} � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 15 Proof of Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since A∞ is separable, there exists a countable dense subset S∞ of sa(A∞) with S∞ ⊆ dom(L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Using Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='23), a diagonal argument shows that there exists a strictly increasing sequence f : N → N such that, for all a ∈ S∞ and for all l � L∞(a), we have limn→∞Haus[A∞] �tf (n) (a|l),{π(a)} � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let now a ∈ dom(L∞), and let l � L∞(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since S∞ is dense in dom(L∞), there exists aε ∈ dom(L∞) such that ∥a − aε∥A∞ < ε 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Note that L∞(aε) < ∞ but in general, there is no relation between L∞(aε) and L∞(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let l = max{L∞(a),L∞(aε)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since it is convergent for the Hausdorff distance Haus[A∞], the sequence �tf (n) (aε|l) � n∈N is Cauchy for Haus[A∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, there exists N ∈ N such that, for all p,q � N, we have Haus[A∞] �tf (p) (aε|l),tf (q) (aε|l) � < ε 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since limn→∞ χ � τf (n) � = 0, there exists N ′ ∈ N such that if n � N ′ then χ � τf (n) � < ε 5(l+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, if n � N ′, then by [35, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='5], Haus[A∞] �tf (n) (a|l),tf (n) (aε|l) � � ∥a − aε∥A∞ +lχ � τf (n) � < 2ε 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let now p,q � max{N,N ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We compute: Haus[A∞] �tf (p) (a|l),tf (q) (a|l) � � Haus[A∞] �tf (p) (a|l),tf (p) (aε|l) � +Haus[A∞] �tf (p) (aε|l),tf (q) (aε|l) � +Haus[A∞] �tf (q) (aε|l),tf (q) (a|l) � < 2ε 5 + ε 5 + 2ε 5 = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, �tf (n) (a|l) � n∈N is Cauchy for Haus[A∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since sa(A∞) is complete, so is the set of all closed subsets of sa(A∞) with the Hausdorff distance Haus[A∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, �tf (n) (a|l) � n∈N converges to some compact subset in sa(A∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In fact, since lim n→∞diam �tf (n) (a|l),A∞ � = 0 by [35, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='5], the sequence �tf (n) (a|l) � n∈N converges to some singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As observed in Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='23), this limit does not depend on l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' we denote it by {π(a)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Again using the same argument, we also note that ∥π(a)∥A∞ = ∥a∥A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since L∞ is lower semicontinuous over A∞, and since by construction, π(a) is the limit in A∞ of any sequence (bn)n∈N with bn ∈ tf (n) (a|L∞(a)) for all n ∈ N, we also conclude that L∞(π(a)) � liminf n→∞ L∞(bn) by lower semicontinuity of L∞, � liminf n→∞ M ·Ln(b) since L∞ � M ·Ln for all n ∈ N, � M ·L∞(a) since Ln(b) � L∞(a), as b ∈ tf (n) (a|L∞(a)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let a,a′ ∈ dom(L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since tf (n) (a|l)+t ·tf (n) � a′��l � ⊆ tf (n) � a + ta′��(1+|t|)l � for all n ∈ N by [35, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='5], we immediately conclude that {π(a)} + t · {π(a′)} ⊆ {π(a + ta′)}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' π is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A similar argument shows that π is a Jordan-Lie morphism over dom(L∞), using [35, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As a linear map π with ∥π(a)∥A∞ = ∥a∥A∞ for all a ∈ dom(L∞), we can uniquely extend π to sa(A∞) as a uniformly continuous map over sa(A∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' this map is of course again a Jordan-Lie morphism from sa(A∞) to sa(A∞) and an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 16 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER A straightforward argument shows that we can uniquely extent π to a continuous Jordan-Lie algebra endomorphism of A∞, and thus π thus extended is a unital ∗-endo- morphism with L∞ ◦π � L∞ over dom(L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We already know that π is an isometry on sa(A∞) and a ∗-morphism, so it is injective on A∞: if π(a) = 0 then π(ℜa) = 0 so ℜa = 0, and π(ℑa) = 0 so ℑa = 0, and thus a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular, since π is now an injective ∗-morphism, it is an isometry on A∞ (rather than just sa(A∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For all ε > 0, there exists N ∈ N such that for all n � N, and for all a ∈ dom(L∞) with L∞(a) � 1, we have Haus[A∞] � {π(a)},tf (n) (a|1) � < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof of Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix µ ∈ S (A∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The set B := � a ∈ dom(L∞) : L∞(a) � 1,µ(a) = 0 � is compact in sa(A∞) by Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, there exists a finite subset F ⊆ B such that Haus[A∞](F,B) < ε 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since F is finite, by Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='24), there exists N ∈ N such that, for all a ∈ F and for all n � N, we have Haus[A∞] � {π(a)},tf (n) (a|L∞(a)) � < ε 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, there exists N ′ ∈ N such that, if n � N ′, then χ(τn) < ε 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let n � max{N,N ′}, a ∈ B and b ∈ tf (n) (a|1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' There exists a′ ∈ F such that ��a − a′��A∞ < ε 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let b′ ∈ tf (n) � a′��1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By [35, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='5], we compute the following expression: ∥π(a)−b∥A∞ � ��π(a)−π(a′) ��A∞ + ��π(a′)−b′��A∞ + ��b′ −b ��A∞ � ��π(a − a′) ��A∞ π is linear + ε 4 by choice of N + ��a − a′��A∞ +χ(τn) by [35] � 2 ��a − a′��A∞ + ε 4 + ε 4 � ε 2 + ε 4 + ε 4 = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We thus have proven our uniform convergence claim over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let now a ∈ dom(L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Then of course, a − µ(a)1 ∈ B, since L∞(a − µ(a)1) � L∞(a) + L∞(µ(a)1) = L∞(a) � 1 (in fact, L∞(a) = L∞(a − µ(a)1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If b ∈ tf (n) (a|1) then b − µ(a)1 ∈ tf (n) � a −µ(a)1 ��1 � by construction, and thus ∥π(a)−b∥A∞ = ��π(a −µ(a)1)−(b −µ(a)1) ��A∞ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, as claimed, Haus[A∞] � {π(a)},tf (n) (a|1) � < ε for all n � max{N,N ′} and for all a ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For all ε > 0, there exists N ∈ N such that, if n � N, then ∀a ∈ dom(L∞) ∃b ∈ dom(Ln) : Ln(b) � L∞(a) and ∥π(a)−b∥A∞ < εL∞(a), ∀b ∈ dom(Ln) ∃a ∈ dom(L∞) : L∞(a) � Ln(b) and ∥π(a)−b∥A∞ < εLn(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof of Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let N ∈ N be chosen as in Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='25), so that for all a ∈ dom(L∞) with L∞(a) � 1, and for all n � N, we have Haus[A∞]({π(a)},tf (n) (a|1)) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let now n � N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If a ∈ dom(L∞)\\R1A∞, and if b ∈ tf (n) (a|L∞(a)), then L∞(a) > 0, Ln(b) � L∞(a), and b L∞(a) ∈ tf (n) � a L∞(a) ���1 � and thus ���π � a L∞(a) � − b L∞(a) ���A∞ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So ∥π(a)−b∥A∞ < εL∞(a), as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let b ∈ dom(Ln) \\ R1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let b′ = b Ln(b), so Ln(b′) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let a′ ∈ tτ−1 f (n) � b′��1 � , so in particular L∞(a′) � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By symmetry, b′ ∈ tf (n) � a′��1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, ��π(a′)−b′��A∞ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 17 Hence, letting a = Ln(b)a′, we conclude that ∥π(a)−b∥A∞ � Ln(b)ε and L∞(a) � Ln(b), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Last, it is immediate that since π(1) = 1, our claim holds whenever L∞(a) = 0 or Ln(b) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', for any a,b ∈ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The map π constructed in Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='24) is a ∗-automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof of Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The map isometry of A∞, hence it is a ∗-monomorphism of A∞, via Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='24), Now, let b ∈ � n∈N dom(Ln), so b ∈ dom(Lm) for some m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus b ∈ dom(L∞) by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let l = Lm(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By assumption, L∞(b) � MLm(b) = Ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let ε > 0 and let N ∈ N given by Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since Ln(b) � ML∞(b) � M2l, for all n � max{N,m}, and there exists an ∈ A∞ with ∥π(an)−b∥A∞ < εM2l (and L∞(a) � Ln(b), which we do not need for this claim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As ε > 0 was arbitrary, the element b lies in the closure of the range of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since A∞ is complete and π is an isometry, the range of π is closed, and we now have shown that the range of π is a closed set containing the total subspace � n∈N dom(Ln) of A∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' consequently, π is a surjection as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus as claimed, π is a ∗-automorphism of A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, by construction, for all a ∈ dom(L∞), as noted in Claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='23), we have L∞(π(a)) � ML∞(a) — in particular, π(a) ∈ dom(L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So π(dom(L∞)) ⊆ dom(L∞) and thus π is a Lipschitz morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This concludes the proof of our theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Limits, for the propinquity, are unique up to full quantum isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' There- fore, the appearance of some map π in Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='22) is to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' However, the map π in Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='22) is quite a bit more general than a full quantum isometry — in fact, it need not be Lipschitz for us to use Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='21) — even though Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='22) shows that it can always be chosen to be so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The map π is really used here as a tool to construct a special kind of bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In general, the function π is not expected to be unique: if Ln is just the restriction to An of L∞ for all n ∈ N, and if θ is a full quantum isometry of (A∞,L∞), then π ◦ θ can be used in place of π, of course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The situation is more delicate when Ln varies, but there will usually be many maps π if there is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='22) characterizes the convergence of inductive sequences in the sense of the propinquity, under the condition of uniform equivalence of the Lipschitz seminorms on the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The condition of uniform equivalence of Lipschitz seminorms is in essence our compatibility condition between the Lipschitz seminorms and the inductive limit structure in Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='22): using the notation of Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='22), as seen in [37], under the hypothesis that dom(Ln) = An ∩dom(L∞), the Lipschitz seminorms Ln and L∞ are equivalent for each n ∈ N, and we require, in the assumptions of Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='22), that we want this equivalence be uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This leads us to several natural questions: does convergence of (An,Ln)n∈N imply some uniform equivalence of the Lipschitz semi- norms Ln (n ∈ N) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' is our assumption redundant)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Does the existence of a bridge builder imply uniform equivalence of the Lipschitz seminorms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Does convergence of an inductive limit for the propinquity imply the existence of a bridge builder without the assumption of uniform equivalence of the Lipschitz seminorms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, does the convergence of (An,Ln)n∈N to (A∞,L∞) for the propinquity imply the convergence of (An,Lk)k�n to (An,L∞) for a fixed n ∈ N, for the propinquity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now will show with two examples that all of the above questions have negative answers, so there is no obvious generalization of Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' First, we see that it is possible to have convergence for the propinquity of an inductive sequence of quantum 18 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER 0 dist 1 n ·dist 1−n−2 0 dist 1 (n +1)·dist 1−(n +1)−2 ··· n → ∞ 0 dist 1 X with dist : x, y ∈ X �→ |x − y| FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Approximating [0,1] with itself by modifying the metric on a small interval at the end (red) compact metric spaces, using the identity as a bridge builder, and yet, not have uniform equivalence of the Lipschitz seminorms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let X = [0,1] with its usual metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If Y ⊆ X with at least two points, then we set LY (f ) = sup � |f (x)−f (y)| |x−y| : x ̸= y,x, y ∈ Y � for all f ∈ C(X ), allowing for ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N, and for all f ∈ C(X ), we set: Ln(f ) = L� 0,1− 1 n2 �(f )+ 1 n L� 1− 1 n2 ,1 �(f ), allowing again for ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let fn : x ∈ [0,1] �−→ � 0 if x � 1− 1 n2 , x −(1− 1 n2 ) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, L[0,1](fn) = 1 for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' On the other hand, Ln(f ) = 0+ 1 n ·1 = 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So there does not exists M > 0 such that L[0,1] � MLn on the common domain of these Lipschitz seminorms (the algebra of Lipschitz functions for the usual metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now prove that (C(X ),Ln)n∈N converges for the propinquity to (C(X ),L[0,1]) — this could be done here just as easily by proving the convergence for the Gromov-Hausdorff distance of X with a sequence of distances which agree with the usual distance on [0,1− 1 n2 ] and is a dilation by a factor n of the usual distance on [1− 1 n2 ,1], but we will keep with our functional analytic perspective here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We thus define, for all n ∈ N, and for all f ,g Lipschitz functions over [0,1] with its usual metric: Tn(f ,g) := max � L[0,1](f ),Ln(g),(n +1) ��f − g �� C(X ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let f ∈ C(X ) with L[0,1](f ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Then Ln(f ) � 1+ 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' From this, we see that Tn � f , 1 1+ 1 n f − 1 n +1 f (0) � � max � 1, n +1 n +1 ��f − f (0)1 �� C(X ) � � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let now g ∈ C(X ) with Ln(g) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus L� 0,1− 1 n2 �(g) � 1 and L� 1− 1 n2 ,1 �(g) � n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular, for all x ∈ [1− 1 n2 ,1], we have ���g(x)− g � 1− 1 n2 ���� < n|x −1+ 1 n2 | � 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let h ∈ C(X ) defined by h(x) = g(x) if x ∈ � 0,1− 1 n2 � , and h(x) = g � 1− 1 n2 � otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, L[0,1](h) � 1 and ��g −h �� C(X ) < 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus Tn(h,g) = 1 = Ln(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 19 1 1 2 2 1 3 3 1 4 4 1 5 5 1 6 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1 n+1 1 2 3 4 5 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1 n+2 1 2 3 4 5 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1 n+3 ··· n → ∞ 0 1 2 3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Approximating N by itself, by merging the first two points at ∞ Therefore, (C(X )⊕C(X ),Tn,p1,p2), with p1 : (f ,g) ∈ C(X )⊕C(X ) �→ f and p2 : (f ,g) ∈ C(X )⊕C(X ) �→ g, is easily seen to be a tunnel whose extent is at most 1 n (the method is analogous to Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='21)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Hence (C(X ),Ln)n∈N converges to (C(X ),L[0,1])n∈N for the propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, the identity map satisfies Condition (2) of Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Nonetheless, there is no M > 0 such that ∀n ∈ N L[0,1] � MLn on the common domain of these seminorms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So convergence in the propinquity does not imply uniform equivalence of the Lipschitz seminorms, even when working with a fixed, Abelian C*-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, we can also ask whether convergence for the propinquity of an inductive se- quence, implies the existence of a bridge builder, and as we shall see in the next example, this is not the case: once again, convergence occurs without uniform equivalence of Lipschitz seminorms (and we prove that we have neither uniform dominance or uniform domination using both examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, we see that (An,Lm)m�n does not converge to (An,L∞) in this case, for any n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let A∞ be the C*-algebra of convergent sequences with values in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N, let An = {(xk)k∈N : (xk)k�n is constant }, so An is a C*-subalgebra of A∞ sharing the unit (1)n∈N of A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For all n ∈ N, and for all (xk)k∈N ∈ An, we set Ln((xk)k∈N) := sup � |xp − xq| |ϕn(p)−ϕn(q)| : p,q ∈ N,p ̸= q � where: ϕn : m ∈ N �→ � 1 m if m > 0, 1+ 1 n if m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Of course, Ln is indeed a seminorm on the finite dimensional C*-subalgebra An of A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We also set L∞((xk)k∈N) = sup � |xp−xq| ��� 1 p+1 − 1 q+1 ��� : p,q ∈ N,p ̸= q � for all (xk)k∈N ∈ A∞, al- lowing for the value ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Of course, � n∈NAn ⊆ dom(L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let x : n ∈ N �→ � 1 if n = 0, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 20 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER By construction, L∞(x) = 2, yet Ln(x) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So there is no M > 0 such that, for all n ∈ N, the inequality MLn � L∞ on dom(Ln) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' On the other hand, limn→∞ Λ∗((An,Ln),(A∞,L∞)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Indeed, let π : (xk)k∈N �→ (x0,x0,x1,x2,x3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=') ∈ A∞, B = π(A∞), and let θ : (xk)k∈N ∈ B �→ (xk+1)k∈N ∈ A∞ — of course, θ is a ∗-isomorphism from B onto A∞ such that π = θ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We define LB(π(x)) = L∞(x) for all x ∈ dom(L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This way, π is easily checked to be a full quantum isometry from (A∞,L∞) to (B,LB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let ε > 0 and let N ∈ N be such that if n � N, then 1 n+1 < ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If x = (xk)k∈N with L∞(x) � 1, and if l = lims→∞ xs, then by construction, |xk −l| 1 k+1 = lim s→∞ |xk − xs| 1 k+1 − 1 s+1 � 1 so |xk −l| � 1 k+1 < ε 2 for all k � N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, if k � N then |xk − xN| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let n � N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let Dn = An ⊕B, and for all (a,b) ∈ dom(An)⊕dom(B), we set: Tn(a,b) := max � Ln(a),LB(b), 1 ε ∥π(a)−b∥B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We also set pn : (a,b) ∈ Dn �→ a ∈ An and qn : (a,b) ∈ Dn �→ θ(b) ∈ A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We are now going to prove that τn := (Dn,TNn,pn,qn) is indeed a tunnel from (An,Ln) to (A∞,L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let a := (xk)k∈N ∈ dom(L∞) with L∞(a) = 1, and let a′ := (x0,x0,x1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=',xN−1,xN,xN,xN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=') ∈ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, Ln(a′) � 1 and ��π(a)− a′��A∞ < ε by our choice of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Also by construc- tion, LB(π(a)) = L∞(a) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus Tn(a′,π(a)) � L∞(a) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So, we have shown that, for any a ∈ dom(L∞) with L∞(a) = 1, there exists an element d := (a′,π(a)) ∈ Dn such that TNn(d) = 1 = L∞(a) and qn(d) = θ(π(a)) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, the map qn is indeed a quantum isometry from (Dn,Tn) to (A∞,L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let now a = (xk)k∈N ∈ dom(Ln) with Ln(a) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By definition, |x1 − x0| � 1 n < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let b = (x1,x1,x2,x3,x4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, b ∈ dom(LB) with LB(b) � Ln(b), and ∥a −b∥A∞ = |x1 − x0| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus again Tn(a,b) = Ln(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So pn : (a,b) ∈ Dn �→ a ∈ An is a quantum isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, (Dn,Tn,pn,qn) is indeed a tunnel from (An,Ln) to (A∞,L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now compute an upper bound on its extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let ϕ ∈ S (Dn) be a state of Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If we set µ : a ∈ An �→ ϕ(a,π(a)), then µ ∈ S (An) is again a state of An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If (a,b) ∈ dom(Tn) with Tn(a,b) � 1, then |ϕ(a,b)−µ◦ pn(a,b)| = |ϕ(a,b)−ϕ(a,π(a))| = |ϕ(0,b −π(a))| � ∥b −π(a)∥A∞ < ε, so indeed Haus �mkTn � (S (Dn),p∗ nS (An)) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' On the other hand, let ν : a ∈ A∞ �→ ϕ′(a,π(a)) where ϕ′ is an extension of ϕ to a state of A∞ ⊕B by the Hahn-Banach theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Once again, it is immediate that mkTn(ϕ,ν◦ qn) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So Haus �mkTn � (S (D),q∗ nS (A∞)) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, for all n � N, the extend of χ(τn) is at most ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We conclude: lim n→∞Λ∗((An,Ln),(A∞,L∞)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' However, for any fixed p ∈ N, it is easy to check, by a similar method, that lim n→∞Λ∗((Ap,Ln),(Ap−1,L∞)) = 0, 21 and since dimAp−1 < dimAp, the sequence (Ap,Ln)n�p does not converge to (Ap,L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The map π we have used here is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In fact, there is no bridge builder in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Indeed, assume that we have a unital ∗-morphism π : A∞ → A∞ such that for all ε > 0, there exists Nπ(ε) ∈ N with the property that if n � Nπ(ε), and if a ∈ dom(L∞), then there exists b ∈ dom(Ln) with Ln(b) � L∞(a) and ∥π(a)−b∥A∞ < ε 2L∞(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix a ∈ dom(L∞) with L∞(a) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let ε > 0 and let n � Nπ(ε) such that 1 n < ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' De- fine (yk)k∈N := π(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Then there exists b := (bk)k∈N ∈ dom(Ln) such that Ln(b) � 1 and ∥π(a)−b∥A∞ < ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By definition of Ln, we thus conclude that |b1 − b0| � 1 n < ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, |y1−y0| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As ε > 0 is arbitrary, we conclude that y1 = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus π can never be surjective — in fact, it is valued in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So no bridge builder exists for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As seen in Example (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='30), convergence of (An,Ln)n∈N to (A∞,L∞) for the propinquity does not imply the convergence of (An,Lp)p∈N to (An,L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We have the following immediate consequence of our work: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let A∞ be a unital separable C*-algebra, such that A∞ = cl(� n∈NAn), where (An)n∈N is an increasing (for ⊆) sequence of C*-subalgebras of A∞, with the unit of A∞ in A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N, let Ln be a Lipschitz seminorm on An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If there exists a bridge builder π : A∞ → A∞ for ((An,Ln)n∈N,(A∞,L∞)) such that π(An) ⊆ An for each n ∈ N, then for all n ∈ N, lim p→∞ p�n Λ∗((An,Lp),(An,L∞)) = 0, and limn→∞ Λspec((An,Ln),(A∞,L∞)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This follows by observing that the restriction of π to An is a bridge builder for ((An,Lp)p�n,(An,L∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our result then follows from Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' CONVERGENCE OF INDUCTIVE SEQUENCES OF METRIC SPECTRAL TRIPLES FOR THE SPECTRAL PROPINQUITY We now study the convergence of certain families of metric spectral triples for the spectral propinquity [47], whose construction we will recall below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We thus begin this section with the definition of a spectral triple, due to Connes, and the foundational concept for noncommutative Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1 ([12, 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A spectral triple (A,H , /D) is given by a unital C*-algebra A of bounded linear operators on a Hilbert space H , and a self-adjoint operator /D defined on some dense subspace dom( /D) of H , such that: (1) {a ∈ A : a ·dom( /D) ⊆ dom( /D),[ /D,a] is bounded } is a dense ∗-algebra in A, (2) /D has compact resolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The operator /D is referred to as the Dirac operator of the spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Preliminaries: The Spectral Propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The spectral propinquity is a distance, up to unitary equivalence, on the class of metric spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If T : D ⊆ E → F is a linear operator defined from a dense subspace D of a normed vector space E to a normed vector space F, then we write: |||T |||F E := sup � ∥T ξ∥F : ξ ∈ D,∥ξ∥E � 1 � allowing for the value ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If F = E, then |||T |||F E is simply denoted by |||T |||E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 22 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A spectral triple (A,H , /D) is metric if the Connes extended pseudo- distance, defined on the state space S (A) of A by: mk /D : ϕ,ψ ∈ S (A) �→ sup � |ϕ(a)−ψ(a)| : a dom( /D) ⊆ dom( /D) and |||[ /D,a]|||H � 1 � is in fact a metric on S (A), which induces the weak-∗ topology on S (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As soon as a spectral triple is metric, it induces a structure of quantum compact metric space on its underlying C*-algebra in a natural manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='4 ([47, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let (A,H , /D) be a spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We set: dom(L /D) := {a ∈ sa(A) : a dom( /D) ⊆ dom( /D) and [ /D,a] is bounded } and for all a ∈ dom(L /D): L /D(a) := |||[ /D,a]|||H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The spectral triple (A,H , /D) is metric if, and only if, (A,L /D) is a quantum compact metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The construction of the spectral propinquity begins with the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Recall from [47] that if (A,H , /D) is a metric spectral triple, and if we set for all ξ ∈ dom( /D): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1) DN /D(ξ) := ∥ξ∥H +∥ /Dξ∥H , dom(L /D) := {a ∈ sa(A) : a dom( /D) ⊆ dom( /D), [ /D,a] is bounded } for all a ∈ dom(L /D): L /D(a) := |||[ /D,a]|||H , then metCor(A,H , /D) := (H ,DN /D,A,L /D,C,0) is an example of a metrical C*-correspondence, in the following sense: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' An A-B-C ∗-correspondence (M ,A,B), for two C*-algebras A and B, is a right Hilbert module M over B (whose B-valued inner product is denoted by 〈·,·〉M ), together with a unital ∗-morphism from A to the C*-algebra of adjoinable B-linear operators over M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6 ([47, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' An (Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='Ω′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='Ωmod,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='Ωinner)-metrical C*-correspondence (M ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='DN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='S),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' where Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='Ωinner � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Ωmod � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' and Ω′ � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' is given by two (Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='Ω′)- quantum compact metric spaces (A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='L) and (B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='S),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' an A-B C*-correspondence (M ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' and a norm DN defined on a dense C-subspace dom(TN) of M ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' such that (1) ∀ω ∈ dom(DN) DN(ω) � ∥ω∥M := ���〈ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='ω〉M ��B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (2) {ω ∈ dom(DN) : DN(ω) � 1} is compact in (M ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='∥·∥M ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (3) for all a ∈ dom(L) and ω ∈ dom(TN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' DN(aω) � Ωmod(∥a∥A +L(a))DN(ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (4) for all ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='η ∈ dom(DN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' max{S(ℜ〈ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='η〉M ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='S(ℑ〈ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='η〉M )} � ΩinnerDN(ω)DN(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular, the norm DN is called a D-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Convention 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In this work, we fix Ωmod � 2 and Ωinner � 1 all throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' All quantum compact metric spaces will be assumed to be in the class of (Ω,Ω′)-quantum compact metric spaces and all metrical C*-correspondences will be assume to be in the class of (Ω,Ω′,Ωmod,Ωinner)-metrical C*-correspondences, unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 23 Note that the compactness condition in Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6) borrows and extends on Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The importance of Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6) is that one can extend the propinquity to metrical C*-correspondences as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' First, we employ a natural notion of morphism between metrical C*-correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='8 ([47, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each j ∈ {1,2}, let Mj = �Mj ,DNj ,Aj ,Lj ,Bj ,Sj � be a metrical C*-correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A metrical quantum isometry (Π,π,θ) from M1 to M2 is a given by: (1) a continuous, surjective C-linear map Π : M1 → M2, (2) a quantum isometry π : (A1,L1) → (A2,L2), (3) a quantum isometry θ : (B1,S1) → (B2,S2), such that (1) ∀a ∈ A ∀ω ∈ M1 Π(aω) = π(a)Π(ω), (2) ∀b ∈ B ∀ω ∈ M2 Π(ω·b) = Π(ω)θ(b), (3) ∀ω,η ∈ M1 θ(〈ω,η〉M1) = 〈Π(ω),Π(η)〉M2, (4) Π(dom(DN1)) ⊆ dom(DN2) and, for all ω ∈ dom(DN2), the equality DN2(ω) = inf �DN1(η) : η ∈ dom(DN1),Π(η) = ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The definition of a distance between metrical C*-correspondences, called the metrical propinquity, relies on a notion of isometric embedding called a tunnel, and is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='9 ([47, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let M1 and M2 be two metrical C*-correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A (metrical) tunnel τ = (J,Π1,Π2) from M1 to M2 is a triple given by a metrical C*- correspondence J, and for each j ∈ {1,2}, a metrical quantum isometry Πj : J �→ Mj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is important to note that our tunnels involve (Ω,Ω′,Ωmod,Ωinner)-C*- metrical correspondences only (as per Convention (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will dispense calling our tun- nels (Ω,Ω′,Ωmod,Ωinner)-tunnels, to keep our notation simple, but it should be stressed that fixing (Ω,Ω′,Ωmod,Ωinner) and staying within the class of (Ω,Ω′,Ωmod,Ωinner)-C*- metrical correspondences is crucial to obtain a metric from tunnels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now proceed by defining the extent of a metrical tunnel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' remarkably this only involves our previous notion of extent of a tunnel between quantum compact metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11 ([47, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let Mj = (Mj ,DNj ,Aj ,Lj ,Bj ,Sj ) be a metrical C*-correspondence, for each j ∈ {1,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let τ = (P,(Π1,π1,θ1),(Π2,π2,θ2)) be a metrical tunnel from M1 to M2, with P = (P,TN,D,LD,E,LE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The extent χ(τ) of a metrical tunnel τ is χ(τ) := max � χ(D,LD,π1,π2),χ(E,TE,θ1,θ2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Given two metric spectral triples, we can thus either take the Gromov-Hausdorff distance between their underlying quantum compact metric spaces, or take the metri- cal propinquity [42, 46] between the metrical C*-correspondence they define, which is defined as the infimum of the extent of every possible metrical tunnels between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' However, the spectral propinquity involves our work on the geometry of quantum dy- namics [43, 44, 47] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We recall the construction of the spectral propinquity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' the new 24 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER quantity called the ε-magnitude was introduced in [47, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='31], but is simpler to express for spectral triples, based on [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='12 ([31, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let (A1,H1, /D1) and (A2,H2, /D2) be two metric spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let τ := � � (P,TN,D,LD,E,S) metrical C*-correspondence , (Π1,π1,θ1) metrical quantum isometry , (Π2,π2,θ2) metrical quantum isometry � � be a metrical tunnel from metCor(A1,H1, /D1) to metCor(A2,H2, /D2), We define the ε-magnitude µ(τ|ε) of τ as the maximum of the extent χ(τ) of τ, and the ε-reach of τ, which is the number: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2) sup ξ∈dom � /D j � DNj (ξ)�1 inf η∈dom( /Dk) DNk(η)�1 sup ω∈dom(TN) TN(ω)�1 0�t� 1 ε ���〈exp(it /D j )ξ,Πj (ω)〉Hj −〈exp(it /Dk)η,Πk(ω)〉Hk ���, for {j,k} = {1,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='13 ([47, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The spectral propinquity between two metric spectral triples (A1,H1, /D1) and (A2,H2, /D2) is Λspec((A1,H1, /D1),(A2,H2, /D2)) := inf �� 2 2 ,ε > 0 : µ(τ|ε) < ε for τ a tunnel from metCor(A1,H1, /D1) to metCor(A2,H2, /D2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The key property of the spectral propinquity is that, for any two metric spectral triples (A1,H1, /D1) and (A2,H2, /D2), we have the following equivalence: Λspec((A1,H1, /D1),(A2,H2, /D2)) = 0 if, and only if, there exists a unitary U : H1 → H2 such that Udom( /D1) = dom( /D2), U /D1 = /D2U on dom( /D1), a ∈ A1 �→UaU ∗ is a ∗-isomorphism from A1 onto A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A nontrivial example of convergence in the sense of the spectral propinquity is pro- vided in [45] with the approximation of spectral triples on quantum tori by spectral triples of certain matrix algebras known as fuzzy tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' These examples include many examples of previously informally stated convergences in mathematical physics, dealing with matrix models and their limits as the dimension of the algebra grows to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Such examples are a major motivation for the construction of the spectral propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Another example on fractals is presented in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, convergence for the spec- tral propinquity implies convergence of the spectra of the Dirac operators and, in an appropriate sense, the convergence of the bounded functional calculi of these operators, among other properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Of course, convergence for the spectral propinquity implies convergence of the underlying quantum compact metric spaces for the propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In this paper, we will construct new examples of convergence for new spectral triples defined over noncommutative solenoids and over Bunce-Deddens algebras, seen as limits of spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Preliminaries: Inductive Limits of Spectral Triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' While the spectral propinquity allows the discussion of convergence of spectral triples defined on vastly different C*- algebras, there are certain more restricted situations where the C*-algebras of a sequence of spectral triples may be related in a manner compatible with the spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In [20], a simple notion of inductive limit for spectral triples is introduced, based on the following encoding of such a compatibility via a natural, and rigid, notion of morphism between spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='14 ([20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' An isometric morphism (π,S) from (A1,H1, /D1) to (A2,H2, /D2) is given by a unital ∗-morphism π : A1 → A2 and a linear isometry S : H1 → H2 such that: (1) π(dom(L1)) ⊆ dom(L2), (2) Sdom( /D1) ⊆ dom( /D2) and S /D1 = /D2S on dom( /D1), (3) ∀a ∈ A1 Sa = π(a)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since S is a linear isometry, H1 can be identified with the closed subspace SH1 of H2 via S at no cost in our definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In that case, /D1 is only defined on H1 ⊆ H2, and we simply require that /D1 is the restriction of /D2 to dom( /D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We also note that if π(a) = 0 for some a ∈ A1, then π(a)S = Sa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since S is an isometry, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So π is actually automatically a ∗-monomorphism, and we thus can also identify A1 with the C*-subalgebra π(A1) of A2, since Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='14) ensures that aH1 ⊆ H1 and [ /D1,a] is identified with P[ /D2,π(a)]P = P[ /D2,π(a)] = [ /D2,π(a)]P where P is the orthogonal projection of H2 onto H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Furthermore, since π is unital, the unit of A2 is contained in A1 with this identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' An inductive sequence of spectral triples, as defined in [20], with a somewhat more involved notation, is simply a sequence of the form ((An,Hn, /Dn),(πn,Sn))n∈N where (An,Hn, /Dn) is a spectral triple and (πn,Sn) is an isometric morphism from (An,Hn, /Dn) to (An+1,Hn+1, /Dn+1), for each n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As we have seen above, we can identify such a sequence with one of the following type, which we will take as our notion of inductive limit of spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let A∞ = cl(� n∈NAn) be a C*-algebra which is the closure of an increas- ing sequence of C*-subalgebras (An)n∈N in A∞, with the unit of A∞ in A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A spectral triple (A∞,H∞, /D∞) is the inductive limit of a sequence (An,Hn, /Dn)n∈N of spectral triples when: (1) H∞ = cl(� n∈N)Hn, where each Hn is a Hilbert subspace of H∞, (2) for each n ∈ N, the restriction of /D∞ to dom( /Dn) is /Dn, (3) for each n ∈ N, the subspace Hn is reducing for An, which is equivalent to AnHn ⊆ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We note, using the notation of Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='15), that the operator which, to any ξ ∈ � n∈N dom( /Dn), associates /Dnξ whenever ξ ∈ dom( /Dn) for any n ∈ N, is indeed well- defined, and shown in [20] to be essentially self-adjoint, so /D∞ is the closure of this operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For our purpose, the following result from [20] will play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='16 ([20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1, partial]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If (An,Hn, /Dn)n∈N is an inductive sequence of spectral triples converging to a spectral triple (A∞,H∞, /D∞), then for any C-valued continuous function f ∈ C0(R) which vanishes at infinity, the sequence (Pn f ( /Dn)Pn)n∈N converges to f ( /D∞) in norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This section is concerned with the question: if a spectral triple is an inductive limit of spectral triples, then what additional assumptions should be made to get a more 26 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER geometric convergence, specifically in the sense of the spectral propinquity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In order to make sense of this question, we will work with metric spectral triples, which give rise to quantum compact metric spaces, and lie within the realm of noncommutative metric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The notion of inductive limit of spectral triples is simpler to define than the spectral propinquity but only applies to rather narrow examples — it is not applicable to fuzzy and quantum tori [45] or the fractals in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is certainly interesting to wonder how much metric information from the spectral triples are continuous with respect to the inductive limit process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In this section, we establish a sufficient condition for the convergence, in the sense of the spectral propinquity, of a sequence of metric spectral triples which already converges to a metric spectral triple in the categorical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This sufficient condition is simply the existence of an appropriate bridge builder which is also a full quantum isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, the main difficulty in establishing convergence for the spectral propinquity, in this context, reduces to proving metric convergence for the propinquity using adequate tunnels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let (A∞,H∞, /D∞) be a metric spectral triple which is the inductive limit of a sequence of metric spectral triples (An,Hn, /Dn), in the sense of Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N, let dom(Ln) := {a ∈ sa(An) : a dom( /Dn) ⊆ dom( /Dn) and [ /Dn,a] is bounded}, and for all a ∈ dom(Ln), define Ln(a) := |||[ /Dn,a]|||Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If there exists a full quantum isometry π : (A∞,L∞) → (A∞,L∞) which is also a bridge builder for ((An,Ln)n∈N,(A∞,L∞)), then lim n→∞Λspec((An,Hn, /Dn),(A∞,H∞, /D∞)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='21), the sequence (An,Ln)n∈N converges to (A∞,L∞) for the propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' More specifically, set, for convenience, ˜ε = ε 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let Nπ ∈ N be given so that, for all n � Nπ, we have: ∀a ∈ dom(L∞) ∃b ∈ dom(Ln) : Ln(b) � L∞(a) and ∥π(a)−b∥A∞ < ˜εL∞(a), ∀b ∈ dom(Ln) ∃a ∈ dom(L∞) : L∞(a) � Ln(b) and ∥π(a)−b∥A∞ < ˜εLn(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N, we constructed in Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='21) a tunnel τn = (Dn,Tn,ψn,θn) with Dn = A∞ ⊕An, and for all (a,b) ∈ dom(L∞)⊕dom(Ln), Tn(a,b) := max � L∞(a),Ln(b), 1 ˜ε ∥π(a)−b∥A∞ � , while ψn : (a,b) ∈ Dn �→ a, θn : (a,b) ∈ Dn �→ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We proved that χ(τn) < ˜ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is immediate, since π is a full quantum isometry, that τ′ n := (Dn,Tn,π◦ψn,θn) is also a tunnel with the same extent as τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N and for all ξ ∈ dom( /Dn), we define DNn(ξ) := ∥ξ∥Hn +∥ /Dnξ∥Hn , following Expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, since DN∞ is a D-norm, the set X∞ = {ξ ∈ dom( /D∞) : DN∞(ξ) � 1} is compact in H∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, there exists a finite subset F ⊆ X∞ of X∞ such that Haus[H∞](X∞,F) < ˜ε 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As /D∞ is the closure of an operator on � n∈NHn by [20], for any ξ ∈ F, there exists a sequence (ξn)n∈N, with ξn ∈ � j∈NHj for all n ∈ N, such that limn→∞ ξn = ξ, and 27 limn→∞ /D∞ξn = /D∞ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since F is finite, there exists NF ∈ N such that if n � NF and ξ ∈ F, then ∥ξ−ξn∥H∞ < ˜ε 3 and ∥ /D∞ξ− /D∞ξn∥H∞ < ˜ε 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Again by Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='15), we also have /D∞ξn = /Dnξn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix n ∈ N,n � N := max{Nπ,NF }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let Mn := H∞ ⊕ Hn, seen as a Dn-(C ⊕ C) C*- correspondence, with the C*-correspondence structure: ∀(a,b) ∈ Dn ∀(ξ,η) ∈ Mn (a,b)◁(ξ,η) := (π(a)ξ,bη), and ∀(ξ,η),(ξ′,η′) ∈ Mn 〈(ξ,ξ′),(η,η′)〉n := � 〈ξ,ξ′〉H∞,〈η,η′〉Hn � ∈ C⊕C, while ∀(t,s) ∈ C⊕C ∀(ξ,η) ∈ Mn (ξ,η)·(t,s) := (tξ,sη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We note that here, C2 is the C*-algebra of C-valued functions over a two points set, and in particular, the norm of (z,w) ∈ C2 is max{|z|,|w|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We then define, for all (ξ,η) ∈ dom( /D∞)⊕dom( /Dn): TNn(ξ,η) := max � DN∞(ξ),DNn(η), 1 ˜ε ��ξ−η ��H∞ � , while we also set Q : (z,w) ∈ C⊕C �→ 1 ˜ε|z − w|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is immediate to see that Q is a Lipschitz seminorm on C ⊕ C (it is, in fact, the Lipschitz seminorm for the metric on the two point set which places these two points exactly ˜ε apart).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, we check that TNn is a D-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Of course, for all (ξ,η) ∈ Mn: TNn(ξ,η) � max{DN∞(ξ),DNn(η)} � max � ∥ξ∥H∞ , ��η ��Hn � = ��(ξ,η) ��Mn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We observe that {(ξ,η) ∈ Mn : TNn(ξ,η) � 1} ⊆ {ξ ∈ dom( /D∞) :DN∞(ξ) � 1}×{η ∈ dom( /Dn) : DNn(η) � 1}, the latter set being compact as a product of two compact sets – since DNn and DN∞ are indeed D-norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since in addition, TNn is lower semicontinuous over Mn as the maximum of three lower semicontinuous functions over this space, the unit ball of TNn is indeed closed, hence compact, in Mn (which is complete).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now check the Leibniz inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If (a,b) ∈ dom(Tn) and (ξ,η) ∈ dom(TNn), then we compute: ��(a,b)◁(ξ,η) ��H∞ = ��π(a)ξ−bη ��H∞ � ∥π(a)−b∥A∞ ∥ξ∥H∞ +∥b∥A∞ ��ξ−η ��H∞ � ˜εTn(a,b)DNn(ξ)+∥(a,b)∥Dn ˜εTNn(ξ,η) � ˜ε �Tn(a,b)+∥(a,b)∥Dn �TNn(ξ,η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' From this, it follows that for all (a,b) ∈ dom(Tn) and for all (ξ,η) ∈ dom(TNn), TNn((a,b)◁(ξ,η)) � �Tn(a,b)+∥(a,b)∥Dn �TNn(ξ,η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 28 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER On the other hand, if (ξ,η),(ξ′,η′) ∈ dom(TNn), we have: Q(〈(ξ,η),(ξ′,η′)〉Mn) = 1 ˜ε ��〈ξ,ξ′〉H∞ −〈η,η′〉H∞ �� � 1 ˜ε ���〈ξ−η,ξ′〉H∞ ��+ ��〈η,ξ′ −η′〉H∞ ��� � 1 ˜ε ���ξ−η ��H∞ ��ξ′��H∞ + ��η ��H∞ ��ξ′ −η′��H∞ � � TNn(ξ,η) ��ξ′��H∞ + ��η ��H∞ TNn(ξ′,η′) � 2TNn(ξ,η)TNn(ξ′,η′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now define the maps: Πn : (ξ,η) ∈ Mn �→ ξ ∈ H∞, and Θn : (ξ,η) ∈ Mn �→ η ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our goal is to show that Υn := �Mn,(Πn,π◦ψn),(Θn,θn) � where Mn := (Mn,TNn,Dn,Tn,C⊕C,Q) is a metrical tunnel, using Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, Πn(a ·ξ,b ·η) = π(a)ξ = π◦ψn(a,b)Πn(ξ,η) and Θn(a ·ξ,b ·η) = bη = θn(a,b)Θn(ξ,η), for all (a,b) ∈ Dn and (ξ,η) ∈ Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let ξ ∈ H∞ with DN∞(ξ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction of F, there exists ξ′ ∈ F such that ��ξ−ξ′��H∞ < ˜ε 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By our choice of N, there exists η(= ξ′ n) ∈ Hn such that DNn(η) � 1+ ˜ε 3 and ��ξ′ −η ��H∞ < ˜ε 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let χ = 1 1+ ˜ε 3 η ∈ Hn, so that DNn(χ) � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, ��ξ−χ ��H∞ � ��ξ−η ��H∞ + ˜ε 3 1+ ˜ε 3 ��η ��H∞ � ��ξ−η ��H∞ + ˜ε 3 1+ ˜ε 3 DNn(η) � ��ξ−ξ′��H∞ + ��ξ′ −η ��H∞ + ˜ε 3 < ˜ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus TNn(ξ,χ) = 1, and therefore, (Πn,π◦ψn) is indeed a metrical quantum isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let η ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, /D∞η = /Dnη, so DN∞(η) = DNn(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, TNn(η,η) = DNn(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Again, we conclude that (Θn,θn) is a metrical quantum isometry as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, Υn is a metrical tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is immediate, of course, that the canonical surjections from C⊕C to C are quantum isometries — the only Lipschitz seminorm on C being the 0 function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So Υn is a metrical tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now compute the extent of Υn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is, by Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11), the maximum of the extent of the tunnel τ′ n, which is at most ˜ε, and the extent of the tunnel (C,0) ←− (C ⊕ C,Q) −→ (C,0), which is immediately computed to be ˜ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So the extent of Υn is ˜ε as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, for all n � N, we have Λ∗met((Hn,DNn,An,Ln,C,0),(H∞,DN∞,A∞,L∞,C,0)) � χ(Υn) = ˜ε < ε, and therefore, lim n→∞Λ∗met((Hn,DNn,An,Ln,C,0),(H∞,DN∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='A∞,L∞,C,0)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 29 It remains to compute an upper bound for the ε-reach of our tunnels Υn (see Defini- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will once again use our finite set F with Haus[H∞](F,X∞) < ˜ε 3 where X∞ is the closed unit ball of DN∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let (vk)k∈N be a sequence of continuous functions on R vanishing at ∞, valued in [0,1], and converging pointwise to 1 over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, (vk( /D∞))k∈N converges to /D∞ in the strong operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since F is finite, there exists k ∈ N such that, for all ξ ∈ F (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3) ∥vk( /D∞)ξ−ξ∥H∞ < ˜ε 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We identify, from now on, /Dn with the linear operator on H∞ whose restriction to Hn is /Dn, and whose restriction to H ⊥ n is 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' thus dom( /Dn) is replaced with dom( /Dn)⊕H ⊥ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We denote by Pn the orthogonal projection of H∞ onto Hn, so that Pn /Dn = /DnPn = /Dn on dom( /Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each t ∈ [0,∞), let ut : s ∈ R �→ exp(its), and for each n ∈ N, we denote ut( /Dn) by U t n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The function ut vk is continuous over R and vanishes at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By Theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='16), since (A∞,H∞, /D∞) is a spectral triple, and the inductive limit of the sequence (An,Hn, /Dn)n∈N of spectral triples, the sequence of operators (Pnut vk( /Dn)Pn)n∈N con- verges in norm to ut vk( /D∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, ut vk( /Dn)Pn = Pnut vk( /Dn)Pn for all n ∈ N by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let F ′ be a finite subset of the compact set � 0, 1 ε � such that Haus[R](F ′, � 0, 1 ε � ) < ˜ε 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since F ′ is finite, there exists Nν ∈ N such that if n � Nν, then for all t ∈ F ′: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='4) ������U t n(vk( /Dn))Pn −U t ∞(vk( /D∞)) ������H∞ < ˜ε 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, we note that if ξ ∈ dom(DNn) with DNn(ξ) � 1, then for all s < t ∈ R: ��U t nξ−U s nξ ��Hn � �t s ���� d dr U r nξ ����Hn dr � �t s ��U r n /Dnξ ��Hn dr � |s − t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, for all s,t ∈ R and ξ ∈ dom(DNn) with DNn(ξ) � 1, we have ��U t nξ−U s nξ ��Hn � |s − t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let n � N ′ := max{Nν,NF }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since /Dn and Pn commute, if ξ ∈ X∞, then DNn(ξ) � DN∞(ξ) and: DNn(νk( /Dn)Pnξ) = ∥vk( /Dn)Pnξ∥H∞ +∥ /Dnvk( /Dn)Pnξ∥H∞ � ∥vk∥C0(R) |||Pn|||H∞DNn(ξ) � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 30 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER For all ξ ∈ X∞ and t ∈ � 0, 1 ˜ε � , let s ∈ F ′ and ξ′ ∈ F such that |s − t| < ˜ε 12, ��ξ−ξ′��H∞ < ˜ε 3, then: ��U t nνk( /Dn)Pnξ−U t ∞ξ ��H∞ � ��U t nνk( /Dn)Pnξ−U s nνk( /Dn)Pnξ ��H∞ �|t−s|< ˜ε 12 since DNn(νk( /Dn)Pnξ)�1 + ��U s nνk( /Dn)Pnξ−U s ∞νk( /D∞)ξ ��H∞ �|||U snνk( /Dn)Pn−U s∞νk( /D∞)|||H∞< ˜ε 12 by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='4) + ��U s ∞νk( /D∞)(ξ−ξ′) ��H∞ �∥ξ−ξ′∥H∞< ˜ε 3 by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3) + ��U s ∞νk( /D∞)ξ′ −U s ∞ξ′��H∞ �∥νk( /D∞)ξ′−ξ′∥H∞< ˜ε 12 + ��U s ∞ξ′ −U t ∞ξ′��H∞ �|s−t|< ˜ε 12 + ��U t ∞ξ′ −U t ∞ξ ��H∞ �∥ξ−ξ′∥H∞< ˜ε 3 < ˜ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let ξ ∈ dom( /D∞) with DN∞(ξ) � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let n � N ′, and set η = νk( /D∞)Pnξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For all t ∈ � 0, 1 ˜ε � , we have, η ∈ dom( /Dn) and DNn(η) � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, inf η∈dom(DNn) DNn(η)�1 sup ω∈dom(TNn) TNn(ω)�1 ���〈U t nη,Θn(ω)〉Hn −〈U t ∞ξ,Πn(ω)〉H∞ ��� � sup ω∈dom(TNn) TNn(ω)�1 ���〈U t nνk( /Dn)Pnξ,Θn(ω)〉Hn −〈U t ∞ξ,Πn(ω)〉H∞ ��� � sup ω∈dom(TNn) TNn(ω)�1 � �� ��U t nνk( /Dn)ξ−U t ∞ξ ��H∞ ∥ω∥Mn + ��U t ∞ξ ��H∞ ∥Θn(ω)−Πn(ω)∥H∞ <˜ε since TNn(ω)�1 � �� < ˜ε+ ˜ε = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, take ξ ∈ Hn, with DNn(ξ) � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, ∥ /D∞ξ∥H∞ = ∥ /Dnξ∥Hn, and U t nξ =U t ∞ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So for all ξ ∈ dom(DNn) with DNn(ξ) � 1, we have, for all t ∈ R: inf η∈dom( /D∞) DN∞(η)�1 sup ω∈dom(TNn) TNn(ω)�1 ���〈U t ∞η,Πn(ω)〉H∞ −〈U t nξ,Θn(ω)〉Hn ��� � ���〈U t ∞ξ,Πn(ω)〉H∞ −〈U t nξ,Θn(ω)〉Hn ��� = 0 < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, for all n � max{N,N ′}, the ε-reach of Υn is no more than ε, and thus the ε-magnitude µ(Υn|ε) of Υn is no more than ε (by Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore for all n � N: Λspec((An,Hn, /Dn),(A∞,H∞, /D∞)) � µ(Υn|ε) < ε, and thus lim n→∞Λspec((An,Hn, /Dn),(A∞,H∞, /D∞)) = 0, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ 31 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A corollary of Theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='17) is that we obtain convergence for the bounded continuous functional calculus for the Dirac operators from the work in [31], which extends Theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' EVEN SPECTRAL TRIPLES ON TWISTED GROUP C ∗-ALGEBRAS We now apply our results of the previous sections to the construction of inductive limits of spectral triples for the spectral propinquity on twisted C*-algebras of discrete groups endowed with length functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular we will prove in this section our third main theorem, Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our approach introduces new metric spectral triples on certain twisted group C*-algebras which generalize the related, though distinct, past constructions using length functions over discrete groups such as the ones in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our main applications would be the construction of new spectral triples over noncom- mutative solenoids and some Bunce-Deddens algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular, we shall prove that the noncommutative solenoids spectral triples are limits of spectral triples over quantum 2-tori for the spectral propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will start with detailing in the next two subsections some background material that will be used to state and prove our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Discrete Groups, Proper Length Functions, 2-Cocycles, and Classical Spectral Triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let G∞ be a discrete group, and let σ be a 2-cocycle over G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let λ be the left regular σ-projective representation of G∞ on ℓ2(G∞), defined by, for all g ∈ G∞ and for all ξ ∈ ℓ2(G∞): λ(g)ξ : h ∈ G∞ �−→ σ(g,g −1h)ξ(g −1h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Of course, each operator λ(g) is unitary for each g ∈ G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let C ∗ red(G∞,σ) be the re- duced C*-algebra of G∞ twisted by σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' the C*-algebra of operators on ℓ2(G∞) gener- ated by � λ(g) : g ∈ G∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For any f ∈ ℓ1(G∞), the operator λ(f ) on ℓ2(G∞) is defined as � g∈G∞ f (g)λ(g) — it is easily checked that ������λ(f ) ������ ℓ2(G∞) � ��f �� ℓ1(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The reduced group C*-algebra C ∗ red(G) is, in particular, C ∗ red(G∞,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In [11], Connes introduced spectral triples (C ∗ red(G∞),ℓ2(G∞),ML) using any proper length function L overG∞, where ML is the operator of multiplication by L, defined on its natural domain in the Hilbert space ℓ2(G∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Connes proved that ������[ML,λ(g)] ������ ℓ2(G∞) = L(g) — which immediately follows from the triangle inequality and the fact that [ML,λ(g)]δe = L(g)σ(g,1)δg , where, for all g ∈ G∞: δg : h ∈ G∞ �→ � 1 if g = h, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It then follows that for the ∗-algebra Cc(G∞) of C-valued functions with finite support, we obtain the inequality, for all f ∈ Cc(G∞): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1) ������� ML,λ(g) ������� ℓ2(G∞) � � g∈G∞ |f (g)|L(g), since λ(g) is unitary for all g ∈ G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Note that by construction, for the multiplication operator by L to have compact resolvent, the spectral projection of this operator on any compact interval must have finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, in particular, the set {δh ∈ ℓ2(G∞) : L(h) � r} must be finite for all r � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In other words, all closed balls in G∞ for L must be finite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', L must indeed be proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' However, natural length functions on G∞ may not be proper, or even give the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' An example of this situation is given when G∞ is the additive group � Z � 1 p ��2 32 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER where: Z � 1 p � := � a pn : n ∈ N,a ∈ Z � , and where p ∈ N is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is natural to regard Z � 1 p � as a subgroup of Q, and thus equip it with the induced length function from the usual absolute value on Q (see Figure (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' However, this length function is not proper — and induces a non-discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We moreover note that Z � 1 p � = � n∈N 1 pn Z, and we would like to capture this inductive limit structure metrically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' while the sequence � 1 pn Z � n∈N converges to Z � 1 p � for the Hausdorff distance induced by | · |, we can not apply this observation directly to the associated twisted C*-algebras since |·| does not define a spectral triple using Connes’ methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let us discuss this situation by returning to a general discrete group G∞ and some 2-cocycle σ on G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now assume that we are given a strictly increasing sequence (Gn)n∈N of subgroups of G∞ such that G∞ = � n∈NGn — in fancier terms, G∞ is the inductive limit of the sequence of groups (Gn)n∈N, which we identify with a sequence of subgroups of G∞ from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We also identify σ with its restriction to Gn for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now have a conundrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If we choose a proper length function L on G∞, then, since G∞ = � n∈NGn with (Gn)n∈N increasing, any finite subset of G∞ is contained in some GN (and thus in all Gn with n � N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This implies that (Gn)n∈N converges to G∞ for the pointed Gromov-Hausdorff distance for proper metric space, where we always use 1 as our base point, and the metrics are induced by L (see [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' On the other hand, as soon as G∞ is infinite — which is the only interesting case to consider when G∞ is the union of countably many groups, otherwise of course G∞ is just Gn for n large enough — not only the diameter of G∞ is infinite — it can not be a closed ball as these are finite — but the subgroups Gn are not close to G∞ for the Hausdorff distance induced by L in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So, we can define the spectral triples (C ∗ red(Gn,σ),ℓ2(Gn),ML) as before since L is proper, but in general, there is no apparent reason why |||[ML,a]|||ℓ2(G∞) is particularly close to |||[ML,a]|||ℓ2(Gn) for a ∈ C ∗ red(Gn,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' On the other hand, there may be length functions on G∞ for which (Gn)n∈N does converge in the Hausdorff distance for these length functions, but these length functions are not proper whenever G∞ is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We are thus led to build a new type of spectral triples which combine these two apparently opposite situations: one where we do not know how to build a spectral triple using a non-proper length with otherwise good metric properties for our purpose, and one with a proper length function which has bad metric property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The following construction is inspired, but different from [19], where a proper length function is constructed as a sum of a non-proper length function with a p-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The Spectral Triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now define our new spectral triples on a particular type of twisted group C*-algebras, which are the subject of our main third theorem, Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11), and its corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' From now on we assume that G∞ is a discrete group endowed with a 2-cocycle σ with values in T := {z ∈ C : |z| = 1}, and that G∞ is the union of a strictly increasing sequence for inclusion, (Gn), of subgroups of G∞ such that G∞ = � n∈NGn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We also assume that we are given a length function LH on G∞, whose restriction to each Gn is proper for each n ∈ N, and with the property that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2) lim n→∞Haus[LH](G∞,Gn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 33 Z � 1 2 � ⊆ Q 3 2 1 0 1 2 3 (A) The geometry of Z � 1 2 � for |·| in Q LH log2 ◦F 0 1 2 3 3 2 1 0 1 2 3 (B) The geometry of Z � 1 2 � using F FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The geometry of Z � 1 2 � In addition we require that we are given a strictly increasing unbounded function scale : N → [0,∞), together with F : G∞ → [0,∞) such that for all g ∉ G0: F(g) = scale(min{n ∈ N : g ∈ Gn}), while F restricted to G0 satisfies: ∀g ∈ G0 F(g) = F(g −1), ∀g,h ∈ G0 F(gh) � max{F(g),F(h)}, ∀g ∈ G0 F(g) ∈ [0,scale(0)], F(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Clearly, the above assumptions provide us with many length functions on G∞ and Gn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' we will use them in our spectral triples constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' One of our main examples for this section will be the noncommutative solenoids, whose fundamental components are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will give more details on this example later in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let d � 2 and p a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let G∞ = � Z � 1 p ��d , and let Gn = � 1 pn Z �d for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We note that G∞ = � n∈NGn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We can then choose LH to be the restriction of any norm on Rd, and scale : n ∈ N → pn ∈ [0,∞), so that: F : g ∈ G∞ �→ scale � min � n ∈ N : g ∈ � 1 pn Z �d�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, for any function f : Gn → C, we denote by M f the operator of multiplication by f on the subspace: dom � M f � := � ξ ∈ ℓ2(Gn) : (h ∈ Gn �→ f (h)ξ(h)) ∈ ℓ2(Gn) � of ℓ2(Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Of course, M f is bounded by ��f �� C(Gn) if f is bounded, and unbounded otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' nonetheless dom � M f � always contains Cc(Gn) and thus is always dense in ℓ2(Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let E be a finite dimensional Hilbert space with inner product 〈·,·〉E and dimE ∈ 2N\\{0}, and let c be a ∗-representation of the Clifford algebra of C2 on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let γ1 = c ��1 0 �� 34 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER and γ2 = c ��0 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For our purpose, we record that for all j,k ∈ {1,2}: γj γk +γkγj = � 2 if j = k, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' There is no particular reason to restrict ourselves to E = C2, though it is the natural choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In this case, we can choose the usual Weyl matrices: γ1 = �1 0 0 −1 � and γ2 = �0 1 1 0 � as the most natural choice for our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N := N∪{∞}, we identify the Hilbert space ℓ2(Gn,E) of E-valued func- tions over Gn (with inner product 〈ξ,η〉ℓ2(Gn,E) := � g∈Gn 〈ξ(g),η(g)〉E) with ℓ2(Gn)⊗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We then let dom( /Dn) := � ξ ∈ ℓ2(Gn,E) : (LH(g)γ1ξ(g)+F(g)γ2ξ(g))g∈Gn ∈ ℓ2(Gn,E) � and on dom( /Dn), we define the Dirac operator: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3) /Dn := MLH ⊗γ1 + MF ⊗γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now prove that (C ∗(Gn,σ),ℓ2(Gn)⊗E, /Dn), as defined above, are indeed spectral triples, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A first step is the computation of the domain of our Dirac operators of Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' To do so, we will need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Recall that a norm ∥·∥R2 on R2 is monotone when it is increasing with respect to the product order on R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' the most important such norm for our purpose will be the max norm x, y ∈ R2 �→ ��(x, y) �� ∞ = max{|x|,|y|};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' we also note that we will often write elements of Rd as simple d-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' With the notation and assumptions of this section, the following identities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (1) For all g ∈ G∞: F(g −1) = F(g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (2) For all g,h ∈ G∞: F(gh) � max �F(g),F(h) � � F(g)+F(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, if ∥·∥R2 is any monotone norm on R2, then g ∈ G∞ �→ ��(LH(g),F(g)) ��R2 is a proper, unbounded length function over G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let g ∈ G∞, and let n ∈ N be the unique natural number such that F(g) = scale(n), or n = 0 if F(g) < scale(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If n = 0 then F(g) = F(g −1) by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If n > 0, then g ∈ Gn and g ∉ Gp for p < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' therefore, g −1 ∈ Gn and g −1 ∉ Gp if p < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' hence, F(g −1) = scale(n) = F(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, take h ∈ G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Again, let m ∈ N be uniquely defined by F(h) = scale(m) or m = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let k = max{m,n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus g,h ∈ Gk and therefore, gh ∈ Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' First, if g,h,gh ∈ G0, then F(gh) � max{F(g),F(h)} by assumption on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Otherwise, k > 0, and we simply observe that either gh ∈ G0 and then F(gh) � scale(0) < scale(k), or gh ∉ G0, and again F(gh) � scale(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' either way we observe: F(gh) � scale(k) = scale(max{n,m}) = max{scale(n),scale(m)} = max{F(g),F(h)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix a monotone norm ∥·∥R2 on R2 and let L : g ∈ G∞ �−→ ��(LH(g),F(g)) ��R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 35 It is then immediate to check that if g,h ∈ G∞, then, since ∥·∥R2 is monotone: ��(LH(gh),F(gh)) ��R2 � ��(LH(g)+LH(h),F(g)+F(h)) ��R2 � ��(LH(g),F(g)) ��R2 +∥(LH(h),F(h))∥R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover ��(LH(g −1),F(g −1)) ��R2 = ��(LH(g),F(g)) ��R2 for all g ∈ G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Finally, if ��(LH(g),F(g)) ��R2 = 0, then LH(g) = 0, which in turns implies g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' On the other hand, F(1) = 0 and LH(1) = 0, so L(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus as claimed, L is a length function on G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let be more specific in our choice of ∥·∥R2, and fix it to be the usual max norm ∥·∥∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' we then rename our length L∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' so L∞(g) := max �LH(g),F(g) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By definition, the following equality between closed balls hold: � g ∈ G∞ : L(g) � scale(n) � = � g ∈ Gn : LH(g) � scale(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since LH is proper on Gn, this set is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So L is indeed proper on G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By assumption, the function scale is unbounded on N and, for all n ∈ N, there exists g ∈ G∞ \\Gn (since (Gn)n∈N is assumed to be strictly increasing), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' F(g) � scale(n), so L is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now return to a general monotone norm ∥·∥R2 on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since all norms on R2 are equivalent, there exists c > 0 such that 1 c ∥·∥∞ � ∥·∥R2 � c ∥·∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, 1 c L∞ � L � cL∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is now easy to check that L is again proper and unbounded on G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This concludes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is quite natural to simply set F(g) = scale(0) for all g ∈ G0 \\ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The difference between such a choice of F, vs any other F′, which meets our assumptions over G0, is a bounded perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We refer to [36] for a discussion on bounded perturbations of spectral triples from the metric perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As seen in the above discussion, the above length function LH will not be proper, so it won’t define a spectral triple by itself, however L is proper, and thus can be used to define a spectral triple on C ∗(G∞,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' However, we take a slightly different route by working with what we shall prove is an even spectral triple, replacing the linear geometry of G∞ with a sort of “two-dimensional” geometry (see Figure (3) for the noncommutative solenoid case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now prove that in the above hypotheses we can indeed define spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We begin with a computation of the domain of the proposed Dirac operators defined in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' With the notation and assumptions of this section, the following assertion holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' for all ξ ∈ E and for all a,b ∈ R: ��(aγ1 +bγ2)ξ ��2 E = � a2 +b2� ∥ξ∥2 E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular, for all n ∈ N, the domain dom( /Dn) of the Dirac operator /Dn is given by � ξ ∈ ℓ2(Gn,E) : � g∈Gn (LH(g)2 +F(g)2) ��ξ(g) ��2 E < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 36 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let ξ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The following identity holds for all a,b ∈ R: ��aγ1ξ+bγ2ξ ��2 E = a2〈γ1ξ,γ1ξ〉E + ab〈γ1ξ,γ2ξ〉E + ab〈γ2ξ,γ1ξ〉E +b2〈γ2ξ,γ2ξ〉E = a2〈γ2 1ξ,ξ〉E + ab〈(γ1γ2 +γ2γ1)ξ,ξ〉E +b2〈γ2 2ξ,ξ〉E = (a2 +b2)∥ξ∥2 E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The computation of the dom( /Dn), for all n ∈ N, follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ We now prove that our Dirac operators are indeed self-adjoint with compact resolvent, and that they can be used to define spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We also establish some useful estimates which will later allow us to prove that our construction gives metric spectral triples over noncommutative solenoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If a is a bounded operator on ℓ2(G∞), we denote by a◦ the operator a⊗1E acting on ℓ2(G∞,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We also define the representation λ of C ∗(G∞,σ) on ℓ2(G∞,E) by setting λE := λ⊗1E, so for all f ∈ Cc(G∞), we have λ(f )◦ := λE(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover (1) For each n ∈ N, define: dom(Ln) := � a ∈ sa � C ∗ red(Gn,σ) � : a◦dom( /Dn) ⊆ dom( /Dn) and [ /Dn,a◦] is bounded � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (2) For all a ∈ dom(Ln) define: Ln(a) := ������[ /Dn,a◦] ������ ℓ2(Gn,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We conclude this subsection by proving that we indeed defined even spectral triples, and lay the groundwork for our third main theorem in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Recall that, by Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3), LH +F is proper and unbounded on G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' With the notation and assumptions of this section, for any fixed n ∈ N, the ordered triple (C ∗ red(Gn,σ),ℓ2(Gn,E), /Dn) is an even spectral triple, where the grading on ℓ2(Gn,E) is given by 1ℓ2(Gn) ⊗iγ1γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' More- over, for all a ∈ dom(Ln): ������[MLH ,a] ������ ℓ2(Gn) � ������[ /Dn,a◦] ������ ℓ2(Gn,E), together with: |||[MF,a]|||ℓ2(Gn) � ������[ /Dn,a◦] ������ ℓ2(Gn,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular, if we define L := LH +F, then for all n ∈ N and all a ∈ dom(Ln): |||[ML,a]|||ℓ2(Gn) � 2 ������[ /Dn,a◦] ������ ℓ2(Gn,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If, for any n ∈ N, the spectral triple (C ∗ red(Gn,σ),ℓ2(Gn), /DL) is metric, then so is (C ∗ red(Gn,σ), ℓ2(Gn,E), /Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will start by showing that, for any fixed n ∈ N, /Dn is self-adjoint with compact resolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fixed any n ∈ N, note that the domain of /Dn contains all finitely supported functions in ℓ2(Gn,E) and is therefore dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, since γ1 and γ2 are self-adjoint, 37 if ξ,η ∈ dom( /Dn), it follows that: 〈 /Dnξ,η〉ℓ2(Gn,E) = � g∈Gn 〈 �LH(g)γ1 +F(g)γ2 � ξ,η〉E = � g∈Gn 〈ξ, �LH(g)γ1 +F(g)γ2 � η〉E = 〈ξ, /Dnη〉ℓ2(Gn,E), so /Dn is also a symmetric operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By using Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='5), we now note that: dom � /D2 n � = � ξ ∈ ℓ2(Gn,E) : � g∈Gn �LH(g)2 +F(g)2�2 ��ξ(g) ��2 E < ∞ � and, over dom � /D2 n � , the Clifford algebra relations imply: /D2 n +1 = � M2 LH + M2 F +1 � ⊗1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now define an operator K on ℓ2(Gn,E) by setting, for all ξ ∈ ℓ2(Gn,E): K ξ : g ∈ Gn �→ 1 � LH(g)2 +F(g)2 +1 ξ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, K is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, if n ∈ N, then LH restricted to Gn is proper and F is bounded over Gn by our hypotheses, so K is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If n = ∞, by our hypotheses, for all r � 0, the set {g ∈ G∞ : F(g) � r} is a subset of Gk for some k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since LH is proper on Gk, the set {g ∈ G∞ : L2 H(g)+F2(g) � r} is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, the eigenspaces of K are all finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It follows easily that K is compact, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In any case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', for all n ∈ N, ( /D2 n +1)K 2ξ = ξ for all ξ ∈ ℓ2(Gn,E), while K 2( /D2 n +1)ξ = ξ for all ξ ∈ dom � /D2 n � , as seen by a direct computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' in particular, we note that K ℓ2(Gn,E) = dom( /Dn) by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='5), for all ξ ∈ ℓ2(Gn,E), we obtain: � g∈Gn �� /DnK ξ(g) ��2 E = � g∈Gn ����� LH(g) � LH(g)2 +F(g)2 +1 (γ1ξ(g))+ F(g) � LH(g)2 +F(g)2 +1 (γ2ξ(g)) ����� 2 E = � g∈Gn LH(g)2 +F(g)2 LH(g)2 +F(g)2 +1 ��ξ(g) ��2 E � ∥ξ∥2 ℓ2(Gn,E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, /DnK is bounded, of norm at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Consequently, ( /Dn ± i)K is also bounded on ℓ2(Gn,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, ( /D ±i)K 2 is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It follows that /D ±i both have compact inverse ( /D ∓i)K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Specifically for our purpose, if ξ ∈ ℓ2(Gn,E), then: ( /Dn +i) � ( /Dn −i)K 2� ξ = ( /D2 n +1)K 2ξ = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, the range of /Dn +i is ℓ2(Gn,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Similarly, the range of /Dn −i is also ℓ2(Gn,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As /Dn is also a symmetric operator defined on a dense domain, we conclude by [53, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2] and [45, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='48] that /Dn is indeed self-adjoint, with compact resolvent (since the inverse of /Dn +i is the compact ( /Dn −i)K 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will now verify the commutator spectral triples condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Note that if g ∈ Gn, then ������[ /Dn,λE(g)] ������ ℓ2(Gn,E) � LH(g)+F(g) = L(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 38 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER Therefore, if f ∈ Cc(Gn), then the operator [ /Dn,λE(f )] is bounded, and in fact, ������[ /Dn,λE(f )] ������ ℓ2(Gn,E) � � g∈Gn |f (g)|L(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We conclude that (C ∗ red(G∞,σ),ℓ2(G∞), /D) is a spectral triple for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will now prove that our spectral triple is metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let a ∈ dom(Ln) for some n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We then note that, (1⊗γ1)[ /Dn,a◦]+[ /Dn,a◦](1⊗γ1) = [MLH ,a]⊗2, which implies: ������[MLH ,a] ������ ℓ2(Gn) � 1 2 ������(1⊗γ1)[ /Dn,a◦]+[ /Dn,a◦](1⊗γ1) ������ ℓ2(Gn,E) � 1 2 �������(1⊗γ1)[ /Dn,a◦] ������ ℓ2(Gn,E) + ������[ /Dn,a◦](1⊗γ1) ������ ℓ2(Gn,E) � � 1 2 �������[ /Dn,a◦] ������ ℓ2(Gn,E) + ������[ /Dn,a◦] ������ ℓ2(Gn,E) � = ������[ /Dn,a◦] ������ ℓ2(Gn,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The same reasoning, with 1⊗γ2 in place of 1⊗γ1, leads to |||[MF,a]|||ℓ2(Gn) � ������[ /Dn,a◦] ������ ℓ2(Gn,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, for all a ∈ dom(Ln), we obtain: |||[ML,a]|||ℓ2(Gn) � ������[MLF ,a] ������ ℓ2(Gn) +|||[MF,a]|||ℓ2(Gn) � 2 ������[ /Dn,a◦] ������ ℓ2(Gn,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular, if (C ∗ red(Gn,σ),ℓ2(Gn), /DL) is a metric spectral triple, then, by [55, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='10], so is (C ∗ red(Gn,σ),ℓ2(Gn), /Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Finally, we will show that our spectral triples are in fact even, with grading given by 1ℓ2(Gn) ⊗ γ where γ := iγ1γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, γ2 is the identity, and γ∗ = γ, so γ is a self-adjoint unitary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' therefore so is 1ℓ2(Gn) ⊗γ, which splits ℓ2(Gn,E) in its two spectral subspaces for 1 and −1, in such a way that λE commutes with 1⊗γ, while /Dn(1⊗γ) = −(1⊗γ) /Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So (C ∗ red(Gn,σ),ℓ2(Gn,E), /Dn) is an even spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' With the notation of Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='7), we note that for each finite n ∈ N, the spectral triple (C ∗ red(Gn),ℓ2(Gn,E), /Dn) is, in some sense, a bounded perturbation of the odd spectral triple (C ∗ red(Gn),ℓ2(Gn),ML), since F is bounded on Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The situation is quite different when n = ∞, of course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Suppose ρ is some other 2-cocycle of G∞, which is equivalent to σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', for some function f : G∞ → T, the following holds: ∀g,h ∈ G∞ ρ(g,h) = f (g)f (h)f (gh)σ(g,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The operator M f is then a unitary which intertwines the left regular σ and ρ projective representation of G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, (AdM f )◦ implements a *-isomorphism from λE(C ∗(G∞,ρ)) onto λE(C ∗(G∞,σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Furthermore, M◦ f commutes with /D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, the spectral triples (C ∗(G∞,σ),ℓ2(G∞,E), /D∞) and (C ∗(G∞,ρ),ℓ2(G∞,E), /D∞) are unitarily equiva- lent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular, whenever one is metric, so is the other, and then they are at distance zero from each others for the spectral propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We begin this section by making some basic identifications that will be used throughout the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will use the notation introduced in the above sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fixed n ∈ N, the C*-algebra C ∗ red(Gn,σ) is technically the closure, in the operator norm, of the linear span of the operators λn(g) defined on ℓ2(Gn) by λn(g)ξ : h ∈ ℓ2(Gn) �→ σ(g,g −1h)ξ(g −1h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' On the other hand, since Gn ⊆ G∞, we obtain a different unitary σ-projective representation of Gn, via the restriction of the σ-projective representation λ of G∞ to Gn on ℓ2(G∞), giving us an alternative C*-algebra generated by {λ(h) : h ∈ Gn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If S ⊆ G∞ is any nonempty subset of G∞, we identify the space ℓ2(S) with {ξ ∈ ℓ2(G∞) : ∀g ∈ G∞ \\S ξ(g) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let Qn ⊆ G∞ be a subset of G∞ such that every right coset of Gn in G∞ is of the form Gnk for a unique k ∈ Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Of course, ℓ2(G∞) = � k∈Qnℓ2(Gnk), where ⊕ is the Hilbert sum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' the closure of the direct sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, we set, for all k ∈ G∞ and ξ ∈ ℓ2(G∞): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='4) ρ(k)ξ : h ∈ ℓ2(G∞) �→ σ(hk,k−1)ξ(hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus defined, ρ is the right regular ˘σ-projective representation of G∞ on ℓ2(G∞), where ˘σ : g,h ∈ G∞ �→ σ(h−1,g −1) is indeed a 2-cocycle of G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If the 2-cocycle σ is normalized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' σ(g,g −1) = 1 for all g ∈ G∞, then ˘σ = σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' we will however not need to work with normalized cocycles here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since σ is a 2-cocycle, we obtain, for all g,h,k ∈ G∞ and ξ ∈ ℓ2(G∞): λ(g)ρ(k)ξ(h) = σ(g,g −1h)ρ(k)ξ(g −1h) = σ(g,g −1h)σ(g −1hk,k−1)ξ(g −1hk) = σ( g =:x ,g −1hk =:y k−1 =:z )σ(g −1hk =y ,k−1 =z )ξ(g −1hk) = σ( g =x ,g −1hk =y )σ(hk =xy ,k−1 =z )ξ(g −1hk) = σ(hk,k−1)(λ(g)ξ)(hk) = ρ(k)λ(g)ξ(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, λ(g) and ρ(k) commute, for all g,k ∈ G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is moreover immediate that ρ(k−1) maps ℓ2(Gnk) onto ℓ2(Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now define the unitary V from ℓ2(G∞) = � k∈Qnℓ2(Gnk) to � k∈Qnℓ2(Gn) by setting, for all ξ = (ξk)k∈Qn ∈ � k∈Qnℓ2(Gnk): V ξ = � ρ(k−1)ξk � k∈Qn ∈ � k∈Qnℓ2(Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, V is unitary, and moreover, for any g ∈ Gn: V λ(g)V ∗(ξk)k∈Qn = (λn(g)ξk)k∈Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, AdV is a ∗-isomorphism from the C*-subalgebra generated by {λ(g) : g ∈ Gn} and the C*-algebra C ∗ red(Gn,σ) which maps λ(g) to λn(g) for all g ∈ Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' From now on, we thus identify C ∗ red(Gn,σ) with the C*-algebra generated by {λ(g) : g ∈ Gn} in C ∗ red(G∞,σ) and work exclusively in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will also identify ℓ2(Gn,E) with � ξ ∈ ℓ2(G∞,E) : ∀g ∈ G∞ \\Gn ξ(g) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 40 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER To complete our picture, we also identify /Dn with the operator defined for ξ = ξn + ξ⊥ n , with ξn ∈ ℓ2(Gn,E)2 and ξ⊥ n ∈ ℓ2(Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='E) = � k∈Qn\\Gnℓ2(Gnk,E), by /Dnξ = /Dnξn ∈ ℓ2(Gn,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We then observe that if Pn is the orthogonal projection from ℓ2(G∞) onto ℓ2(Gn), we have, for all ξ ∈ dom( /Dn) and for all g ∈ G∞: P◦ n /Dnξ(g) = � (LH(g)⊗γ1 +F(g)⊗γ2)ξ(g) if g ∈ Gn, 0 otherwise, = /D∞P◦ nξ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We thus have shown that P◦ ndom( /Dn) ⊆ dom( /D∞) and P◦ n /Dn = /D∞P◦ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, P◦ n /D∞P◦ n = /Dn and thus, for all a ∈ dom(Ln) we compute the following expression, using the fact that [Pn,a] = 0,: P◦ n[ /D∞,a◦]P◦ n = P◦ n /D∞a◦P◦ n −P◦ na◦ /D∞P◦ n = P◦ n /D∞P◦ na◦ − a◦P◦ n /D∞P◦ n = /Dna◦ − a◦ /Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So we have, for all a ∈ dom(L∞): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='5) Ln(a) = ������[ /Dn,a◦] ������ ℓ2(Gn,E) = ������P◦ n[ /D∞,a◦]P◦ n ������ ℓ2(G∞,E) � L∞(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' With all of the above identifications, we thus have a natural unital ∗-morphism from C ∗ red(Gn,σ) into C ∗ red(G∞,σ) which becomes just the natural inclusion, and λ(g)ℓ2(Gnk) ⊆ ℓ2(Gnk) for each g ∈ Gn and k ∈ G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By linearity and continuity, we conclude that if a ∈ C ∗ red(Gn,σ), then aℓ2(Gnk) ⊆ ℓ2(Gnk) for all k ∈ G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We also note that [ /D∞,a◦]ℓ2(Gnk,E) ⊆ ℓ2(Gnk,E) for all k ∈ G∞ and a ∈ dom(Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will work for the rest of this section with the above identifications and their basic properties without further mention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our main theorem in this section involves, in particular, a strong result about the convergence of some of the quantum compact metric spaces induced by our spectral triples: namely, we obtain some convergence in the sense of the Lipschitz distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The Lipschitz distance LipD, extended to noncommutative metric geometry in [37], is defined between any two quantum compact metric spaces (A,LA) and (B,LB), by LipD((A,LA),(B,LB)) := inf � ln(k) : ∃π : (A,LA) → (B,LB) Lipschitz *-isomorphism with 1 k LA � LB ◦π � kLA � , with the convention that inf� = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus LipD is finite only between quantum compact metric spaces built over isomorphic C*-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As shown in [37], the Lipschitz distance dominates the Gromov-Hausdorff propinquity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' in fact, closed balls for the Lipschitz distance are compact in the propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular, if A is a unital C*-algebra, and if L1 and L2 are two Lipschitz seminorms over A with the same domain, then the identity is bi-Lipschitz, and we do obtain, by definition: LipD((A,L1),(A,L2)) � ln(C) if 1 C L1 � L2 � CL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now prove our main result about inductive limits of discrete groups and the convergence, for the spectral propinquity, of their spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Note that below we use the notation established in Definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 41 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' With the notation and assumptions of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2, if (C ∗ red(Gn,σ),ℓ2(Gn,E), /Dn) is a metric spectral triple for all n ∈ N, and if {a ∈ dom(Ln) : Ln(a) � 1} = cl({a ∈ Cc(Gn) : Ln(a) � 1}), then lim n→∞Λspec � (C ∗ red(Gn,σ),ℓ2(Gn,E), /Dn),(C ∗ red(G∞,σ),ℓ2(G∞,E), /D∞) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, for any fixed k ∈ N, the sequence (C ∗ red(Gk,σ),Ln)n�k converges in the Lips- chitz distance LipD to the quantum compact metric space (C ∗ red(Gk,σ),L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We shall check that the identity automorphism of C ∗ red(G∞,σ) satisfies the hypoth- esis of Theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Obviously, the identity is a full quantum isometry of (C ∗ red(G∞,σ),L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let C = 2qdiam(C ∗(G∞,σ),L∞) — note that since G∞ ̸= {1}, we have C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let tr : a ∈ C ∗ red(G∞,σ) �→ 〈aδ1,δ1〉ℓ2(G∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' tr is a tracial state of C ∗(G∞,σ) which maps a ∈ Cc(G∞) to a(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix ε ∈ � 0, C 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since (C ∗ red(G∞,σ),L∞) is a quantum compact metric space by assump- tion, the set X∞ := {a ∈ dom(L∞) : L∞(a) � 1,tr(a) = 0} is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, there exists a finite ε-dense subset X ε ∞ ⊆ X∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since X∞ = cl({a ∈ Cc(G∞) : L∞(a) � 1,tr(a) = 0}), we can moreover assume that X ε ∞ ⊆ Cc(G∞) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since X ε ∞ is finite and each of its element has finite support, there exists a finite subset S ⊆ G∞ which contains the support of all the elements in X ε ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since G∞ = � n∈NGn and (Gn)n∈N is increasing, there exists N1 ∈ N such that, for all n � N1, we have S ⊆ Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus X ε ∞ ⊆ Cc(Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, by Expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='5), we also obtain Ln(a) � L∞(a) for all a ∈ X ε ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In summary, ∀a ∈ X∞ ∃b ∈ X ε ∞ ⊆ Cc(Gn) ⊆ C ∗ red(Gn,σ) : ∥a −b∥C∗ red(G∞,σ) < ε and Ln(a) � L∞(a) � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If a ∈ dom(L∞), then there exists b ∈ X ε ∞ such that ∥a −tr(a)−b∥C∗ red(G∞,σ) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Of course, b +tr(a) ∈ C ∗ red(Gn,σ) and Ln(b +tr(a)) = Ln(b) � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By homogeneity, it follows that for all a ∈ dom(L∞), and for all n � N1, there exists b ∈ dom(Ln) such that ∥a −b∥C∗ red(G∞,σ) < εL∞(a) and Ln(b) � L∞(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, using our assumption of Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2), there exists N2 ∈ N, with N2 � N1, such that Haus[LH](G∞,Gn) < ε C 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each right coset c of Gn in G∞, let k ∈ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since the distance for LH from k ∈ G∞ to Gn is strictly less than ε C2 , there exists g ∈ Gn such that LH(g −1k) < ε C2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Setting kc = g −1k, we have by definition of right cosets that c = Gnkc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, there exists a subset Qn ⊆ G∞ of G∞ such that, if k ∈ Qn then LH(k) < ε C2 , and if c is a right coset of Gn in G∞, then there exists a unique k ∈ Qn such that c = Gnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let n � N2 and let b ∈ Cc(Gn) ⊆ C ∗ red(Gn,σ) with b(1) = tr(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Note that b ∈ dom(L∞)∩dom(Ln) so, in particular, both Ln(b) and L∞(b) are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We thus have ℓ2(G∞) = ⊕k∈Qnℓ2(Gnk), where ⊕ is the Hilbert sum (the closure of the sum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If h ∈ Gn, then, by definition of a right coset, λ(h)ℓ2(Gnk) ⊆ ℓ2(Gnk) for all k ∈ Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As /D∞ (dom( /Dn)) ⊆ ℓ2(Gnk,E) as well for all k ∈ Qn, we conclude that 42 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER [ /D∞,b◦] � ℓ2(Gnk,E) � ⊆ ℓ2(Gnk,E) — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' b, /D∞ and its commutators are all block di- agonal in this decomposition of ℓ2(G∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It follows that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6) ������[ /D∞,b◦] ������ ℓ2(G∞,E) = sup k∈Qn ������[ /Dn,b◦] ������ ℓ2(Gnk,E), allowing for any of the above norm to be infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, the restriction of /D∞ to dom( /Dn) is exactly /Dn, so: ������[ /D∞,b◦] ������ ℓ2(Gn,E) = ������[ /Dn,b◦] ������ ℓ2(Gn,E) = Ln(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, fix k ∈ Qn and k ∉ Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By assumption, and using repeatedly that (Gp)p∈N is increasing, we observe that F(gk) = F(k) for all g ∈ Gn: Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3) implies that F(gk) � F(k) since F(g) � n < F(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' on the other hand, if p ∈ {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=',m −1} where F(k) = scale(m), noting that m > 0 since k ∉ G0, then gk ∈ Gp implies k = g −1gk ∈ Gn, which is a contradiction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' hence F(kg) = scale(m), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, the operator MF is constant on ℓ2(Gnk), and thus [MF,b] = 0 on ℓ2(Gnk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So, ������[ /D∞,b◦] ������ ℓ2(Gnk,E) = ������[MLH ,b] ������ ℓ2(Gnk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will use the ˘σ-projective right representation of G∞ on ℓ2(G∞), as defined in Expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, the restriction of ρ(k) to ℓ2(Gn) (which we will keep denoting by ρ(k)) is a unitary onto ℓ2(Gnk) (with inverse the restriction to ℓ2(Gnk) of its adjoint ρ(k)∗ = ρ(k−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='7) ������[MLH ,b] ������ ℓ2(Gnk) = ������[MLH ,b]ρ(k) ������ℓ2(Gnk) ℓ2(Gn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Next, a simple computation shows (like with λ) that the unitary ρ(k) maps dom � MLH � to itself, and for all ξ ∈ ℓ2(G∞) and h ∈ G∞: [MLH ,ρ(k)]ξ(h) = (LH(h)−LH(hk))σ(hk,k−1)ξ(hk) so ������[MLH ,ρ(k)]ξ ������ ℓ2(Gn) � suph∈Gn |LH(h)−LH(hk)|∥ξ∥ℓ2(Gn) � LH(k)∥ξ∥ℓ2(Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Choos- ing ξ = δ1, we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='8) ������[MLH ,ρ(k)] ������ ℓ2(Gn) = LH(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, since ρ(k) commutes with λ(g) for all g ∈ G∞, we conclude [b,ρ(k)] = 0, and thus, on dom � MLH � [MLH ,b]ρ(k) = MLH bρ(k)−bMLH ρ(k) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='9) = MLH ρ(k)b [b,ρ(k)]=0 −bρ(k)MLH −b[MLH ,ρ(k)] = [MLH ,ρ(k)]b +ρ(k)MLH b − ρ(k)b [ρ(k),b]=0 MLH −b[MLH ,ρ(k)] = [MLH ,ρ(k)]b +ρ(k)[MLH ,b]−b[MLH ,ρ(k)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 43 Therefore, by Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='7), ������[ /D∞,b◦] ������ ℓ2(Gnk,E) = ������[MLH ,b]ρ(k) ������ℓ2(Gnk) ℓ2(Gn) � ������[MLH ,ρ(k)]b ������ℓ2(Gnk) ℓ2(Gn) + ������ρ(k)[MLH ,b] ������ℓ2(Gnk) ℓ2(Gn) + ������b[MLH ,ρ(k)] ������ℓ2(Gnk) ℓ2(Gn) by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='9) � ������[MLH ,ρ(k)] ������ℓ2(Gnk) ℓ2(Gn) �LH (k) by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='8) ∥b∥C∗(Gn,σ) + ������ρ(k) ������ℓ2(Gnk) ℓ2(Gn) =1 as ρ(k) is unitary ������[MLH ,b] ������ ℓ2(Gn) �Ln(b) by Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='7) +∥b∥C∗(Gn,σ) ������[MLH ,ρ(k)] ������ℓ2(Gnk) ℓ2(Gn) �LH (k) by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='8) � LH(k)∥b∥C∗ red(Gn,σ) � C 2 L∞(b) +Ln(b)+∥b∥C∗ red(Gn,σ)LH(k) � ε C 2 C 2 L∞(b)+Ln(b)+ C 2 L∞(b) ε C 2 � Ln(b)+ ε C L∞(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By Expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6), we thus get L∞(b) � Ln(b)+ ε C L∞(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, we have shown that since ε ∈ � 0, C 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='10) ∀b ∈ Cc(Gn) tr(b) = 0 =⇒ L∞(b) � 1 1− ε C Ln(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let b ∈ Cc(Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We then easily compute: L∞(b) = L∞(b −tr(b)1) � 1 1− ε C Ln(b −tr(b)1) = 1 1− ε C Ln(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let a ∈ dom(Ln) with Ln(a) � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By assumption, there exists a sequence (ak)k∈N converging in C ∗ red(Gn,σ) to a such that Ln(ak) � 1 and ak ∈ Cc(Gn) for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We thus have, by lower semicontinuity of Ln, and Expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='10): L∞(a) � liminf k→∞ L∞(ak) � 1 1− ε C liminf k→∞ Ln(ak) � 1 1− ε C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, we have shown that, for all n � N, if a ∈ dom(Ln), then a ∈ dom(L∞), and more- over, ∀a ∈ dom(Ln) L∞(a) � 1 1− ε C Ln(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is immediate by construction that Ln � L∞ on dom(Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus we have proven that for all n � N and k � n, we have Lk � L∞ � 1 1− ε C Lk(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As a byproduct of this, we have shown that limk→∞LipD((C ∗(Gn,σ),Lk),(C ∗(Gn,σ),L∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now pause to note that, thanks to our identifications discussed prior to this theo- rem, and the observation that dom(Ln) ⊆ dom(L∞) which we have just now established, (C ∗ red(Gn,σ),ℓ2(Gn)⊗E, /Dn)n∈N is an inductive sequence of spectral triples in the sense of [20], where the ∗-morphisms from C ∗ red(Gn,σ) to C ∗ red(Gn+1,σ) and the linear isometry 44 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER from ℓ2(Gn) to ℓ2(Gn+1) are just the inclusion maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover (C ∗ red(G∞,σ),ℓ2(G∞,E), /D∞) is indeed the inductive limit of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now note that since L∞ � � 1 1− ε C � Ln and ε ∈ � 0, C 2 � , we have qdiam � C ∗ red(Gn,σ),Ln � � � 1 1− ε C � qdiam � C ∗ red(G∞,σ),L∞ � = C 2 2(C −ε) � C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let b ∈ dom(Ln), and let a = � 1− ε C � b ∈ dom(L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We then compute: ∥b − a∥C∗ red(G∞,σ) = ���b − � 1− ε C � b ��� C∗ red(G∞,σ) � ε C ∥b∥C∗ red(G∞,σ) � ε C qdiam � C ∗ red(Gn,σ),Ln �Ln(b) � ε C C Ln(b) = εLn(b), while L∞(a) = L∞ �� 1− ε C � b � � 1 1− ε C Ln �� 1− ε C � b � = Ln(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Hence, if n � N2, then: ∀a ∈ dom(L∞) ∃b ∈ dom(Ln) : Ln(b) � L∞(a) and ∥b − a∥C∗ red(G∞,σ) < εL∞(a), ∀b ∈ dom(Ln) ∃a ∈ dom(L∞) : L∞(a) � Ln(b) and ∥a −b∥C∗ red(G∞,σ) < εLn(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, by Theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='17), we conclude that lim n→∞Λspec((C ∗ red(Gn,σ),ℓ2(Gn,E), /Dn),(C ∗ red(G∞,σ),ℓ2(G∞,E), /D∞)) = 0, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ We now wish to apply Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11) to the family in Example (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1), as well as to the Bunce-Deddens algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, we shall now focus on Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' So from now on we assume that G∞ is Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore we will employ the additive notation for the groups Gn (n ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since Abelian groups are amenable, we will also from now on identify C ∗ red(Gn,σ) with C ∗(Gn,σ) for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A key condition for Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11) is always met when working with Abelian groups, as seen in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' With the assumptions and notation of Subsection (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2), for any n ∈ N, if Gn is Abelian, then we have that {a ∈ dom(Ln) : Ln(a) � 1} = cl({a ∈ Cc(Gn) : Ln(a) � 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since Ln is lower semicontinuous, we get cl({a ∈ dom(Ln)∩Cc(Gn) : Ln(a) � 1}) ⊆ {a ∈ dom(Ln) : Ln(a) � 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We now prove that when Gn is Abelian, the converse inclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let � Gn be the Pontryagin dual of Gn (we will use the multiplicative notation for � Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The dual action β of � Gn on C ∗(Gn,σ) is unitarily implemented by defining, for each z ∈ � Gn, the unitary vz of ℓ2(Gn,E) which is given by, for all ξ ∈ ℓ2(Gn)⊗E: vzξ : g ∈ Gn �−→ z(g)ξ(g)(= z(−g)ξ(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 45 It is easily checked that z ∈ � Gn �→ vz is a unitary representation of � Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We then note that: ∀z ∈ � Gn vzλE(g)(vz)∗ = βzλE(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, /Dn commutes with vz for all z ∈ � Gn, so β acts by full quantum isometries on (C ∗(Gn,σ),Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let µ be the Haar probability measure on � Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As seen in [32, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1],[57, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2], there exists a sequence (ϕk)k∈N of non-negative functions over � Gn, each obtained as a linear combination of characters of � Gn (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' of the form z ∈ � Gn �→ z(g) for some g ∈ Gn, by Pontryagin duality), such that � � Gn ϕk dµ = 1 for all k ∈ N, and (ϕk)k∈N converges, in the sense of distributions, to the Dirac measure at 1 ∈ � Gn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', for all f ∈ C(� Gn), lim k→∞ � � Gn f (z)ϕk(z)dµ(z) = f (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We define, for each k ∈ N, the continuous linear endomorphism: βϕk : a ∈ C ∗(Gn,σ) �→ � � Gn βz(a)ϕk(z)dµ(z), acting on C ∗(Gn,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since the dual action is strongly continuous, we conclude that, for all a ∈ C ∗(Gn,σ): lim k→∞ ��βϕk (a)− a �� C∗(Gn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since Ln is lower semicontinuous, ϕk � 0 and � � G∞ ϕk dµ = 1 for all k ∈ N, and β acts by quantum isometries, we also get, for all a ∈ dom(Ln), Ln � βϕk (a) � � � � Gn ϕ(z)Ln(a)dµ(z) = Ln(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As a quick digression, lower semicontinuity also implies that Ln(a) � liminfk→∞Ln(βϕk(a)), so altogether we have shown that Ln(a) = liminfk→∞Ln(βϕk (a))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each k ∈ N, as ϕk is a linear combination of characters of � Gn, there exists a finite subset F ⊆ Gn and a function t : F → C such that ϕk : z ∈ � Gn �→ � g∈F t(g)z(g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' the range of βϕk is then the finite dimensional subspace of Cc(Gn) consisting of the functions supported on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For our purpose, the main observations here are that, given a ∈ dom(Ln), and ε > 0, there exists K ∈ N such that if k � K , then ��a −βϕk (a) �� C∗(Gn,σ) < ε and Ln(βϕk (a)) � Ln(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In particular, again since Ln is lower semi-continuous, it follows that: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11) {a ∈ dom(L /D) : L /D(a) � 1} = cl({a ∈ dom(L /D)∩Cc(Gn) : L /D(a) � 1}), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' With the notation of the proof of Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='12), fix ϕ ∈ S (C ∗(Gn,σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since, for all k ∈ N, we have � � Gn ϕk dµ = 1, we conclude that βϕk is a unital map, and thus sup ���a −βϕk (a) �� C∗(Gn) : a ∈ dom(Ln),Ln(a) � 1 � = sup ���a −βϕk (a) �� C∗(Gn) : a ∈ dom(Ln),Ln(a) � 1,µ(a) = 0 � where the second supremum is indeed finite since X = {a ∈ dom(Ln) : Ln(a) � 1,µ(a) = 0} is compact and we take the supremum of a continuous function over this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In fact, Arzelà-Ascoli theorem can be applied here to prove that the convergence of (βϕk )k∈N to 46 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER the identity on X is uniform, though we here offer a simple ε 3-type argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' First, note that for all a,b ∈ C ∗(G∞), and for all k ∈ N, ��βϕk (a)−βϕk (b) �� C∗(G∞) � � � G∞ ∥a −b∥C∗(G∞) ϕk(z)dµ(z) = ∥a −b∥C∗(G∞) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, for all ε > 0, there exists a finite ε 3-dense subset Xε of X ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' as Xε is finite, there exists K ∈ N such that, for all k � K and for all a ∈ Xε, then ��a −βϕk (a) �� C∗(G∞) < ε 3, as seen above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' therefore for all k � K , we have ��a −βϕk (a) �� C∗(G∞) � ��a − a′�� C∗(G∞) + ��a′ −βϕk (a′) �� C∗(G∞) + ��βϕk (a′ − a) �� C∗(G∞) < ε 3 + ε 3 + ε 3 = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This proves that indeed, (βϕk )k∈N converges uniformly to the identity over X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We will prove that some of the spectral triples introduced in Subsection (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2) are metric by invoking a property central to the work in [9, 50], called bounded doubling, which we now recall in the formulation of [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='14 ([9, 50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A proper length function L on a discrete group G satisfies the bounded doubling property when there exists θ > 1 and c > 0 such that, for all r � 1: ��� g ∈ G : L(g) � θ ·r ��� � c ��� g ∈ G : L(g) � r ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The bounded doubling property indeed ensures the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The spectral triples constructed in Subsection (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='2) are metric if the proper length function L := max{LH,F} has the bounded doubling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We note that Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3) proves that L is a proper unbounded length function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By [9, 50], since all our groups are Abelian hence nilpotent, for any µ ∈ S (C ∗(Gn),σ), the set � a ∈ Cc(Gn) : |||[ML,a]|||ℓ2(Gn) � 1,µ(a) = 0 � is totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since |||[ML,·]|||ℓ2(Gn) � Ln on Cc(Gn), we thus conclude that � a ∈ Cc(Gn) : Ln(a) � 1,µ(a) = 0 � ⊆ � a ∈ Cc(Gn) : |||[ML,a]|||ℓ2(Gn) � 1,µ(a) = 0 � and thus � a ∈ Cc(Gn) : Ln(a) � 1,µ(a) = 0 � is also totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='12), we also have: {a ∈ dom(Ln) : Ln(a) � 1,µ(a) = 0} = cl �� a ∈ Cc(Gn) : Ln(a) � 1,µ(a) = 0 �� so {a ∈ dom(Ln) : Ln(a) � 1,µ(a) = 0} is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus by Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='9), Ln is a Lipschitz seminorm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' our spectral triples are metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ We are now ready to establish the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let G = � n∈NGn be an Abelian discrete group, arising as the union of a strictly increasing sequence (Gn)n∈N of subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let σ be a 2-cocycle of G and LH a length function on G such that lim n→∞Haus[LH](Gn,G) = 0, and whose restriction to Gn is proper for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Assume scale : N → [0,∞) is a strictly increasing, unbounded function such that, if we set F : g ∈ G �−→ scale(min{n ∈ N : g ∈ Gn}) 47 then the proper length function L := max{LH,F} has the bounded doubling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Then, for any Hermitian space E, lim n→∞Λspec((C ∗(G,σ),ℓ2(G)⊗E, /D),(C ∗(Gn,σ),ℓ2(Gn)⊗E, /Dn)) = 0, where /D = MLH ⊗γ1 +MF ⊗γ2 on � ξ ∈ ℓ2(G)⊗E : � g∈G(LH(g)2 +F(g)2) ��ξ(g) ��2 E < ∞ � , with γ1,γ2 unitaries of E such that, for all j,k ∈ {1,2}: γj γk +γkγj = � 2 if j = k, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' ℓ2(Gn)⊗E is identified with the subspace of Gn-supported vectors in ℓ2(G)⊗E, /Dn is the restriction of /D to dom( /D)∩ � ℓ2(Gn)⊗E � , C ∗(G,σ) and C ∗(Gn,σ) act via their left regular σ-projective representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Our theorem follows from Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We first note that Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='15) proves that all our spectral triples are metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='12), since G∞ is Abelian, we conclude that, for all n ∈ N, {a ∈ dom(Ln) : Ln(a) � 1} = cl({a ∈ Cc(Gn) : Ln(a) � 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since all hypotheses of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11) are met, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ In particular, for the noncommutative solenoids of Example (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='1), we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix a prime number p ∈ N and d ∈ N\\{0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For each n ∈ N, let Gn := � 1 pn Z �d and G∞ := � Z � 1 p ��d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix a 2-cocycle σ on G∞ such that ∀g ∈ G∞ σ(g,−g) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let LH be the restriction to G∞ of some norm on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We define F by setting, for all g ∈ G∞: F(g) := min � pn : g ∈ � 1 pn �d� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let E be an even dimensional hermitian space, with γ1,γ2 be two unitaries on E such that, for all j,k ∈ {1,2}: γj γk +γkγj = � 2 if j = k, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If we define, for all n ∈ N, the operator /Dn := MLH ⊗γ1 + MF ⊗γ2 on dom( /Dn) on the domain dom( /Dn) := � ξ ∈ ℓ2(Gn,E) : � g∈Gn (LH(g)2 +F(g)2) ��ξ(g) ��2 E < ∞ � , then, for all n ∈ N, the triple (C ∗(Gn,σ),ℓ2(Gn,E), /Dn) is a metric spectral triple, and: lim n→∞Λspec((C ∗(Gn,σ),ℓ2(Gn,E), /Dn),(C ∗(G∞,σ),ℓ2(G∞,E), /D∞)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 48 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER Moreover, for each n ∈ N, the sequence (C ∗(Gn,σ),Lk)k�n of quantum compact metric spaces converge to (C ∗(Gn,σ),L∞) in the Lipschitz distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We first establish the bounded doubling property of certain related length func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Fix a prime number p and d � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For all g ∈ G∞, let L′(g) = max � ��g ��Rd ,p min � n∈N:g∈ � 1 pn Z �d �� , where the norm we choose on Rd for this proof is the max norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3), the function L′ is an unbounded proper length function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='7), we have that |||[ML′,·]|||ℓ2(Gn) � Ln on C(Gn) for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By [11], the triple (C ∗(Gn,σ),ℓ2(Gn),ML′) is a spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Assume L′(g) � pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since g ∈ � 1 pn Z �d , we can write g = � aj pn � 1�j�d for a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=',ad ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since ��g ��Rd � pn, we also have a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=',ad ∈ [−p2n,p2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Conversely, if g = � aj pn � 1�j�d with −p2n � a j � p2n for all j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=',d}, then L′(g) � pn by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Hence, the closed ball of center (0,0) and radius pn has cardinal (2p2n +1)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Consequently: ��� g ∈ G∞ : L′(g) � pn+1��� = (2p2n+2 +1)d � (2p2n+2 + p2)d = p2d(2p2n +1)d � p2d ��� g ∈ G∞ : L′(g) � pn���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, L′ is a proper unbounded length with the bounded doubling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let LH be any norm on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since all the norms on Rd are equivalent, there exists C > 0 such that 1 C LH � ∥·∥Rd � CLH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Then 1 C (max{LH,F}) � L′ � C max{LH,F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, ��� g ∈ G∞ : max �LH(g),F(g) � � pn+1��� � C 2p2d ��� g ∈ G∞ : max �LH(g),F(g) � � pn���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Write L := max{LH,F} on Cc(Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We thus have shown that L, which is unbounded and proper by Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3), also has the bounded doubling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='12), since G∞ is Abelian, we conclude that ∀n ∈ N {a ∈ dom(Ln) : Ln(a) � 1} = cl({a ∈ Cc(Gn) : Ln(a) � 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, our corollary follows from Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ We can choose somewhat different length functions over � Z � 1 p ��d , by varying not only LH, but also F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For instance, Corollary (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='17) remains valid if we replace F by F′ : (g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=',gd) ∈ G∞ �→ maxd j=1 |g j |p, where |·|p is now the p-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The resulting length function max{LH,F′} has the bounded doubling property, as seen by applying [19, Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='17] up to an equivalence of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We also note that for this construction to give us something different from Corollary (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='17), we require that LH(g) < F′(g) for at least one g ∈ Zd \\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In general, the difference is only up to a bounded perturbation of the underlying Dirac operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 49 Another interesting family of C*-algebras to which our work applies are certain Bunce- Deddens algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let P be the set of all sequences (αn)n∈N of nonzero natural numbers such that αn+1 αn is a prime number for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For any integer m ∈ Z, we denote the quotient group Z⧸mZ simply by Z⧸m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let α := (αn)n∈N ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If n ∈ N, then αn divides αn+1, and thus the map ρn : (m mod αn+1) ∈ Z⧸αn+1 → (m mod αn) ∈ Z⧸αn where x mod y is the equivalence class of x ∈ Z modulo y ∈ Z \\ {0}, is a well-defined surjective group morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The projective limit of the projective sequence Z⧸α0 ρ0 ←−−− Z⧸α1 ρ1 ←−−− Z⧸α2 ρ2 ←−−− Z⧸α3 ρ4 ←−−− ··· is denoted by Z⧸α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By construction, we observe that: Z⧸α = � (zn)n∈N ∈ ∞ � j=0 Z⧸αn : ρn(zn+1) = zn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We endow Z⧸α with its topology as a projective space of compact spaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' with the topology induced by the product topology on �∞ j=0Z⧸αn, which is compact by Tychonoff theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We identify, for any m ∈ N\\{0}, the Pontryagin dual � Z⧸m of Z⧸m with the subgroup of T of m-th roots of unity in the obvious manner — while of course, Z⧸m is self-dual, this identification will be helpful to our presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The Pontryagin dual Z(α) := � Z⧸α of Z⧸α is thus, by contravariant functoriality, the limit of the inductive sequence: � Z⧸α0 j0 −−−→ � Z⧸α1 j1 −−−→ � Z⧸α2 j2 −−−→ � Z⧸α3 j3 −−−→ ··· where j1,j2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', are simply the injection maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Of course, by construction: Z(α) = � ζ ∈ T : ∃n ∈ N ζαn = 1 � , where T = {u ∈ C : |u| = 1} is the circle group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' moreover Z(α) is a discrete group as the dual of a compact group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' we do not endow it with the topology inherited as a subset of T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The Pontryagin duality pairing between Z(α) and its dual Z⧸α is given for all ζ ∈ Z(α) and for all z := (zn)n∈N ∈ Z⧸α by ζz := limn→∞ ζzn, noting that the sequence (ζzn)n∈N is eventually constant, by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In the special case when α = (p,p2,p3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='), the group Z(α) is the Prüfer group Z(p∞) and the group Z⧸α is the group Zp of p-adic integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let α := (αn)n∈N ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let LH be a length function over the circle group T restricted to Z(α) such that limn→∞Haus[LH] � � Z⧸αn ,Z(α) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For all ζ ∈ Z(α), we define F(ζ) := min � p ∈ N : ζp = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let ∥·∥R2 be any monotone norm on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The function L : ζ ∈ Z(α) �→ ∥(LH(ζ),F(ζ))∥R2 50 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER is a proper unbounded length function over Z(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Moreover, L has the bounded doubling property if, and only if, the sequence � αn+1 αn � n∈N is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' First, it is easy to see that, for all ζ ∈ Z(α)\\{1}, F(ζ) = p min � n∈N:ζ∈� Z⧸αn � , while F(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, by Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3), we already know that L is a proper unbounded length function on Z(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For now, let us assume ∥·∥R2 is the max norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For any ρ > 0, we write B[ρ] the cardinality of the closed ball centered at (1,0) ∈ Z(α)×Z of radius ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For any d ∈ N, we compute the following expression: B [αd] = |{ζ ∈ Z(α) : L(ζ) � αd}| = ���� � ζ ∈ � Z⧸αd ����� = αd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let R � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Then, there exists d ∈ N such that αd � R � αd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We note that since B[R] � B[αd+1] < ∞, our length function L is indeed proper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' we also note that since B[R] � B[αd] = αd � 2d, the length function L is also unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, assume that M := supn∈N αn+1 αn < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We then compute: B[2R] � B[2αd+1] � B[αd+2] = αd+2 = αd+2 αd+1 αd+1 αd αd � M2αd = M2B[αd] � M2B[R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, our length L has the bounding doubling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, if we allow for a different choice of monotone norm for ∥·∥R2, then, as all norms on R2 are equivalent, the resulting length function still has the property of bounded doubling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, assume instead that supn∈N αn+1 αn = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let n ∈ N, and let rn = αn+1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We then note, using our above computation, that B[2rn] = αn+1 = αn+1 αn B[rn], and thus αn+1 αn = B[2rn] B[rn] for all n ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' therefore, our length L does not actually have the bounded doubling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let α = (αn)n∈N be a sequence of nonzero natural numbers such that � αn+1 αn � n∈N is a bounded sequence of prime numbers, and let Z(α) := � ζ ∈ C : ∃n ∈ N ζαn = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Define: G∞ := Z(α)×Z and ∀n ∈ N Gn := � Z⧸αn ×Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Gn = {(ζ,z) ∈ G∞ : z ∈ Z,ζαn = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let σ be a 2-cocycle of G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let LZ be the restriction of any continuous length function on T to Z(α), and define LH : (u,z) ∈ G∞ �→ LZ (u)+|z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' For all ζ ∈ Z(α), set: F(ζ) := min{n ∈ N : un = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Let E be a Hermitian vector space, and let γ1,γ2 be unitaries such that γ1γ2 = −γ2γ1 and γ2 1 = γ2 2 = 1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If we set, for all n ∈ N, /Dn := MLH ⊗γ1 + MF ⊗γ2, 51 then for all n ∈ N, the spectral triple (C ∗(Gn,σ),ℓ2(Gn)⊗E, /Dn) is metric, and lim n→∞Λspec �� C ∗(Gn,σ),ℓ2(Gn)⊗E, /Dn � , � C ∗(Z(α)×Z,σ),ℓ2(Z(α)×Z)⊗E, /D∞ �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A straightforward computation shows that |·| is proper with the bounded doubling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By [19, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='7] applied to the proper unbounded lengths | · | and LZ , we conclude that L := (ζ,z) ∈ G∞ �→ LZ (ζ)+F(ζ)+|m| has the bounded doubling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since LZ is continuous on T, it induces the usual topology on T (as a subset of C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Therefore, the topology of the Hausdorff distance Haus[LH] is the Vietoris topology for the usual topology of T, and thus the same as the topology induced by Haus[T], when T is endowed with the restriction of the usual metric on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It then follows that: lim n→∞Haus[LH] � � Z⧸αn,Z(α) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' As all the other assumptions are now met, we conclude that our corollary holds, by Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' □ The map ϖ : z ∈ Z �→ (z mod αn)n∈N ∈ Z⧸α is an injective *-morphism of group with dense range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, we define the following automorphism of Z(α): τ : u ∈ Z(α) �→ u +ϖ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' The C*-crossed-product C(Z(α))⋊τ Z is the Bunce-Deddens algebra associated to the “supernatural” number n := � p|{n∈N: αn+1 αn =p}|� p prime .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' It is also *-isomorphic to C ∗(Z(α) × Z,σ), as defined above, when σ is the 2-cocycle defined by setting, for all (ζ,z),(η, y) ∈ G∞: σ((ζ,z),(η, y)) := ηz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Indeed, this isomorphism can be obtained by using [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We begin with the observa- tion that Bunce-Deddens algebras [8] are C*-crossed products [54, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Now, let us briefly explain the construction of this isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Since the natural inclusion j : Z(α) → T is a character of Z(α), it is given by the pairing with an element in Z⧸α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' this element is precisely our ϖ(1) defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' In our case, we note that λ(1,1)λζ,0λ∗ (1,1) = ζ−1λζ,0 for all ζ ∈ Z(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' If f ∈ Cc �Z⧸α � , we denote its Fourier transform by �f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' specifically �f : ζ ∈ Z(α) �→ � z∈Z⧸α f (z)ζ−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A straightforward computation shows that � τ(f )(ζ) = ζ−1 �f (ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, we conclude that λ(1,1)λ( �f )λ∗ (1,1) = λ � � τ(f ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A similar computation invoking the inverse Fourier transform can be done by using the canonical generators of the C*-crossed product C �Z⧸α � ⋊τ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' By universality of the C*-crossed-product and the twisted group C*-algebra (here, since our groups are Abelian, these algebras agree with their image by their left regular representations), we conclude the description of our isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, we have constructed metric spectral triples over Bunce-Deddens algebra for bounded supernatural numbers, and these triples are limits of sequences of metric spectral triples for the spectral propinquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 52 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER In particular, C ∗(Z(α)×Z,σ) is seen to be the inductive limit and the limit for the propinquity, with the quantum metrics described here, of the C*-algebras C ∗ � � Z⧸αn ×Z,σ � as n ∈ N approaches ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Notably, C ∗ � � Z⧸αn ×Z,σ � is actually *-isomorphic to the C*-algebra of continuous sections of a vector bundle over the circle T with fibers the algebras of square αn-matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' This situation is of course reminiscent of the fact that in particular, Bunce-Deddens algebras are AT algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' However, starting from the usual description of Bunce-Deddens algebras as AT algebras led to difficulties in [6], where the quantum metrics on the Bunce-Deddens algebra do not arise from a spectral triple, and the convergence is only proven in the sense of Rieffel’s quantum Gromov- Hausdorff distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Thus, for Bunce-Deddens algebras associated with supernatural numbers consisting of only finitely many prime numbers, we have now constructed metric spectral triples which actually capture their inductive limit structure within our geometric framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' We hope that Theorems (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='16) will prove useful in constructing other examples of metric spectral triples over twisted group C*-algebras for interesting inductive limits of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' REFERENCES [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Aiello, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Guido, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Isola, Spectral triples for noncommutative solenoidal spaces from self-coverings, 448 (2017), 1378–1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Austad and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Luef, Modulation spaces as a smooth structure in noncommutative geometry, Banach J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 15 (2021) 15–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Aguilar and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Kaad, The Podle´s sphere as a spectral metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 133 (2018), 260–278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Aguilar, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Latrèmolière, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Rainone, Bunce-Deddens algebras as quantum Gromov-Hausdorff distance limits of circle algebras, Integral Equations Operator Theory 94 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 2, 42 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [5] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Aguilar and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Latrémolière, Quantum ultrametrics on AF algebras and the Gromov-Hausdorff propin- quity, Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 231 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 2, 149–194, ArXiv: 1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='07114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [6] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Aguilar, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Latrémolière, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Rainone, Bunce-Deddens algebras as quantum Gromov-Hausorff distance limits of circle algebras, Integral Equations Operator Theory 94 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, Paper no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 2, 42pp, ArXiv: 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='07676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Antonescu and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Christensen, Spectral triples for af C∗-algebras and metrics on the Cantor set, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Theory 56 (2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, 17–46, arXiv: 0309044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Bunce and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Deddens, A family of simple C∗-algebras related to weighted shift operators, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Functional Analysis 19 (1975), 13–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Christ and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Rieffel, Nilpotent group C∗-algebras-algebras as compact quantum metric spaces, Canad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 60 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, 77–94, arXiv: 1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='00980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Christensen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Ivan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Lapidus, Dirac operators and spectral triples for some fractal sets built on curves, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 217 (2008), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, 42–78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Connes, Compact metric spaces, Fredholm modules and hyperfiniteness, Ergodic Theory Dynam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Systems 9 (1989), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 2, 207–220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [12] , Noncommutative geometry, Academic Press, San Diego, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [13] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Dabrowski, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Landi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Sitarz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' van Suijlekom, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Varilly, The Dirac operator on SUq(2), Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 259 (2005), 729–759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [14] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Davidson, C∗−Algebras by example, Fields Institute Monographs, American Mathematical Society, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Da¸browski and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Sitarz, Curved noncommutative torus and Gauß-Bonnet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 013518 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [16] , An asymmetric noncommutative torus, SIGMA 11 (2015), 11 pages, arXiv: 1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='4645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [17] David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Edwards, The structure of superspace, Studies in topology (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' North Carolina, Charlotte, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' dedicated to Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Polish Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' ), 1975, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 121–133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' MR 0401069 [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Exel, The Bunce-Deddens algebras as crossed products by partial automorphisms, Bol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Brasil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=') 25 (1994), 173–179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Farsi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Basa Landry, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Larsen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Packer, Spectral triples for noncommutative solenoids and a Wiener’s lemma, (2022), ArXiv: 2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='07470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 53 [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Floricel and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Ghorbanpour, On inductive limit spectral triples, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 147 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 8, 3611–3619, ArXiv: 1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='09621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [21] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Gabriel and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Grensing, Ergodic actions and spectral triples, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Theory 76 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 2, 307–334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Gromov, Groups of polynomial growth and expanding maps, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Hautes Études Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 53 (1981), 53–78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [23] , Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, BirkhÀuser, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Hawkins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Skalski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' White, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Zacharias, On spectral triples on crossed products arising from equicontinuous actions, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Scand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 113 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 2, 262–291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [25] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Herron, Gromov-Hausdorff distance for pointed metric spaces, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 24 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, 1–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [26] Antoine Julien and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Putnam, Spectral triples for subshifts, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 270 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 3, 1031–1063, arXiv: 1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [27] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Kantorovich, On one effective method of solving certain classes of extremal problems, Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Nauk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' USSR 28 (1940), 212–215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [28] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Kantorovich and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Rubinstein, On the space of completely additive functions, Vestnik Leningrad Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Mekh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' i Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 13 (1958), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 7, 52–59, In Russian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [29] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Landry, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Lapidus, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Latrémolière, Metric approximations of the spectral triple on the Sierpinki gasket and other fractals, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 385 (2021), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 107771, 43 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Lapidus and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Sarhad, Dirac operators and geodesic metric on the harmonic Sierpi´nskigasket and other fractal sets, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Noncommut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 8 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 4, 947–985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [31] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Latrémolière, Continuity of the spectrum of Dirac operators of spectral triples for the spectral propinquity, Submitted (2022), 49 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [32] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Latrémolière, Approximation of the quantum tori by finite quantum tori for the quantum Gromov- Hausdorff distance, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 223 (2005), 365–395, math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='OA/0310214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [33] , Convergence of fuzzy tori and quantum tori for the quantum Gromov-Hausdorff propinquity: an explicit approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', Münster J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 8 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, 57–98, arXiv: math/1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='0069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [34] , Curved noncommutative tori as Leibniz compact quantum metric spaces, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 56 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 12, 123503, 16 pages, arXiv: 1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='08771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [35] , The dual Gromov-Hausdorff propinquity, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Pures Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 103 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 2, 303–351, arXiv: 1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='0104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [36] , Quantum metric spaces and the Gromov-Hausdorff propinquity, Noncommutative geometry and optimal transport, Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 676, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=', 2015, arXiv: 150604341, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 47–133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [37] , Equivalence of quantum metrics with a common domain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 443 (2016), 1179– 1195, arXiv: 1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='00755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [38] , The quantum Gromov-Hausdorff propinquity, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 368 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, 365–411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [39] , A compactness theorem for the dual Gromov-Hausdorff propinquity, Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 66 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 5, 1707–1753, arXiv: 1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='06121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [40] , The triangle inequality and the dual Gromov-Hausdorff propinquity, Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 66 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, 297–313, arXiv: 1404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [41] , Convergence of Cauchy sequences for the covariant Gromov-Hausdorff propinquity, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 469 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, 378–404, arXiv: 1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='04721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [42] , The modular Gromov-Hausdorff propinquity, Dissertationes Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 544 (2019), 1–70, arXiv: 1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='04881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [43] , Convergence of Heisenberg modules for the modular Gromov-Hausdorff propinquity, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Operator Theory 84 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, 211–237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [44] , The covariant Gromov-Hausdorff propinquity, Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 251 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 2, 135–169, arXiv: 1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='11229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [45] , Convergence of spectral triples on fuzzy tori to spectral triples on quantum tori, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 388 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 2, 1049–1128, arXiv: 2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='03729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [46] , The dual-modular Gromov-Hausdorff propinquity and completeness, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Noncomm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Geometry 115 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, 347–398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [47] , The Gromov-Hausdorff propinquity for metric spectral triples, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 404 (2022), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 108393, 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [48] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Latrémolière and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Packer, Noncommutative solenoids and the Gromov-Hausdorff propinquity, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 145 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 5, 1179–1195, arXiv: 1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='02707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [49] , Noncommutative solenoids, New York J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 24A (2018), 155–191, arXiv: 1110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='6227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [50] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Long and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Wu, Twisted bounded-dilation group C∗-algebras as C∗-metric algebras, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' China Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 64 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 3, 547–572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 54 CARLA FARSI, FRÉDÉRIC LATRÉMOLIÈRE, AND JUDITH PACKER [51] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Ozawa and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Rieffel, Hyperbolic group C∗-algebras and free product C∗-algebras as compact quantum metric spaces, Canad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 57 (2005), 1056–1079, arXiv: math/0302310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [52] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Packer, Twisted group C∗-algebras corresponding to nilpotent discrete groups, Mathematica Scandinavica 64 (1989), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 1, 109–122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [53] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Reed and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Simon, Functional analysis, Methods of modern Mathematical Physics, Academic Press, San Diego, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [54] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Riedel, Classification of the C∗-algebras associated with minimal rotations, Pacific J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 101 (1982), 153–161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [55] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Rieffel, Metrics on states from actions of compact groups, Doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 3 (1998), 215–229, math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='OA/9807084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [56] , Metrics on state spaces, Doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 4 (1999), 559–600, math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='OA/9906151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [57] , Gromov-Hausdorff distance for quantum metric spaces, Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 168 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' 796, 1–65, math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='OA/0011063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' [58] , Leibniz seminorms for "matrix algebras converge to the sphere", Clay Mathematics Proceedings 11 (2010), 543–578, arXiv: 0707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='3229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content=' Email address: carla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='farsi@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='edu DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COLORADO AT BOULDER, BOULDER CO 80309-0395 Email address: frederic@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='edu URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='edu/~frederic DEPARTMENT OF MATHEMATICS, UNIVERSITY OF DENVER, DENVER CO 80208 Email address: judith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='jesudason@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfb_eW/content/2301.00274v1.pdf'} +page_content='edu DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COLORADO AT BOULDER, 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file mode 100644 index 0000000000000000000000000000000000000000..71049a6bcc2735d4fb9e905e18a758f28d38d23f --- /dev/null +++ b/49AyT4oBgHgl3EQfQPZI/content/tmp_files/2301.00040v1.pdf.txt @@ -0,0 +1,4021 @@ +Sensitivity Analysis with the R2-calculus +Tobias Freidling and Qingyuan Zhao +Statistical Laboratory, DPMMS, University of Cambridge, United Kingdom. +E-mail: taf40@cam.ac.uk +Summary. Causal inference necessarily relies upon untestable assumptions; hence, +it is crucial to assess the robustness of obtained results to violations of identification +assumptions. However, such sensitivity analysis is only occasionally undertaken in +practice, as many existing methods only apply to relatively simple models and their +results are often difficult to interpret. We take a more flexible approach to sensitivity +analysis and view it as a constrained stochastic optimization problem. +We focus +on linear models with an unmeasured confounder and a potential instrument. We +show how the R2-calculus—a set of algebraic rules that relates different (partial) R2- +values and correlations—can be applied to identify the bias of the k-class estimators +and construct sensitivity models flexibly. We further show that the heuristic “plug-in” +sensitivity interval may not have any confidence guarantees; instead, we propose a +boostrap approach to construct sensitivity intervals which perform well in numerical +simulations. We illustrate the proposed methods with a real study on the causal effect +of education on earnings and provide user-friendly visualization tools. +Keywords: Causal inference; Instrumental variables; k-class estimator; Linear +models; Partial identification; Stochastic optimization +1. +Introduction +In many scientific disciplines, provisional causal knowledge is predominantly gen- +erated from observational data as randomized controlled experiments are often in- +feasible or too costly. Because the treatment is not randomly assigned in an obser- +vational study, any causal conclusions must rely on untestable assumptions, such +as absence of unmeasured confounders or validity of instrumental variables. Hence, +the causal inference is inherently sensitive to violations of any identification and +modelling assumptions, so reseachers are advised to investigate the robustness of +their results. +The importance of sensitivity analysis has been emphasized in guidelines for +designing and reporting observational studies (Vandenbroucke et al., 2007; PCORI +Methodology Committee, 2021). +For instance, the STROBE guidelines caution +that “taking [observed] confounders into account is crucial in observational studies, +but readers should not assume that analyses adjusted for [observed] confounders +establish the ‘causal part’ of an association” (p. 1638). They recommend to conduct +sensitivity analyses as they are “helpful to investigate the influence of choices made +in the statistical analysis, or to investigate the robustness of the findings to missing +data or possible biases” (p. 1647). +arXiv:2301.00040v1 [stat.ME] 30 Dec 2022 + +Sensitivity Analysis with the R2-calculus +2 +However, sensitivity analysis is still rarely being conducted in actual studies, +leaving other researchers difficult to assess the robustness of their empirical findings. +In medicine, Thabane et al. (2013) did a spot check on the January 2012 editions of +major medical journals and found that only 26.7% (36 out of 135) of the articles that +included some statistical analysis also performed sensitivity analysis. In nutrition +research, de Souza et al. (2016) found that, in a representative sample of 100 articles +from 2013 to 2015, merely 18% of them conducted some sensitivity analysis. In +political science, Cinelli and Hazlett (2020) found that only 4 out of 64 observational +studies published in three leading journals in 2017 conducted a formal sensitivity +analysis beyond just some model specification checks. +There are several reasons for the hesitant uptake of sensitivity analysis in prac- +tice. First, it is not straightforward to define a reasonable model for sensitivity +analysis, even for the familiar setting of one treatment variable, one outcome vari- +able, and multiple baseline covariates that has been studied since the seminal work +of Cornfield et al. (1959). For example, Lin et al. (1998) assume an unmeasured +confounder U independent of the measured covariates X conditional on the treat- +ment. However, Hernan and Robins (1999) point out that this cannot be generally +true as conditioning on the treatment opens a collider path between U and X. For +more complicated settings such as instrumental variables (IV), specifying a good +sensitivity model is even more difficult and the literature on sensitivity analysis is +considerably smaller. Second, many methods for sensitivity analysis were devel- +oped under simple settings where closed-form solutions are available. This results +in a limited scope of applicability. Finally, it is often not easy for practitioners to +understand and communicate the results of a sensitivity analysis. +In general, (non-Bayesian) sensitivity analysis can be broadly categorized into +point identified and partially identified approaches. The former requires a precise +specification of the confounding mechanism, so that the causal effect of interest is +still identified; see for instance Rosenbaum and Rubin (1983), Imbens (2003), and +VanderWeele and Arah (2011) for the usual observational study design, Scharfstein +et al. (1999) for longitudinal studies with dropouts, and Altonji et al. (2005) for +instrumental variables. On the other hand, the partially identified approach con- +siders the union of many point identified sensitivity models, so the causal effect is +only partially identified. Examples include the first sensitivity analysis by Corn- +field et al. (1959), the approach developed by Rosenbaum (1987, 2002) based on +randomization tests, the E-value proposed by Ding and VanderWeele (2016) that +generalizes the Cornfield bound, the generalization of Scharfstein et al. (1999) by +Vansteelandt et al. (2006), bounds on the average treatment effect under Rosen- +baum’s sensitivity model by Yadlowsky et al. (2022) and the marginal sensitivity +model studied in Zhao et al. (2019) and Dorn and Guo (2022). +In our experience, the partially identified approach is more flexible and usually +aligns with practical demand better. This is why we adopt it in this article. We +limit our discussion to linear regression and linear instrumental variable models, +but the methodology we develope below is quite general and can potentially be ex- +tended to other models. Compared with previous work, a crucial distinction is that +we do not require the partially identified region (or, as in Rosenbaum’s sensitivity + +Sensitivity Analysis with the R2-calculus +3 +analysis, an upper bound of the randomization p-value) to have a closed form so- +lution. Instead, we leverage a novel perspective on sensitivity analysis through the +lens of constrained stochastic optimization. This is elaborated next. +1.1. +A General Framework for Sensitivity Analysis +Consider an i.i.d. sample (Vi, Ui)n +i=1 from some population, but only the vari- +ables (Vi)n +i=1 are observed. +Denote the joint probability distribution of (Vi, Ui) +as P = PV,U. Depending on the assumptions on the data generating process, the +distribution P may be restricted to be within a parametric, semi-parametric or +non-parametric family. The marginal distribution of V and the distribution of U +conditional on V are denoted by PV and PU|V , respectively. +We are interested in estimating and conducting inference for some functional +β = β(PV,U). For example, suppose V = (D, Y, X) includes a treatment variable +D, an outcome Y , and some covariates X. We may be interested in estimating +the causal effect of D on Y , which would be point identified if there are no other +confounders given (X, U) and U is observed. However, since U is not observed, β +may only be partially identified if we restrict the “strength of confounding” for U +in some sense. +In many cases, β can be expressed as a function of two types of parameters, +θ = θ(PV ) and ψ = ψ(PV,U). The former only depends on the marginal distribution +of V and can therefore be estimated from the observed variables; the latter addi- +tionally depends on the distribution of U and thus cannot be directly estimated. +Adopting a Bayesian perspective, Gustafson (2005) and Daniels and Hogan (2008) +advocate the use of a separable parameterization, meaning that ψ = ψ(PU|V ) only +depends on the conditional distribution PU|V . In this set-up, no information about +ψ can be learnt from the observed data, which has several advantages in deriving +bounds or making Bayesian inference. However, requiring a separable parameteriza- +tion could be too restrictive in our experience and we will not make this assumption +below. +Since U is unobserved, the parameter ψ and thus the functional β cannot be +identified from the observed data. A point identified sensitivity analysis assumes +that ψ is given, for example by eliciting the opinion of a domain expert. In this +sense, the primary analysis can be viewed as a special case of a point identified +sensitivity analysis, where ψ takes the value (conventionally 0) that corresponds to +the unobserved variable U being ”ignorable”. +To assess the robustness of the primary analysis, a partially identified sensitivity +analysis assumes that ψ belongs to a set Ψ = Ψ(θ). Comparing to point identified +models, this is appealing because it is much easier for domain experts to specify +a possible range of ψ than a specific value. However, under the weaker condition +ψ ∈ Ψ, the functional β is only partially identified; we call the corresponding set of +β-values the partially identified region (PIR): +PIR(PV ) := +� +β(θ(PV ), ψ): ψ ∈ Ψ(θ(PV )) +� +. +(1) +The condition ψ ∈ Ψ(θ) in (1) implies a constraint on the joint distribution PV,U. + +Sensitivity Analysis with the R2-calculus +4 +For this reason, we will refer to Ψ as the sensitivity model. +In general, the partially identified region can be quite complicated and difficult +to infer. However, this can be simplified in the case where β is real-valued and +one-dimensional by seeking to solve the following optimization problems: +min / max β(θ(PV ), ψ), +subject to ψ ∈ Ψ(θ(PV )), +(2) +where the distribution PV is fixed. As both the objective and the feasible set in (2) +depend on the unknown PV we can sample from, this is an instance of stochastic +optimization or stochastic programming (Shapiro et al., 2009). A natural, plug-in +estimator of the optimal values of this problem can be obtained by solving +min / max β(ˆθ, ψ), +subject to ψ ∈ Ψ(ˆθ), +(3) +where ˆθ is an estimator of θ based on the observed data. This can be viewed as a +generalization of the sample average approximation (SAA) estimator in stochastic +optimization (Shapiro et al., 2009, chap. 5). Thus, a general recipe for partially +identified sensitivity analysis is the following: +(i) The functional β of interest is expressed in terms of the identifiable parameters +θ = θ(PV ) and the sensitivity parameters ψ = ψ(PV,U); +(ii) The set of constraints ψ ∈ Ψ(θ) is specified by consulting domain experts; +(iii) The optimal values of the stochastic program (2) are estimated either by first +obtaining a closed-form solution to (2) and then estimating that quantity, or +by directly solving the plug-in problem (3); +(iv) Suitable methods are then used to quantify the uncertainty of the estimators +in the previous step. +In this article, we will focus on sensitivity analysis for linear regression and linear +instrumental variables models in which θ and ψ are low-dimensional. Nevertheless, +the general framework outlined above may also be suitable for problems involving +high- or infinite-dimensional parameters; see Section 8 for more discussion. +1.2. +Interpretable Sensitivity Models using the R2-Calculus +In practice, the usefulness of the partially identified region in (1) or the optimal +values of (2) depends crucially on the interpretability of the sensitivity model Ψ. +This is where the R2-calculus can be extremely useful. In short, the R2-value R2 +Y ∼X, +also known as coefficient of determination, measures how much variance of Y can +be explained by linear combinations of X. An R2-value close to 1 indicates that +X can explain a large degree of the variance of Y ; on the other hand, values close +to 0 indicate that the linear dependence between Y and X is weak. The partial +R2-value R2 +Y ∼X|Z naturally extends this idea and measures how much variance of +Y can be explained by X given Z. +Due to their straightforward interpretation, R2- and partial R2-values are widely +used to help practitioners interpret the results of sensitivity analyses. For instance, + +Sensitivity Analysis with the R2-calculus +5 +Imbens (2003) uses them in sensitivity analysis for regression models with a discrete +treatment variable and this idea is recently extended by Veitch and Zaveri (2020); +Small (2007) measures the amount of violations to the instrumental variable as- +sumptions by using R2-values. Cinelli and Hazlett (2020) take this idea further and +parameterize the bias of the linear regression estimator by solely using R2-values. +Other parameterizations that are not fully based on R2-values can be found in +Hosman et al. (2010) and Oster (2019). +In this article, we extend this line of work and make several novel contributions: +• We use partial correlations (or R-values) instead of R2-values (which are just +squared R-values) to parameterize the sensitivity model, so the direction of the +confounder effect is naturally captured. In contrast, previous works either use +worst-case bounds implied by R2-values (Cinelli and Hazlett, 2020) or directly +specify the sign of the bias in an additional sensitivity parameter (Zhang and +Ding, 2022). +• We provide a list of algebraic relations between R- and R2- values. We give a +proof of this R2-calculus from a general Hilbert space perspective which may +be of independent interest. +• We give a general bias formula for the family of k-class estimators which in- +cludes the ordinary least squares estimator and the two-stage least squares +estimator. This allows us to provide a unified framework of sensitivity analy- +sis for linear regression and instrumental variables models. +• Facilitated by the R2-calculus and the general bias formula, we allow users to +specify very flexible constraints Ψ on the sensitivity parameters. For example, +we allow constraints that compare explanatory capability (in terms of the R2- +value) of some unmeasured confounder U with that of a measured covariate. +• We show that the simple method of fixing the sample R2-value related to the +unmeasured variable U in the sensitivity analysis, as proposed by Cinelli and +Hazlett (2020), may not provide confidence statements in the frequentist sense. +Instead, we propose a bootstrap approach to obtain sensitivity intervals. +• We provide a suite of user-friendly plots to visualize the results of the sensi- +tivity analysis. +1.3. +Organization of the Paper +Section 2 describes the R2-calculus, a collection of algebraic rules that relate (par- +tial) R2-values and correlations. Section 3 provides a general bias formula for the +k-class estimator in presence of one unmeasured confounder and discusses extensions +to multiple unmeasured confounders. +Section 4 uses the R2-calculus to develop multiple ways for practitioners to spec- +ify the constraints in Ψ(θ) based on domain knowledge. Specifically, we provide +comparative bounds on the sensitivity parameters that correspond to deviations +from the no unmeasured confounders and the instrumental variable assumptions. + +Sensitivity Analysis with the R2-calculus +6 +Section 5 reviews some approaches to construct sensitivity intervals that contains +β or the PIR with high probability. We show that directly specifying sample R2- +values as sensitivity parameter may not provide frequentist guarantees and propose +an approach based on the bootstrap. +Section 6 applies our proposed sensitivity analysis method to a famous study +in labour economics by Card (1993). We consider both the linear regression and +instrumental variable estimators and compare the results obtained by imposing +different sensitivity models. +Section 7 introduces sensitivity contour plots that +help to investigate how the choice of constraints affects the PIR. These plots are +illustrated with the real data example. Finally, Section 8 concludes this article with +a discussion of our method and an outlook on future research. +Readers who are more interested in applying the proposed method and interpret- +ing its results may wish to skip Sections 2 to 5 initially. Proofs for some theoretical +results in this article and a detailed description of the optimization algorithm can +be found in the Appendix. +2. +R2-calculus +We first give a summary of the R2-calculus – a set of widely used algebraic rules +which concern the coefficient of determination (also called R2-value) and related +quantities. Although these rules are often introduced together with the multivariate +normal distribution (see e.g. Anderson, 1958, sec. 2.5), they are purely algebraic and +rely on no distributional assumptions. In fact, this calculus not only applies to the +R2- and R-values in the population but also to their counterparts in the sample, +which will be denoted by ˆR2 and ˆR below; see Appendix A.3. For brevity, we will +only state the definitions and results for the population values. +Let Y be a random variable, let X and Z be two random vectors, and suppose +they all have finite variances. Without loss of generality, we suppose that all random +variables and vectors have mean equal to zero. Otherwise, we can replace them with +their centred versions; see Appendix A.3. We use Y ⊥⊥ X | Z to denote that Y and +X are independent conditional on Z as defined in Dawid (1979). Furthermore, the +residual of Y after partialing/regressing out X is given by +Y ⊥X := Y − XT var(X)−1 cov(X, Y ). +The variance of Y ⊥X equals that of the residual in the linear regression of Y on X, +which motivates the notation σ2 +Y ∼X = var(Y ⊥X); let σY ∼X denote its square root. +Definition 1. Suppose σ2 +Y ∼Z > 0. The R2-value of Y on X is defined as +R2 +Y ∼X := 1 − σ2 +Y ∼X +σ2 +Y +. +The partial R2-value and f2-value of Y on X given Z are defined as +R2 +Y ∼X|Z := R2 +Y ∼X+Z − R2 +Y ∼Z +1 − R2 +Y ∼Z +and +f2 +Y ∼X|Z := +R2 +Y ∼X|Z +1 − R2 +Y ∼X|Z +, + +Sensitivity Analysis with the R2-calculus +7 +respectively. If X is one-dimensional and σ2 +X∼Z > 0, the partial R- and f-value +(Cohen, 1977) are defined as +RY ∼X|Z := corr(Y ⊥Z, X⊥Z), +and +fY ∼X|Z := +RY ∼X|Z +� +1 − R2 +Y ∼X|Z +. +The marginal f2-, R- and f-values can be further defined by using an “empty” Z +in the definitions above; details are omitted. +The partial R2 takes values in [0, 1] and is a measure of how well the variables in +X can be linearly combined to explain the variation in Y after already using linear +combinations of Z. Values close to 1 indicate high explanatory capability. This +simple interpretation makes the R2-value a popular tool to assess the goodness of +fit of a linear model. The partial f2 is a monotone transformation of the partial R2 +and takes values in [0, ∞]. The partial R-value captures not only the strength but +also the direction of dependence between Y and X after partialing out Z. +The next result justifies calling R2 +Y ∼X|Z a partial R2-value and shows that the +definitions of R2- and R-value are consistent. It follows from the Gram-Schmidt +orthogonalization. +Lemma 1. In the setting of Definition 1, R2 +Y ∼X|Z = R2 +Y ⊥Z∼X⊥Z holds true. More- +over, if X is one-dimensional, then R2 +Y ∼X|Z = (RY ∼X|Z)2. +The next Proposition collects several useful results about R2-values. +Proposition 1 (R2-calculus). In the setting above, let W be another random vector. +Assume σ2 +Y ∼X+W+Z > 0. Further, suppose σ2 +X∼W+Z > 0 and σ2 +W∼X+Z > 0 when +X and/or W are one-dimensional. Then, the following rules hold: +[i] Independence: if Y ⊥⊥ X, then R2 +Y ∼X = 0; +[ii] Independent additivity: if X ⊥⊥ W, then R2 +Y ∼X+W = R2 +Y ∼X + R2 +Y ∼W ; +[iii] Decomposition of unexplained variance: +1 − R2 +Y ∼X+W = (1 − R2 +Y ∼X)(1 − R2 +Y ∼W|X); +[iv] Recursion of partial correlation: if X and W are one-dimensional, then +RY ∼X|W = +RY ∼X − RY ∼W RX∼W +� +1 − R2 +Y ∼W +� +1 − R2 +X∼W +; +[v] Reduction of partial correlation: if X is one-dimensional and Y ⊥⊥ W, then +RY ∼X|W = +RY ∼X +� +1 − R2 +X∼W +; + +Sensitivity Analysis with the R2-calculus +8 +[vi] Three-variable identity: if both X and W are one-dimensional, then +fY ∼X|W +� +1 − R2 +Y ∼W|X = fY ∼X +� +1 − R2 +X∼W − RY ∼W|XRX∼W . +Remark 1. All of the relationships above also hold when Z is partialed out (i.e. +if “ |Z” is appended to the subscripts of all R-, R2-, and f-values) and the inde- +pendence assumptions are conditional on Z. Rules [i], [ii] and [v] remain true if +(conditional) independence condition is replaced by (partial) uncorrelatedness. A +more succint sufficient condition for the positive partial variance requirements is +that the covariance matrix of (Y, X, Z, W) has full rank. +Remark 2. Rule [vi] may appear unintuitive at first. To see how this identity may +come up, consider three random variables Y , X and W. There are in total three +marginal R-values, RY ∼X, RY ∼W and RX∼W , and three partial R-values, RY ∼X|W , +RY ∼W|X and RX∼W|Y . Rule [iv] shows that the partial R-values can be determined +by all the marginal values. In other words, there are only three degrees of freedom +among the six R-values. This implies that there must be an equality constraint +relating RY ∼X, RX∼W , RY ∼X|W , and RY ∼W|X. +Remark 3. The (partial) R2- and R-value can be defined in a more general Hilbert +space setting. The corresponding rules of the R2-calculus also hold true, yielding +Proposition 1 as a corollary. See Appendix A. +3. +Bias of the k-class Estimator +Our main goal in this article is to outline a unified approach to sensitivity analysis +in linear structural equation models that leverages the R2-calculus. To this end, +we will focus on the case with a one-dimensional treatment D and a continuous +outcome Y . We would like to estimate the causal effect of D on Y , which will be +denoted as β. We may also observe some covariates X and a potential instrumental +variable Z. Let V = (D, Y, X, Z) be the observed variables. +In a sensitivity analysis, we are worried about some unmeasured variables U +that confound the causal effect of D on Y . This can potentially be addressed by +finding an instrumental variable Z for the treatment D, but this instrumental vari- +able may itself be invalid; readers who are unfamiliar with instrumental variables +are referred to Section 4.2 for its definition in the context of linear models. Below +we will derive a bias formula for the usual linear regression and instrumental vari- +able estimators, which essentially determines the objective functional β(θ, ψ) in the +stochastic optimization problem (2). +3.1. +A Single Unmeasured Confounder +We start with the case of a one-dimensional unmeasured confounder U and work +with the so-called k-class estimators as defined below. + +Sensitivity Analysis with the R2-calculus +9 +Definition 2. Suppose var(D⊥X) > var(D⊥X,Z) > 0. +The k-class estimand is +given by +βk := +� +� +� +� +� +� +� +� +� +cov(D⊥X, Y ⊥X) − k cov(D⊥X,Z, Y ⊥X,Z) +var(D⊥X) − k var(D⊥X,Z) +, +if − ∞ < k ≤ 1, +cov(D⊥X,Z, Y ⊥X,Z) +var(D⊥X,Z) +, +if k = −∞. +The k-class estimator is defined by replacing variance/covariance and the residuals +in the equation above by their sample counterparts. +The family of k-class estimators was introduced by Theil (1958) and Nagar (1959) +to interpolate the ordinary least squares (OLS) estimator and the two-stage least +squares (TSLS) estimator. +It provides a convenient representation for a unified +analysis. To see the interpolation, the OLS estimand that adjusts for X is given by +βY ∼D|X := cov(Y ⊥X, D⊥X) +var(D⊥X) +. +The TSLS estimand (also called the Wald ratio) that uses Z as an instrumental +variable and X as exogenous covariates is given by +βD∼Z|X, Y ∼Z|X := cov(Y ⊥X, Z⊥X) +cov(D⊥X, Z⊥X). +They are special cases of the k-class estimands according to the following result. +Proposition 2. In the setting of Definition 2, +β1 = βD∼Z|X, Y ∼Z|X, +β0 = βY ∼D|X, +and +lim +k→−∞ βk = β−∞ = βY ∼D|X,Z. +Remark 4. Another important estimator contained in the k-class is the limited +information maximum likelihood of Anderson and Rubin (1949), where k needs +to be estimated from the data. +Other examples can be found in Davidson and +MacKinnon (1993, p. 649) and Koles´ar et al. (2015). Also related is the anchor +regression estimator recently introduced by Rothenh¨ausler et al. (2021) that aims +to gain robustness under distributional shifts. +The target functional β = β(PV,U) we consider is the OLS estimand βY ∼D|X,Z,U +which adjusts for X, Z, and the unmeasured confounder U. When Y is causally +determined by a linear structural equation containing D, X, Z and U, the causal +effect of D on Y is precisely given by β = βY ∼D|X,Z,U; see Figure 1 for an illustration +of the data-generating process. When the true structural relationship is not linear, +βY ∼D|X,Z,U may still be interpreted as a kind of weighted average treatment effect +under additional assumptions (Angrist and Pischke, 2009, p. 75). +Because U is not observed, β cannot be consistently estimated without further +assumptions on the relationship between U and V . +The difference between the +estimand βk and the target β is quantified by the next result. + +Sensitivity Analysis with the R2-calculus +10 +Theorem 1. Suppose σ2 +D∼X > σ2 +D∼X+Z > σ2 +D∼X+Z+U > 0 and let k ∈ (−∞, 1] be +fixed. Then, +βk − β = +� +fY ∼Z|X,D RD∼Z|X +1 − k + k R2 +D∼Z|X ++ RY ∼U|X,Z,D fD∼U|X,Z +� +σY ∼X+Z+D +σD∼X+Z +. +(4) +For k = −∞, equation (4) holds by taking the limit k → −∞ on the right-hand +side. +Equation (4) generalizes previous bias formulas for the OLS estimator to the +entire family of k-class estimators; see Remark 5 below. Interestingly, this more +general formula can be easily derived by applying the OLS bias formula twice; see +equation (5) below. Because the bias of any k-class estimand can be written as +a function of RY ∼U|X,Z,D and RD∼U|X,Z, we will refer to them as the primitive +sensitivity parameters. +Corollary 1 in the appendix contains specialized bias formulas for the common +estimands in Proposition 2. The next proposition states the causal identification +assumption under which these estimands are unbiased. +Proposition 3. In the setting of Theorem 1, the following statements are true: +(i) If RD∼U|X,Z = 0 or RY ∼U|X,Z,D = 0, then β = βY ∼D|X,Z. +(ii) If R2 +D∼U+Z|X = 0 or R2 +Y ∼U+Z|X,D = 0, then β = βY ∼D|X,Z = βY ∼D|X. +(iii) If RZ∼U|X = 0 and RY ∼Z|X,D,U = 0, then β = βD∼Z|X, Y ∼Z|X. +3.2. +Proof Sketch of Theorem 1 +By expanding the difference between the k-class and the target estimands and ap- +plying the R2-calculus to the first term, we deduce +βk − βY ∼D|X,Z,U = βk − βY ∼D|X,Z + βY ∼D|X,Z − βY ∼D|X,Z,U +(5) += +� +βY ∼D|X − βY ∼D|X,Z +� +1 − k +� +1 − R2 +D∼Z|X +� ++ +� +βY ∼D|X,Z − βY ∼D|X,Z,U +� +. +Equation (4) can then be derived by applying the following Lemma twice; see Ap- +pendix B.1. +Lemma 2. Let Y, D and W be random variables, X be a random vector, and +suppose σ2 +D∼X+W > 0. Then +βY ∼D|X − βY ∼D|X,W = RY ∼W|X,D fD∼W|X +σY ∼X+D +σD∼X +. +Remark 5. To our knowledge, Lemma 2 first appeared in Cochran (1938) and was +later generalized by Cox (2007). In the context of sensitivity analysis, it has already +been used by Frank (2000), Hosman et al. (2010) and Cinelli and Hazlett (2020). +The bias formula in the last paper can be obtained by taking k → −∞ in (4). +Remark 6. Heuristically, the true causal effect β should not depend on the choice +of k. This can also been seen from equation (5). + +Sensitivity Analysis with the R2-calculus +11 +3.3. +Multiple Unmeasured Confounders +The assumption that the unmeasured confounder U is one-dimensional has kept +the algebra tractable thus far. +In order to obtain a bias formula with multiple +confounders, a generalization of Lemma 2 is required. For instance, when W is +l-dimensional, we can repeatedly apply Lemma 2 to the following telescoping series: +βY ∼D|X − βY ∼D|X,W = +l +� +j=1 +βY ∼D|X,W[j−1] − βY ∼D|X,W[j] += +l +� +j=1 +RY ∼Wj|X,D,W[j−1] fD∼Wj|X,W[j−1] +� +� +� +�1 − R2 +Y ∼W[j−1]|X,D +1 − R2 +D∼W[j−1]|X +σY ∼X+D +σD∼X +, +(6) +where [j] := {1, . . . , j} and [0] := ∅. +By using an expansion similar to (5), we +may identify the bias in linear regression and instrumental variables models with +multiple unmeasured confounders; of course, more sensitivity parameters will be +required. Such extensions are explored in Appendix B.2. +Alternatively, Lemma 2 provides an upper bound on |βY ∼D|X − βY ∼D|X,W | that +can be immediately generalized to multi-dimensional W as stated in the next re- +sult. Heuristically, this is because the confounding effects of several unmeasured +variables can negate each other; see Cinelli and Hazlett (2020, sec. 4.5). To our +knowledge, this result is first obtained by Hosman et al. (2010); we simplify their +proof substantially using the R2-calculus in Appendix B.2. +Lemma 3. Let Y and D be random variables, let X and W be random vectors. +Assume that σ2 +D∼X+W > 0 and that the covariance matrix var(W ⊥X,D) is positive +definite. Then, +��βY ∼D|X − βY ∼D|X,W +�� ≤ +� +R2 +Y ∼W|D,X f2 +D∼W|X +σ2 +Y ∼X+D +σ2 +D∼X +. +(7) +Returning to the k-class estimator, when the unmeasured confounder U is multi- +dimensional, we may still apply the expansion in equation (5). The first term on +its right-hand side does not involve U and the second term is bounded by (7). This +immediately implies a bound on the bias of the k-class estimand. +4. +Interpretable and Flexible Constraints +Theorem 1 in the previous section has established the dependece of the objective β +on two primitive sensitivity parameters: RD∼U|X,Z and RY ∼U|X,Z,D. In this section, +we develop different ways to specify interpretable constraints on these parameters +by extending ideas in previous work, most notably Cinelli and Hazlett (2020). +The key idea is to compare the R2-value of the unmeasured confounder with that +of an observed covariate. To facilitate this comparison, we assume that the random +vector X ∈ Rp can be partitioned into +X = ( ˙X, ˜X), +˙X ∈ R ˙p, ˜X ∈ R˜p such that ˙X ⊥⊥ U | ˜X, Z. +(8) + +Sensitivity Analysis with the R2-calculus +12 +Z +D +U +˙X +Y +˜X +1 +2 +3 +4 +Fig. 1. Causal diagram for regression and instrumental variables. Directed edges repre- +sent causal effects and bidirected edges represent dependence due to unmeasured com- +mon causes. +In Figure 1, we give a causal graphical model that fulfills (8); other possibilities +may be verified by the familiar d-sepration (Pearl, 2009). We further denote [ ˙p] := +{1, . . . , ˙p}. For I ⊆ [ ˙p] and ˙X ∈ R ˙p, define ˙XI := ( ˙Xi)i∈I and Ic := [ ˙p] \ I. Finally, +let ˙X−j := ˙X{j}c for any j ∈ [ ˙p]. +Table 1 summarizes the constraints on sensitivity parameters considered in this +work; it may be helpful to visualize the relations parameterized by these constraints +using the causal diagram in Figure 1. In principle, the constraints in Table 1 can +be combined arbitrarily. In particular, one may specify several comparative bounds +using different sets of covariates, although specifying too many bounds may leave +the sensitivity model infeasible. Next, we show how these bounds naturally arise +from the sensitivity analysis for the OLS and TSLS estimators. +4.1. +Ordinary Least Squares +The OLS estimand β−∞ = βY ∼D|X,Z identifies the causal effect β = βY ∼D|X,Z,U if +the causal diagram in Figure 1 does not contain U → D or U → Y , or equivalent, +if RD∼U|X,Z = 0 or RY ∼U|X,Z,D = 0. Some of the sensitivity models in Table 1 +directly bound them; others bound related R2-values that can be linked to the +primitive parameters by the R2-calculus. Such relations are elaborated below. +4.1.1. +Constraints on U → D +First of all, we may directly specify a bound on the primitive sensitivity parameter +RD∼U|X,Z ∈ [Bl +UD, Bu +UD] ⊆ [−1, 1]. +(9) +This constraint means that the correlation between D and U, after accounting for +linear effects of X and Z, lies within the interval [Bl +UD, Bu +UD]. + +Sensitivity Analysis with the R2-calculus +13 +Table 1. Specification of constraints: When the user specifies bounds on the sensi- +tivity parameters, the corresponding constraints in the last column are added to the +stochastic optimization (2). When bounds on U ↔ Z and/or Z → Y are chosen, the +TSLS-related equality constraints (17) and (18) also need to be included. +Edge +Sensitivity model +Optimization constraint +1 U → D +1. +RD∼U|X,Z ∈ [Bl +UD, Bu +UD] +(9) +2. +R2 +D∼U| ˜ +X, ˙XI,Z ≤ bUDR2 +D∼ ˙XJ| ˜ +X, ˙XI,Z +(10) +2 U → Y +1. +RY ∼U|X,Z,D ∈ [Bl +UY , Bu +UY ] +(11) +2. +R2 +Y ∼U| ˜ +X, ˙XI,Z ≤ bUY R2 +Y ∼ ˙XJ| ˜ +X, ˙XI,Z +(14), (15) +3. +R2 +Y ∼U| ˜ +X, ˙XI,Z,D ≤ bUY R2 +Y ∼ ˙XJ| ˜ +X, ˙XI,Z,D +(13), (15), (16) +3 U ↔ Z +1. +RZ∼U|X ∈ [Bl +UZ, Bu +UZ] +(19) +2. +R2 +Z∼U| ˜ +X, ˙X−j ≤ bUZR2 +Z∼ ˙Xj| ˜ +X, ˙X−j +(20) +4 Z → Y +1. +RY ∼Z|X,U,D ∈ [Bl +ZY , Bu +ZY ] +(21) +2. +R2 +Y ∼Z|X,U,D ≤ bZY R2 +Y ∼ ˙Xj| ˜ +X, ˙X−j,Z,U,D +(22), (23) +Alternatively (or in addition to the previous bound), we can specify the following +comparative bound that is arguably more interpretable: +R2 +D∼U| ˜ +X, ˙XI,Z ≤ bUDR2 +D∼ ˙XJ| ˜ +X, ˙XI,Z, I ⊂ [ ˙p], J ⊆ Ic, bUD ≥ 0. +This inequality means that the unmeasured confounder U can explain at most +bUD times as much variance of D as +˙XJ does, after accounting for the effect of +( ˜X, ˙XI, Z) on D. For practical purposes, a good choice of the comparison sets is +J = {j} and I = Jc. We can relate RD∼U| ˜ +X, ˙XI,Z in the last bound to RD∼U|X,Z via +the R2-calculus. By using ˙XIc ⊥⊥ U | ˜X, ˙XI, Z (which follows from the assumption +in (8)) and applying the reduction of partial correlation with Y ≡ U, X ≡ D, +Z ≡ ( ˜X, ˙XI, Z) and W ≡ ˙XIc, we have +R2 +D∼U|X,Z +[v] += +R2 +D∼U| ˜ +X, ˙XI,Z +1 − R2 +D∼ ˙XIc| ˜ +X, ˙XI,Z +≤ bUD +R2 +D∼ ˙XJ| ˜ +X, ˙XI,Z +1 − R2 +D∼ ˙XIc| ˜ +X, ˙XI,Z +. +(10) +4.1.2. +Constraints on U → Y +Similarly to U → D, we may specify a direct bound: +RY ∼U|X,Z,D ∈ [Bl +UY , Bu +UY ] ⊆ [−1, 1]. +(11) +Alternatively, we may use comparative bounds. +Here we consider two types of +bounds depending on whether D is regressed out: +R2 +Y ∼U| ˜ +X, ˙XI,Z ≤ bUY R2 +Y ∼ ˙XJ| ˜ +X, ˙XI,Z, +(12) +R2 +Y ∼U| ˜ +X, ˙XI,Z,D ≤ bUY R2 +Y ∼ ˙XJ| ˜ +X, ˙XI,Z,D, +(13) + +Sensitivity Analysis with the R2-calculus +14 +where I ⊂ [ ˙p], J ⊆ Ic, bUY ≥ 0. When comparing the explanatory capability of +the variables U and ˙XJ, it is natural to regress out all other variables. However, +regressing out D, a potential common child of X and U, may introduce dependence +between U and Y ; this is essentially the point made by Hernan and Robins (1999) in +their criticism of Lin et al. (1998). Thus, we consider both the comparative bound +(12) without D and the bound (13) with D. For (12), we may apply rule [v] as in +(10) and obtain +R2 +Y ∼U|X,Z +[v] += +R2 +Y ∼U| ˜ +X, ˙XI,Z +1 − R2 +Y ∼ ˙XIc| ˜ +X, ˙XI,Z +≤ bUY +R2 +Y ∼ ˙XJ| ˜ +X, ˙XI,Z +1 − R2 +Y ∼ ˙XIc| ˜ +X, ˙XI,Z +. +(14) +However, we cannot regress out D in (14) because D may be a collider in the path +˙XIc → D ← U. Instead, we can link it to RY ∼U|X,Z,D via the R2-calculus: +RY ∼U|X,Z,D +[iv] += RY ∼U|X,Z − RY ∼D|X,Z RD∼U|X,Z +� +1 − R2 +Y ∼D|X,Z +� +1 − R2 +D∼U|X,Z +. +(15) +Hence, the first type of comparative bound can be represented as the inequality +constraint (14) and the equality constraint (15) in the optimization problem (2). +The second type of comparative bounds partials out D and involves two addi- +tional sensitivity parameters: RY ∼U|X,Z and RY ∼U| ˜ +X, ˙XI,Z,D. To link them to the +primitive sensitivity parameters, we may use equation (15) and +RY ∼U|X,Z = +1 +� +1 − R2 +Y ∼ ˙XIc| ˜ +X, ˙XI,Z +� +RY ∼D| ˜ +X, ˙XI,ZRD∼U|X,Z +� +1 − R2 +D∼ ˙XIc| ˜ +X, ˙XI,Z ++ RY ∼U| ˜ +X, ˙XI,Z,D +� +1 − R2 +Y ∼D| ˜ +X, ˙XI,Z +� +1 − R2 +D∼U|X,Z(1 − R2 +D∼ ˙XIc| ˜ +X, ˙XI,Z) +� +(16) +as an equality constraint. The derivation of (16) is deferred to Appendix C.1. +4.2. +Two-stage Least Squares +The method of instrumental variables (IV) is commonly used to overcome unmea- +sured confounding. Here we only provide a very brief introduction to it; the reader +is referred to Wooldridge (2010) for a more comprehensive discussion. +A variable Z is called an instrument for D if (i) it is an independent predictor +of D, (ii) it is exogenous in the sense that Z is conditionally independent of the +unmeasured confounder U and (iii) it has no direct effect on the outcome Y that is +not mediated by D. In linear models, these conditions can be expressed as +(i) RZ∼D|X ̸= 0, +(ii) RZ∼U|X = 0, +(iii) RY ∼Z|X,U,D = 0. +Proposition 3(iii) suggests that under these conditions, the target β = βY ∼D|X,Z,U +is identified by the TSLS estimand β1 = βD∼Z|X, Y ∼Z|X. + +Sensitivity Analysis with the R2-calculus +15 +As the last two conditions above involve the unmeasured confounder U and +thus cannot be verified, a sensitivity analysis for TSLS would specify bounds on +the sensitivity parameters RZ∼U|X and RY ∼Z|X,U,D. To use the bias formula in +Theorem 1, we need to link them to the primitive sensitivity parameters RD∼U|X,Z +and RY ∼U|X,Z,D. +To achieve this, we apply the three-variable identity [vi] with +Y ≡ Y , X ≡ Z, W ≡ U and Z ≡ (X, D) to obtain +fY ∼Z|X,U,D +� +1 − R2 +Y ∼U|X,Z,D = fY ∼Z|X,D +� +1 − R2 +Z∼U|X,D − RY ∼U|X,Z,DRZ∼U|X,D, +(17) +and with Y ≡ U, X ≡ Z, W ≡ D and Z ≡ X to obtain +fZ∼U|X,D +� +1 − R2 +D∼U|X,Z = fZ∼U|X +� +1 − R2 +D∼Z|X − RD∼Z|XRD∼U|X,Z. +(18) +These are then added to the stochastic program (2) as equality constraints. +4.2.1. +Constraints on U ↔ Z +The sensitivity parameter RZ∼U|X can be constrained by directly providing a range +of plausible values, i.e. +RZ∼U|X ∈ [Bl +UZ, Bu +UZ] ⊆ [−1, 1]. +(19) +Alternatively, we allow practitioners to specify the following comparative bound +R2 +Z∼U| ˜ +X, ˙X−j ≤ bUZR2 +Z∼ ˙Xj| ˜ +X, ˙X−j, j ∈ [ ˙p], bUZ ≥ 0. +Using the conditional independence assumption (8), this can be shown to be equiv- +alent to (see Appendix C.2) +R2 +Z∼U|X ≤ bUZ R2 +Z∼ ˙Xj| ˜ +X, ˙X−j +1 − R2 +Z∼ ˙Xj| ˜ +X, ˙X−j +1 − bUZ R4 +Z∼ ˙Xj| ˜ +X, ˙X−j +. +(20) +4.2.2. +Constraints on Z → Y +We can bound the sensitivity parameter RY ∼Z|X,U,D by specifying the direct bound +RY ∼Z|X,U,D ∈ [Bl +ZY , Bu +ZY ] ⊆ [−1, 1]. +(21) +Furthermore, we allow the following comparative bound +R2 +Y ∼Z|X,U,D ≤ bZY R2 +Y ∼ ˙Xj| ˜ +X, ˙X−j,Z,U,D, j ∈ [ ˙p], bZY ≥ 0. +(22) +This last bound is unusual in the sense that the sets of variables that are regressed +out are different in the two partial R2-values. It is difficult to specify compara- +tive bounds for the exclusion restriction as the corresponding sensitivity parame- +ter RY ∼Z|X,U,D partials out U. Therefore, we cannot directly compare U to an + +Sensitivity Analysis with the R2-calculus +16 +observed covariate, e.g. ˙Xj, and the right-hand side of the bound cannot be esti- +mated. For this reason, we resort to the adjustment set in (22) because we can +connect RY ∼ ˙Xj| ˜ +X, ˙X−j,Z,U,D to the primitive sensitivity parameters via the following +equality constraint +fY ∼ ˙Xj| ˜ +X, ˙X−j,Z,U,D +� +1 − R2 +Y ∼U|X,Z,D = +� +fY ∼ ˙Xj| ˜ +X, ˙X−j,Z,D +� +1 − R2 +D∼U|X,Z ++ RY ∼U|X,Z,D RD∼ ˙Xj| ˜ +X, ˙X−j,Z RD∼U|X,Z +��� +1 − R2 +D∼U|X,Z(1 − R2 +D∼ ˙Xj| ˜ +X, ˙X−j,Z). +(23) +See Appendix C.2 for the derivation. +5. +Sensitivity Intervals +So far, we have derived the objective function β = β(θ, ψ) of the stochastic program +(2) in Section 3 and a rich set of constraints ψ ∈ Ψ(θ) in Section 4. As θ only involves +partial correlations and the standard deviation of regression residuals, we can plug +in an empirical estimator of θ to obtain a point estimator of the optimal value of (2). +In other words, we only need to solve the optimization problem in (3) to estimate +the lower and upper bounds of the partially identified region. +Complications arise when we would like to construct an interval estimator S of +β with certain statistical guarantees. In the general setup presented in Section 1.1 +and for a given 0 < α < 1, we call S a (1 − α)-sensitivity interval of β if +PV +� +β(θ(PV ), ψ) ∈ S +� +≥ 1 − α +for all +PV and ψ ∈ Ψ(θ(PV )), +and S a (1 − α)-sensitivity interval of the partially identified region if +PV +� +PIR(PV ) ⊆ S +� +≥ 1 − α +for all +PV . +Obviously, the second notion of confidence is stronger. For a more detailed discus- +sion on confidence statements in partially identified problems including issues with +asymptotic sensitivity intervals, the reader is referred to Imbens and Manski (2004), +Stoye (2009) and Molinari (2020). +Next we review several methods to construct sensitivity intervals. To obtain an +interval estimator of β in a sensitivity analysis of the OLS, a heuristic approach, as +suggested by Cinelli and Hazlett (2020), is to treat U as observed and use the usual +confidence interval +� +�ˆβY ∼D|X,Z + +� +− ˆRY ∼U|X,Z,D ˆfD∼U|X,Z ± qα +√n +� +� +� +�1 − ˆR2 +Y ∼U|X,Z,D +1 − ˆR2 +D∼U|X,Z +� +ˆσY ∼X+Z+D +ˆσD∼X+Z +� +�, +where qα is the (1 − α/2)-quantile of the standard normal distribution. Here it is +assumed that a domain expert can specify ˆψ = ( ˆRY ∼U|X,Z,D ˆRD∼U|X,Z) even though + +Sensitivity Analysis with the R2-calculus +17 +U cannot observed. For the partially identified problem, a seemingly reasonable idea +is to minimize/maximize the confidence bounds over ˆψ ∈ Ψ(ˆθ). +However, a closer look at this heuristic shows that it achieves no obvious confi- +dence guarantees. This is because the sensitivity parameter ˆψ depends on the data +and thus its value changes when another sample is drawn. If ˆψ is almost certainly +contained in Ψ(ˆθ), i.e. P( ˆψ ∈ Ψ(ˆθ)) = 1, this heuristic interval would actually be a +sensitivity interval for β. However, this is only possible if the sensitivity model Ψ +is non-informative (e.g. RD∼U|X,Z ∈ [−1, 1]). Numerical simulations in Appendix +E confirm this intuitive argument; in particular, the heuristic interval has cover- +age 50% in one setting and above 99% in another, where the nominal coverage is +1 − α = 90%. +To account for the uncertainty in estimating the feasible set Ψ(θ), Tudball et al. +(2022) propose to solve the optimization problem (3) with a relaxed constraint +ψ ∈ ˜Ψ(ˆθ), where ˜Ψ(ˆθ) is constructed to contain Ψ(θ) with high probability. However, +several technical difficulties prevent us from directly applying their method to our +problem. +A third approach to construct sensitivity interval is to use the bootstrap (Efron +and Tibshirani, 1994). More specifically, we can compute a collection of estimators +ˆˆθ using resamples of the observable data, solve the plug-in optimization problem +(3) with ˆθ = ˆˆθ, and then use the bootstrap distribution to construct one-sided +confidence intervals [βl +min, ∞) and (−∞, βu +max] with level (1 − α/2) for the minimal +and maximal values, respectively. Different procedures may be employed in the +last step. For instance, percentile bootstrap takes the α/2 and 1 − α/2 quantile +of the bootstrap distribution to construct the respective confidence interval. Other +options include the basic (or reverse percentile) bootstrap, studentized bootstrap, +and bias-corrected bootstrap; see Davison and Hinkley (1997, chap. 5) for more +detail. Finally, a sensitivity interval with nominal confidence level (1 − α) may be +constructed as [βl +min, βu +max]. +For the sensitivity analysis problems described in this article, simulation studies +in Appendix E suggest that the percentile bootstrap performs better than the basic +boostrap. In two simulation studies with nominal confidence level 90%, we found +that the percentile bootstrap intervals covers the partially identified region around +90% and the true parameter, which equals the lower end of the PIR under the +specified sensitivity model, around 95% of the time. +The empirical coverage of +basic bootstrap intervals is about 10% below the nominal level when the sample +size is n = 200; this gap closes as n increases. +Although a rigorous asymptotic analysis of the different bootstrap procedures is +beyond the scope of this article, we offer some heuristics on why the boostrap is +expected to “work” here. First, Shapiro (1991) provides an asymptotic theory for +stochastic optimization and shows that the plug-in estimator of the optimal value +of certain stochastic programs is asymptotically linear; see also Shapiro et al. (2009, +chap. 5). Although our optimization problem (2) involves unknown parameters θ in +the constraints and thus does not fall in the class of problems considered by Shapiro +(1991), one may hope that the theory there extends to the problem considered here. + +Sensitivity Analysis with the R2-calculus +18 +Second, due to optimization over the sample, the plug-in estimator is always biased, +even though the bias may be small asymptotically. With just a moderate sample +size, our simulations also show that the bootstrap distribution of the optimal value +estimators is quite skewed; see Figure 7 in Appendix E. It is plausible that the finite +sample effects of bias and skewness in the bootstrap distribution cancel out each +other for the percentile bootstrap. Finally, Zhao et al. (2019) provide an alternative +justification for the percentile bootstrap in partially identified sensitivity analysis +by using the generalized minimax inequality. However, their proof requires a fixed +constraint set Ψ and thus cannot be directly applied to the problem here. +Remark 7. Although the probability of the estimated constraint set Ψ(ˆθ) being +empty should converge to zero as the sample size grows, this can occasionally occur +with moderate sample sizes. Our implementation of the bootstrap procedures takes +a conservative approach and sets the optimal value to ∞ or −∞ depending on which +end of the PIR is considered. +6. +Data Example +We demonstrate the practicality of the proposed method using a prominent study +of the economic return of schooling. +The dataset was compiled by Card (1993) +from the National Longitudinal Survey of Young Men (NLSYM) and contains a +sample of 3010 young men at the age of 14 to 24 in 1966 who were followed up until +1981. Card uses several linear models to estimate the causal effect of education, +measured by years of schooling and denoted as D, on the logarithm of earnings, +denoted as Y . For brevity, we only consider the most parsimonious model used by +Card which includes, as covariates for adjustment and denoted as X, years of labour +force experience and its square, and indicators for living in the southern USA, being +black and living in a metropolitan area. +Card (1993) recognizes that many researchers are reluctant to interpret the es- +tablished positive correlation between education and earnings as a positive causal +effect due to the large number of potential unmeasured confounders. In our analysis, +we will consider the possibility that an unmeasured variable U, which represents the +motivation of the young men, may influence both schooling and salary. To address +this issue, Card suggests to use an instrumental variable, namely the indicator for +growing up in proximity to a 4-year college; this is denoted as Z below. Nonethe- +less, proximity to college may not be a valid instrumental variable. For example, +growing up near a college may be correlated with a higher socioeconomic status, +more career opportunities, or stronger motivation. A more detailed discussion of +the identification assumptions can be found in Card (1993). +For the purpose of sensitivity analysis, we assume that being black and living in +the southern USA are not directly related with motivation and treat them as ˙X; +the remaining covariates are regarded as ˜X in the sensitivity analysis. We assume +that this partition satisfies the conditional independence in (8). In this example, we +use comparative bounds to express our beliefs about the effects of the unmeasured +confounder U on Y and D. +We assume that motivation can explain at most 4 +times as much variation in the level of education as being black (denoted as ˙Xj) + +Sensitivity Analysis with the R2-calculus +19 +does after accounting for all other observed covariates, and that motivation can +explain at most 5 times as much variation in log-earnings as being black does after +accounting for the other covariates and education: +(B1) +R2 +D∼U| ˜ +X, ˙X−j,Z ≤ 4 R2 +D∼ ˙Xj| ˜ +X, ˙X−j,Z, +(B2) R2 +Y ∼U| ˜ +X, ˙X−j,Z,D ≤ 5 R2 +Y ∼ ˙Xj| ˜ +X, ˙X−j,Z,D. +The bounds (B1) and (B2) address deviations from the identification assumptions of +a linear regression. Likewise, we can also specify deviations from the instrumental +variable assumptions. We suppose that motivation U can explain at most half as +much variation in Z (college proximity) as ˙Xj (black) can after accounting for the +effects of ( ˜X, ˙X−j). Furthermore, we assume that college proximity Z can explain +at most 10 % as much variance in log-earnings after excluding effects of (X, U, D) as +being black can explain log-earnings after excluding the effects of ( ˜X, ˙X−j, Z, U, D). +These assumptions translate to +(B3) R2 +Z∼U| ˜ +X, ˙X−j ≤ 0.5 R2 +Z∼ ˙Xj| ˜ +X, ˙X−j, +(B4) R2 +Y ∼Z|X,U,D ≤ 0.1 R2 +Y ∼ ˙Xj| ˜ +X, ˙X−j,Z,U,D. +When the bound (B1) is not part of the constraints, we additionally require +RD∼U|X,Z ∈ [−0.98, 0.98]. +(24) +This ensures that RD∼U|X,Z is bounded away from −1 and 1 and that the partially +identified range has finite length. +Figure 2 shows the OLS estimates that adjust/do not adjust for Z, the TSLS +estimate, and their corresponding 95% confidence intervals. The same plot shows +the estimated partially identified regions and 95% sensitivity intervals (obtained by +the percentile bootstrap) for five different sensitivity models that involve different +combinations of the bounds (B1) to (B4). +Both the OLS and the TSLS estimates suggest a statistically significant positive +effect of education on earnings. In the first sensitivity model in Figure 2, we relax +the assumption of no unmeasured confounders, which would be required if the OLS +estimate is interpreted causally, and assume that the effects of U on D and Y are +bounded by (B1) and (B2), respectively. The sensitivity interval remains positive in +this case. In other cases, the estimated partially identified regions and the sensitivity +intervals become very wide whenever (B1) is not part of the constraints. +This +is because the other constraints, except the loose bound in (24), do not bound +|RD∼U|X,Z| away from 1, so the association between D and Y may be entirely +driven by the unmeasured confounder U. In fact, the PIR would have an infinite +length if (24) was not imposed. Therefore, just specifying deviations from the IV- +assumptions, as in (B3) and (B4), is not sufficient to ensure that the PIR is finite in +this dataset. Moreover, comparing the first and last sensitivity model in Figure 2, +we notice that imposing the IV-related bounds (B3) and (B4) on top of (B1) and +(B2) does not shorten the estimated PIR and sensitivity intervals. These findings +suggest that the results of Card (1993) are more robust towards deviations from the +OLS than from the IV assumptions. + +Sensitivity Analysis with the R2-calculus +20 +-0.5 +0.0 +0.5 +1.0 +OLS adj. +OLS unadj. +TSLS +(B1), (B2) +(B3), (B4) +(B1), (B3), (B4) +(B2), (B3), (B4) +(B1) - (B4) +Fig. 2. +Three estimation strategies and five sensitivity models for the causal effect β: +Point estimates/estimates of the PIR (blue); 95% confidence/sensitivity intervals (black). +7. +Sensitivity Contour Plots +This section presents graphical tools to further aid the interpretation of sensitivity +analysis. The main idea is to plot the estimated lower or upper bound of the PIR +against the sensitivity parameters or the parameters in the comparative bounds. +Contour lines in this plot allow practitioners to identify the magnitude of unmea- +sured confounding (or violations of the instrumental variables assumptions) needed +to alter the conclusion of the study qualitatively. This idea dates back at least to +Imbens (2003); our method below refines the proposal in Cinelli and Hazlett (2020) +and Zhang and Ding (2022). The contour plots will be illustrated using the real +data example in the previous section. +7.1. +b-contour Plot +For comparative bounds, the b-factor (such as bUD in (10)) controls how tightly the +corresponding sensitivity parameter is constrained. Hence, it is important to gain +a practical understanding of b. The b-sensitivity contour plot shows the estimated +lower/upper end of the PIR on a grid of b-factors. +In Figure 3, we consider the sensitivity model with the bounds (B1) and (B2) +and investigate our choice (bUD, bUY ) = (4, 5) above. +The plot shows that the +estimated lower end of the PIR is still positive even for more conservative values +such as (bUD, bUY ) = (6, 10) or (bUD, bUY ) = (10, 5). Thus, a substantial deviation +from the OLS-related assumptions is needed to alter the sign of the estimate. + +Sensitivity Analysis with the R2-calculus +21 +-0.04 +-0.02 +0.01 +0.03 +0.05 +0.07 +0 +(4, 5) +0 +5 +10 +15 +0 +2 +4 +6 +8 +10 +12 +bUD +bUY +Fig. 3. b-sensitivity contours for (B1), (B2). +0.03 +0.04 +0.05 +0.06 +0.07 +(4, 0.1) +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0 +2 +4 +6 +8 +bUD +bZY +Fig. 4. b-sensitivity contours for (B1)-(B4). +Figure 4 considers the sensitivity model using the constraints (B1) to (B4) with +changing (bUD, bZY ). This plot confirms our observation in Section 6 that imposing +the IV-related bounds (B3) and (B4) does not change the estimated lower bound +substantially when (B1) and (B2) are already present. In the terminology of con- +strained optimization, this means that the “shadow prices” for (B3) and (B4) are +small. +7.2. +R-contour Plot +We may also directly plot the estimated lower/upper end of the PIR against the +sensitivity parameters RD∼U|X,Z and RY ∼U|X,Z,D. This idea has been adopted in +several previous articles already (Imbens, 2003; Blackwell, 2014; Veitch and Zaveri, +2020). +For such R-contour plots, the key challenge is to benchmark or calibrate the +R-values. This was often done informally. For example, Cinelli and Hazlett (2020) +consider a model without potential instrument Z, use sensitivity contours parame- +terized by R2 +D∼U|X and R2 +Y ∼U|X,D and add (a ˆR2 +D∼Xj|X−j, a ˆR2 +Y ∼Xj|X−j,D) for certain +choices of a > 0 and j ∈ [p] to the plot. Thus, they aim to provide context for +plausible values of the sensitivity parameters; the underlying idea is similar to the +comparative bounds in Section 4. However, this method of benchmarking is not +entirely honest because different sets of covariates are conditioned on. Moreover, +regressing out a potential collider D may leave ˆR2 +Y ∼Xj|X−j,D difficult to interpret. +Here, we revise the contour plot in Cinelli and Hazlett (2020) by using the +R2-calculus. +To this end, we first construct benchmarking points for RD∼U|X,Z +and RY ∼U|X,Z. +Applying the reduction of partial correlation (rule [v]) and the + +Sensitivity Analysis with the R2-calculus +22 +black +2x black +5x black +south +2x south +5x south +-0.8 +-0.4 +0.0 +0.4 +0.8 +-0.8 +-0.4 +0.0 +0.4 +0.8 +RD~U|X,Z +RY~U|X,Z,D +-0.08 +-0.06 +-0.04 +-0.02 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +0.22 +Fig. 5. R-sensitivity contours for the lower end of the estimated PIR: Our comparison +points (black dots) and Cinelli and Hazlett’s comparison points (green triangles). +conditional independence U ⊥⊥ ˙Xj | ˜X, ˙X−j, Z, j ∈ [ ˙p], we obtain +RD∼U|X,Z = +RD∼U| ˜ +X, ˙X−j,Z +� +1 − R2 +D∼ ˙Xj| ˜ +X, ˙X−j,Z +and +RY ∼U|X,Z = +RY ∼U| ˜ +X, ˙X−j,Z +� +1 − R2 +Y ∼ ˙Xj| ˜ +X, ˙X−j,Z +, +which can be directly compared to, for any j ∈ [ ˙p], +ˆRD∼ ˙Xj| ˜ +X, ˙X−j,Z +� +1 − ˆR2 +D∼ ˙Xj| ˜ +X, ˙X−j +and +ˆRY ∼ ˙Xj| ˜ +X, ˙X−j +� +1 − ˆR2 +Y ∼ ˙Xj| ˜ +X, ˙X−j,Z +. +Moreover, we can multiply these values by a factor of √bR to compare the ex- +planatory capability of U (in terms of its R2-value) to bR times the explanatory +capability of the measured covariate ˙Xj. Finally, we may use the bijection between +(RD∼U|X,Z, RY ∼U|X,Z) and (RD∼U|X,Z, RY ∼U|X,Z,D) in (15) to map the benchmarks +to the scale used by the R-contour plot. +To illustrate the proposal, Figure 5 shows the R-contour plot for the estimated +lower end of the PIR and adds benchmarks corresponding to black and living in the +southern USA. We observe that, even if the unmeasured confounder was five times +as strong as black in terms of their capability of explaining the variation of D and +Y , the estimator would still be positive. Figure 5 further contrasts our comparison +points with the benchmarks proposed in Cinelli and Hazlett; in our experience, the +difference between the two methods is usually not significant. +Finally, we illustrate the utility of the R-contour plot as a way to visualize +the feasible set Ψ. Sensitivity analysis with multiple bounds often entails a non- + +Sensitivity Analysis with the R2-calculus +23 +RZ~U|X , RY~Z|X,U,D ∈ [-0.03 , 0.03] +RZ~U|X , RY~Z|X,U,D ∈ [-0.04 , 0.04] +RZ~U|X , RY~Z|X,U,D ∈ [-0.01 , 0.01] +RZ~U|X , RY~Z|X,U,D ∈ [-0.02 , 0.02] +-1.0 +-0.5 +0.0 +0.5 +1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +-0.5 +0.0 +0.5 +1.0 +-0.5 +0.0 +0.5 +1.0 +RD~U|X,Z +RY~U|X,Z,D +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Fig. 6. R-sensitivity contours for the lower end of the estimated PIR: The red lines corre- +spond to the values of RD∼U|X,Z and RY ∼U|X,Z,D that conform with the IV-assumptions. +intuitive, complex set of constraints. Consider the following sensitivity model +RZ∼U|X, RY ∼Z|X,U,D ∈ [−r, r], +r ∈ {0.01, 0.02, 0.03, 0.04}, +R2 +Y ∼U| ˜ +X, ˙X−j,Z,D ≤ 5 R2 +Y ∼ ˙Xj| ˜ +X, ˙X−j,Z,D, +RD∼U|X,Z ∈ [−0.99, 0.99], +where r parameterizes the degree of deviation from the instrumental variables as- +sumptions; the covariate ˙Xj is the indicator for black. +Figure 6 shows the estimated feasible set Ψ(ˆθ) for different values of r. +For +r = 0.01, the feasible set is small and concentrated around the lines that correspond +to RD∼U|X,Z = RY ∼U|X,Z,D = 0 (the instrumental variable is valid). As r increases, +the feasible set becomes larger as expected. +The curved shape of the region of +feasible values is a result of the comparative bound on U → Y and the associated +constraints (15) and (16). Moreover, we observe that β assumes its most extreme +values as RD∼U|X,Z approaches 1. +This highlights the importance of bounding +RD∼U|X,Z away from −1 and 1 to ensure that the PIR has finite length. +8. +Discussion and Outlook +Thus far, we have sidestepped the issue of numerically computing the solution to the +constrained stochastic optimization problem (3). In fact, standard algorithms fail +to reliably solve the problem due to the complexity of the constraints. Therefore, +we develop a grid search algorithm which leverages the structure of the objective +and the equality constraints. The details can be found in Appendix D. + +Sensitivity Analysis with the R2-calculus +24 +Two insights underlie the methodological development in this article. First, sen- +sitivity analysis (or more generally, any one-dimensional partially identified prob- +lem) may be viewed as a constrained stochastic program and we can leverage meth- +ods developed in stochastic optimization. Second, the R2-calculus provides a pa- +rameterization of the bias of any k-class estimator and a systematic approach to +specify interpretable sensitivity models. +Partial identification has attracted considerable attention in econometrics and +causal inference since Manski (1990) and Balke and Pearl (1997); see Manski (2003); +Imbens and Manski (2004); Vansteelandt et al. (2006); Chernozhukov et al. (2007); +Aronow and Lee (2013); Richardson et al. (2014); Miratrix et al. (2018); Molinari +(2020). Existing methods typically assume a closed-form solution to the stochas- +tic program (2) (the lower/upper end of the PIR) and that the plug-in estimator is +asymptotically normal. As such results are only known for relatively simple models, +these methods only have limited utility in practice. The constrained optimization +perspective of partial identification is only beginning to get embraced in the litera- +ture (Kaido et al., 2019; Hu et al., 2021; Padh et al., 2022). +Our article further shows the need for a more complete, asymptotic theory of the +optimal value of a general stochastic program. This may allow one to extend the +methodology developed here to sensitivity models with high- or infinite-dimensional +parameters. In particular, a theory for the bootstrap distribution of the optimal +value estimator is required to clarify when and which bootstrap procedures provide +asymptotically correct sensitivity intervals. +The R2-values, R-values and generalizations thereof are popular for the calibra- +tion of sensitivity analysis. They have been recently used in the sensitivity analysis +for linear models with multiple treatments (Zheng et al., 2021), mediation analy- +sis (Zhang and Ding, 2022), missing values (Colnet et al., 2022) and models with +factor-structured outcomes (Zheng et al., 2022). In these works, certain algebraic +relationships about R2-values and benchmarking techniques such as contour plots +and robustness values are frequently used. Thus, the R2-calculus summarized in +this article may also benefit the calibration of other sensitivity models. Our proof of +the R2-calculus in general Hilbert spaces suggests that it may be useful in nonlinear +models, too. See Chernozhukov et al. (2022) for related work in partially linear +and semiparametric models using the Riesz-Frechet representation of certain causal +parameters. +The rules of the R2-calculus are purely algebraic and can therefore be applied in +any linear structural equation model – with or without unmeasured variables. This +raises the question: can sensitivity analysis be automated for reasonable sensitivity +models defined by direct and comparative bounds? Such an algorithm would be +immensely useful in applied research, but given the substantial amount of algebra +needed for the relatively simple models considered here, the required work would +be extremely challenging. + +Sensitivity Analysis with the R2-calculus +25 +Acknowledgments +Tobias Freidling is supported by a PhD studentship from GlaxoSmithKline Re- +search & Development. Qingyuan Zhao is partly supported by the EPSRC grant +EP/V049968/1. +References +Altonji, J. G., Elder, T. E. and Taber, C. R. (2005) An Evaluation of Instrumental +Variable Strategies for Estimating the Effects of Catholic Schooling. Journal of +Human Resources, XL, 791–821. +Anderson, T. W. (1958) An introduction to multivariate statistical analysis. New +York; London: Wiley Publications in Mathematical Statistics, 1st edn. +Anderson, T. W. and Rubin, H. (1949) Estimation of the parameters of a single +equation in a complete system of stochastic equations. The Annals of Mathemat- +ical Statistics, 20, 46–63. +Angrist, J. D. and Pischke, J.-S. (2009) Mostly harmless econometrics: An empiri- +cist’s companion. Princeton University Press. +Aronow, P. M. and Lee, D. K. K. (2013) Interval estimation of population means +under unknown but bounded probabilities of sample selection. Biometrika, 100, +235–240. +Balke, A. and Pearl, J. (1997) Bounds on Treatment Effects from Studies with +Imperfect Compliance. Journal of the American Statistical Association, 92, 1171– +1176. +Blackwell, M. (2014) A Selection Bias Approach to Sensitivity Analysis for Causal +Effects. Political Analysis, 22, 169–182. +Card, D. (1993) Using Geographic Variation in College Proximity to Estimate the +Return to Schooling. Tech. Rep. w4483, National Bureau of Economic Research, +Cambridge, MA. +Chernozhukov, V., Cinelli, C., Newey, W., Sharma, A. and Syrgkanis, V. +(2022) Long Story Short: Omitted Variable Bias in Causal Machine Learning. +arXiv:2112.13398. +Chernozhukov, V., Hong, H. and Tamer, E. (2007) Estimation and Confidence +Regions for Parameter Sets in Econometric Models. +Econometrica, 75, 1243– +1284. +Cinelli, C. and Hazlett, C. (2020) Making sense of sensitivity: extending omit- +ted variable bias. Journal of the Royal Statistical Society: Series B (Statistical +Methodology), 82, 39–67. + +Sensitivity Analysis with the R2-calculus +26 +Cochran, W. G. (1938) The Omission or Addition of an Independent Variate in +Multiple Linear Regression. Supplement to the Journal of the Royal Statistical +Society, 5, 171. +Cohen, J. (1977) Statistical power analysis for the behavioral sciences. New York: +Academic Press, rev. edn. +Colnet, B., Josse, J., Scornet, E. and Varoquaux, G. (2022) Causal effect on a target +population: a sensitivity analysis to handle missing covariates. arXiv:2105.06435. +Cornfield, J., Haenszel, W., Hammond, E. C., Lilienfeld, A. M., Shimkin, M. B. and +Wynder, E. L. (1959) Smoking and lung cancer: recent evidence and a discussion +of some questions. Journal of the National Cancer institute, 22, 173–203. +Cox, D. R. (2007) On a generalization of a result of W. G. Cochran. Biometrika, +94, 755–759. +Daniels, M. J. and Hogan, J. W. (2008) Missing Data in Longitudinal Stud- +ies: Strategies for Bayesian Modeling and Sensitivity Analysis. Chapman and +Hall/CRC. +Davidson, R. and MacKinnon, J. G. (1993) Estimation and inference in economet- +rics. New York: Oxford University Press. +Davison, A. C. and Hinkley, D. V. (1997) Bootstrap Methods and their Application. +Cambridge University Press, 1st edn. +Dawid, A. P. (1979) Conditional independence in statistical theory. Journal of the +Royal Statistical Society: Series B (Statistical Methodology), 41, 1–15. +Ding, P. and VanderWeele, T. J. (2016) Sensitivity Analysis Without Assumptions. +Epidemiology, 27, 368–377. +Dorn, J. and Guo, K. (2022) Sharp Sensitivity Analysis for Inverse Propensity +Weighting via Quantile Balancing. Journal of the American Statistical Associa- +tion, 1–13. +Efron, B. and Tibshirani, R. (1994) An Introduction to the Bootstrap. Chapman +and Hall/CRC. +Frank, K. A. (2000) Impact of a Confounding Variable on a Regression Coefficient. +Sociological Methods & Research, 29, 147–194. +Gustafson, P. (2005) On Model Expansion, Model Contraction, Identifiability and +Prior Information: Two Illustrative Scenarios Involving Mismeasured Variables. +Statistical Science, 20. +Halmos, P. R. (2000) Introduction to Hilbert space and the theory of spectral multi- +plicity. Providence, R.I.: AMS Chelsea, 2nd edn. +Hernan, M. A. and Robins, J. M. (1999) Letter to the editor of biometrics. Biomet- +rics, 55, 1316–1317. + +Sensitivity Analysis with the R2-calculus +27 +Hosman, C. A., Hansen, B. B. and Holland, P. W. (2010) The sensitivity of linear +regression coefficients’ confidence limits to the omission of a confounder. +The +Annals of Applied Statistics, 4. +Hu, Y., Wu, Y., Zhang, L. and Wu, X. (2021) A Generative Adversarial Framework +for Bounding Confounded Causal Effects. Proceedings of the AAAI Conference +on Artificial Intelligence, 35, 12104–12112. +Imbens, G. W. (2003) Sensitivity to exogeneity assumptions in program evaluation. +American Economic Review, 93, 126–132. +Imbens, G. W. and Manski, C. F. (2004) Confidence Intervals for Partially Identified +Parameters. Econometrica, 72, 1845–1857. +Kaido, H., Molinari, F. and Stoye, J. (2019) Confidence intervals for projections of +partially identified parameters. Econometrica, 87, 1397–1432. +Koles´ar, M., Chetty, R., Friedman, J., Glaeser, E. and Imbens, G. W. (2015) Iden- +tification and inference with many invalid instruments. Journal of Business & +Economic Statistics, 33, 474–484. +Lin, D. Y., Psaty, B. M. and Kronmal, R. A. (1998) Assessing the Sensitivity of +Regression Results to Unmeasured Confounders in Observational Studies. Bio- +metrics, 54, 948–963. +Manski, C. F. (1990) Nonparametric Bounds on Treatment Effects. The American +Economic Review, 80, 319–323. +— (2003) Partial Identification of Probability Distributions. +Springer Series in +Statistics. New York: Springer. +Miratrix, L. W., Wager, S. and Zubizarreta, J. R. (2018) Shape-constrained par- +tial identification of a population mean under unknown probabilities of sample +selection. Biometrika, 105, 103–114. +Molinari, F. (2020) Chapter 5 - Microeconometrics with Partial Identification. In +Handbook of Econometrics, vol. 7, 355–486. Elsevier. +Nagar, A. L. (1959) The Bias and Moment Matrix of the General k-Class Estimators +of the Parameters in Simultaneous Equations. Econometrica, 27, 575–595. +Oster, E. (2019) Unobservable Selection and Coefficient Stability: Theory and Ev- +idence. Journal of Business & Economic Statistics, 37, 187–204. +Padh, +K., +Zeitler, +J., +Watson, +D., +Kusner, +M., +Silva, +R. and Kilbertus, +N. (2022) Stochastic Causal Programming for Bounding Treatment Effects. +arXiv:2202.10806. +PCORI Methodology Committee (2021) The PCORI Methodology Report. +https://www.pcori.org/sites/default/files/PCORI-Methodology-Report.pdf. + +Sensitivity Analysis with the R2-calculus +28 +Pearl, J. (2009) Causality: Models, Reasoning, and Inference. Cambridge University +Press, 2 edn. +Richardson, A., Hudgens, M. G., Gilbert, P. B. and Fine, J. P. (2014) Nonparametric +Bounds and Sensitivity Analysis of Treatment Effects. Statistical Science, 29, +596–618. +Rosenbaum, P. R. (1987) Sensitivity analysis for certain permutation inferences in +matched observational studies. Biometrika, 74, 13–26. +— (2002) Observational Studies. Springer Series in Statistics. New York: Springer. +Rosenbaum, P. R. and Rubin, D. B. (1983) Assessing Sensitivity to an Unobserved +Binary Covariate in an Observational Study with Binary Outcome. Journal of +the Royal Statistical Society: Series B (Statistical Methodology), 45, 212–218. +Rothenh¨ausler, D., Meinshausen, N., B¨uhlmann, P. and Peters, J. (2021) Anchor +regression: Heterogeneous data meet causality. Journal of the Royal Statistical +Society: Series B (Statistical Methodology), 83, 215–246. +Scharfstein, D. O., Rotnitzky, A. and Robins, J. M. (1999) Adjusting for Nonig- +norable Drop-Out Using Semiparametric Nonresponse Models. +Journal of the +American Statistical Association, 94, 1096–1120. +Shapiro, A. (1991) Asymptotic analysis of stochastic programs. Annals of Opera- +tions Research, 30, 169–186. +Shapiro, A., Dentcheva, D. and Ruszczy´nski, A. (2009) Lectures on Stochastic Pro- +gramming: Modeling and Theory. Society for Industrial and Applied Mathemat- +ics. +Small, D. S. (2007) Sensitivity Analysis for Instrumental Variables Regression With +Overidentifying Restrictions. +Journal of the American Statistical Association, +102, 1049–1058. +de Souza, R. J., Eisen, R. B., Perera, S., Bantoto, B., Bawor, M., Dennis, B. B., +Samaan, Z. and Thabane, L. (2016) Best (but oft-forgotten) practices: sensitiv- +ity analyses in randomized controlled trials. The American Journal of Clinical +Nutrition, 103, 5–17. +Stoye, J. (2009) More on Confidence Intervals for Partially Identified Parameters. +Econometrica, 77, 1299–1315. +Thabane, L., Mbuagbaw, L., Zhang, S., Samaan, Z., Marcucci, M., Ye, C., Thabane, +M., Giangregorio, L., Dennis, B., Kosa, D., Debono, V. B., Dillenburg, R., Fruci, +V., Bawor, M., Lee, J., Wells, G. and Goldsmith, C. H. (2013) A tutorial on +sensitivity analyses in clinical trials: the what, why, when and how. BMC Medical +Research Methodology, 13, 92. +Theil, H. (1958) Economist Forecasts and Policy. North-Holland Publishing Com- +pany, Amsterdam-London. + +Sensitivity Analysis with the R2-calculus +29 +Tudball, M. J., Hughes, R. A., Tilling, K., Bowden, J. and Zhao, Q. (2022) Sample- +constrained partial identification with application to selection bias. Biometrika, +in press. +Vandenbroucke, J. P., von Elm, E., Altman, D. G., Gøtzsche, P. C., Mulrow, C. D., +Pocock, S. J., Poole, C., Schlesselman, J. J. and Egger, M. (2007) Strengthening +the Reporting of Observational Studies in Epidemiology (STROBE): Explanation +and Elaboration. PLoS Medicine, 4, e297. +VanderWeele, T. J. and Arah, O. A. (2011) Bias Formulas for Sensitivity Analysis of +Unmeasured Confounding for General Outcomes, Treatments, and Confounders. +Epidemiology, 22, 42–52. +Vansteelandt, S., Goetghebeur, E., Kenward, M. G. and Molenberghs, G. (2006) +Ignorance and Uncertainty Regions as Inferential Tools in a Sensitivity Analysis. +Statistica Sinica, 16, 953–979. +Veitch, V. and Zaveri, A. (2020) Sense and Sensitivity Analysis: Simple Post-Hoc +Analysis of Bias Due to Unobserved Confounding. In Advances in Neural Infor- +mation Processing Systems, vol. 33, 10999–11009. Curran Associates, Inc. +Wooldridge, J. M. (2010) Econometric analysis of cross section and panel data. +Cambridge, Mass: MIT Press, 2nd edn. +Yadlowsky, S., Namkoong, H., Basu, S., Duchi, J. and Tian, L. (2022) Bounds on +the conditional and average treatment effect with unobserved confounding factors. +The Annals of Statistics, 50, 2587–2615. +Zhang, M. and Ding, P. (2022) Interpretable sensitivity analysis for the Baron- +Kenny approach to mediation with unmeasured confounding. arXiv:2205.08030. +Zhao, Q., Small, D. S. and Bhattacharya, B. B. (2019) Sensitivity analysis for +inverse probability weighting estimators via the percentile bootstrap. Journal of +the Royal Statistical Society: Series B (Statistical Methodology), 81, 735–761. +Zheng, J., D’Amour, A. and Franks, A. (2021) Copula-based Sensitivity Analy- +sis for Multi-Treatment Causal Inference with Unobserved Confounding. arXiv: +2102.09412. +Zheng, J., Wu, J., D’Amour, A. and Franks, A. (2022) Sensitivity to Unobserved +Confounding in Studies with Factor-structured Outcomes. arXiv: 2208.06552. + +Sensitivity Analysis with the R2-calculus +30 +A. +Hilbert Space R2-calculus and Proofs +The algebraic rules of the R2-calculus – both the population version in Proposition +1 and its sample counterpart – are not specific to linear models. In fact, all rela- +tionships fundamentally stem from the geometry of projections in Hilbert spaces. +For this reason, the definitions of R2- and R-values can be generalized and the +corresponding algebraic rules can be proven in more generality. +This section is organized as follows. +First, we recall some results on Hilbert +space theory (Halmos, 2000, sec. 26-29) and define generalized (partial) R2- and R- +values. Then, we prove Hilbert space generalizations to Lemma 1 and Proposition +1. Finally, in Section A.3, we explain how the R2-calculus for linear models directly +follows from the more general result and provides more details on the assumptions +and notation involved. +A.1. +Hilbert Space R2-value +Let (H, ⟨·, ·⟩) be a Hilbert space over the field K of real or complex numbers; denote +its associated norm as ∥·∥ and let X, Y, Z ⊆ H be closed linear subspaces. The +Minkowski sum of Y and X is given by X + Y := {x + y: x ∈ X, y ∈ Y}. For x ∈ X +and y ∈ Y, we write x ⊥ y, if ⟨x, y⟩ = 0, x ⊥ Y, if x ⊥ y for all y ∈ Y, and X ⊥ Y, if +x ⊥ Y for all x ∈ X. For every element h ∈ H, there are unique x ∈ X and x⊥ ∈ H +such that x ⊥ x⊥ and h = x + x⊥. +Definition 3. The projection on X is the operator PX : H → X defined by the +assignment h = x + x⊥ �→ x. The projection off X is the operator QX : H → H +defined by h = x + x⊥ �→ x⊥. +Clearly, the projection on and off X add up to the identity operator, i.e. PX + +QX = Id. +Furthermore, we introduce the notations y⊥X := QX y and Y⊥X := +{y⊥X : y ∈ Y}. +The space Y⊥X is a closed linear subspace of H; thus, the pro- +jections PY⊥X and QY⊥X are well-defined. They can be used to define conditional +orthogonality: Y ⊥ X | Z ⇔ Y⊥Z ⊥ X ⊥Z. +Lemma 4. +(i) PX and QX are linear, self-adjoint, and idempotent operators. +(ii) If X ⊥ Y, PX+Y = PX + PY and QX+Y = QX QY. +(iii) PX+Y = PX + PY⊥X and QX+Y = QX QY⊥X = QY⊥X QX . +(iv) If h1, h2 ∈ H and h1 ⊥ h2, ∥h1 + h2∥2 = ∥h1∥2 + ∥h2∥2. +Proof. +(i) See Halmos (2000, sec. 26, Thm. 1). +(ii) See Halmos (2000, sec. +28, Thm. +2) for the proof of PX+Y = PX + PY. +According to Halmos (2000, sec. 29, Thm. 1), PX PY = 0 holds due to X ⊥ Y. +Hence, the second statement directly follows +QX+Y = Id − PX − PY = (Id − PX )(Id − PY) = QX QY. + +Sensitivity Analysis with the R2-calculus +31 +(iii) We rewrite the direct sum X + Y as follows +X + Y = {x + y: x ∈ X, y ∈ Y} = {x + PX y + QX y: x ∈ X, y ∈ Y} += {x + QX y: x ∈ X, y ∈ Y} = X + Y⊥X . +Since X and Y⊥X are orthogonal by definition, the statement directly follows +from (ii). +(iv) See Halmos (2000, sec. 4, Thm. 3). +Any one-dimensional linear subspace X can be expressed as X = {λ x: λ ∈ K}, +where x is an arbitrary element in X \{0}. Hence, we can identify a one-dimensional +subspace with any non-zero element contained in it. +Definition 4 (Hilbert space R2- and R-value). Let X, Y, Z ⊆ H be closed linear +subspaces. Assume Y is one-dimensional, let y ∈ Y \ {0} and suppose ∥y⊥Z∥2 > 0. +The R2-value of Y on X is defined as +R2 +Y∼X := 1 − ∥y⊥X ∥2 +∥y∥2 +. +The partial R2-value of Y on X given Z is defined as +R2 +Y∼X|Z := R2 +Y∼X+Z − R2 +Y∼Z +1 − R2 +Y∼Z +. +If X is one-dimensional, x ∈ X \ {0} and ∥x⊥Z∥2 > 0, the partial R-value is defined +as +RY∼X|Z := +⟨y⊥Z, x⊥Z⟩ +∥y⊥Z∥ ∥x⊥Z∥. +The corresponding (partial) f2- and f-values are defined analogously to Defini- +tion 1. The choice of the non-zero elements y and x does not change the (partial) +R2- and R-values due to the normalization. +Therefore, all quantities above are +well-defined. +A.2. +Proofs of Results in Section 2 +In this subsection, we state and prove the generalized versions of Lemma 1 and +Proposition 1. +Lemma 5. In the setting of Definition 4, R2 +Y∼X|Z = R2 +Y⊥Z∼X ⊥Z holds true. More- +over, if X is a one-dimensional subspace, then R2 +Y∼X|Z = (RY∼X|Z)2. + +Sensitivity Analysis with the R2-calculus +32 +Proof. The first statement of the lemma follows from some elementary algebraic +manipulations and Lemma 4 (iii) +R2 +Y∼X|Z = R2 +Y∼X+Z − R2 +Y∼Z +1 − R2 +Y∼Z +� +1 − ∥y⊥X+Z∥2 +∥y∥2 +− 1 + ∥y⊥Z∥2 +∥y∥2 +� �∥y⊥Z∥2 +∥y∥2 += 1 − ∥y⊥X+Z∥2 +∥y⊥Z∥2 +(iii) += 1 − ∥QX ⊥Z y⊥Z∥2 +∥y⊥Z∥2 += R2 +Y⊥Z∼X ⊥Z. +To prove the second part of the lemma, we assume that X is one-dimensional and +choose x ∈ X \ {0}. If X ⊥Z = 0, the projection on X ⊥Z is 0; otherwise, it is given +by +PX ⊥Zh = ⟨h, x⊥Z⟩x⊥Z +∥x⊥Z∥2 +, +for h ∈ H. +(25) +This can be easily checked: PX ⊥Z is linear and its image is contained in X ⊥Z. +Moreover, we compute +�⟨h, x⊥Z⟩ x⊥Z +∥x⊥Z∥2 +, h − ⟨h, x⊥Z⟩ x⊥Z +∥x⊥Z∥2 +� += ⟨h, x⊥Z⟩2 +∥x⊥Z∥2 − ⟨h, x⊥Z⟩2∥x⊥Z∥2 +∥x⊥Z∥4 += 0. +Following from the first part of the proof and Lemma 4 (iv), we infer +R2 +Y∼X|Z = 1 − ∥QX ⊥Z y⊥Z∥2 +∥y⊥Z∥2 +(iv) += ∥PX ⊥Z y⊥Z∥2 +∥y⊥Z∥2 +. +Directly plugging in the formula for the projection on X ⊥Z yields the second state- +ment of the lemma +R2 +Y∼X|Z = ∥⟨y⊥Z, x⊥Z⟩ x⊥Z∥2 +∥y⊥Z∥2 ∥x⊥Z∥4 += ⟨y⊥Z, x⊥Z⟩2∥x⊥Z∥2 +∥y⊥Z∥2 ∥x⊥Z∥4 += +� +RY∼X|Z +�2. +Proposition 4 (Hilbert space R2-calculus). In the setting of Definition 4, let W +be another closed linear subspace. Assume ∥Y⊥X+W+Z∥2 > 0. Further suppose +∥X ⊥W+Z∥2 >0 and ∥W⊥X+Z∥2 > 0 when X and/or W are one-dimensional sub- +spaces. Then, the following rules hold +[i] Orthogonality: if Y ⊥ X, R2 +Y∼X = 0; +[ii] Orthogonal additivity: if X ⊥ W, R2 +Y∼X+W = R2 +Y∼X + R2 +Y∼W; +[iii] Decomposition of unexplained variation: +1 − R2 +Y∼X+W = (1 − R2 +Y∼X )(1 − R2 +Y∼W|X ); +[iv] Recursion of partial R-value: if X and W are one-dimensional, +RY∼X|W = +RY∼X − RY∼WRX∼W +� +1 − R2 +Y∼W +� +1 − R2 +X∼W +; + +Sensitivity Analysis with the R2-calculus +33 +[v] Reduction of partial R-value: if X is one-dimensional and Y ⊥ W, +RY∼X|W = +RY∼X +� +1 − R2 +X∼W +; +[vi] Three-dimensional restriction: if X and W are one-dimensional, +fY∼X|W +� +1 − R2 +Y∼W|X = fY∼X +� +1 − R2 +X∼W − RY∼W|X RX∼W. +All of the relationships above also hold when Z is partialed out (i.e. if “ |Z” is +appended to the subscripts of all R-, R2-, and f-values) and the orthogonality as- +sumptions are conditional on Z. +Proof. +[i] Since Y⊥Z and X ⊥Z are orthogonal, QX ⊥Zy⊥Z = y⊥Z. Hence, +R2 +Y∼X|Z = 1 − ∥y⊥Z∥2 +∥y⊥Z∥2 = 0. +[ii] Lemma 5 and its proof yield +R2 +Y∼X+W|Z = R2 +Y⊥Z∼X ⊥Z+W⊥Z = ∥PX ⊥Z+W⊥Z y⊥Z∥2 +∥y⊥Z∥2 +. +Following from Lemma 4 (ii) and (iv), we get +R2 +Y∼X+W|Z +(ii) += ∥PX ⊥Zy⊥Z + PW⊥Z y⊥Z∥2 +∥y⊥Z∥2 +(iv) += ∥PX ⊥Z y⊥Z∥2 +∥y⊥Z∥2 ++ ∥PW⊥Z y⊥Z∥2 +∥y⊥Z∥2 += R2 +Y∼X|Z + R2 +Y∼W|Z. +[iii] The statement directly follows from the definition of the partial R2-value +� +1 − R2 +Y∼X|Z +� � +1 − R2 +Y∼W|X+Z +� += ∥y⊥X+Z∥2 +∥y⊥Z∥2 +∥y⊥W+X+Z∥2 +∥y⊥X+Z∥2 += ∥y⊥W+X+Z∥2 +∥y⊥Z∥2 += 1 − R2 +Y∼W+X|Z. +[iv] Plugging in the definition of the partial R-value into the right-hand side, we +get +RHS = RY∼X|Z − RY∼W|Z RX∼W|Z +� +1 − R2 +Y∼W|Z +� +1 − R2 +X∼W|Z += +� ⟨y⊥Z, x⊥Z⟩ +∥y⊥Z∥∥x⊥Z∥ − ⟨y⊥Z, w⊥Z⟩ +∥y⊥Z∥∥w⊥Z∥ +⟨x⊥Z, w⊥Z⟩ +∥x⊥Z∥∥w⊥Z∥ +� � �∥y⊥W+Z∥ +∥y⊥Z∥ +∥x⊥W+Z∥ +∥x⊥Z∥ +� += +⟨y⊥Z, x⊥Z⟩ +∥y⊥W+Z∥∥x⊥W+Z∥ − +⟨y⊥Z, w⊥Z⟩ ⟨x⊥Z, w⊥Z⟩ +∥w⊥Z∥2∥y⊥W+Z∥∥x⊥W+Z∥. + +Sensitivity Analysis with the R2-calculus +34 +Recalling the formula (25) for the projection operator on a one-dimensional +subspace, we can reformulate the upper equation further +RHS = +� +y⊥Z, x⊥Z − ⟨x⊥Z,w⊥Z⟩ w⊥Z +∥w⊥Z∥2 +� +∥y⊥W+Z∥∥x⊥W+Z∥ += ⟨y⊥Z, QW⊥Z x⊥Z⟩ +∥y⊥W+Z∥∥x⊥W+Z∥ +(iii) += +⟨y⊥W+Z, x⊥W+Z⟩ +∥y⊥W+Z∥∥x⊥W+Z∥ = RY∼X|W+Z = LHS, +where the third equality follows from Lemma 4 (iii). +[v] Let (w⊥Z +j +)j∈{1,...,J}, be an orthonormal basis of W⊥Z. The subspace spanned +by the first j vectors is denoted by W⊥Z +j +:= span{w⊥Z +1 +, . . . , w⊥Z +j +}. Due to +rule [i] and Y ⊥ W | Z, R2 +Y∼Wj|Z = 0 and R2 +Y∼Wj+1|Wj+Z = 0 hold for all +j ∈ {1, . . . , J − 1}. By induction, we prove the statement +RY∼X|Z+Wj = +RY∼X|Z +� +1 − R2 +X∼Wj|Z +, +for all j ∈ {1, . . . , J}. +For the base case, we apply rule [iv] and RY∼W1|Z = 0 as follows +RY∼X|W1+Z +[iv] += +RY∼X|Z − RY∼W1|Z RX∼W1|Z +� +1 − R2 +Y∼W1|Z +� +1 − R2 +X∼W1|Z += +RY∼X|Z +� +1 − R2 +X∼W1|Z +. +The induction step uses rule [iv] and simplifies the resulting expression via +RY∼Wj+1|Wj+Z = 0, the induction hypothesis and rule [iii]: +RY∼X|Wj+1+Z +[iv] += RY∼X|Wj+Z − RY∼Wj+1|Wj+Z RX∼Wj+1|Wj+Z +� +1 − R2 +Y∼Wj+1|Wj+Z +� +1 − R2 +X∼Wj+1|Wj+Z += +RY∼X|Wj+Z +� +1 − R2 +X∼Wj+1|Wj+Z += +RY∼X|Z +� +1 − R2 +X∼Wj|Z +� +1 − R2 +X∼Wj+1|Wj+Z +[iii] += +RY∼X|Z +� +1 − R2 +X∼Wj+1|Z +. +[vi] First, we apply rule [iv] to RY∼X|W+Z and RY∼W|X+Z +RY∼X|W+Z = RY∼X|Z − RY∼W|Z RX∼W|Z +� +1 − R2 +Y∼W|Z +� +1 − R2 +X∼W|Z +, +RY∼W|X+Z = RY∼W|Z − RY∼X|Z RX∼W|Z +� +1 − R2 +Y∼X|Z +� +1 − R2 +X∼W|Z +, + +Sensitivity Analysis with the R2-calculus +35 +and compute +RY∼X|W+Z +� +1 − R2 +Y∼W|Z + RY∼W|X+ZRX∼W|Z +� +1 − R2 +Y∼X|Z += +1 +� +1 − R2 +X∼W|Z +� +RY∼X|Z − RY∼W|ZRX∼W|Z ++ RY∼W|ZRX∼W|Z − RY∼X|ZR2 +W∼X|Z +� += RY∼X|Z +� +1 − R2 +X∼W|Z. +Next, we divide both sides of the equation by +� +1 − R2 +Y∼X|Z and rearrange it +which results in +RY∼X|W+Z +� +1 − R2 +Y∼W|Z +� +1 − R2 +Y∼X|Z += fY∼X|Z +� +1 − R2 +X∼W|Z − RY∼W|X+ZRX∼W|Z. +According to rule [iii], we obtain +(1−R2 +Y∼X|Z)(1−R2 +Y∼W|X+Z) = 1−R2 +Y∼X+W|Z = (1−R2 +Y∼W|Z)(1−R2 +Y∼X|W+Z) +and thus +1 − R2 +Y∼W|Z +1 − R2 +Y∼X|Z += +1 − R2 +Y∼W|X+Z +1 − R2 +Y∼X|W+Z +. +Plugging this relationship into the left-hand side of the upper equation, we +arrive at +fY∼X|W+Z +� +1 − R2 +Y∼W|X+Z = fY∼X|Z +� +1 − R2 +X∼W|Z − RY∼W|X+Z RX∼W|Z. +A.3. +R2-calculus for Linear Models +The R2-calculus for linear models as presented in the main text is a special case +of the R2-calculus for Hilbert spaces. To be consistent with the standard notation +for R2-values in linear models in the main text, we make two slight changes to +the Hilbert space notation. First, a random vector denotes the linear space that +is spanned by its components. Analogously, for an i.i.d. sample of size n for a +p-dimensional random vector X, we use the matrix X ∈ Rn×p to denote the row- +space. Second, we replace the plus-sign with a comma for partialed out variables. +For instance, we write R2 +Y ∼X|W,Z instead of R2 +Y ∼X|W+Z in the main text. +Denote the space of square-integrable random variables L2 := {X : E[X2] < ∞}. + +Sensitivity Analysis with the R2-calculus +36 +We define the following four Hilbert spaces with associated inner products +H := L2, +⟨X, Y ⟩H := E[XY ], +H0 := +� +X ∈ L2 : E[X] = 0 +� +, +⟨X, Y ⟩H0 := cov(X, Y ), +ˆH := Rn, +⟨x, y⟩ ˆ +H +:= n−1xT y, +ˆH0 := +� +x ∈ Rn : ¯x = 0 +� +, +⟨x, y⟩ ˆ +H0 := � +cov(x, y), +where ¯x denotes the empirical mean of x. The population R2-calculus for linear mod- +els as stated in the main text follows from choosing the Hilbert space (H0, ⟨·, ·⟩H0) +in Lemma 5 and Proposition 4. Likewise, we use ( ˆH0, ⟨·, ·⟩ ˆ +H0) for the empirical R2- +calculus. Since we choose the scaling n−1 in the empirical covariance, the estimators +of covariance, variance and standard deviation are not unbiased. To account for the +loss of degrees of freedom through estimation of the mean and potentially partialing +out a p-dimensional subspace, the factor (n−p−1)−1 must be used. We choose the +scaling n−1 instead to accord with the textbook definition of the empirical R2-value +(Davidson and MacKinnon, 1993, chap. 1). Besides, for a sufficiently large sample +size n the difference will be negligible. +In the main text, we made the assumption that the random variables and the +observations are centred and thus are elements of H0. If this does not hold, we can +redefine the population R2-value via the inner product ⟨·, ·⟩H as follows +R2 +Y ∼X := 1 − E[(Y ⊥X)2] +E[Y 2] +. +Similarly, we replace the inner product in the definition of partial R2-, R-, f2- and +f-values. +This formulation contains the definition of R2-value in the main text +as a special case because, for centred random variables, ⟨·, ·⟩H and the covariance +are equal. Furthermore, if we treat the constant 1 as an additional covariate, the +following relationship holds +R2 +Y −E[Y ]∼X−E[X] = R2 +Y ∼X|1. +Hence, centring random variables is equivalent to partialing out the effect of the +constant, and thus always observed, covariate. As our focus lies on the explanatory +capability of the non-constant covariates, we always partial out 1 or equivalently +centre the observed variables. +The same arguments also apply to the empirical +R2-value and centring the samples. +B. +Proofs of Results in Section 3 +Without loss of generality, we assume that all random variables/vectors are cen- +tred; moreover, we only state and prove the population version of the results. As +explained in Appendix A.3, the sample and non-centred counterparts of the results +and proofs follow by the same arguments but choosing a different Hilbert space and +inner product. + +Sensitivity Analysis with the R2-calculus +37 +B.1. +A Single Unmeasured Confounder +Proof of Proposition 2. First, we rewrite the partialing out of Z in terms of a +projection operation, cf. Lemma 4 (ii); then, we use linearity of the covariance and +Lemma 4 (iv) to simplify the numerator and denominator, respectively: +β1 = cov(D⊥X, Y ⊥X) − cov(D⊥X, QZ⊥XY ⊥X) +var(D⊥X) − var(QZ⊥XD⊥X) += cov(D⊥X, PZ⊥XY ⊥X) +var(PZ⊥XD⊥X) +. +Since Z⊥X is one-dimensional, the projection PZ⊥X is given by (25). Plugging this +relationship into the equation above yields +β1 = +cov +� +D⊥X, cov(Z⊥X,Y ⊥X) +var(⊥X) +Z⊥X� +var +� +cov(D⊥X,Z⊥X) +var(Z⊥X) +Z⊥X +� += cov(Z⊥X, Y ⊥X) +cov(Z⊥X, D⊥X) = βD∼Z|X, Y ∼Z|X +which proves the first result. The second and third statements directly follow from +the definition of the k-class estimand. +Proof of Lemma 2. First, we express the estimands βY ∼D|X and βY ∼D|X,W in +terms of standard deviations and correlations and replace the terms with the R- +and σ-notation +βY ∼D|X − βY ∼D|X,W += corr(Y ⊥X, D⊥X)sd(Y ⊥X)sd(D⊥X) +sd(D⊥X)2 +− corr(Y ⊥X,W, D⊥X,W )sd(Y ⊥X,W )sd(D⊥X,W ) +sd(D⊥X,W )2 += RY ∼D|X +σY ∼X +σD∼X +− RY ∼D|X,W +σY ∼X+W +σD∼X+W +. +Next, we extract the common factor σY ∼X+D/σD∼X by applying the formula for +decomposition of unexplained variance [iii] four times. We then rewrite the differ- +ence so that it is expressed in terms of RY ∼W|X,D instead of RY ∼D|X,W . To this +end, we subsequently replace RY ∼D|X,W and RY ∼W|X via the recursion of partial +correlation formula [iv]. In summary, we get +βY ∼D|X − βY ∼D|X,W +[iii] += +� +� +RY ∼D|X +� +1 − R2 +Y ∼D|X +− RY ∼D|X,W +� +1 − R2 +Y ∼W|X +� +1 − R2 +Y ∼D|X +� +1 − R2 +D∼W|X +� +� σY ∼X+D +σD∼X +[iv] += +� +�fY ∼D|X − RY ∼D|X − RY ∼W|X RD∼W|X +� +1 − R2 +Y ∼D|X +� +1 − R2 +D∼W|X +� +� +� σY ∼X+D +σD∼X +[iv] += +� +fY ∼D|X − +1 +� +1 − R2 +Y ∼D|X +� +1 − R2 +D∼W|X +� +� +RY ∼D|X − RD∼W|X +�� +1 − R2 +Y ∼D|X +� +1 − R2 +D∼W|XRY ∼W|X,D + RY ∼D|X RD∼W|X +��� +σY ∼X+D +σD∼X + +Sensitivity Analysis with the R2-calculus +38 += +� +fY ∼D|X +� +1 − +1 +1 − R2 +D∼W|X ++ +R2 +D∼W|X +1 − R2 +D∼W|X +� ++fD∼W|X RY ∼W|X,D +� +σY ∼X+D +σD∼X += fD∼W|X RY ∼W|X,D +σY ∼X+D +σD∼X +. +Proof of Theorem 1. Throughout this proof, all quantities partial out X which +is indicated by either the subscript “|X” or the superscript “⊥X”. In order to +shorten the notation, we only indicate partialing out X in the estimands and drop +the X-dependence in the other quantities. +First, we focus on the difference between the k-class estimand βk and the OLS +estimand βY ∼D|X,Z that adjusts for X and Z; multiplying the respective denomi- +nators yields +βk − βY ∼D|X,Z = cov(D, Y ) − k cov(D⊥Z, Y ⊥Z) +var(D) − k var(D⊥Z) +− cov(Y ⊥Z, D⊥Z) +var(D⊥Z) += cov(D, Y ) var(D⊥Z) − k cov(D⊥Z, Y ⊥Z) var(D⊥Z) +var(D⊥Z) var(D) − k var(D⊥Z)2 ++ − cov(D⊥Z, Y ⊥Z) var(D) + k cov(D⊥Z, Y ⊥Z) var(D⊥Z) +var(D⊥Z) var(D) − k var(D⊥Z)2 += cov(D, Y ) var(D⊥Z) − cov(D⊥Z, Y ⊥Z) var(D) +var(D⊥Z) var(D) − k var(D⊥Z)2 += cov(D, Y ) var(D⊥Z) − cov(D⊥Z, Y ⊥Z) var(D) +var(D⊥Z) var(D) +� +1 − k var(D⊥Z)/ var(D) +� . +Next, we simplify the last expression by using 1 − R2 +D∼Z = var(D⊥Z)/ var(Z). This +results in a formula which involves the difference of the OLS estimands βY ∼D|X and +βY ∼D|X,Z: +βk − βY ∼D|X,Z = +1 +1 − k +� +1 − R2 +D∼Z +� +�cov(D, Y ) +var(D) +− cov(D⊥Z, Y ⊥Z) +var(D⊥Z) +� += +1 +1 − k +� +1 − R2 +D∼Z +� � +βY ∼D|X − βY ∼D|X,Z +� +. +We can now use the last result to express the difference βk −β as a telescoping sum: +βk − β = βk − βY ∼D|X,Z + βY ∼D|X,Z − βY ∼D|X,Z,U += +1 +1 − k +� +1 − R2 +D∼Z +� � +βY ∼D|X − βY ∼D|X,Z +� ++ +� +βY ∼D|X,Z − βY ∼D|X,Z,U +� +. +This representation includes two differences of OLS estimands; hence, Lemma 2 +can be applied twice. For the first summand, we use X ≡ X and W ≡ Z; for the + +Sensitivity Analysis with the R2-calculus +39 +second, X ≡ (X, Z) and W ≡ U. Thus, we get +βk − β = +1 +1 − k +� +1 − R2 +D∼Z +�RY ∼Z|D fD∼Z +σY ∼D +σD ++ RY ∼U|Z,D fD∼U|Z +σY ∼Z+D +σD∼Z +. +Finally, we can simplify the expression above by extracting a common factor of +σY ∼Z+D/σD∼Z. We use the definition of the (partial) R2-value, e.g. 1 − R2 +Y ∼Z|D = +σ2 +Y ∼Z+D/σ2 +Y ∼D, and deduce that +βk − β = +� +RY ∼Z|D fD∼Z +1 − k +� +1 − R2 +D∼Z +� +� +1 − R2 +D∼Z +� +1 − R2 +Y ∼Z|D ++ RY ∼U|Z,D fD∼U|Z +�σY ∼Z+D +σD∼Z += +� fY ∼Z|D RD∼Z +1 − k + k R2 +D∼Z ++ RY ∼U|Z,D fD∼U|Z +� σY ∼Z+D +σD∼Z +, +which concludes the proof. +Corollary 1. In the setting of Theorem 1, the following are true +(i) Adjusted Regression: +βY ∼D|X,Z − β = RY ∼U|X,Z,D fD∼U|X,Z +σY ∼X+Z+D +σD∼X+Z +; +(ii) Unadjusted Regression: +βY ∼D|X − β = +� +fY ∼Z|X,D RD∼Z|X + RY ∼U|X,Z,D fD∼U|X,Z +� σY ∼X+Z+D +σD∼X+Z +; +(iii) Instrumental Variable: +βD∼Z|X, Y ∼Z|X − β = +�fY ∼Z|X,D +RD∼Z|X ++ RY ∼U|X,Z,D fD∼U|X,Z +� σY ∼X+Z+D +σD∼X+Z +. +Proof. The statements are a direct consequence of Theorem 1 and Proposition 2. +Proof of Proposition 3. +(i) This statement directly follows from taking the limit k → −∞ and setting +RY ∼U|X,Z,DfD∼U|X,Z = 0 in equation (4). +(ii) We apply the decomposition of unexplained variance rule +1 − R2 +D∼U+Z|X +[iii] += +� +1 − R2 +D∼Z|X +�� +1 − R2 +D∼U|X,Z +� +, +1 − R2 +Y ∼U+Z|X,D +[iii] += +� +1 − R2 +Y ∼Z|X,D +�� +1 − R2 +Y ∼U|X,Z,Z +� +, + +Sensitivity Analysis with the R2-calculus +40 +which yields the implications +R2 +D∼U+Z|X = 0 +⇒ +RD∼Z|X = 0, +RD∼U|X,Z = 0; +R2 +Y ∼U+Z|X,D = 0 +⇒ +RY ∼Z|X,D = 0, +RY ∼U|X,Z,D = 0. +Then, the unbiasedness of βY ∼D|X,Z and βY ∼D|X follows from Corollary 1 or +Theorem 1 with k = 0. +(iii) In order to connect the IV-related sensitivity parameters to RD∼U|X,Z and +RY ∼U|X,Z,D, we apply the three-variable identity [vi] with Y ≡ Y , X ≡ Z, +W ≡ U and Z ≡ (X, D) as well as Y ≡ U, X ≡ Z, W ≡ D and Z ≡ X. We +obtain +fY ∼Z|X,U,D +� +1 − R2 +Y ∼U|X,Z,D = fY ∼Z|X,D +� +1 − R2 +Z∼U|X,D +− RY ∼U|X,Z,DRZ∼U|X,D, +fZ∼U|X,D +� +1 − R2 +D∼U|X,Z = fZ∼U|X +� +1 − R2 +D∼Z|X − RD∼Z|XRD∼U|X,Z. +If we set RZ∼U|X = 0 and RY ∼Z|X,U,D = 0 in the equations above and simplify +them, we get the relationship +fD∼U|X,Z RY ∼U|X,Z,D = −fY ∼Z|X,D +RD∼Z|X +. +Due to Corollary 1 or Theorem 1 with k = 1, this implies βD∼Z|X, Y ∼Z|X = β. +B.2. +Multiple Unmeasured Confounders +Proof of Lemma 3. Analogously to the proof of Theorem 1, we only indicate +partialing out X in the estimands and drop the X-dependence in the other quantities +for ease of notation. +We define the vector λ as follows +λ = var(W ⊥D)−1 cov(W ⊥D, Y ⊥D). +It equals the regression coefficients of W in the linear model Y ∼ D + X + W. +In order to reduce the number of dimensions of W, we introduce a new random +variable W ∗ := λT W. Since it captures all linear influence of W on Y , the estimands +βY ∼D|X,W and βY ∼D|X,W ∗ are equal. To formally prove this result, we let A denote +either Y or D and show that A⊥W ∗ = A⊥W . By definition of λ and some algebraic + +Sensitivity Analysis with the R2-calculus +41 +manipulations we derive +A⊥W ∗ = A − (W ∗)T var(W ∗)−1 cov(W ∗, A) += A − W T var(W ⊥D)−1 cov(W ⊥D, Y ⊥D) +� +var(W ⊥D)−1 cov(W ⊥D, Y ⊥D) +�−1 +× var(W)−1� +var(W ⊥D)−1 cov(W ⊥D, Y ⊥D) +�−T +× cov(W ⊥D, Y ⊥D)T var(W ⊥D)−T cov(W, A) += A − W T var(W)−1 cov(W, A) = A⊥W . +Choosing Y and D for A, we get +βY ∼D|X,W ∗ = cov(Y ⊥W ∗, D⊥W ∗) +var(D⊥W ∗) += cov(Y ⊥W , D⊥W ) +var(D⊥W ) += βY ∼D|X,W . +Since W ∗ is one-dimensional, we can use Lemma 2 to find a precise characterization +for the difference between the OLS estimand that does not and does adjust for W: +βY ∼D|X − βY ∼D|X,W = βY ∼D|X − βY ∼D|X,W ∗ = RY ∼W ∗|D fD∼W ∗ σY ∼D +σD +. +(26) +Moreover, the explanatory capabilities of W and W ∗ for Y are identical. According +to Lemma 4 (iii), we infer +Y ⊥D,W = Q(D,W)Y = QD⊥W QW Y = QD⊥W ∗Y ⊥W ∗ = Y ⊥D,W ∗ +which yields +R2 +Y ∼W|D = 1 − var(Y ⊥D,W ) +var(Y ⊥D) += 1 − var(Y ⊥D,W ∗) +var(Y ⊥D) += R2 +Y ∼W ∗|D. +The new random variable W ∗ fully captures the effect of W on Y but does not +capture the entire effect of W on D due to the reduced dimension, i.e. R2 +D∼W ≥ +R2 +D∼W ∗. To prove this result, we rewrite D⊥W using Lemma 4 (iii) as follows +D⊥W = QPW ∗W+QW ∗W D = Q(W ∗,QW ∗W)D = QW ⊥W ∗D⊥W ∗. +Based on this equation, Lemma 4 (iv) yields the inequality +var(D⊥W ) ≤ var(QW ⊥W ∗D⊥W ∗) + var(PW ⊥W ∗D⊥W ∗) = var(D⊥W ∗), +which implies +R2 +D∼W = 1 − var(D⊥W ) +var(D) +≥ 1 − var(D⊥W ∗) +var(D) += R2 +D∼W ∗. +Returning to (26), we use the equality and inequality derived for the R2-values con- +cerning W ∗ → Y and W ∗ → D, respectively. Since f2 is a monotone transformation +of R2, we have +|βY ∼D|X − βY ∼D|X,W |2 ≤ R2 +Y ∼W|D,X f2 +D∼W|X +σ2 +Y ∼D+X +σ2 +D∼X +. + +Sensitivity Analysis with the R2-calculus +42 +In presence of multiple unmeasured confounders, finding an interpretable chara- +terization of the difference βY ∼D|X,Z − βY ∼D|X,Z,U becomes more complicated. In +the main text, we use a telescoping expansion and repeatedly apply Lemma 2 to ob- +tain equation (6). The sensitivity parameters in this characterization, however, are +not symmetric in the set of partialed out variables which impedes their interpreta- +tion. Under the additional assumption that the components of U are conditionally +independent given (X, Z), a symmetric representation can be obtained. +The following result is closely related to Wright’s path analysis. +Our proof, +however, only relies on the algebraic relationships of the R2-calculus and does not +consult the underlying DAG. +Lemma 6. Assume the setting of Lemma 3 and further suppose that all components +of W are conditionally independent given X. Then, +βY ∼D|X − βY ∼D|X,W = +l +� +j=1 +βY ∼Wj|X,D,W−j βWj∼D|X, +(27) +where W−j = (W1, . . . , Wj−1, Wj+1, . . . , Wl). +Proof. For ease of notation, we only indicate partialing out X in the estimands and +drop the X-dependence in the other quantities. +Due to the conditional independence assumption and Lemma 4 (ii), we can +decompose Y as follows +Y = Y ⊥D,W + PD⊥W Y + +l +� +j=1 +PWjY. +Plugging this relationship into the definition of βY ∼D|X, using linearity of the co- +variance and the formula for projections on a one-dimensional space (25) yields +βY ∼D|X = cov(Y, D) +var(D) += 0 + cov(PD⊥W Y, D) +var(D) ++ +l +� +j=1 +cov(PWjY, D) +var(D) += +1 +var(D) +� +�cov +�cov(Y, D⊥W ) +var(D⊥W ) +D⊥W , D +� ++ +l +� +j=1 +cov +�cov(Y, Wj) +var(Wj) +Wj, D +�� +� += cov(Y ⊥W , D⊥W ) +var(D) ++ +l +� +j=1 +cov(Y, Wj) +var(Wj) +cov(D, Wj) +var(D) += βY ∼D|X,W +σ2 +D∼W +σ2 +D ++ +l +� +j=1 +RY ∼Wj +σY +σWj +βWj∼D. +By applying the definition of the R2-value, we derive +βY ∼D|X − βY ∼D|X,W = −βY ∼D|X,W R2 +D∼W + +l +� +j=1 +RY ∼Wj +σY +σWj +βWj∼D. + +Sensitivity Analysis with the R2-calculus +43 +Next, we use rule [ii] of the R2-calculus – independent additivity – on R2 +D∼W and +rewrite βY ∼D|X,W in terms of R-values and σ-values, i.e. standard deviations: +βY ∼D|X − βY ∼D|X,W +[ii] += −RY ∼D|W +σY ∼W +σD∼W +l +� +j=1 +R2 +D∼Wj + +l +� +j=1 +RY ∼Wj +σY +σWj +βWj∼D += +l +� +j=1 +βWj∼D +� +RY ∼Wj +σY +σWj +− +σD +RD∼WjσWj +RY ∼D|W +σY ∼W +σD∼W +R2 +D∼Wj +� +In order to extract the factor σY ∼D+W−j/σWj∼D+W−j, we apply rule [iii] – decom- +position of unexplained variance – six times and arrive at +βY ∼D|X − βY ∼D|X,W +[iii] += +l +� +j=1 +βWj∼D +σY ∼D+W−j +σWj∼D+W−j +� +RY ∼Wj +� +� +� +�1 − R2 +Wj∼D+W−j +1 − R2 +Y ∼D+W−j +− RY ∼D|W RD∼Wj +� +� +� +�(1 − R2 +Wj∼D+W−j)(1 − R2 +Y ∼Wj|W−j) +(1 − R2 +D∼W )(1 − R2 +Y ∼D|W−j) +� +. +(28) +We concentrate on the term in brackets, denoted by Tj. Invoking rule [v] – reduction +of partial correlation – and the (conditional) independence assumption, we infer +RY ∼Wj +[v] += RY ∼Wj|W−j +� +1 − R2 +Y ∼W−j, +RD∼Wj +[v] += RD∼Wj|W−j +� +1 − R2 +D∼W−j, +R2 +Wj∼D+W−j = R2 +Wj∼D|W−j. +We insert these relationships into the expression of Tj and simplify it via rule [iii]. +Then, we apply rule [iv] – recursion of partial correlation – on RY ∼D|W and simplify +the resulting expression +Tj = RY ∼Wj|W−j +� +1 − R2 +Wj∼D|W−j +� +� +� +� 1 − R2 +Y ∼W−j +1 − R2 +Y ∼D+W−j +− RY ∼D|W RD∼Wj|W−j +� +1 − R2 +D∼W−j +� +� +� +�(1 − R2 +Wj∼D|W−j)(1 − R2 +Y ∼Wj|W−j) +(1 − R2 +D∼W )(1 − R2 +Y ∼D|W−j) +[iii] += RY ∼Wj|W−j +� +� +� +�1 − R2 +D∼Wj|W−j +1 − R2 +Y ∼D|W−j +− RY ∼D|W RD∼Wj|W−j +� +� +� +�1 − R2 +Y ∼Wj|W−j +1 − R2 +Y ∼D|W−j +[iv] += RY ∼Wj|W−j +� +� +� +�1 − R2 +D∼Wj|W−j +1 − R2 +Y ∼D|W−j +−RD∼Wj|W−j +RY ∼D|W−j−RY ∼Wj|W−jRD∼Wj|W−j +� +1 − R2 +Y ∼D|W−j +� +1 − R2 +D∼Wj|W−j + +Sensitivity Analysis with the R2-calculus +44 += +RY ∼Wj|W−j(1 − R2 +D∼Wj|W−j) − RY ∼D|W−jRD∼Wj|W−j − RY ∼Wj|W−jR2 +D∼Wj|W−j +� +1 − R2 +Y ∼D|W−j +� +1 − R2 +D∼Wj|W−j += RY ∼Wj|W−j − RY ∼D|W−j RWj∼D|W−j +� +1 − R2 +Y ∼D|W−j +� +1 − R2 +Wj∼D|W−j += RY ∼Wj|D,W−j. +Returning to equation (28), we plug in Tj = RY ∼Wj|D,W−j and thus finish the proof +βY ∼D|X − βY ∼D|X,W = +l +� +j=1 +βWj∼D +σY ∼D+W−j +σWj∼D+W−j +RY ∼Wj|D,W−j += +l +� +j=1 +βY ∼Wj|D,W−j βWj∼D. +Lemma 6 helps us express the bias of OLS and k-class estimands in terms of par- +tial R-values which serve as sensitivity parameters. Whether these are intuitive, de- +pends on the causal structure of the underlying DAG. In the case of two independent +unmeasured variables U1 and U2 which confound or mediate β – the direct effect of +D on Y –, the sensitivity parameters (RD∼U1, RD∼U2) and (RY ∼U1|D,U2, RY ∼U2|D,U1) +are indeed intuitive. The former tuple targets the dependence between D and U, +the latter tuple focuses on the direct effects of U on Y regressing out the remaining +variables. +The following theorem demonstrates how Lemma 6 can be applied. We identify +the bias of the k-class estimand in terms of the intuitive sensitivity parameters. +Theorem 2. Assume the setting of Theorem 1 and let U = (U1, U2) be a two- +dimensional random vector. Further, suppose U1 ⊥⊥ U2 | X, Z holds. Then, +βk − β = +� +fY ∼Z|X,D RD∼Z|X +1 − k + k R2 +D∼Z|X ++ +2 +� +j=1 +Rj fj +� +1 − f2 +j f2 +−j + +� +R−j +� +1−R2 +j +1−R2 +−j − Rj fj f−j +�2 +� +σY ∼X+Z+D +σD∼X+Z +, +where Rj and fj abbreviate RY ∼Uj|X,Z,D,U−j and fD∼Uj|X,Z, respectively, for j ∈{1, 2}. +Proof. Similarly to the proof of Theorem 1, we expand the difference βk − β as a +telescoping sum +βk − β = (βk − βY ∼D|X,Z) + (βY ∼D|X,Z − βY ∼D|X,Z,U), + +Sensitivity Analysis with the R2-calculus +45 +which allows us to deal with the two summands separately. Following from the +same arguments, the first summand equals +1 +1 − k(1 − R2 +D∼Z|X)(βY ∼D|X − βY ∼D|X,Z) = fY ∼Z|X,D RD∼Z|X +1 − k + k R2 +D∼Z|X +σY ∼X+Z+D +σD∼X+Z +. +From here onwards, partialing out X and Z is only indicated in the estimands. In +order to rewrite and simplify the second summand, we invoke Lemma 6 and the +rule on decomposition of unexplained variance +βY ∼D|X,Z − βY ∼D|X,Z,U = +2 +� +j=1 +RY ∼Uj|D,U−j +σY ∼D+U−j +σUj∼D+U−j +RD∼Uj +σUj +σD +[iii] += +2 +� +j=1 +RY ∼Uj|D,U−jRD∼Uj +� +� +� +� 1 − R2 +Y ∼U−j|D +1 − R2 +Uj∼D+U−j +σY ∼D +σD +. +(29) +Due to rule [i] and the conditional independence assumption, RU1∼U2 = 0 holds. +This result can be used to rewrite RUj∼U−j|D via the recursive partial correlation +formula [iv]; moreover, we use the decomposition of unexplained variance [iii] on +1 − R2 +Ui∼D+U−i as follows +RUj∼U−j|D +[iv] += RUj∼U−j − RUj∼D RU−j∼D +� +1 − R2 +Uj∼D +� +1 − R2 +U−j∼D += −fD∼Uj fD∼U−j, +(30) +1 − R2 +Ui∼D+U−i +[iii] += (1 − R2 +D∼Ui)(1 − R2 +Ui∼U−i|D). +Inserting these relationships into (29), we find +βY ∼D|X,Z − βY ∼D|X,Z,U = +2 +� +j=1 +RY ∼Uj|D,U−j fD∼Uj +� +� +� +� 1 − R2 +Y ∼U−j|D +1 − f2 +D∼U1f2 +D∼U2 +σY ∼D +σD += +2 +� +j=1 +Rj fj +� +1 − R2 +Y ∼U−j|D +1 − f2 +1 f2 +2 +σY ∼D +σD +. +(31) +Lastly, we aim to express +� +1 − R2 +Y ∼U−j|D in terms of the other sensitivity param- +eters. To this end, we use the three-variable identity [vi] with Y ≡ Y , X ≡ U−j, +W ≡ Uj and Z ≡ D, where we replace RUj∼U−j|D according to (30). +R−j +� +1 − R2 +−j +� +1 − R2 +j − f1f2 Rj +[vi] += fY ∼U−j|D +� +1 − f2 +1 f2 +2 . +By definition, the identity +√ +1 − R2 = 1/ +� +1 + f2 holds true for any (partial) R2 + +Sensitivity Analysis with the R2-calculus +46 +and its corresponding f2. Thus, we get +� +1 − R2 +Y ∼U−j|D = +� +����1 + +� +R−j +√ +1−R2 +−j +� +1 − R2 +j − f1f2 Rj +�2 +1 − f2 +1 f2 +2 +� +���� +−1/2 +. +Substituting the +� +1 − R2 +Y ∼U−j|D term in (31) for the expression above proves the +form of the second summand that was required. +C. +Derivation of Constraints in Section 4 +C.1. +Ordinary Least Squares +Specifying a comparative bound on U → Y that partials out D involves two addi- +tional sensitivity parameters, RY ∼U|X,Z and RY ∼U| ˜ +X, ˙XI,Z,D. The former is related +to RY ∼U|X,Z,D via equation (15); hence, it remains to find a relationship that con- +nects RY ∼U| ˜ +X, ˙XI,Z,D to the other sensitivity parameters. To this end, we employ +rule [v] – reduction of partial correlation – and the recursive partial correlation +formula [iv] for RY ∼U| ˜ +X, ˙XI,Z,D and infer +RY ∼U|X,Z +[v] += +RY ∼U| ˜ +X, ˙XI,Z +� +1 − R2 +Y ∼ ˙XIc| ˜ +X, ˙XI,Z +[iv] += +1 +� +1 − R2 +Y ∼ ˙XIc| ˜ +X, ˙XI,Z +� +RY ∼D| ˜ +X, ˙XI,ZRD∼U| ˜ +X, ˙XI,Z ++ RY ∼U| ˜ +X, ˙XI,Z,D +� +1 − R2 +Y ∼D| ˜ +X, ˙XI,Z +� +1 − R2 +D∼U| ˜ +X, ˙XI,Z +� +. +This equation contains the unknown quantity RD∼U| ˜ +X, ˙XI,Z which can be expressed +in terms of RD∼U|X,Z via rule [v] +RD∼U|X,Z +[v] += +RD∼U| ˜ +X, ˙XI,Z +� +1 − R2 +D∼ ˙XIc| ˜ +X, ˙XI,Z +. +Plugging this relationship into the equation above, we arrive at the constraint +RY ∼U|X,Z = +1 +� +1 − R2 +Y ∼ ˙XIc| ˜ +X, ˙XI,Z +� +RY ∼D| ˜ +X, ˙XI,Z +� +1 − R2 +D∼ ˙XIc| ˜ +X, ˙XI,ZRD∼U|X,Z ++ RY ∼U| ˜ +X, ˙XI,Z,D +� +1 − R2 +Y ∼D| ˜ +X, ˙XI,Z +� +1 − R2 +D∼U|X,Z +� +1 − R2 +D∼ ˙XIc| ˜ +X, ˙XI,Z +� +� +. + +Sensitivity Analysis with the R2-calculus +47 +C.2. +Two-stage Least Squares +The comparative bound on U ↔ Z is in fact equivalent to a bound on the sensitivity +parameter RZ∼U|X. +First, we relate RZ∼U| ˜ +X, ˙X−j to RZ∼U|X via the conditional +independence assumption. Recursion of partial correlation [iv] yields +0 = RU∼ ˙Xj| ˜ +XI, ˙X−j,Z +[iv] += +RU∼ ˙Xj| ˜ +X, ˙X−j − RZ∼ ˙Xj| ˜ +X, ˙X−jRZ∼U| ˜ +X, ˙X−j +� +1 − R2 +Z∼ ˙Xj| ˜ +X, ˙X−j +� +1 − R2 +Z∼U| ˜ +X, ˙X−j +⇔ +RU∼ ˙Xj| ˜ +X, ˙X−j = RZ∼ ˙Xj| ˜ +X, ˙X−jRZ∼U| ˜ +X, ˙X−j. +Employing this relationship and rule [iv] again, we find +RZ∼U|X +[iv] += +RZ∼U| ˜ +X, ˙X−j − RZ∼ ˙Xj| ˜ +X, ˙X−jRU∼ ˙Xj| ˜ +X, ˙X−j +� +1 − R2 +Z∼ ˙Xj| ˜ +X, ˙X−j +� +1 − R2 +U∼ ˙Xj| ˜ +X, ˙X−j += RZ∼U| ˜ +X, ˙X−j +� +� +� +� +1 − R2 +Z∼ ˙Xj| ˜ +X, ˙X−j +1 − R2 +Z∼ ˙Xj| ˜ +X, ˙X−jR2 +Z∼U| ˜ +X, ˙X−j +. +As the right-hand side above is monotone in RZ∼U| ˜ +X, ˙Xj, we conclude +R2 +Z∼U|X ≤ bUZ R2 +Z∼ ˙Xj| ˜ +X, ˙X−j +1 − R2 +Z∼ ˙Xj| ˜ +X, ˙X−j +1 − bUZ R4 +Z∼ ˙Xj| ˜ +X, ˙X−j +. +If a practitioner specifies a comparative bound on Z → Y , we need to connect +RY ∼ ˙Xj| ˜ +X, ˙X−j,U,D to RD∼U|X,Z and RY ∼U|X,Z,D. To this end, we employ rule [vi] – +the three-variable identity – with Y ≡ Y , X ≡ ˙Xj, W ≡ U and Z ≡ ( ˜X, ˜X−j, D) +which yields +fY ∼ ˙Xj| ˜ +X, ˙X−j,Z,U,D +� +1 − R2 +Y ∼U|X,Z,D +[vi] += fY ∼ ˙Xj| ˜ +X, ˙X−j,Z,D +� +1 − R2 +U∼ ˙Xj| ˜ +X, ˙X−j,Z,D − RY ∼U|X,Z,D RU∼ ˙Xj| ˜ +X, ˙X−j,Z,D. +Furthermore, we use the conditional independence U ⊥⊥ +˙Xj | ˜X, ˙X−j, Z both to +simplify the following recursive partial correlation formula [iv] and to apply the +reduction of partial correlation formula [v] on RD∼U|X,Z +RU∼ ˙Xj| ˜ +X, ˙X−j,Z,D +[iv] += +RU∼ ˙Xj| ˜ +X, ˙X−j,Z − RD∼ ˙Xj| ˜ +X, ˙X−j,Z RD∼U| ˜ +X, ˙X−j,Z +� +1 − R2 +D∼ ˙Xj| ˜ +X, ˙X−j,Z +� +1 − R2 +D∼U| ˜ +X, ˙X−j,Z += −fD∼ ˙Xj| ˜ +X, ˙X−j,Z fD∼U| ˜ +X, ˙X−j,Z, +RD∼U| ˜ +X, ˙X−j,Z +[v] += RD∼U|X,Z +� +1 − R2 +D∼ ˙Xj| ˜ +X, ˙X−j,Z. + +Sensitivity Analysis with the R2-calculus +48 +Inserting these two relationships in the three-variable identity above and cancelling +some terms, we arrive at +fY ∼ ˙Xj| ˜ +X, ˙X−j,Z,U,D +� +1 − R2 +Y ∼U|X,Z,D = +� +fY ∼ ˙Xj| ˜ +X, ˙X−j,Z,D +� +1 − R2 +D∼U|X,Z ++ RY ∼U|X,Z,D RD∼ ˙Xj| ˜ +X, ˙X−j,Z RD∼U|X,Z +��� +1 − R2 +D∼U|X,Z(1 − R2 +D∼ ˙Xj| ˜ +X, ˙X−j,Z). +D. +Solving the Optimization Problem +Since users can specify any number and kind of bounds on the sensitivity parameters, +the resulting constraint set Ψ(ˆθ) is potentially very complex. It may be non-convex +and can contain multiple non-linear equality- and inequality constraints. This only +leaves few standard optimization algorithms to compute a global solution for (2). +These, however, often require careful choice of hyper-parameters and sometimes fail +to solve the problem. For this reason, we propose an adapted grid search algorithm +that is more robust and tailored to our specific optimization problem by exploiting +the structure of β. First, we characterize the set of potential minimizers and max- +imizers; then, we explain how we can use monotonicity of equality constraints to +reduce the number of dimensions of the grid search algorithm; finally, we give the +pseudocode of the algorithm and discuss its computational complexity. +D.1. +Characterization of the Solution +According to Theorem 1, the objective β is identified in terms of the sensitivity +parameters (ψ1, ψ2) = (RD∼U|X,Z, RY ∼U|X,Z,D). Due to its monotonicity in ψ2, the +objective β attains its optimal values on a subset of the boundary of Ψ(ˆθ). In order +to show this, we characterize the feasible set as +Ψ(ˆθ) = +� +ψ1 : Pψ1̸=∅ +Pψ1, where +Pψ1 = {ψ2 : (ψ1, ψ2) ∈ Ψ(ˆθ)}. +For every fixed ψ1 such that Pψ1 ̸= ∅, the objective β is a linear function in ψ2. +This implies that, for any ψ2 ∈ Pψ1, we obtain +β(ˆθ, ψ1, min Pψ1) ⋚ β(ˆθ, ψ1, ψ2) ⋚ β(ˆθ, ψ1, max Pψ1), +where the direction of the inequalities depends on the sign of ψ1. Therefore, ψ-values +that minimize/maximize β are contained in +Ψ∗(ˆθ) := +� +ψ1 : Pψ1̸=∅ +{min Pψ1, max Pψ1}, +(32) +which is a subset of the boundary of Ψ(ˆθ). +Therefore, it suffices to discretize the set Ψ∗(ˆθ) instead of Ψ(ˆθ) to find an ap- +proximate solution to the optimization problem. + +Sensitivity Analysis with the R2-calculus +49 +D.2. +Transfering Bounds via Monotonicity +Regular grid search algorithms are highly computationally expensive as their com- +plexity grows exponentially in the number of unknown parameters. Yet, the high +computational costs can be significantly reduced by leveraging the monotonicity of +many equality constraints. We illustrate this with an example. +Example 1. Suppose a practitioner specifies the following direct constraint on U → D +and comparative constraint on U → Y : +RD∼U|X,Z ∈ [−0.5, 0.5], +R2 +Y ∼U| ˜ +X, ˙X−j,Z ≤ 2R2 +Y ∼ ˙Xj| ˜ +X, ˙X−j,Z ⇔ R2 +Y ∼U|X,Z ≤ 2 f2 +Y ∼ ˙Xj| ˜ +X, ˙X−j,Z, +(33) +where the latter equivalence is due to (14). +In addition, the unknown parame- +ters RD∼U|X,Z, RY ∼U|X,Z,D and RY ∼U|X,Z are constrained by the recursive partial +correlation formula +RY ∼U|X,Z,D +[iv] += RY ∼U|X,Z − RY ∼D|X,Z RD∼U|X,Z +� +1 − R2 +Y ∼D|X,Z +� +1 − R2 +D∼U|X,Z +. +(34) +Note that, for any fixed RD∼U|X,Z value, RY ∼U|X,Z,D is a linear, and hence, mono- +tone function of RY ∼U|X,Z. +In this setting, brute-force grid search creates a three-dimensional grid of points +– one dimension per unknown partial R-value – and only keeps those that (approxi- +mately) conform with (33) and (34). (Partial R- and f-values that only depend on V +are estimated.) The remaining points are projected onto the (RD∼U|X,Z, RY ∼U|X,Z,D)- +plane and, for every fixed RD∼U|X,Z, we can find the smallest/largest value of +RY ∼U|X,Z,D to approximate Ψ∗(ˆθ). Hence, in this example, the complexity of brute- +force grid search is cubic in the number of points per dimension. +Our algorithm, on the other hand, only needs to create a one-dimensional grid +of RD∼U|X,Z values, i.e. discretize [−0.5, 0.5]. For every such value, we can compute +the smallest value for RY ∼U|X,Z,D by plugging RY ∼U|X,Z = − +√ +2 | ˆfY ∼ ˙Xj| ˜ +X, ˙X−j,Z| +into (34) directly and likewise for the largest value. Therefore, the complexity only +grows linearly in the number of points per dimension. +This principle of using monotonicity of the equality constraints can reduce the +dimension of the grid and applies beyond the above example. In fact, when only +bounds on U → D and U → Y are given, we solely require a one-dimensional grid. +Hence, the computational complexity of generating equally spaced points in Ψ∗(ˆθ) +grows linearly in the number of grid points. In the most general case, when any +(finite) number and kind of bound can be specified, only a three-dimensional grid is +needed. Hence, the worst complexity of the point-generation algorithm is cubic in +the number of points per dimension. Evaluating the objective over Ψ∗(ˆθ) has linear +complexity in any case. +D.3. +Adapted Grid-Search Algorithm +Our proposed algorithm first constructs a set of equally spaced points that are +(approximately) contained in Ψ∗(ˆθ); then, it evaluates β over this set and takes the + +Sensitivity Analysis with the R2-calculus +50 +Algorithm 1: Grid approximation of Ψ∗(ˆθ) +Input: lower and upper bounds given by Al, Au, Bl, Bu, Dl, Du, Eu, El, +Ml, Mu, Ol, Ou, bZY +Output: vectors A, L and U +1 al ← max{Al}; +au← min{Au} +2 ml← max{Ml}; +mu← min{Mu} +3 Initialize A, L, U ∈ RNa +4 for i ∈ [Na] do +5 +Ai ← al + (i − 1) (au − al)/(Na − 1) +6 +dl ← max{hd(Ai, El, c2, c3, c4), Dl} +/* Pushing bounds onto b */ +7 +du ← min{hd(Ai, Eu, c2, c3, c4), Du} +8 +bl ← max{hb(Ai, dl), Bl} +9 +bu ← min{hb(Ai, du), Bu} +10 +if bl > bu then +11 +Ai ← NA +12 +Li ← NA +13 +Ui ← NA +14 +else +15 +fgl← hfg(Ai, Ml) +/* Pushing bounds onto g */ +16 +fgu← hfg(Ai, Mu) +17 +gl ← fgl/ +� +1 + f2 +gl +18 +gu ← fgu/ +� +1 + f2gu +19 +found ← False +/* Finding Li */ +20 +for j ∈ [Nb] and not found do +21 +Bij ← bl + (j − 1) (bu − bl)/(Nb − 1) +22 +fq ← hfq(Ai, Bij, c7) +/* Computing bounds on o */ +23 +ol ← max{− +� +bZY · f2q /(1 + f2q ), Ol} +24 +ou ← min{ +� +bZY · f2q /(1 + f2q ), Ou} +25 +if ol <= ou then +26 +for k ∈ [Ng] and not found do +27 +Gik ← gl + (j − 1) (gu − gl)/(Ng − 1) +28 +fo ← hfo(Bij, Gik) +29 +o ← fo/ +� +1 + f2o +30 +found ← found ∨(ol ≤ o ∧ o ≤ ou) +31 +if found then +32 +Li ← Bij +33 +if not found then +34 +Li ← NA +35 +found ← False +/* Finding Ui */ +36 +. . . +/* Analogously to Li but the Bij decrease */ +37 return A, L, U + +Sensitivity Analysis with the R2-calculus +51 +minimum/maximum of the obtained β-values. The latter step is straightforward +whereas the former is complex when multiple interlocking constraints are present. +In order to keep the the notation short, we introduce some abbreviations: +a = RD∼U|X,Z, +b = RY ∼U|X,Z,D, +d = RY ∼U|X,Z, +e = RY ∼U| ˜ +X, ˙XI,Z,D, +g = RZ∼U|X,D, +m = RZ∼U|X, +o = RY ∼Z|X,U,D, +q = RY ∼ ˙Xj| ˜ +X, ˙X−j,Z,U,D, +c1 = RY ∼D|X,Z, +c2 = RY ∼D| ˜ +X, ˙XI,Z, +c3 = RD∼ ˙XIc| ˜ +X, ˙XI,Z, +c4 = RD∼ ˙XIc| ˜ +X, ˙XI,Z, +c5 = RD∼Z|X, +c6 = RY ∼Z|X,D, +c7 = RY ∼ ˙Xj| ˜ +X, ˙X−j,Z,D, +hb(a, d) = +d − c3 a +� +1 − c2 +3 +√ +1 − a2 , +hd(a, e, c2, c3, c4) = +1 +� +1 − c2 +4 +� +e +� +1 − c2 +2 +� +1 − a2(1 − c2 +3) + c2 +� +1 − c2 +3 a2 +� +, +hfg(a, m) = +1 +√ +1 − a2 +�� +1 − c2 +5 · fm − c5 a +� +, +hfo(b, g) = +1 +√ +1 − b2 +�� +1 − g2 · fc6 − b g +� +, +hfq(a, b, c7) = +√ +1 − a2 · fc7 + c7 a b +√ +1 − b2� +1 − a2(1 − c2 +7) +. +With a slight abuse of notation, the parameter e and the associated constants +c2, c3 and c4 as well as q and the associated constant c7 may be scalars or vectors +depending on the number of (13)- and (22)-constraints, respectively. The notation +fs is a shorthand for the f-transformation of some scalar or vector, that is fs = +s/ +√ +1 − s2. The functions hb and hd are abbreviations for the right-hand sides of +the equations (15) and (16), hfo and hfg stem from (17) and (18) and hfq states +(23) in the new notation. Inserting vectors instead of scalars into the functions is +interpreted as componentwise evaluation. +In order to compute a set of points that is approximately contained in Ψ∗(ˆθ), we +first discretize the interval of all possible a-values and construct the vector A ∈ RNa +which contains Na equally spaced points. +(This corresponds to discretizing the +interval [min{ψ1 : Pψ1 ̸= ∅}, max{ψ1 : Pψ1 ̸= ∅}].) Second, we construct the vectors +L, U ∈ RNa which approximate the corresponding minima and maxima of β at the +respective a-value. Thus, we can create the points +{(Ai, Li): i ∈ [Na]} ∪ {(Ai, Ui): i ∈ [Na]}, +which are (approximately) contained and equally spaced in Ψ∗(ˆθ). Evaluating the +objective β over this set has complexity O(Na). +In case that only bounds on U → D and U → Y are specified, the computa- +tional complexity of generating A, L and U grows linearly in Na and the computed +points are actually elements of Ψ∗(ˆθ) instead of merely approximating it. The two +types of bounds on U → D specify direct constraints on a. Hence, denoting the + +Sensitivity Analysis with the R2-calculus +52 +vectors of upper and lower bounds on a stemming from (9) and (10) Al and Au, we +can construct A by equally spacing Na points in the interval [max{Al}, min{Au}]. +Bounds on U → Y directly constrain b (11), d (14) and e (13). Crucially, for every +fixed a-value Ai, the functions hd and hb are linear in e and d, respectively. Hence, +we can transfer bounds on e onto d and, thus, update bounds on d; likewise, we can +then push forward the bounds on d onto b and compute Li and Ui. +In case that at least one bound on U ↔ Z or Z → Y is specified, A can be +constructed in the same way as before whereas L and U are more computationally +involved. We again use the observation that many h-functions are monotone in one +argument in order to ”push forward” bounds. For fixed a-value, we can transfer +bounds on m onto g; for fixed a- and b- value, we can compute bounds on o; for fixed +a- and g-value, we can compute the corresponding o-value. We construct Li and Ui +by discretizing the range of possible b-values (after successively pushing bounds on +e and d onto b) into Nb points and searching for the smallest/largest feasible value. +To test whether a given b-value is feasible, we construct a sequence of Ng values +of g and check whether there is at least one value such that the bounds on o are +satisfied. Therefore, the computational complexity of constructing A, L and U is +O(Na · Nb · Ng). +Algorithm 1 contains the pseudocode of the algorithm to generate A, L and U. +It concerns the case where at least one bound on U ↔ Z or Z → Y is specified. +Otherwise, we could directly set Li ← bl and Ui ← bu in line 15. +A full implementation of the algorithm will be made available in a public Github +repository soon. +E. +Simulation Study +We investigate the empirical coverage of sensitivity intervals computed with the +bootstrap in two scenarios: a regression model with one additional covariate and +an instrumental variable model. In both set-ups, we set the nominal level to 90 %. +E.1. +Linear Regression Simulation +We generate a sample of n i.i.d. random vectors (εU, εX, εX, εY )T ∼ N(0, Id) and +compute the variables in the model using the following linear structural equations: +U := εU, +X := εX, +D := X + U + εD, +Y := D + 2X + U + εY . +Based on these structural equations, we derive the covariance matrix of the involved +random variables +var +� +� +� +� +� +U +X +D +Y +� +� +� +� +� += +� +� +� +� +1 +0 +1 +2 +0 +1 +1 +3 +1 +1 +3 +6 +2 +3 +6 +15 +� +� +� +� . + +Sensitivity Analysis with the R2-calculus +53 +It can be used to compute (partial) R-values as well as the bias βY ∼D|X − β = 1/2. +If the comparative constraints +R2 +D∼U ≤ R2 +D∼X, +R2 +Y ∼U ≤ 4 +9 R2 +Y ∼X +are specified, the partially identified region is [1, (3 + +√ +3)/2]. Hence, the true value +β = 1 equals the lower end of the PIR. The bounds above are sharp in the sense +that the lower end of the partially identified range can only be reached when both +inequalities are active, i.e. hold with equality. +In order to construct sensitivity intervals, we generate bootstrap samples of the +observed data and solve the corresponding optimization problems. Then, we use +either percentile or basic bootstrap (Davison and Hinkley, 1997, chap. 5) to compute +the lower and upper end of the sensitivity interval. This approach is compared to +the heuristic sensitivity intervals of Cinelli and Hazlett (2020) as well as the oracle +90% confidence interval, which could be computed if U was observed. +We simulate data for different sample sizes n and repeat each such experi- +ment 1000 times to compute the empirical coverage and length of the sensitiv- +ity/confidence intervals. More specifically, we evaluate the empirical coverage of β +and the PIR for different sensitivity intervals and adapt the notion of length. In or- +der to account for the fact that the length of typical confidence intervals approaches +zero as n → ∞ whereas the length of valid sensitivity intervals is lower bounded by +the length of the PIR, we use the distance between the lower end of an interval and +1, when it covers 1, as length. +The results of this simulation study are summarized in Table 2. Percentile boot- +strap exhibits coverage of PIR close to the envisaged level of 90%; its coverage of +β is close to 95%. The latter is expected as the true value of β is the lower end of +the PIR. By comparison, the empirical coverage of sensitivity intervals constructed +via basic bootstrap is 5 to 10 percentage points below the required level. Hence, +we use percentile bootstrap to construct sensitivity intervals in the data example +in the main text. Moreover, this simulation study illustrates that Cinelli and Ha- +zlett’s heuristic sensitivity intervals do not possess frequentist coverage guarantees: +the empirical coverage of the PIR is consistently below 50%. Finally, we see that +sensitivity intervals are substantially longer than the oracle confidence interval. We +attribute the increased length to the uncertainty stemming from estimating the con- +straints. In this simulation study, we did not encounter cases where the estimated +constraint set was empty on a bootstrap sample. +In order to investigate the coverage of bootstrap sensitivity intervals more closely, +we consider the distribution of the estimated upper and lower end of the PIR as +well as the corresponding bootstrap distributions. Figure 7 depicts the estimates +of these distributions based on 1000 repitions of the experiment. For small sample +sizes n, we notice that the bootstrap distribution is both biased and skewed. Both +phenomena diminish as n grows so that the bootstrap distribution approximates the +target distribution more closely. This is in line with the observation that coverage +improves for larger sample sizes, especially for basic bootstrap. + +Sensitivity Analysis with the R2-calculus +54 +Table 2. Simulation results of the linear regression example. +n +Method +Coverage +Length +β +PIR +Mean +Median +200 +Percentile bootstrap +95.5% +92.6% +2.800 +0.748 +Basic bootstrap +86.1% +78.9% +0.533 +0.310 +Heuristic +73.7% +47.6% +0.339 +0.213 +Oracle +90.2% +- +0.127 +0.124 +500 +Percentile bootstrap +96.1% +92.7% +0.431 +0.320 +Basic bootstrap +88.4% +81.1% +0.247 +0.197 +Heuristic +71.7% +44.0% +0.160 +0.126 +Oracle +88.8% +- +0.079 +0.077 +1000 +Percentile bootstrap +94.1% +90.6% +0.240 +0.207 +Basic bootstrap +87.5% +82.6 % +0.172 +0.144 +Heuristic +69.5% +45.7% +0.111 +0.089 +Oracle +90.2% +- +0.054 +0.053 +2000 +Percentile bootstrap +95.8% +91.7% +0.148 +0.135 +Basic bootstrap +90.8% +85.9% +0.117 +0.106 +Heuristic +70.9% +42.8% +0.073 +0.063 +Oracle +90.5% +- +0.039 +0.037 +E.2. +Linear Instrumental Variable Simulation +We generate 100 i.i.d. samples from the distribution (εU, εZ, εD, εY )T ∼ N(0, Id) +and compute the variables of the model as follows +U := εU, +Z := εZ, +D := Z + U + εD, +Y := D + U + εY . +This data-generating process fulfills the instrumental variable assumptions which +renders β = 1 point identified. Hence, a sensitivity interval where the IV-related sen- +sitivity parameters are set to zero ought to be comparable with the confidence inter- +val that is based on the asymptotic normality of the TSLS estimator. In order to use +Algorithm 1, we slightly relax the IV assumptions requiring RZ∼U|X, RY ∼Z|X,U,D ∈ +[−0.002, 0.002] and further set RD∼U|X,Z ∈ [−0.999, 0.999] to bound it away from +−1 and 1. +We compute the empirical coverage and length of sensitivity intervals constructed +via percentile and basic bootstrap, the heuristic sensitivity intervals and the oracle +confidence intervals over 500 repitions of the experiment. Due to the high compu- +tational costs, we conduct this simulation study only for sample size n = 100. +The results of this experiment are stated in Table 3. We notice that the bootstrap +sensitivity intervals are on par with the oracle confidence interval, both in terms + +Sensitivity Analysis with the R2-calculus +55 +n = 1000 +n = 2000 +n = 200 +n = 500 +0 +1 +2 +3 +0 +1 +2 +3 +0 +2 +4 +6 +0 +2 +4 +6 +PIR lower +PIR lower - Boot +PIR upper +PIR upper - Boot +Fig. 7. +Empirical distribution of the lower and upper end of the PIR as well as the corre- +sponding bootstrap distributions. +of coverage and length. By contrast, the heuristic sensitivity intervals exhibit very +high coverage but their length is too long to be informative in practice. In this +simulation study, 24 of the 500 · 500 = 250, 000 constructed bootstrap samples led +to an empty constraint set. In these cases, we set the solution of the optimization +problem to −∞ and ∞, respectively. +F. +Choice of Hyper-parameters +In Table 4, we list the hyper-parameters of Algorithm 1 that were used for different +data analyses. The mesh size of the grid is the same in every dimension, that is +Table 3. Simulation results of the instrumental variable +example. +Method +Coverage +Length +Mean +Median +Percentile bootstrap +91.2% +0.338 +0.301 +Basic bootstrap +94.8% +0.266 +0.240 +Heuristic +99.0% +1.085 +0.581 +Oracle +92.0% +0.290 +0.257 + +Sensitivity Analysis with the R2-calculus +56 +Table 4. +Hyper-parameters for different +plots and simulation examples. +Ngrid +Nb-contour +Nboot +Figure 2 +200 +- +500 +Figure 3 +200 +30 +- +Figure 4 +150 +30 +- +Figure 5 +400 +- +- +Figure 6 +300 +- +- +Table 2 +200 +- +500 +Table 3 +100 +- +500 +the numbers of points considered per dimension Na, Nb, and Ng are equal. We +define Ngrid := Na = Nb = Ng. The number of points per dimension for b-contour +plots and the number of bootstrap samples are denoted by Nb-contour and Nboot, +respectively. +In the simulation study and data example in this work, we found that the PIR +estimates change only marginally for values of Ngrid larger than 200. We recommend +to consider at least 100 points per grid dimension, i.e. Ngrid = 100. The rough struc- +ture of the b-contours often becomes apparent for Nb-contour as low as 10. Due to the +computational costs of the optimization problem, we choose a relatively low number +of bootstrap samples Nboot = 500. The simulation studies empirically confirm that +percentile bootstrap sensitivity intervals achieve good coverage nonetheless. + diff --git a/49AyT4oBgHgl3EQfQPZI/content/tmp_files/load_file.txt b/49AyT4oBgHgl3EQfQPZI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ecd7da764b92d1c5a616eb46c1462c66c1c9151f --- /dev/null +++ b/49AyT4oBgHgl3EQfQPZI/content/tmp_files/load_file.txt @@ -0,0 +1,1861 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf,len=1860 +page_content='Sensitivity Analysis with the R2-calculus Tobias Freidling and Qingyuan Zhao Statistical Laboratory, DPMMS, University of Cambridge, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' E-mail: taf40@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='uk Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Causal inference necessarily relies upon untestable assumptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' hence, it is crucial to assess the robustness of obtained results to violations of identification assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' However, such sensitivity analysis is only occasionally undertaken in practice, as many existing methods only apply to relatively simple models and their results are often difficult to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We take a more flexible approach to sensitivity analysis and view it as a constrained stochastic optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We focus on linear models with an unmeasured confounder and a potential instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We show how the R2-calculus—a set of algebraic rules that relates different (partial) R2- values and correlations—can be applied to identify the bias of the k-class estimators and construct sensitivity models flexibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We further show that the heuristic “plug-in” sensitivity interval may not have any confidence guarantees;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' instead, we propose a boostrap approach to construct sensitivity intervals which perform well in numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We illustrate the proposed methods with a real study on the causal effect of education on earnings and provide user-friendly visualization tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Keywords: Causal inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Instrumental variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' k-class estimator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Linear models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Partial identification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Stochastic optimization 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Introduction In many scientific disciplines, provisional causal knowledge is predominantly gen- erated from observational data as randomized controlled experiments are often in- feasible or too costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Because the treatment is not randomly assigned in an obser- vational study, any causal conclusions must rely on untestable assumptions, such as absence of unmeasured confounders or validity of instrumental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, the causal inference is inherently sensitive to violations of any identification and modelling assumptions, so reseachers are advised to investigate the robustness of their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The importance of sensitivity analysis has been emphasized in guidelines for designing and reporting observational studies (Vandenbroucke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' PCORI Methodology Committee, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For instance, the STROBE guidelines caution that “taking [observed] confounders into account is crucial in observational studies, but readers should not assume that analyses adjusted for [observed] confounders establish the ‘causal part’ of an association” (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 1638).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' They recommend to conduct sensitivity analyses as they are “helpful to investigate the influence of choices made in the statistical analysis, or to investigate the robustness of the findings to missing data or possible biases” (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 1647).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='00040v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='ME] 30 Dec 2022 Sensitivity Analysis with the R2-calculus 2 However, sensitivity analysis is still rarely being conducted in actual studies, leaving other researchers difficult to assess the robustness of their empirical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In medicine, Thabane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2013) did a spot check on the January 2012 editions of major medical journals and found that only 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='7% (36 out of 135) of the articles that included some statistical analysis also performed sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In nutrition research, de Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2016) found that, in a representative sample of 100 articles from 2013 to 2015, merely 18% of them conducted some sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In political science, Cinelli and Hazlett (2020) found that only 4 out of 64 observational studies published in three leading journals in 2017 conducted a formal sensitivity analysis beyond just some model specification checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' There are several reasons for the hesitant uptake of sensitivity analysis in prac- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' First, it is not straightforward to define a reasonable model for sensitivity analysis, even for the familiar setting of one treatment variable, one outcome vari- able, and multiple baseline covariates that has been studied since the seminal work of Cornfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For example, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1998) assume an unmeasured confounder U independent of the measured covariates X conditional on the treat- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' However, Hernan and Robins (1999) point out that this cannot be generally true as conditioning on the treatment opens a collider path between U and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For more complicated settings such as instrumental variables (IV), specifying a good sensitivity model is even more difficult and the literature on sensitivity analysis is considerably smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Second, many methods for sensitivity analysis were devel- oped under simple settings where closed-form solutions are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This results in a limited scope of applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Finally, it is often not easy for practitioners to understand and communicate the results of a sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In general, (non-Bayesian) sensitivity analysis can be broadly categorized into point identified and partially identified approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The former requires a precise specification of the confounding mechanism, so that the causal effect of interest is still identified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see for instance Rosenbaum and Rubin (1983), Imbens (2003), and VanderWeele and Arah (2011) for the usual observational study design, Scharfstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1999) for longitudinal studies with dropouts, and Altonji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2005) for instrumental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' On the other hand, the partially identified approach con- siders the union of many point identified sensitivity models, so the causal effect is only partially identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Examples include the first sensitivity analysis by Corn- field et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1959), the approach developed by Rosenbaum (1987, 2002) based on randomization tests, the E-value proposed by Ding and VanderWeele (2016) that generalizes the Cornfield bound, the generalization of Scharfstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1999) by Vansteelandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2006), bounds on the average treatment effect under Rosen- baum’s sensitivity model by Yadlowsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2022) and the marginal sensitivity model studied in Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2019) and Dorn and Guo (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In our experience, the partially identified approach is more flexible and usually aligns with practical demand better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This is why we adopt it in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We limit our discussion to linear regression and linear instrumental variable models, but the methodology we develope below is quite general and can potentially be ex- tended to other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Compared with previous work, a crucial distinction is that we do not require the partially identified region (or, as in Rosenbaum’s sensitivity Sensitivity Analysis with the R2-calculus 3 analysis, an upper bound of the randomization p-value) to have a closed form so- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Instead, we leverage a novel perspective on sensitivity analysis through the lens of constrained stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This is elaborated next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A General Framework for Sensitivity Analysis Consider an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' sample (Vi, Ui)n i=1 from some population, but only the vari- ables (Vi)n i=1 are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Denote the joint probability distribution of (Vi, Ui) as P = PV,U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Depending on the assumptions on the data generating process, the distribution P may be restricted to be within a parametric, semi-parametric or non-parametric family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The marginal distribution of V and the distribution of U conditional on V are denoted by PV and PU|V , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We are interested in estimating and conducting inference for some functional β = β(PV,U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For example, suppose V = (D, Y, X) includes a treatment variable D, an outcome Y , and some covariates X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We may be interested in estimating the causal effect of D on Y , which would be point identified if there are no other confounders given (X, U) and U is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' However, since U is not observed, β may only be partially identified if we restrict the “strength of confounding” for U in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In many cases, β can be expressed as a function of two types of parameters, θ = θ(PV ) and ψ = ψ(PV,U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The former only depends on the marginal distribution of V and can therefore be estimated from the observed variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' the latter addi- tionally depends on the distribution of U and thus cannot be directly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Adopting a Bayesian perspective, Gustafson (2005) and Daniels and Hogan (2008) advocate the use of a separable parameterization, meaning that ψ = ψ(PU|V ) only depends on the conditional distribution PU|V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In this set-up, no information about ψ can be learnt from the observed data, which has several advantages in deriving bounds or making Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' However, requiring a separable parameteriza- tion could be too restrictive in our experience and we will not make this assumption below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Since U is unobserved, the parameter ψ and thus the functional β cannot be identified from the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A point identified sensitivity analysis assumes that ψ is given, for example by eliciting the opinion of a domain expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In this sense, the primary analysis can be viewed as a special case of a point identified sensitivity analysis, where ψ takes the value (conventionally 0) that corresponds to the unobserved variable U being ”ignorable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To assess the robustness of the primary analysis, a partially identified sensitivity analysis assumes that ψ belongs to a set Ψ = Ψ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Comparing to point identified models, this is appealing because it is much easier for domain experts to specify a possible range of ψ than a specific value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' However, under the weaker condition ψ ∈ Ψ, the functional β is only partially identified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' we call the corresponding set of β-values the partially identified region (PIR): PIR(PV ) := � β(θ(PV ), ψ): ψ ∈ Ψ(θ(PV )) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1) The condition ψ ∈ Ψ(θ) in (1) implies a constraint on the joint distribution PV,U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 4 For this reason, we will refer to Ψ as the sensitivity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In general, the partially identified region can be quite complicated and difficult to infer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' However, this can be simplified in the case where β is real-valued and one-dimensional by seeking to solve the following optimization problems: min / max β(θ(PV ), ψ), subject to ψ ∈ Ψ(θ(PV )), (2) where the distribution PV is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' As both the objective and the feasible set in (2) depend on the unknown PV we can sample from, this is an instance of stochastic optimization or stochastic programming (Shapiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A natural, plug-in estimator of the optimal values of this problem can be obtained by solving min / max β(ˆθ, ψ), subject to ψ ∈ Ψ(ˆθ), (3) where ˆθ is an estimator of θ based on the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This can be viewed as a generalization of the sample average approximation (SAA) estimator in stochastic optimization (Shapiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', 2009, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Thus, a general recipe for partially identified sensitivity analysis is the following: (i) The functional β of interest is expressed in terms of the identifiable parameters θ = θ(PV ) and the sensitivity parameters ψ = ψ(PV,U);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (ii) The set of constraints ψ ∈ Ψ(θ) is specified by consulting domain experts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (iii) The optimal values of the stochastic program (2) are estimated either by first obtaining a closed-form solution to (2) and then estimating that quantity, or by directly solving the plug-in problem (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (iv) Suitable methods are then used to quantify the uncertainty of the estimators in the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In this article, we will focus on sensitivity analysis for linear regression and linear instrumental variables models in which θ and ψ are low-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Nevertheless, the general framework outlined above may also be suitable for problems involving high- or infinite-dimensional parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see Section 8 for more discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Interpretable Sensitivity Models using the R2-Calculus In practice, the usefulness of the partially identified region in (1) or the optimal values of (2) depends crucially on the interpretability of the sensitivity model Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This is where the R2-calculus can be extremely useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In short, the R2-value R2 Y ∼X, also known as coefficient of determination, measures how much variance of Y can be explained by linear combinations of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' An R2-value close to 1 indicates that X can explain a large degree of the variance of Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' on the other hand, values close to 0 indicate that the linear dependence between Y and X is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The partial R2-value R2 Y ∼X|Z naturally extends this idea and measures how much variance of Y can be explained by X given Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Due to their straightforward interpretation, R2- and partial R2-values are widely used to help practitioners interpret the results of sensitivity analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For instance, Sensitivity Analysis with the R2-calculus 5 Imbens (2003) uses them in sensitivity analysis for regression models with a discrete treatment variable and this idea is recently extended by Veitch and Zaveri (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Small (2007) measures the amount of violations to the instrumental variable as- sumptions by using R2-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Cinelli and Hazlett (2020) take this idea further and parameterize the bias of the linear regression estimator by solely using R2-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Other parameterizations that are not fully based on R2-values can be found in Hosman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2010) and Oster (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In this article, we extend this line of work and make several novel contributions: We use partial correlations (or R-values) instead of R2-values (which are just squared R-values) to parameterize the sensitivity model, so the direction of the confounder effect is naturally captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In contrast, previous works either use worst-case bounds implied by R2-values (Cinelli and Hazlett, 2020) or directly specify the sign of the bias in an additional sensitivity parameter (Zhang and Ding, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We provide a list of algebraic relations between R- and R2- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We give a proof of this R2-calculus from a general Hilbert space perspective which may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We give a general bias formula for the family of k-class estimators which in- cludes the ordinary least squares estimator and the two-stage least squares estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This allows us to provide a unified framework of sensitivity analy- sis for linear regression and instrumental variables models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Facilitated by the R2-calculus and the general bias formula, we allow users to specify very flexible constraints Ψ on the sensitivity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For example, we allow constraints that compare explanatory capability (in terms of the R2- value) of some unmeasured confounder U with that of a measured covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We show that the simple method of fixing the sample R2-value related to the unmeasured variable U in the sensitivity analysis, as proposed by Cinelli and Hazlett (2020), may not provide confidence statements in the frequentist sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Instead, we propose a bootstrap approach to obtain sensitivity intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We provide a suite of user-friendly plots to visualize the results of the sensi- tivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Organization of the Paper Section 2 describes the R2-calculus, a collection of algebraic rules that relate (par- tial) R2-values and correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Section 3 provides a general bias formula for the k-class estimator in presence of one unmeasured confounder and discusses extensions to multiple unmeasured confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Section 4 uses the R2-calculus to develop multiple ways for practitioners to spec- ify the constraints in Ψ(θ) based on domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Specifically, we provide comparative bounds on the sensitivity parameters that correspond to deviations from the no unmeasured confounders and the instrumental variable assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 6 Section 5 reviews some approaches to construct sensitivity intervals that contains β or the PIR with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We show that directly specifying sample R2- values as sensitivity parameter may not provide frequentist guarantees and propose an approach based on the bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Section 6 applies our proposed sensitivity analysis method to a famous study in labour economics by Card (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We consider both the linear regression and instrumental variable estimators and compare the results obtained by imposing different sensitivity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Section 7 introduces sensitivity contour plots that help to investigate how the choice of constraints affects the PIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' These plots are illustrated with the real data example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Finally, Section 8 concludes this article with a discussion of our method and an outlook on future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Readers who are more interested in applying the proposed method and interpret- ing its results may wish to skip Sections 2 to 5 initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proofs for some theoretical results in this article and a detailed description of the optimization algorithm can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R2-calculus We first give a summary of the R2-calculus – a set of widely used algebraic rules which concern the coefficient of determination (also called R2-value) and related quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Although these rules are often introduced together with the multivariate normal distribution (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Anderson, 1958, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5), they are purely algebraic and rely on no distributional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In fact, this calculus not only applies to the R2- and R-values in the population but also to their counterparts in the sample, which will be denoted by ˆR2 and ˆR below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For brevity, we will only state the definitions and results for the population values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Let Y be a random variable, let X and Z be two random vectors, and suppose they all have finite variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Without loss of generality, we suppose that all random variables and vectors have mean equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Otherwise, we can replace them with their centred versions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We use Y ⊥⊥ X | Z to denote that Y and X are independent conditional on Z as defined in Dawid (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Furthermore, the residual of Y after partialing/regressing out X is given by Y ⊥X := Y − XT var(X)−1 cov(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The variance of Y ⊥X equals that of the residual in the linear regression of Y on X, which motivates the notation σ2 Y ∼X = var(Y ⊥X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' let σY ∼X denote its square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Suppose σ2 Y ∼Z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The R2-value of Y on X is defined as R2 Y ∼X := 1 − σ2 Y ∼X σ2 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The partial R2-value and f2-value of Y on X given Z are defined as R2 Y ∼X|Z := R2 Y ∼X+Z − R2 Y ∼Z 1 − R2 Y ∼Z and f2 Y ∼X|Z := R2 Y ∼X|Z 1 − R2 Y ∼X|Z , Sensitivity Analysis with the R2-calculus 7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' If X is one-dimensional and σ2 X∼Z > 0, the partial R- and f-value (Cohen, 1977) are defined as RY ∼X|Z := corr(Y ⊥Z, X⊥Z), and fY ∼X|Z := RY ∼X|Z � 1 − R2 Y ∼X|Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The marginal f2-, R- and f-values can be further defined by using an “empty” Z in the definitions above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' details are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The partial R2 takes values in [0, 1] and is a measure of how well the variables in X can be linearly combined to explain the variation in Y after already using linear combinations of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Values close to 1 indicate high explanatory capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This simple interpretation makes the R2-value a popular tool to assess the goodness of fit of a linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The partial f2 is a monotone transformation of the partial R2 and takes values in [0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The partial R-value captures not only the strength but also the direction of dependence between Y and X after partialing out Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The next result justifies calling R2 Y ∼X|Z a partial R2-value and shows that the definitions of R2- and R-value are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' It follows from the Gram-Schmidt orthogonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the setting of Definition 1, R2 Y ∼X|Z = R2 Y ⊥Z∼X⊥Z holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' More- over, if X is one-dimensional, then R2 Y ∼X|Z = (RY ∼X|Z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The next Proposition collects several useful results about R2-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proposition 1 (R2-calculus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the setting above, let W be another random vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Assume σ2 Y ∼X+W+Z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Further, suppose σ2 X∼W+Z > 0 and σ2 W∼X+Z > 0 when X and/or W are one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Then, the following rules hold: [i] Independence: if Y ⊥⊥ X, then R2 Y ∼X = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [ii] Independent additivity: if X ⊥⊥ W, then R2 Y ∼X+W = R2 Y ∼X + R2 Y ∼W ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [iii] Decomposition of unexplained variance: 1 − R2 Y ∼X+W = (1 − R2 Y ∼X)(1 − R2 Y ∼W|X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [iv] Recursion of partial correlation: if X and W are one-dimensional, then RY ∼X|W = RY ∼X − RY ∼W RX∼W � 1 − R2 Y ∼W � 1 − R2 X∼W ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [v] Reduction of partial correlation: if X is one-dimensional and Y ⊥⊥ W, then RY ∼X|W = RY ∼X � 1 − R2 X∼W ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 8 [vi] Three-variable identity: if both X and W are one-dimensional, then fY ∼X|W � 1 − R2 Y ∼W|X = fY ∼X � 1 − R2 X∼W − RY ∼W|XRX∼W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' All of the relationships above also hold when Z is partialed out (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' if “ |Z” is appended to the subscripts of all R-, R2-, and f-values) and the inde- pendence assumptions are conditional on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Rules [i], [ii] and [v] remain true if (conditional) independence condition is replaced by (partial) uncorrelatedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A more succint sufficient condition for the positive partial variance requirements is that the covariance matrix of (Y, X, Z, W) has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Rule [vi] may appear unintuitive at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To see how this identity may come up, consider three random variables Y , X and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' There are in total three marginal R-values, RY ∼X, RY ∼W and RX∼W , and three partial R-values, RY ∼X|W , RY ∼W|X and RX∼W|Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Rule [iv] shows that the partial R-values can be determined by all the marginal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In other words, there are only three degrees of freedom among the six R-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This implies that there must be an equality constraint relating RY ∼X, RX∼W , RY ∼X|W , and RY ∼W|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The (partial) R2- and R-value can be defined in a more general Hilbert space setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The corresponding rules of the R2-calculus also hold true, yielding Proposition 1 as a corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Bias of the k-class Estimator Our main goal in this article is to outline a unified approach to sensitivity analysis in linear structural equation models that leverages the R2-calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To this end, we will focus on the case with a one-dimensional treatment D and a continuous outcome Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We would like to estimate the causal effect of D on Y , which will be denoted as β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We may also observe some covariates X and a potential instrumental variable Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Let V = (D, Y, X, Z) be the observed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In a sensitivity analysis, we are worried about some unmeasured variables U that confound the causal effect of D on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This can potentially be addressed by finding an instrumental variable Z for the treatment D, but this instrumental vari- able may itself be invalid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' readers who are unfamiliar with instrumental variables are referred to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2 for its definition in the context of linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Below we will derive a bias formula for the usual linear regression and instrumental vari- able estimators, which essentially determines the objective functional β(θ, ψ) in the stochastic optimization problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A Single Unmeasured Confounder We start with the case of a one-dimensional unmeasured confounder U and work with the so-called k-class estimators as defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 9 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Suppose var(D⊥X) > var(D⊥X,Z) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The k-class estimand is given by βk := � � � � � � � � � cov(D⊥X, Y ⊥X) − k cov(D⊥X,Z, Y ⊥X,Z) var(D⊥X) − k var(D⊥X,Z) , if − ∞ < k ≤ 1, cov(D⊥X,Z, Y ⊥X,Z) var(D⊥X,Z) , if k = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The k-class estimator is defined by replacing variance/covariance and the residuals in the equation above by their sample counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The family of k-class estimators was introduced by Theil (1958) and Nagar (1959) to interpolate the ordinary least squares (OLS) estimator and the two-stage least squares (TSLS) estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' It provides a convenient representation for a unified analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To see the interpolation, the OLS estimand that adjusts for X is given by βY ∼D|X := cov(Y ⊥X, D⊥X) var(D⊥X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The TSLS estimand (also called the Wald ratio) that uses Z as an instrumental variable and X as exogenous covariates is given by βD∼Z|X, Y ∼Z|X := cov(Y ⊥X, Z⊥X) cov(D⊥X, Z⊥X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' They are special cases of the k-class estimands according to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the setting of Definition 2, β1 = βD∼Z|X, Y ∼Z|X, β0 = βY ∼D|X, and lim k→−∞ βk = β−∞ = βY ∼D|X,Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Another important estimator contained in the k-class is the limited information maximum likelihood of Anderson and Rubin (1949), where k needs to be estimated from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Other examples can be found in Davidson and MacKinnon (1993, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 649) and Koles´ar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Also related is the anchor regression estimator recently introduced by Rothenh¨ausler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2021) that aims to gain robustness under distributional shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The target functional β = β(PV,U) we consider is the OLS estimand βY ∼D|X,Z,U which adjusts for X, Z, and the unmeasured confounder U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' When Y is causally determined by a linear structural equation containing D, X, Z and U, the causal effect of D on Y is precisely given by β = βY ∼D|X,Z,U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see Figure 1 for an illustration of the data-generating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' When the true structural relationship is not linear, βY ∼D|X,Z,U may still be interpreted as a kind of weighted average treatment effect under additional assumptions (Angrist and Pischke, 2009, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Because U is not observed, β cannot be consistently estimated without further assumptions on the relationship between U and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The difference between the estimand βk and the target β is quantified by the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 10 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Suppose σ2 D∼X > σ2 D∼X+Z > σ2 D∼X+Z+U > 0 and let k ∈ (−∞, 1] be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Then, βk − β = � fY ∼Z|X,D RD∼Z|X 1 − k + k R2 D∼Z|X + RY ∼U|X,Z,D fD∼U|X,Z � σY ∼X+Z+D σD∼X+Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (4) For k = −∞, equation (4) holds by taking the limit k → −∞ on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Equation (4) generalizes previous bias formulas for the OLS estimator to the entire family of k-class estimators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see Remark 5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Interestingly, this more general formula can be easily derived by applying the OLS bias formula twice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see equation (5) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Because the bias of any k-class estimand can be written as a function of RY ∼U|X,Z,D and RD∼U|X,Z, we will refer to them as the primitive sensitivity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Corollary 1 in the appendix contains specialized bias formulas for the common estimands in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The next proposition states the causal identification assumption under which these estimands are unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the setting of Theorem 1, the following statements are true: (i) If RD∼U|X,Z = 0 or RY ∼U|X,Z,D = 0, then β = βY ∼D|X,Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (ii) If R2 D∼U+Z|X = 0 or R2 Y ∼U+Z|X,D = 0, then β = βY ∼D|X,Z = βY ∼D|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (iii) If RZ∼U|X = 0 and RY ∼Z|X,D,U = 0, then β = βD∼Z|X, Y ∼Z|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proof Sketch of Theorem 1 By expanding the difference between the k-class and the target estimands and ap- plying the R2-calculus to the first term, we deduce βk − βY ∼D|X,Z,U = βk − βY ∼D|X,Z + βY ∼D|X,Z − βY ∼D|X,Z,U (5) = � βY ∼D|X − βY ∼D|X,Z � 1 − k � 1 − R2 D∼Z|X � + � βY ∼D|X,Z − βY ∼D|X,Z,U � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Equation (4) can then be derived by applying the following Lemma twice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see Ap- pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Let Y, D and W be random variables, X be a random vector, and suppose σ2 D∼X+W > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Then βY ∼D|X − βY ∼D|X,W = RY ∼W|X,D fD∼W|X σY ∼X+D σD∼X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To our knowledge, Lemma 2 first appeared in Cochran (1938) and was later generalized by Cox (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the context of sensitivity analysis, it has already been used by Frank (2000), Hosman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2010) and Cinelli and Hazlett (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The bias formula in the last paper can be obtained by taking k → −∞ in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Heuristically, the true causal effect β should not depend on the choice of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This can also been seen from equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Multiple Unmeasured Confounders The assumption that the unmeasured confounder U is one-dimensional has kept the algebra tractable thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In order to obtain a bias formula with multiple confounders, a generalization of Lemma 2 is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For instance, when W is l-dimensional, we can repeatedly apply Lemma 2 to the following telescoping series: βY ∼D|X − βY ∼D|X,W = l � j=1 βY ∼D|X,W[j−1] − βY ∼D|X,W[j] = l � j=1 RY ∼Wj|X,D,W[j−1] fD∼Wj|X,W[j−1] � � � �1 − R2 Y ∼W[j−1]|X,D 1 − R2 D∼W[j−1]|X σY ∼X+D σD∼X , (6) where [j] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' , j} and [0] := ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' By using an expansion similar to (5), we may identify the bias in linear regression and instrumental variables models with multiple unmeasured confounders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' of course, more sensitivity parameters will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Such extensions are explored in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Alternatively, Lemma 2 provides an upper bound on |βY ∼D|X − βY ∼D|X,W | that can be immediately generalized to multi-dimensional W as stated in the next re- sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Heuristically, this is because the confounding effects of several unmeasured variables can negate each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see Cinelli and Hazlett (2020, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To our knowledge, this result is first obtained by Hosman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' we simplify their proof substantially using the R2-calculus in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Let Y and D be random variables, let X and W be random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Assume that σ2 D∼X+W > 0 and that the covariance matrix var(W ⊥X,D) is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Then, ��βY ∼D|X − βY ∼D|X,W �� ≤ � R2 Y ∼W|D,X f2 D∼W|X σ2 Y ∼X+D σ2 D∼X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (7) Returning to the k-class estimator, when the unmeasured confounder U is multi- dimensional, we may still apply the expansion in equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The first term on its right-hand side does not involve U and the second term is bounded by (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This immediately implies a bound on the bias of the k-class estimand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Interpretable and Flexible Constraints Theorem 1 in the previous section has established the dependece of the objective β on two primitive sensitivity parameters: RD∼U|X,Z and RY ∼U|X,Z,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In this section, we develop different ways to specify interpretable constraints on these parameters by extending ideas in previous work, most notably Cinelli and Hazlett (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The key idea is to compare the R2-value of the unmeasured confounder with that of an observed covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To facilitate this comparison, we assume that the random vector X ∈ Rp can be partitioned into X = ( ˙X, ˜X), ˙X ∈ R ˙p, ˜X ∈ R˜p such that ˙X ⊥⊥ U | ˜X, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (8) Sensitivity Analysis with the R2-calculus 12 Z D U ˙X Y ˜X 1 2 3 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Causal diagram for regression and instrumental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Directed edges repre- sent causal effects and bidirected edges represent dependence due to unmeasured com- mon causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In Figure 1, we give a causal graphical model that fulfills (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' other possibilities may be verified by the familiar d-sepration (Pearl, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We further denote [ ˙p] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' , ˙p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For I ⊆ [ ˙p] and ˙X ∈ R ˙p, define ˙XI := ( ˙Xi)i∈I and Ic := [ ˙p] \\ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Finally, let ˙X−j := ˙X{j}c for any j ∈ [ ˙p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Table 1 summarizes the constraints on sensitivity parameters considered in this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' it may be helpful to visualize the relations parameterized by these constraints using the causal diagram in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In principle, the constraints in Table 1 can be combined arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In particular, one may specify several comparative bounds using different sets of covariates, although specifying too many bounds may leave the sensitivity model infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Next, we show how these bounds naturally arise from the sensitivity analysis for the OLS and TSLS estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Ordinary Least Squares The OLS estimand β−∞ = βY ∼D|X,Z identifies the causal effect β = βY ∼D|X,Z,U if the causal diagram in Figure 1 does not contain U → D or U → Y , or equivalent, if RD∼U|X,Z = 0 or RY ∼U|X,Z,D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Some of the sensitivity models in Table 1 directly bound them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' others bound related R2-values that can be linked to the primitive parameters by the R2-calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Such relations are elaborated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Constraints on U → D First of all, we may directly specify a bound on the primitive sensitivity parameter RD∼U|X,Z ∈ [Bl UD, Bu UD] ⊆ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (9) This constraint means that the correlation between D and U, after accounting for linear effects of X and Z, lies within the interval [Bl UD, Bu UD].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 13 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Specification of constraints: When the user specifies bounds on the sensi- tivity parameters, the corresponding constraints in the last column are added to the stochastic optimization (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' When bounds on U ↔ Z and/or Z → Y are chosen, the TSLS-related equality constraints (17) and (18) also need to be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Edge Sensitivity model Optimization constraint 1 U → D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' RD∼U|X,Z ∈ [Bl UD, Bu UD] (9) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R2 D∼U| ˜ X, ˙XI,Z ≤ bUDR2 D∼ ˙XJ| ˜ X, ˙XI,Z (10) 2 U → Y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' RY ∼U|X,Z,D ∈ [Bl UY , Bu UY ] (11) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R2 Y ∼U| ˜ X, ˙XI,Z ≤ bUY R2 Y ∼ ˙XJ| ˜ X, ˙XI,Z (14), (15) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R2 Y ∼U| ˜ X, ˙XI,Z,D ≤ bUY R2 Y ∼ ˙XJ| ˜ X, ˙XI,Z,D (13), (15), (16) 3 U ↔ Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' RZ∼U|X ∈ [Bl UZ, Bu UZ] (19) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R2 Z∼U| ˜ X, ˙X−j ≤ bUZR2 Z∼ ˙Xj| ˜ X, ˙X−j (20) 4 Z → Y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' RY ∼Z|X,U,D ∈ [Bl ZY , Bu ZY ] (21) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R2 Y ∼Z|X,U,D ≤ bZY R2 Y ∼ ˙Xj| ˜ X, ˙X−j,Z,U,D (22), (23) Alternatively (or in addition to the previous bound), we can specify the following comparative bound that is arguably more interpretable: R2 D∼U| ˜ X, ˙XI,Z ≤ bUDR2 D∼ ˙XJ| ˜ X, ˙XI,Z, I ⊂ [ ˙p], J ⊆ Ic, bUD ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This inequality means that the unmeasured confounder U can explain at most bUD times as much variance of D as ˙XJ does, after accounting for the effect of ( ˜X, ˙XI, Z) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For practical purposes, a good choice of the comparison sets is J = {j} and I = Jc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We can relate RD∼U| ˜ X, ˙XI,Z in the last bound to RD∼U|X,Z via the R2-calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' By using ˙XIc ⊥⊥ U | ˜X, ˙XI, Z (which follows from the assumption in (8)) and applying the reduction of partial correlation with Y ≡ U, X ≡ D, Z ≡ ( ˜X, ˙XI, Z) and W ≡ ˙XIc, we have R2 D∼U|X,Z [v] = R2 D∼U| ˜ X, ˙XI,Z 1 − R2 D∼ ˙XIc| ˜ X, ˙XI,Z ≤ bUD R2 D∼ ˙XJ| ˜ X, ˙XI,Z 1 − R2 D∼ ˙XIc| ˜ X, ˙XI,Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (10) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Constraints on U → Y Similarly to U → D, we may specify a direct bound: RY ∼U|X,Z,D ∈ [Bl UY , Bu UY ] ⊆ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (11) Alternatively, we may use comparative bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Here we consider two types of bounds depending on whether D is regressed out: R2 Y ∼U| ˜ X, ˙XI,Z ≤ bUY R2 Y ∼ ˙XJ| ˜ X, ˙XI,Z, (12) R2 Y ∼U| ˜ X, ˙XI,Z,D ≤ bUY R2 Y ∼ ˙XJ| ˜ X, ˙XI,Z,D, (13) Sensitivity Analysis with the R2-calculus 14 where I ⊂ [ ˙p], J ⊆ Ic, bUY ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' When comparing the explanatory capability of the variables U and ˙XJ, it is natural to regress out all other variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' However, regressing out D, a potential common child of X and U, may introduce dependence between U and Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' this is essentially the point made by Hernan and Robins (1999) in their criticism of Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Thus, we consider both the comparative bound (12) without D and the bound (13) with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For (12), we may apply rule [v] as in (10) and obtain R2 Y ∼U|X,Z [v] = R2 Y ∼U| ˜ X, ˙XI,Z 1 − R2 Y ∼ ˙XIc| ˜ X, ˙XI,Z ≤ bUY R2 Y ∼ ˙XJ| ˜ X, ˙XI,Z 1 − R2 Y ∼ ˙XIc| ˜ X, ˙XI,Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (14) However, we cannot regress out D in (14) because D may be a collider in the path ˙XIc → D ← U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Instead, we can link it to RY ∼U|X,Z,D via the R2-calculus: RY ∼U|X,Z,D [iv] = RY ∼U|X,Z − RY ∼D|X,Z RD∼U|X,Z � 1 − R2 Y ∼D|X,Z � 1 − R2 D∼U|X,Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (15) Hence, the first type of comparative bound can be represented as the inequality constraint (14) and the equality constraint (15) in the optimization problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The second type of comparative bounds partials out D and involves two addi- tional sensitivity parameters: RY ∼U|X,Z and RY ∼U| ˜ X, ˙XI,Z,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To link them to the primitive sensitivity parameters, we may use equation (15) and RY ∼U|X,Z = 1 � 1 − R2 Y ∼ ˙XIc| ˜ X, ˙XI,Z � RY ∼D| ˜ X, ˙XI,ZRD∼U|X,Z � 1 − R2 D∼ ˙XIc| ˜ X, ˙XI,Z + RY ∼U| ˜ X, ˙XI,Z,D � 1 − R2 Y ∼D| ˜ X, ˙XI,Z � 1 − R2 D∼U|X,Z(1 − R2 D∼ ˙XIc| ˜ X, ˙XI,Z) � (16) as an equality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The derivation of (16) is deferred to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Two-stage Least Squares The method of instrumental variables (IV) is commonly used to overcome unmea- sured confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Here we only provide a very brief introduction to it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' the reader is referred to Wooldridge (2010) for a more comprehensive discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A variable Z is called an instrument for D if (i) it is an independent predictor of D, (ii) it is exogenous in the sense that Z is conditionally independent of the unmeasured confounder U and (iii) it has no direct effect on the outcome Y that is not mediated by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In linear models, these conditions can be expressed as (i) RZ∼D|X ̸= 0, (ii) RZ∼U|X = 0, (iii) RY ∼Z|X,U,D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proposition 3(iii) suggests that under these conditions, the target β = βY ∼D|X,Z,U is identified by the TSLS estimand β1 = βD∼Z|X, Y ∼Z|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 15 As the last two conditions above involve the unmeasured confounder U and thus cannot be verified, a sensitivity analysis for TSLS would specify bounds on the sensitivity parameters RZ∼U|X and RY ∼Z|X,U,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To use the bias formula in Theorem 1, we need to link them to the primitive sensitivity parameters RD∼U|X,Z and RY ∼U|X,Z,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To achieve this, we apply the three-variable identity [vi] with Y ≡ Y , X ≡ Z, W ≡ U and Z ≡ (X, D) to obtain fY ∼Z|X,U,D � 1 − R2 Y ∼U|X,Z,D = fY ∼Z|X,D � 1 − R2 Z∼U|X,D − RY ∼U|X,Z,DRZ∼U|X,D, (17) and with Y ≡ U, X ≡ Z, W ≡ D and Z ≡ X to obtain fZ∼U|X,D � 1 − R2 D∼U|X,Z = fZ∼U|X � 1 − R2 D∼Z|X − RD∼Z|XRD∼U|X,Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (18) These are then added to the stochastic program (2) as equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Constraints on U ↔ Z The sensitivity parameter RZ∼U|X can be constrained by directly providing a range of plausible values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' RZ∼U|X ∈ [Bl UZ, Bu UZ] ⊆ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (19) Alternatively, we allow practitioners to specify the following comparative bound R2 Z∼U| ˜ X, ˙X−j ≤ bUZR2 Z∼ ˙Xj| ˜ X, ˙X−j, j ∈ [ ˙p], bUZ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Using the conditional independence assumption (8), this can be shown to be equiv- alent to (see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2) R2 Z∼U|X ≤ bUZ R2 Z∼ ˙Xj| ˜ X, ˙X−j 1 − R2 Z∼ ˙Xj| ˜ X, ˙X−j 1 − bUZ R4 Z∼ ˙Xj| ˜ X, ˙X−j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (20) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Constraints on Z → Y We can bound the sensitivity parameter RY ∼Z|X,U,D by specifying the direct bound RY ∼Z|X,U,D ∈ [Bl ZY , Bu ZY ] ⊆ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (21) Furthermore, we allow the following comparative bound R2 Y ∼Z|X,U,D ≤ bZY R2 Y ∼ ˙Xj| ˜ X, ˙X−j,Z,U,D, j ∈ [ ˙p], bZY ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (22) This last bound is unusual in the sense that the sets of variables that are regressed out are different in the two partial R2-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' It is difficult to specify compara- tive bounds for the exclusion restriction as the corresponding sensitivity parame- ter RY ∼Z|X,U,D partials out U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Therefore, we cannot directly compare U to an Sensitivity Analysis with the R2-calculus 16 observed covariate, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙Xj, and the right-hand side of the bound cannot be esti- mated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For this reason, we resort to the adjustment set in (22) because we can connect RY ∼ ˙Xj| ˜ X, ˙X−j,Z,U,D to the primitive sensitivity parameters via the following equality constraint fY ∼ ˙Xj| ˜ X, ˙X−j,Z,U,D � 1 − R2 Y ∼U|X,Z,D = � fY ∼ ˙Xj| ˜ X, ˙X−j,Z,D � 1 − R2 D∼U|X,Z + RY ∼U|X,Z,D RD∼ ˙Xj| ˜ X, ˙X−j,Z RD∼U|X,Z ��� 1 − R2 D∼U|X,Z(1 − R2 D∼ ˙Xj| ˜ X, ˙X−j,Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (23) See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2 for the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Intervals So far, we have derived the objective function β = β(θ, ψ) of the stochastic program (2) in Section 3 and a rich set of constraints ψ ∈ Ψ(θ) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' As θ only involves partial correlations and the standard deviation of regression residuals, we can plug in an empirical estimator of θ to obtain a point estimator of the optimal value of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In other words, we only need to solve the optimization problem in (3) to estimate the lower and upper bounds of the partially identified region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Complications arise when we would like to construct an interval estimator S of β with certain statistical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the general setup presented in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 and for a given 0 < α < 1, we call S a (1 − α)-sensitivity interval of β if PV � β(θ(PV ), ψ) ∈ S � ≥ 1 − α for all PV and ψ ∈ Ψ(θ(PV )), and S a (1 − α)-sensitivity interval of the partially identified region if PV � PIR(PV ) ⊆ S � ≥ 1 − α for all PV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Obviously, the second notion of confidence is stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For a more detailed discus- sion on confidence statements in partially identified problems including issues with asymptotic sensitivity intervals, the reader is referred to Imbens and Manski (2004), Stoye (2009) and Molinari (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Next we review several methods to construct sensitivity intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To obtain an interval estimator of β in a sensitivity analysis of the OLS, a heuristic approach, as suggested by Cinelli and Hazlett (2020), is to treat U as observed and use the usual confidence interval � �ˆβY ∼D|X,Z + � − ˆRY ∼U|X,Z,D ˆfD∼U|X,Z ± qα √n � � � �1 − ˆR2 Y ∼U|X,Z,D 1 − ˆR2 D∼U|X,Z � ˆσY ∼X+Z+D ˆσD∼X+Z � �, where qα is the (1 − α/2)-quantile of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Here it is assumed that a domain expert can specify ˆψ = ( ˆRY ∼U|X,Z,D ˆRD∼U|X,Z) even though Sensitivity Analysis with the R2-calculus 17 U cannot observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For the partially identified problem, a seemingly reasonable idea is to minimize/maximize the confidence bounds over ˆψ ∈ Ψ(ˆθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' However, a closer look at this heuristic shows that it achieves no obvious confi- dence guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This is because the sensitivity parameter ˆψ depends on the data and thus its value changes when another sample is drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' If ˆψ is almost certainly contained in Ψ(ˆθ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' P( ˆψ ∈ Ψ(ˆθ)) = 1, this heuristic interval would actually be a sensitivity interval for β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' However, this is only possible if the sensitivity model Ψ is non-informative (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' RD∼U|X,Z ∈ [−1, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Numerical simulations in Appendix E confirm this intuitive argument;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' in particular, the heuristic interval has cover- age 50% in one setting and above 99% in another, where the nominal coverage is 1 − α = 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To account for the uncertainty in estimating the feasible set Ψ(θ), Tudball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2022) propose to solve the optimization problem (3) with a relaxed constraint ψ ∈ ˜Ψ(ˆθ), where ˜Ψ(ˆθ) is constructed to contain Ψ(θ) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' However, several technical difficulties prevent us from directly applying their method to our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A third approach to construct sensitivity interval is to use the bootstrap (Efron and Tibshirani, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' More specifically, we can compute a collection of estimators ˆˆθ using resamples of the observable data, solve the plug-in optimization problem (3) with ˆθ = ˆˆθ, and then use the bootstrap distribution to construct one-sided confidence intervals [βl min, ∞) and (−∞, βu max] with level (1 − α/2) for the minimal and maximal values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Different procedures may be employed in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For instance, percentile bootstrap takes the α/2 and 1 − α/2 quantile of the bootstrap distribution to construct the respective confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Other options include the basic (or reverse percentile) bootstrap, studentized bootstrap, and bias-corrected bootstrap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see Davison and Hinkley (1997, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 5) for more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Finally, a sensitivity interval with nominal confidence level (1 − α) may be constructed as [βl min, βu max].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For the sensitivity analysis problems described in this article, simulation studies in Appendix E suggest that the percentile bootstrap performs better than the basic boostrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In two simulation studies with nominal confidence level 90%, we found that the percentile bootstrap intervals covers the partially identified region around 90% and the true parameter, which equals the lower end of the PIR under the specified sensitivity model, around 95% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The empirical coverage of basic bootstrap intervals is about 10% below the nominal level when the sample size is n = 200;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' this gap closes as n increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Although a rigorous asymptotic analysis of the different bootstrap procedures is beyond the scope of this article, we offer some heuristics on why the boostrap is expected to “work” here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' First, Shapiro (1991) provides an asymptotic theory for stochastic optimization and shows that the plug-in estimator of the optimal value of certain stochastic programs is asymptotically linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see also Shapiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2009, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Although our optimization problem (2) involves unknown parameters θ in the constraints and thus does not fall in the class of problems considered by Shapiro (1991), one may hope that the theory there extends to the problem considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 18 Second, due to optimization over the sample, the plug-in estimator is always biased, even though the bias may be small asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' With just a moderate sample size, our simulations also show that the bootstrap distribution of the optimal value estimators is quite skewed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see Figure 7 in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' It is plausible that the finite sample effects of bias and skewness in the bootstrap distribution cancel out each other for the percentile bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Finally, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2019) provide an alternative justification for the percentile bootstrap in partially identified sensitivity analysis by using the generalized minimax inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' However, their proof requires a fixed constraint set Ψ and thus cannot be directly applied to the problem here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Although the probability of the estimated constraint set Ψ(ˆθ) being empty should converge to zero as the sample size grows, this can occasionally occur with moderate sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Our implementation of the bootstrap procedures takes a conservative approach and sets the optimal value to ∞ or −∞ depending on which end of the PIR is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Data Example We demonstrate the practicality of the proposed method using a prominent study of the economic return of schooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The dataset was compiled by Card (1993) from the National Longitudinal Survey of Young Men (NLSYM) and contains a sample of 3010 young men at the age of 14 to 24 in 1966 who were followed up until 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Card uses several linear models to estimate the causal effect of education, measured by years of schooling and denoted as D, on the logarithm of earnings, denoted as Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For brevity, we only consider the most parsimonious model used by Card which includes, as covariates for adjustment and denoted as X, years of labour force experience and its square, and indicators for living in the southern USA, being black and living in a metropolitan area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Card (1993) recognizes that many researchers are reluctant to interpret the es- tablished positive correlation between education and earnings as a positive causal effect due to the large number of potential unmeasured confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In our analysis, we will consider the possibility that an unmeasured variable U, which represents the motivation of the young men, may influence both schooling and salary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To address this issue, Card suggests to use an instrumental variable, namely the indicator for growing up in proximity to a 4-year college;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' this is denoted as Z below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Nonethe- less, proximity to college may not be a valid instrumental variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For example, growing up near a college may be correlated with a higher socioeconomic status, more career opportunities, or stronger motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A more detailed discussion of the identification assumptions can be found in Card (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For the purpose of sensitivity analysis, we assume that being black and living in the southern USA are not directly related with motivation and treat them as ˙X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' the remaining covariates are regarded as ˜X in the sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We assume that this partition satisfies the conditional independence in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In this example, we use comparative bounds to express our beliefs about the effects of the unmeasured confounder U on Y and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We assume that motivation can explain at most 4 times as much variation in the level of education as being black (denoted as ˙Xj) Sensitivity Analysis with the R2-calculus 19 does after accounting for all other observed covariates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and that motivation can explain at most 5 times as much variation in log-earnings as being black does after accounting for the other covariates and education: (B1) R2 D∼U| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z ≤ 4 R2 D∼ ˙Xj| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (B2) R2 Y ∼U| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D ≤ 5 R2 Y ∼ ˙Xj| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The bounds (B1) and (B2) address deviations from the identification assumptions of a linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Likewise, we can also specify deviations from the instrumental variable assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We suppose that motivation U can explain at most half as much variation in Z (college proximity) as ˙Xj (black) can after accounting for the effects of ( ˜X, ˙X−j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Furthermore, we assume that college proximity Z can explain at most 10 % as much variance in log-earnings after excluding effects of (X, U, D) as being black can explain log-earnings after excluding the effects of ( ˜X, ˙X−j, Z, U, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' These assumptions translate to (B3) R2 Z∼U| ˜ X, ˙X−j ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5 R2 Z∼ ˙Xj| ˜ X, ˙X−j, (B4) R2 Y ∼Z|X,U,D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 R2 Y ∼ ˙Xj| ˜ X, ˙X−j,Z,U,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' When the bound (B1) is not part of the constraints, we additionally require RD∼U|X,Z ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='98, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (24) This ensures that RD∼U|X,Z is bounded away from −1 and 1 and that the partially identified range has finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Figure 2 shows the OLS estimates that adjust/do not adjust for Z, the TSLS estimate, and their corresponding 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The same plot shows the estimated partially identified regions and 95% sensitivity intervals (obtained by the percentile bootstrap) for five different sensitivity models that involve different combinations of the bounds (B1) to (B4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Both the OLS and the TSLS estimates suggest a statistically significant positive effect of education on earnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the first sensitivity model in Figure 2, we relax the assumption of no unmeasured confounders, which would be required if the OLS estimate is interpreted causally, and assume that the effects of U on D and Y are bounded by (B1) and (B2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The sensitivity interval remains positive in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In other cases, the estimated partially identified regions and the sensitivity intervals become very wide whenever (B1) is not part of the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This is because the other constraints, except the loose bound in (24), do not bound |RD∼U|X,Z| away from 1, so the association between D and Y may be entirely driven by the unmeasured confounder U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In fact, the PIR would have an infinite length if (24) was not imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Therefore, just specifying deviations from the IV- assumptions, as in (B3) and (B4), is not sufficient to ensure that the PIR is finite in this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Moreover, comparing the first and last sensitivity model in Figure 2, we notice that imposing the IV-related bounds (B3) and (B4) on top of (B1) and (B2) does not shorten the estimated PIR and sensitivity intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' These findings suggest that the results of Card (1993) are more robust towards deviations from the OLS than from the IV assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 OLS adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' OLS unadj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' TSLS (B1), (B2) (B3), (B4) (B1), (B3), (B4) (B2), (B3), (B4) (B1) - (B4) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Three estimation strategies and five sensitivity models for the causal effect β: Point estimates/estimates of the PIR (blue);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 95% confidence/sensitivity intervals (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Contour Plots This section presents graphical tools to further aid the interpretation of sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The main idea is to plot the estimated lower or upper bound of the PIR against the sensitivity parameters or the parameters in the comparative bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Contour lines in this plot allow practitioners to identify the magnitude of unmea- sured confounding (or violations of the instrumental variables assumptions) needed to alter the conclusion of the study qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This idea dates back at least to Imbens (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' our method below refines the proposal in Cinelli and Hazlett (2020) and Zhang and Ding (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The contour plots will be illustrated using the real data example in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' b-contour Plot For comparative bounds, the b-factor (such as bUD in (10)) controls how tightly the corresponding sensitivity parameter is constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, it is important to gain a practical understanding of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The b-sensitivity contour plot shows the estimated lower/upper end of the PIR on a grid of b-factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In Figure 3, we consider the sensitivity model with the bounds (B1) and (B2) and investigate our choice (bUD, bUY ) = (4, 5) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The plot shows that the estimated lower end of the PIR is still positive even for more conservative values such as (bUD, bUY ) = (6, 10) or (bUD, bUY ) = (10, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Thus, a substantial deviation from the OLS-related assumptions is needed to alter the sign of the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='07 0 (4, 5) 0 5 10 15 0 2 4 6 8 10 12 bUD bUY Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' b-sensitivity contours for (B1), (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='07 (4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='12 0 2 4 6 8 bUD bZY Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' b-sensitivity contours for (B1)-(B4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Figure 4 considers the sensitivity model using the constraints (B1) to (B4) with changing (bUD, bZY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This plot confirms our observation in Section 6 that imposing the IV-related bounds (B3) and (B4) does not change the estimated lower bound substantially when (B1) and (B2) are already present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the terminology of con- strained optimization, this means that the “shadow prices” for (B3) and (B4) are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R-contour Plot We may also directly plot the estimated lower/upper end of the PIR against the sensitivity parameters RD∼U|X,Z and RY ∼U|X,Z,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This idea has been adopted in several previous articles already (Imbens, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Blackwell, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Veitch and Zaveri, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For such R-contour plots, the key challenge is to benchmark or calibrate the R-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This was often done informally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For example, Cinelli and Hazlett (2020) consider a model without potential instrument Z, use sensitivity contours parame- terized by R2 D∼U|X and R2 Y ∼U|X,D and add (a ˆR2 D∼Xj|X−j, a ˆR2 Y ∼Xj|X−j,D) for certain choices of a > 0 and j ∈ [p] to the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Thus, they aim to provide context for plausible values of the sensitivity parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' the underlying idea is similar to the comparative bounds in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' However, this method of benchmarking is not entirely honest because different sets of covariates are conditioned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Moreover, regressing out a potential collider D may leave ˆR2 Y ∼Xj|X−j,D difficult to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Here, we revise the contour plot in Cinelli and Hazlett (2020) by using the R2-calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To this end, we first construct benchmarking points for RD∼U|X,Z and RY ∼U|X,Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Applying the reduction of partial correlation (rule [v]) and the Sensitivity Analysis with the R2-calculus 22 black 2x black 5x black south 2x south 5x south 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='8 RD~U|X,Z RY~U|X,Z,D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R-sensitivity contours for the lower end of the estimated PIR: Our comparison points (black dots) and Cinelli and Hazlett’s comparison points (green triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' conditional independence U ⊥⊥ ˙Xj | ˜X, ˙X−j, Z, j ∈ [ ˙p], we obtain RD∼U|X,Z = RD∼U| ˜ X, ˙X−j,Z � 1 − R2 D∼ ˙Xj| ˜ X, ˙X−j,Z and RY ∼U|X,Z = RY ∼U| ˜ X, ˙X−j,Z � 1 − R2 Y ∼ ˙Xj| ˜ X, ˙X−j,Z , which can be directly compared to, for any j ∈ [ ˙p], ˆRD∼ ˙Xj| ˜ X, ˙X−j,Z � 1 − ˆR2 D∼ ˙Xj| ˜ X, ˙X−j and ˆRY ∼ ˙Xj| ˜ X, ˙X−j � 1 − ˆR2 Y ∼ ˙Xj| ˜ X, ˙X−j,Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Moreover, we can multiply these values by a factor of √bR to compare the ex- planatory capability of U (in terms of its R2-value) to bR times the explanatory capability of the measured covariate ˙Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Finally, we may use the bijection between (RD∼U|X,Z, RY ∼U|X,Z) and (RD∼U|X,Z, RY ∼U|X,Z,D) in (15) to map the benchmarks to the scale used by the R-contour plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To illustrate the proposal, Figure 5 shows the R-contour plot for the estimated lower end of the PIR and adds benchmarks corresponding to black and living in the southern USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We observe that, even if the unmeasured confounder was five times as strong as black in terms of their capability of explaining the variation of D and Y , the estimator would still be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Figure 5 further contrasts our comparison points with the benchmarks proposed in Cinelli and Hazlett;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' in our experience, the difference between the two methods is usually not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Finally, we illustrate the utility of the R-contour plot as a way to visualize the feasible set Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity analysis with multiple bounds often entails a non- Sensitivity Analysis with the R2-calculus 23 RZ~U|X , RY~Z|X,U,D ∈ [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='03 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='03] RZ~U|X , RY~Z|X,U,D ∈ [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='04 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='04] RZ~U|X , RY~Z|X,U,D ∈ [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='01 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='01] RZ~U|X , RY~Z|X,U,D ∈ [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='02 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='02] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0 RD~U|X,Z RY~U|X,Z,D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R-sensitivity contours for the lower end of the estimated PIR: The red lines corre- spond to the values of RD∼U|X,Z and RY ∼U|X,Z,D that conform with the IV-assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' intuitive, complex set of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Consider the following sensitivity model RZ∼U|X, RY ∼Z|X,U,D ∈ [−r, r], r ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='04}, R2 Y ∼U| ˜ X, ˙X−j,Z,D ≤ 5 R2 Y ∼ ˙Xj| ˜ X, ˙X−j,Z,D, RD∼U|X,Z ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='99, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='99], where r parameterizes the degree of deviation from the instrumental variables as- sumptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' the covariate ˙Xj is the indicator for black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Figure 6 shows the estimated feasible set Ψ(ˆθ) for different values of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='01, the feasible set is small and concentrated around the lines that correspond to RD∼U|X,Z = RY ∼U|X,Z,D = 0 (the instrumental variable is valid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' As r increases, the feasible set becomes larger as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The curved shape of the region of feasible values is a result of the comparative bound on U → Y and the associated constraints (15) and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Moreover, we observe that β assumes its most extreme values as RD∼U|X,Z approaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This highlights the importance of bounding RD∼U|X,Z away from −1 and 1 to ensure that the PIR has finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Discussion and Outlook Thus far, we have sidestepped the issue of numerically computing the solution to the constrained stochastic optimization problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In fact, standard algorithms fail to reliably solve the problem due to the complexity of the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Therefore, we develop a grid search algorithm which leverages the structure of the objective and the equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The details can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 24 Two insights underlie the methodological development in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' First, sen- sitivity analysis (or more generally, any one-dimensional partially identified prob- lem) may be viewed as a constrained stochastic program and we can leverage meth- ods developed in stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Second, the R2-calculus provides a pa- rameterization of the bias of any k-class estimator and a systematic approach to specify interpretable sensitivity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Partial identification has attracted considerable attention in econometrics and causal inference since Manski (1990) and Balke and Pearl (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' see Manski (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Imbens and Manski (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Vansteelandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Aronow and Lee (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Miratrix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Molinari (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Existing methods typically assume a closed-form solution to the stochas- tic program (2) (the lower/upper end of the PIR) and that the plug-in estimator is asymptotically normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' As such results are only known for relatively simple models, these methods only have limited utility in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The constrained optimization perspective of partial identification is only beginning to get embraced in the litera- ture (Kaido et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Padh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Our article further shows the need for a more complete, asymptotic theory of the optimal value of a general stochastic program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This may allow one to extend the methodology developed here to sensitivity models with high- or infinite-dimensional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In particular, a theory for the bootstrap distribution of the optimal value estimator is required to clarify when and which bootstrap procedures provide asymptotically correct sensitivity intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The R2-values, R-values and generalizations thereof are popular for the calibra- tion of sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' They have been recently used in the sensitivity analysis for linear models with multiple treatments (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', 2021), mediation analy- sis (Zhang and Ding, 2022), missing values (Colnet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', 2022) and models with factor-structured outcomes (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In these works, certain algebraic relationships about R2-values and benchmarking techniques such as contour plots and robustness values are frequently used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Thus, the R2-calculus summarized in this article may also benefit the calibration of other sensitivity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Our proof of the R2-calculus in general Hilbert spaces suggests that it may be useful in nonlinear models, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' See Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2022) for related work in partially linear and semiparametric models using the Riesz-Frechet representation of certain causal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The rules of the R2-calculus are purely algebraic and can therefore be applied in any linear structural equation model – with or without unmeasured variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This raises the question: can sensitivity analysis be automated for reasonable sensitivity models defined by direct and comparative bounds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Such an algorithm would be immensely useful in applied research, but given the substantial amount of algebra needed for the relatively simple models considered here, the required work would be extremely challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 25 Acknowledgments Tobias Freidling is supported by a PhD studentship from GlaxoSmithKline Re- search & Development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Qingyuan Zhao is partly supported by the EPSRC grant EP/V049968/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' References Altonji, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Elder, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Taber, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2005) An Evaluation of Instrumental Variable Strategies for Estimating the Effects of Catholic Schooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of Human Resources, XL, 791–821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Anderson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1958) An introduction to multivariate statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' New York;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' London: Wiley Publications in Mathematical Statistics, 1st edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Anderson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Rubin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1949) Estimation of the parameters of a single equation in a complete system of stochastic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The Annals of Mathemat- ical Statistics, 20, 46–63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Angrist, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Pischke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2009) Mostly harmless econometrics: An empiri- cist’s companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Princeton University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Aronow, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2013) Interval estimation of population means under unknown but bounded probabilities of sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Biometrika, 100, 235–240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Balke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Pearl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1997) Bounds on Treatment Effects from Studies with Imperfect Compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of the American Statistical Association, 92, 1171– 1176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Blackwell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2014) A Selection Bias Approach to Sensitivity Analysis for Causal Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Political Analysis, 22, 169–182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Card, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1993) Using Geographic Variation in College Proximity to Estimate the Return to Schooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' w4483, National Bureau of Economic Research, Cambridge, MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Chernozhukov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Cinelli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Newey, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Sharma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Syrgkanis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2022) Long Story Short: Omitted Variable Bias in Causal Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='13398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Chernozhukov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Hong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Tamer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2007) Estimation and Confidence Regions for Parameter Sets in Econometric Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Econometrica, 75, 1243– 1284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Cinelli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Hazlett, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2020) Making sense of sensitivity: extending omit- ted variable bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Statistical Methodology), 82, 39–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 26 Cochran, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1938) The Omission or Addition of an Independent Variate in Multiple Linear Regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Supplement to the Journal of the Royal Statistical Society, 5, 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Cohen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1977) Statistical power analysis for the behavioral sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' New York: Academic Press, rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Colnet, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Josse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Scornet, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Varoquaux, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2022) Causal effect on a target population: a sensitivity analysis to handle missing covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='06435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Cornfield, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Haenszel, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Hammond, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Lilienfeld, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Shimkin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Wynder, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1959) Smoking and lung cancer: recent evidence and a discussion of some questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of the National Cancer institute, 22, 173–203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Cox, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2007) On a generalization of a result of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Cochran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Biometrika, 94, 755–759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Daniels, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Hogan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2008) Missing Data in Longitudinal Stud- ies: Strategies for Bayesian Modeling and Sensitivity Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Chapman and Hall/CRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Davidson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and MacKinnon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1993) Estimation and inference in economet- rics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' New York: Oxford University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Davison, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Hinkley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1997) Bootstrap Methods and their Application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Cambridge University Press, 1st edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Dawid, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1979) Conditional independence in statistical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Statistical Methodology), 41, 1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Ding, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and VanderWeele, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2016) Sensitivity Analysis Without Assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Epidemiology, 27, 368–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Dorn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Guo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2022) Sharp Sensitivity Analysis for Inverse Propensity Weighting via Quantile Balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of the American Statistical Associa- tion, 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Efron, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Tibshirani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1994) An Introduction to the Bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Chapman and Hall/CRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Frank, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2000) Impact of a Confounding Variable on a Regression Coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sociological Methods & Research, 29, 147–194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Gustafson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2005) On Model Expansion, Model Contraction, Identifiability and Prior Information: Two Illustrative Scenarios Involving Mismeasured Variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Statistical Science, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Halmos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2000) Introduction to Hilbert space and the theory of spectral multi- plicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Providence, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=': AMS Chelsea, 2nd edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hernan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Robins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1999) Letter to the editor of biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Biomet- rics, 55, 1316–1317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 27 Hosman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Hansen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Holland, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2010) The sensitivity of linear regression coefficients’ confidence limits to the omission of a confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The Annals of Applied Statistics, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2021) A Generative Adversarial Framework for Bounding Confounded Causal Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proceedings of the AAAI Conference on Artificial Intelligence, 35, 12104–12112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Imbens, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2003) Sensitivity to exogeneity assumptions in program evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' American Economic Review, 93, 126–132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Imbens, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Manski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2004) Confidence Intervals for Partially Identified Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Econometrica, 72, 1845–1857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Kaido, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Molinari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Stoye, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2019) Confidence intervals for projections of partially identified parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Econometrica, 87, 1397–1432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Koles´ar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Chetty, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Friedman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Glaeser, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Imbens, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2015) Iden- tification and inference with many invalid instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of Business & Economic Statistics, 33, 474–484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Lin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Psaty, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Kronmal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1998) Assessing the Sensitivity of Regression Results to Unmeasured Confounders in Observational Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Bio- metrics, 54, 948–963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Manski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1990) Nonparametric Bounds on Treatment Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The American Economic Review, 80, 319–323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' — (2003) Partial Identification of Probability Distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Springer Series in Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' New York: Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Miratrix, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Wager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Zubizarreta, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2018) Shape-constrained par- tial identification of a population mean under unknown probabilities of sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Biometrika, 105, 103–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Molinari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2020) Chapter 5 - Microeconometrics with Partial Identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In Handbook of Econometrics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 7, 355–486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Nagar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1959) The Bias and Moment Matrix of the General k-Class Estimators of the Parameters in Simultaneous Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Econometrica, 27, 575–595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Oster, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2019) Unobservable Selection and Coefficient Stability: Theory and Ev- idence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of Business & Economic Statistics, 37, 187–204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Padh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Zeitler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Watson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Kusner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Silva, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Kilbertus, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2022) Stochastic Causal Programming for Bounding Treatment Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='10806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' PCORI Methodology Committee (2021) The PCORI Methodology Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='pcori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='org/sites/default/files/PCORI-Methodology-Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 28 Pearl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2009) Causality: Models, Reasoning, and Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Cambridge University Press, 2 edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Richardson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Hudgens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Gilbert, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Fine, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2014) Nonparametric Bounds and Sensitivity Analysis of Treatment Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Statistical Science, 29, 596–618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Rosenbaum, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1987) Sensitivity analysis for certain permutation inferences in matched observational studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Biometrika, 74, 13–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' — (2002) Observational Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Springer Series in Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' New York: Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Rosenbaum, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Rubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1983) Assessing Sensitivity to an Unobserved Binary Covariate in an Observational Study with Binary Outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Statistical Methodology), 45, 212–218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Rothenh¨ausler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Meinshausen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', B¨uhlmann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Peters, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2021) Anchor regression: Heterogeneous data meet causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Statistical Methodology), 83, 215–246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Scharfstein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Rotnitzky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Robins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1999) Adjusting for Nonig- norable Drop-Out Using Semiparametric Nonresponse Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of the American Statistical Association, 94, 1096–1120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Shapiro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1991) Asymptotic analysis of stochastic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Annals of Opera- tions Research, 30, 169–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Shapiro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Dentcheva, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Ruszczy´nski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2009) Lectures on Stochastic Pro- gramming: Modeling and Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Society for Industrial and Applied Mathemat- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Small, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2007) Sensitivity Analysis for Instrumental Variables Regression With Overidentifying Restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of the American Statistical Association, 102, 1049–1058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' de Souza, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Eisen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Perera, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Bantoto, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Bawor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Dennis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Samaan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Thabane, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2016) Best (but oft-forgotten) practices: sensitiv- ity analyses in randomized controlled trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The American Journal of Clinical Nutrition, 103, 5–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Stoye, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2009) More on Confidence Intervals for Partially Identified Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Econometrica, 77, 1299–1315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Thabane, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Mbuagbaw, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Samaan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Marcucci, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Ye, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Thabane, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Giangregorio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Dennis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Kosa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Debono, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Dillenburg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Fruci, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Bawor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Wells, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Goldsmith, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2013) A tutorial on sensitivity analyses in clinical trials: the what, why, when and how.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' BMC Medical Research Methodology, 13, 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Theil, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (1958) Economist Forecasts and Policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' North-Holland Publishing Com- pany, Amsterdam-London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 29 Tudball, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Hughes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Tilling, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Bowden, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2022) Sample- constrained partial identification with application to selection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Biometrika, in press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Vandenbroucke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', von Elm, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Altman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Gøtzsche, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Mulrow, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Pocock, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Poole, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Schlesselman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Egger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2007) Strengthening the Reporting of Observational Studies in Epidemiology (STROBE): Explanation and Elaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' PLoS Medicine, 4, e297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' VanderWeele, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Arah, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2011) Bias Formulas for Sensitivity Analysis of Unmeasured Confounding for General Outcomes, Treatments, and Confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Epidemiology, 22, 42–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Vansteelandt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Goetghebeur, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Kenward, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Molenberghs, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2006) Ignorance and Uncertainty Regions as Inferential Tools in a Sensitivity Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Statistica Sinica, 16, 953–979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Veitch, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Zaveri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2020) Sense and Sensitivity Analysis: Simple Post-Hoc Analysis of Bias Due to Unobserved Confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In Advances in Neural Infor- mation Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 33, 10999–11009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Wooldridge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2010) Econometric analysis of cross section and panel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Cambridge, Mass: MIT Press, 2nd edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Yadlowsky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Namkoong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Basu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Duchi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Tian, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2022) Bounds on the conditional and average treatment effect with unobserved confounding factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The Annals of Statistics, 50, 2587–2615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Ding, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2022) Interpretable sensitivity analysis for the Baron- Kenny approach to mediation with unmeasured confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='08030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Small, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Bhattacharya, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2019) Sensitivity analysis for inverse probability weighting estimators via the percentile bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Statistical Methodology), 81, 735–761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', D’Amour, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Franks, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2021) Copula-based Sensitivity Analy- sis for Multi-Treatment Causal Inference with Unobserved Confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' arXiv: 2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='09412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=', D’Amour, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' and Franks, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (2022) Sensitivity to Unobserved Confounding in Studies with Factor-structured Outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' arXiv: 2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='06552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 30 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hilbert Space R2-calculus and Proofs The algebraic rules of the R2-calculus – both the population version in Proposition 1 and its sample counterpart – are not specific to linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In fact, all rela- tionships fundamentally stem from the geometry of projections in Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For this reason, the definitions of R2- and R-values can be generalized and the corresponding algebraic rules can be proven in more generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This section is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' First, we recall some results on Hilbert space theory (Halmos, 2000, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 26-29) and define generalized (partial) R2- and R- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Then, we prove Hilbert space generalizations to Lemma 1 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Finally, in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='3, we explain how the R2-calculus for linear models directly follows from the more general result and provides more details on the assumptions and notation involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hilbert Space R2-value Let (H, ⟨·, ·⟩) be a Hilbert space over the field K of real or complex numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' denote its associated norm as ∥·∥ and let X, Y, Z ⊆ H be closed linear subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The Minkowski sum of Y and X is given by X + Y := {x + y: x ∈ X, y ∈ Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For x ∈ X and y ∈ Y, we write x ⊥ y, if ⟨x, y⟩ = 0, x ⊥ Y, if x ⊥ y for all y ∈ Y, and X ⊥ Y, if x ⊥ Y for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For every element h ∈ H, there are unique x ∈ X and x⊥ ∈ H such that x ⊥ x⊥ and h = x + x⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The projection on X is the operator PX : H → X defined by the assignment h = x + x⊥ �→ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The projection off X is the operator QX : H → H defined by h = x + x⊥ �→ x⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Clearly, the projection on and off X add up to the identity operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' PX + QX = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Furthermore, we introduce the notations y⊥X := QX y and Y⊥X := {y⊥X : y ∈ Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The space Y⊥X is a closed linear subspace of H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' thus, the pro- jections PY⊥X and QY⊥X are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' They can be used to define conditional orthogonality: Y ⊥ X | Z ⇔ Y⊥Z ⊥ X ⊥Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (i) PX and QX are linear, self-adjoint, and idempotent operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (ii) If X ⊥ Y, PX+Y = PX + PY and QX+Y = QX QY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (iii) PX+Y = PX + PY⊥X and QX+Y = QX QY⊥X = QY⊥X QX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (iv) If h1, h2 ∈ H and h1 ⊥ h2, ∥h1 + h2∥2 = ∥h1∥2 + ∥h2∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (i) See Halmos (2000, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 26, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (ii) See Halmos (2000, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 28, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 2) for the proof of PX+Y = PX + PY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' According to Halmos (2000, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 29, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 1), PX PY = 0 holds due to X ⊥ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, the second statement directly follows QX+Y = Id − PX − PY = (Id − PX )(Id − PY) = QX QY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 31 (iii) We rewrite the direct sum X + Y as follows X + Y = {x + y: x ∈ X, y ∈ Y} = {x + PX y + QX y: x ∈ X, y ∈ Y} = {x + QX y: x ∈ X, y ∈ Y} = X + Y⊥X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Since X and Y⊥X are orthogonal by definition, the statement directly follows from (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (iv) See Halmos (2000, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Any one-dimensional linear subspace X can be expressed as X = {λ x: λ ∈ K}, where x is an arbitrary element in X \\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, we can identify a one-dimensional subspace with any non-zero element contained in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Definition 4 (Hilbert space R2- and R-value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Let X, Y, Z ⊆ H be closed linear subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Assume Y is one-dimensional, let y ∈ Y \\ {0} and suppose ∥y⊥Z∥2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The R2-value of Y on X is defined as R2 Y∼X := 1 − ∥y⊥X ∥2 ∥y∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The partial R2-value of Y on X given Z is defined as R2 Y∼X|Z := R2 Y∼X+Z − R2 Y∼Z 1 − R2 Y∼Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' If X is one-dimensional, x ∈ X \\ {0} and ∥x⊥Z∥2 > 0, the partial R-value is defined as RY∼X|Z := ⟨y⊥Z, x⊥Z⟩ ∥y⊥Z∥ ∥x⊥Z∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The corresponding (partial) f2- and f-values are defined analogously to Defini- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The choice of the non-zero elements y and x does not change the (partial) R2- and R-values due to the normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Therefore, all quantities above are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proofs of Results in Section 2 In this subsection, we state and prove the generalized versions of Lemma 1 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the setting of Definition 4, R2 Y∼X|Z = R2 Y⊥Z∼X ⊥Z holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' More- over, if X is a one-dimensional subspace, then R2 Y∼X|Z = (RY∼X|Z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 32 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The first statement of the lemma follows from some elementary algebraic manipulations and Lemma 4 (iii) R2 Y∼X|Z = R2 Y∼X+Z − R2 Y∼Z 1 − R2 Y∼Z � 1 − ∥y⊥X+Z∥2 ∥y∥2 − 1 + ∥y⊥Z∥2 ∥y∥2 � �∥y⊥Z∥2 ∥y∥2 = 1 − ∥y⊥X+Z∥2 ∥y⊥Z∥2 (iii) = 1 − ∥QX ⊥Z y⊥Z∥2 ∥y⊥Z∥2 = R2 Y⊥Z∼X ⊥Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To prove the second part of the lemma, we assume that X is one-dimensional and choose x ∈ X \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' If X ⊥Z = 0, the projection on X ⊥Z is 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' otherwise, it is given by PX ⊥Zh = ⟨h, x⊥Z⟩x⊥Z ∥x⊥Z∥2 , for h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (25) This can be easily checked: PX ⊥Z is linear and its image is contained in X ⊥Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Moreover, we compute �⟨h, x⊥Z⟩ x⊥Z ∥x⊥Z∥2 , h − ⟨h, x⊥Z⟩ x⊥Z ∥x⊥Z∥2 � = ⟨h, x⊥Z⟩2 ∥x⊥Z∥2 − ⟨h, x⊥Z⟩2∥x⊥Z∥2 ∥x⊥Z∥4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Following from the first part of the proof and Lemma 4 (iv), we infer R2 Y∼X|Z = 1 − ∥QX ⊥Z y⊥Z∥2 ∥y⊥Z∥2 (iv) = ∥PX ⊥Z y⊥Z∥2 ∥y⊥Z∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Directly plugging in the formula for the projection on X ⊥Z yields the second state- ment of the lemma R2 Y∼X|Z = ∥⟨y⊥Z, x⊥Z⟩ x⊥Z∥2 ∥y⊥Z∥2 ∥x⊥Z∥4 = ⟨y⊥Z, x⊥Z⟩2∥x⊥Z∥2 ∥y⊥Z∥2 ∥x⊥Z∥4 = � RY∼X|Z �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proposition 4 (Hilbert space R2-calculus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the setting of Definition 4, let W be another closed linear subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Assume ∥Y⊥X+W+Z∥2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Further suppose ∥X ⊥W+Z∥2 >0 and ∥W⊥X+Z∥2 > 0 when X and/or W are one-dimensional sub- spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Then, the following rules hold [i] Orthogonality: if Y ⊥ X, R2 Y∼X = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [ii] Orthogonal additivity: if X ⊥ W, R2 Y∼X+W = R2 Y∼X + R2 Y∼W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [iii] Decomposition of unexplained variation: 1 − R2 Y∼X+W = (1 − R2 Y∼X )(1 − R2 Y∼W|X );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [iv] Recursion of partial R-value: if X and W are one-dimensional, RY∼X|W = RY∼X − RY∼WRX∼W � 1 − R2 Y∼W � 1 − R2 X∼W ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 33 [v] Reduction of partial R-value: if X is one-dimensional and Y ⊥ W, RY∼X|W = RY∼X � 1 − R2 X∼W ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [vi] Three-dimensional restriction: if X and W are one-dimensional, fY∼X|W � 1 − R2 Y∼W|X = fY∼X � 1 − R2 X∼W − RY∼W|X RX∼W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' All of the relationships above also hold when Z is partialed out (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' if “ |Z” is appended to the subscripts of all R-, R2-, and f-values) and the orthogonality as- sumptions are conditional on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [i] Since Y⊥Z and X ⊥Z are orthogonal, QX ⊥Zy⊥Z = y⊥Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, R2 Y∼X|Z = 1 − ∥y⊥Z∥2 ∥y⊥Z∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [ii] Lemma 5 and its proof yield R2 Y∼X+W|Z = R2 Y⊥Z∼X ⊥Z+W⊥Z = ∥PX ⊥Z+W⊥Z y⊥Z∥2 ∥y⊥Z∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Following from Lemma 4 (ii) and (iv), we get R2 Y∼X+W|Z (ii) = ∥PX ⊥Zy⊥Z + PW⊥Z y⊥Z∥2 ∥y⊥Z∥2 (iv) = ∥PX ⊥Z y⊥Z∥2 ∥y⊥Z∥2 + ∥PW⊥Z y⊥Z∥2 ∥y⊥Z∥2 = R2 Y∼X|Z + R2 Y∼W|Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [iii] The statement directly follows from the definition of the partial R2-value � 1 − R2 Y∼X|Z � � 1 − R2 Y∼W|X+Z � = ∥y⊥X+Z∥2 ∥y⊥Z∥2 ∥y⊥W+X+Z∥2 ∥y⊥X+Z∥2 = ∥y⊥W+X+Z∥2 ∥y⊥Z∥2 = 1 − R2 Y∼W+X|Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [iv] Plugging in the definition of the partial R-value into the right-hand side, we get RHS = RY∼X|Z − RY∼W|Z RX∼W|Z � 1 − R2 Y∼W|Z � 1 − R2 X∼W|Z = � ⟨y⊥Z, x⊥Z⟩ ∥y⊥Z∥∥x⊥Z∥ − ⟨y⊥Z, w⊥Z⟩ ∥y⊥Z∥∥w⊥Z∥ ⟨x⊥Z, w⊥Z⟩ ∥x⊥Z∥∥w⊥Z∥ � � �∥y⊥W+Z∥ ∥y⊥Z∥ ∥x⊥W+Z∥ ∥x⊥Z∥ � = ⟨y⊥Z, x⊥Z⟩ ∥y⊥W+Z∥∥x⊥W+Z∥ − ⟨y⊥Z, w⊥Z⟩ ⟨x⊥Z, w⊥Z⟩ ∥w⊥Z∥2∥y⊥W+Z∥∥x⊥W+Z∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 34 Recalling the formula (25) for the projection operator on a one-dimensional subspace, we can reformulate the upper equation further RHS = � y⊥Z, x⊥Z − ⟨x⊥Z,w⊥Z⟩ w⊥Z ∥w⊥Z∥2 � ∥y⊥W+Z∥∥x⊥W+Z∥ = ⟨y⊥Z, QW⊥Z x⊥Z⟩ ∥y⊥W+Z∥∥x⊥W+Z∥ (iii) = ⟨y⊥W+Z, x⊥W+Z⟩ ∥y⊥W+Z∥∥x⊥W+Z∥ = RY∼X|W+Z = LHS, where the third equality follows from Lemma 4 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [v] Let (w⊥Z j )j∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=',J}, be an orthonormal basis of W⊥Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The subspace spanned by the first j vectors is denoted by W⊥Z j := span{w⊥Z 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' , w⊥Z j }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Due to rule [i] and Y ⊥ W | Z, R2 Y∼Wj|Z = 0 and R2 Y∼Wj+1|Wj+Z = 0 hold for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' , J − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' By induction, we prove the statement RY∼X|Z+Wj = RY∼X|Z � 1 − R2 X∼Wj|Z , for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' , J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For the base case, we apply rule [iv] and RY∼W1|Z = 0 as follows RY∼X|W1+Z [iv] = RY∼X|Z − RY∼W1|Z RX∼W1|Z � 1 − R2 Y∼W1|Z � 1 − R2 X∼W1|Z = RY∼X|Z � 1 − R2 X∼W1|Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The induction step uses rule [iv] and simplifies the resulting expression via RY∼Wj+1|Wj+Z = 0, the induction hypothesis and rule [iii]: RY∼X|Wj+1+Z [iv] = RY∼X|Wj+Z − RY∼Wj+1|Wj+Z RX∼Wj+1|Wj+Z � 1 − R2 Y∼Wj+1|Wj+Z � 1 − R2 X∼Wj+1|Wj+Z = RY∼X|Wj+Z � 1 − R2 X∼Wj+1|Wj+Z = RY∼X|Z � 1 − R2 X∼Wj|Z � 1 − R2 X∼Wj+1|Wj+Z [iii] = RY∼X|Z � 1 − R2 X∼Wj+1|Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' [vi] First, we apply rule [iv] to RY∼X|W+Z and RY∼W|X+Z RY∼X|W+Z = RY∼X|Z − RY∼W|Z RX∼W|Z � 1 − R2 Y∼W|Z � 1 − R2 X∼W|Z , RY∼W|X+Z = RY∼W|Z − RY∼X|Z RX∼W|Z � 1 − R2 Y∼X|Z � 1 − R2 X∼W|Z , Sensitivity Analysis with the R2-calculus 35 and compute RY∼X|W+Z � 1 − R2 Y∼W|Z + RY∼W|X+ZRX∼W|Z � 1 − R2 Y∼X|Z = 1 � 1 − R2 X∼W|Z � RY∼X|Z − RY∼W|ZRX∼W|Z + RY∼W|ZRX∼W|Z − RY∼X|ZR2 W∼X|Z � = RY∼X|Z � 1 − R2 X∼W|Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Next, we divide both sides of the equation by � 1 − R2 Y∼X|Z and rearrange it which results in RY∼X|W+Z � 1 − R2 Y∼W|Z � 1 − R2 Y∼X|Z = fY∼X|Z � 1 − R2 X∼W|Z − RY∼W|X+ZRX∼W|Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' According to rule [iii], we obtain (1−R2 Y∼X|Z)(1−R2 Y∼W|X+Z) = 1−R2 Y∼X+W|Z = (1−R2 Y∼W|Z)(1−R2 Y∼X|W+Z) and thus 1 − R2 Y∼W|Z 1 − R2 Y∼X|Z = 1 − R2 Y∼W|X+Z 1 − R2 Y∼X|W+Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Plugging this relationship into the left-hand side of the upper equation, we arrive at fY∼X|W+Z � 1 − R2 Y∼W|X+Z = fY∼X|Z � 1 − R2 X∼W|Z − RY∼W|X+Z RX∼W|Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R2-calculus for Linear Models The R2-calculus for linear models as presented in the main text is a special case of the R2-calculus for Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To be consistent with the standard notation for R2-values in linear models in the main text, we make two slight changes to the Hilbert space notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' First, a random vector denotes the linear space that is spanned by its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Analogously, for an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' sample of size n for a p-dimensional random vector X, we use the matrix X ∈ Rn×p to denote the row- space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Second, we replace the plus-sign with a comma for partialed out variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For instance, we write R2 Y ∼X|W,Z instead of R2 Y ∼X|W+Z in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Denote the space of square-integrable random variables L2 := {X : E[X2] < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 36 We define the following four Hilbert spaces with associated inner products H := L2, ⟨X, Y ⟩H := E[XY ], H0 := � X ∈ L2 : E[X] = 0 � , ⟨X, Y ⟩H0 := cov(X, Y ), ˆH := Rn, ⟨x, y⟩ ˆ H := n−1xT y, ˆH0 := � x ∈ Rn : ¯x = 0 � , ⟨x, y⟩ ˆ H0 := � cov(x, y), where ¯x denotes the empirical mean of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The population R2-calculus for linear mod- els as stated in the main text follows from choosing the Hilbert space (H0, ⟨·, ·⟩H0) in Lemma 5 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Likewise, we use ( ˆH0, ⟨·, ·⟩ ˆ H0) for the empirical R2- calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Since we choose the scaling n−1 in the empirical covariance, the estimators of covariance, variance and standard deviation are not unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To account for the loss of degrees of freedom through estimation of the mean and potentially partialing out a p-dimensional subspace, the factor (n−p−1)−1 must be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We choose the scaling n−1 instead to accord with the textbook definition of the empirical R2-value (Davidson and MacKinnon, 1993, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Besides, for a sufficiently large sample size n the difference will be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the main text, we made the assumption that the random variables and the observations are centred and thus are elements of H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' If this does not hold, we can redefine the population R2-value via the inner product ⟨·, ·⟩H as follows R2 Y ∼X := 1 − E[(Y ⊥X)2] E[Y 2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Similarly, we replace the inner product in the definition of partial R2-, R-, f2- and f-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This formulation contains the definition of R2-value in the main text as a special case because, for centred random variables, ⟨·, ·⟩H and the covariance are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Furthermore, if we treat the constant 1 as an additional covariate, the following relationship holds R2 Y −E[Y ]∼X−E[X] = R2 Y ∼X|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, centring random variables is equivalent to partialing out the effect of the constant, and thus always observed, covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' As our focus lies on the explanatory capability of the non-constant covariates, we always partial out 1 or equivalently centre the observed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The same arguments also apply to the empirical R2-value and centring the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proofs of Results in Section 3 Without loss of generality, we assume that all random variables/vectors are cen- tred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' moreover, we only state and prove the population version of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' As explained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='3, the sample and non-centred counterparts of the results and proofs follow by the same arguments but choosing a different Hilbert space and inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 37 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A Single Unmeasured Confounder Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' First, we rewrite the partialing out of Z in terms of a projection operation, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Lemma 4 (ii);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' then, we use linearity of the covariance and Lemma 4 (iv) to simplify the numerator and denominator, respectively: β1 = cov(D⊥X, Y ⊥X) − cov(D⊥X, QZ⊥XY ⊥X) var(D⊥X) − var(QZ⊥XD⊥X) = cov(D⊥X, PZ⊥XY ⊥X) var(PZ⊥XD⊥X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Since Z⊥X is one-dimensional, the projection PZ⊥X is given by (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Plugging this relationship into the equation above yields β1 = cov � D⊥X, cov(Z⊥X,Y ⊥X) var(⊥X) Z⊥X� var � cov(D⊥X,Z⊥X) var(Z⊥X) Z⊥X � = cov(Z⊥X, Y ⊥X) cov(Z⊥X, D⊥X) = βD∼Z|X, Y ∼Z|X which proves the first result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The second and third statements directly follow from the definition of the k-class estimand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' First, we express the estimands βY ∼D|X and βY ∼D|X,W in terms of standard deviations and correlations and replace the terms with the R- and σ-notation βY ∼D|X − βY ∼D|X,W = corr(Y ⊥X, D⊥X)sd(Y ⊥X)sd(D⊥X) sd(D⊥X)2 − corr(Y ⊥X,W, D⊥X,W )sd(Y ⊥X,W )sd(D⊥X,W ) sd(D⊥X,W )2 = RY ∼D|X σY ∼X σD∼X − RY ∼D|X,W σY ∼X+W σD∼X+W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Next, we extract the common factor σY ∼X+D/σD∼X by applying the formula for decomposition of unexplained variance [iii] four times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We then rewrite the differ- ence so that it is expressed in terms of RY ∼W|X,D instead of RY ∼D|X,W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To this end, we subsequently replace RY ∼D|X,W and RY ∼W|X via the recursion of partial correlation formula [iv].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' we get βY ∼D|X − βY ∼D|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='W [iii] = � � RY ∼D|X � 1 − R2 Y ∼D|X − RY ∼D|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='W � 1 − R2 Y ∼W|X � 1 − R2 Y ∼D|X � 1 − R2 D∼W|X � � σY ∼X+D σD∼X [iv] = � �fY ∼D|X − RY ∼D|X − RY ∼W|X RD∼W|X � 1 − R2 Y ∼D|X � 1 − R2 D∼W|X � � � σY ∼X+D σD∼X [iv] = � fY ∼D|X − 1 � 1 − R2 Y ∼D|X � 1 − R2 D∼W|X � � RY ∼D|X − RD∼W|X �� 1 − R2 Y ∼D|X � 1 − R2 D∼W|XRY ∼W|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D + RY ∼D|X RD∼W|X ��� σY ∼X+D σD∼X Sensitivity Analysis with the R2-calculus 38 = � fY ∼D|X � 1 − 1 1 − R2 D∼W|X + R2 D∼W|X 1 − R2 D∼W|X � +fD∼W|X RY ∼W|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D � σY ∼X+D σD∼X = fD∼W|X RY ∼W|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D σY ∼X+D σD∼X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Throughout this proof, all quantities partial out X which is indicated by either the subscript “|X” or the superscript “⊥X”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In order to shorten the notation, we only indicate partialing out X in the estimands and drop the X-dependence in the other quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' First, we focus on the difference between the k-class estimand βk and the OLS estimand βY ∼D|X,Z that adjusts for X and Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' multiplying the respective denomi- nators yields βk − βY ∼D|X,Z = cov(D, Y ) − k cov(D⊥Z, Y ⊥Z) var(D) − k var(D⊥Z) − cov(Y ⊥Z, D⊥Z) var(D⊥Z) = cov(D, Y ) var(D⊥Z) − k cov(D⊥Z, Y ⊥Z) var(D⊥Z) var(D⊥Z) var(D) − k var(D⊥Z)2 + − cov(D⊥Z, Y ⊥Z) var(D) + k cov(D⊥Z, Y ⊥Z) var(D⊥Z) var(D⊥Z) var(D) − k var(D⊥Z)2 = cov(D, Y ) var(D⊥Z) − cov(D⊥Z, Y ⊥Z) var(D) var(D⊥Z) var(D) − k var(D⊥Z)2 = cov(D, Y ) var(D⊥Z) − cov(D⊥Z, Y ⊥Z) var(D) var(D⊥Z) var(D) � 1 − k var(D⊥Z)/ var(D) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Next, we simplify the last expression by using 1 − R2 D∼Z = var(D⊥Z)/ var(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This results in a formula which involves the difference of the OLS estimands βY ∼D|X and βY ∼D|X,Z: βk − βY ∼D|X,Z = 1 1 − k � 1 − R2 D∼Z � �cov(D, Y ) var(D) − cov(D⊥Z, Y ⊥Z) var(D⊥Z) � = 1 1 − k � 1 − R2 D∼Z � � βY ∼D|X − βY ∼D|X,Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We can now use the last result to express the difference βk −β as a telescoping sum: βk − β = βk − βY ∼D|X,Z + βY ∼D|X,Z − βY ∼D|X,Z,U = 1 1 − k � 1 − R2 D∼Z � � βY ∼D|X − βY ∼D|X,Z � + � βY ∼D|X,Z − βY ∼D|X,Z,U � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This representation includes two differences of OLS estimands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' hence, Lemma 2 can be applied twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For the first summand, we use X ≡ X and W ≡ Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' for the Sensitivity Analysis with the R2-calculus 39 second, X ≡ (X, Z) and W ≡ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Thus, we get βk − β = 1 1 − k � 1 − R2 D∼Z �RY ∼Z|D fD∼Z σY ∼D σD + RY ∼U|Z,D fD∼U|Z σY ∼Z+D σD∼Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Finally, we can simplify the expression above by extracting a common factor of σY ∼Z+D/σD∼Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We use the definition of the (partial) R2-value, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 1 − R2 Y ∼Z|D = σ2 Y ∼Z+D/σ2 Y ∼D, and deduce that βk − β = � RY ∼Z|D fD∼Z 1 − k � 1 − R2 D∼Z � � 1 − R2 D∼Z � 1 − R2 Y ∼Z|D + RY ∼U|Z,D fD∼U|Z �σY ∼Z+D σD∼Z = � fY ∼Z|D RD∼Z 1 − k + k R2 D∼Z + RY ∼U|Z,D fD∼U|Z � σY ∼Z+D σD∼Z , which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the setting of Theorem 1, the following are true (i) Adjusted Regression: βY ∼D|X,Z − β = RY ∼U|X,Z,D fD∼U|X,Z σY ∼X+Z+D σD∼X+Z ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (ii) Unadjusted Regression: βY ∼D|X − β = � fY ∼Z|X,D RD∼Z|X + RY ∼U|X,Z,D fD∼U|X,Z � σY ∼X+Z+D σD∼X+Z ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (iii) Instrumental Variable: βD∼Z|X, Y ∼Z|X − β = �fY ∼Z|X,D RD∼Z|X + RY ∼U|X,Z,D fD∼U|X,Z � σY ∼X+Z+D σD∼X+Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The statements are a direct consequence of Theorem 1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (i) This statement directly follows from taking the limit k → −∞ and setting RY ∼U|X,Z,DfD∼U|X,Z = 0 in equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (ii) We apply the decomposition of unexplained variance rule 1 − R2 D∼U+Z|X [iii] = � 1 − R2 D∼Z|X �� 1 − R2 D∼U|X,Z � , 1 − R2 Y ∼U+Z|X,D [iii] = � 1 − R2 Y ∼Z|X,D �� 1 − R2 Y ∼U|X,Z,Z � , Sensitivity Analysis with the R2-calculus 40 which yields the implications R2 D∼U+Z|X = 0 ⇒ RD∼Z|X = 0, RD∼U|X,Z = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R2 Y ∼U+Z|X,D = 0 ⇒ RY ∼Z|X,D = 0, RY ∼U|X,Z,D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Then, the unbiasedness of βY ∼D|X,Z and βY ∼D|X follows from Corollary 1 or Theorem 1 with k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (iii) In order to connect the IV-related sensitivity parameters to RD∼U|X,Z and RY ∼U|X,Z,D, we apply the three-variable identity [vi] with Y ≡ Y , X ≡ Z, W ≡ U and Z ≡ (X, D) as well as Y ≡ U, X ≡ Z, W ≡ D and Z ≡ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We obtain fY ∼Z|X,U,D � 1 − R2 Y ∼U|X,Z,D = fY ∼Z|X,D � 1 − R2 Z∼U|X,D − RY ∼U|X,Z,DRZ∼U|X,D, fZ∼U|X,D � 1 − R2 D∼U|X,Z = fZ∼U|X � 1 − R2 D∼Z|X − RD∼Z|XRD∼U|X,Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' If we set RZ∼U|X = 0 and RY ∼Z|X,U,D = 0 in the equations above and simplify them, we get the relationship fD∼U|X,Z RY ∼U|X,Z,D = −fY ∼Z|X,D RD∼Z|X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Due to Corollary 1 or Theorem 1 with k = 1, this implies βD∼Z|X, Y ∼Z|X = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Multiple Unmeasured Confounders Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Analogously to the proof of Theorem 1, we only indicate partialing out X in the estimands and drop the X-dependence in the other quantities for ease of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We define the vector λ as follows λ = var(W ⊥D)−1 cov(W ⊥D, Y ⊥D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' It equals the regression coefficients of W in the linear model Y ∼ D + X + W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In order to reduce the number of dimensions of W, we introduce a new random variable W ∗ := λT W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Since it captures all linear influence of W on Y , the estimands βY ∼D|X,W and βY ∼D|X,W ∗ are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To formally prove this result, we let A denote either Y or D and show that A⊥W ∗ = A⊥W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' By definition of λ and some algebraic Sensitivity Analysis with the R2-calculus 41 manipulations we derive A⊥W ∗ = A − (W ∗)T var(W ∗)−1 cov(W ∗, A) = A − W T var(W ⊥D)−1 cov(W ⊥D, Y ⊥D) � var(W ⊥D)−1 cov(W ⊥D, Y ⊥D) �−1 × var(W)−1� var(W ⊥D)−1 cov(W ⊥D, Y ⊥D) �−T × cov(W ⊥D, Y ⊥D)T var(W ⊥D)−T cov(W, A) = A − W T var(W)−1 cov(W, A) = A⊥W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Choosing Y and D for A, we get βY ∼D|X,W ∗ = cov(Y ⊥W ∗, D⊥W ∗) var(D⊥W ∗) = cov(Y ⊥W , D⊥W ) var(D⊥W ) = βY ∼D|X,W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Since W ∗ is one-dimensional, we can use Lemma 2 to find a precise characterization for the difference between the OLS estimand that does not and does adjust for W: βY ∼D|X − βY ∼D|X,W = βY ∼D|X − βY ∼D|X,W ∗ = RY ∼W ∗|D fD∼W ∗ σY ∼D σD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (26) Moreover, the explanatory capabilities of W and W ∗ for Y are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' According to Lemma 4 (iii), we infer Y ⊥D,W = Q(D,W)Y = QD⊥W QW Y = QD⊥W ∗Y ⊥W ∗ = Y ⊥D,W ∗ which yields R2 Y ∼W|D = 1 − var(Y ⊥D,W ) var(Y ⊥D) = 1 − var(Y ⊥D,W ∗) var(Y ⊥D) = R2 Y ∼W ∗|D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The new random variable W ∗ fully captures the effect of W on Y but does not capture the entire effect of W on D due to the reduced dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R2 D∼W ≥ R2 D∼W ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To prove this result, we rewrite D⊥W using Lemma 4 (iii) as follows D⊥W = QPW ∗W+QW ∗W D = Q(W ∗,QW ∗W)D = QW ⊥W ∗D⊥W ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Based on this equation, Lemma 4 (iv) yields the inequality var(D⊥W ) ≤ var(QW ⊥W ∗D⊥W ∗) + var(PW ⊥W ∗D⊥W ∗) = var(D⊥W ∗), which implies R2 D∼W = 1 − var(D⊥W ) var(D) ≥ 1 − var(D⊥W ∗) var(D) = R2 D∼W ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Returning to (26), we use the equality and inequality derived for the R2-values con- cerning W ∗ → Y and W ∗ → D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Since f2 is a monotone transformation of R2, we have |βY ∼D|X − βY ∼D|X,W |2 ≤ R2 Y ∼W|D,X f2 D∼W|X σ2 Y ∼D+X σ2 D∼X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 42 In presence of multiple unmeasured confounders, finding an interpretable chara- terization of the difference βY ∼D|X,Z − βY ∼D|X,Z,U becomes more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the main text, we use a telescoping expansion and repeatedly apply Lemma 2 to ob- tain equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The sensitivity parameters in this characterization, however, are not symmetric in the set of partialed out variables which impedes their interpreta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Under the additional assumption that the components of U are conditionally independent given (X, Z), a symmetric representation can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The following result is closely related to Wright’s path analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Our proof, however, only relies on the algebraic relationships of the R2-calculus and does not consult the underlying DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Assume the setting of Lemma 3 and further suppose that all components of W are conditionally independent given X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Then, βY ∼D|X − βY ∼D|X,W = l � j=1 βY ∼Wj|X,D,W−j βWj∼D|X, (27) where W−j = (W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' , Wj−1, Wj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' , Wl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For ease of notation, we only indicate partialing out X in the estimands and drop the X-dependence in the other quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Due to the conditional independence assumption and Lemma 4 (ii), we can decompose Y as follows Y = Y ⊥D,W + PD⊥W Y + l � j=1 PWjY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Plugging this relationship into the definition of βY ∼D|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' using linearity of the co- variance and the formula for projections on a one-dimensional space (25) yields βY ∼D|X = cov(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' D) var(D) = 0 + cov(PD⊥W Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' D) var(D) + l � j=1 cov(PWjY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' D) var(D) = 1 var(D) � �cov �cov(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' D⊥W ) var(D⊥W ) D⊥W ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' D � + l � j=1 cov �cov(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Wj) var(Wj) Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' D �� � = cov(Y ⊥W ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' D⊥W ) var(D) + l � j=1 cov(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Wj) var(Wj) cov(D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Wj) var(D) = βY ∼D|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='W σ2 D∼W σ2 D + l � j=1 RY ∼Wj σY σWj βWj∼D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' By applying the definition of the R2-value, we derive βY ∼D|X − βY ∼D|X,W = −βY ∼D|X,W R2 D∼W + l � j=1 RY ∼Wj σY σWj βWj∼D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 43 Next, we use rule [ii] of the R2-calculus – independent additivity – on R2 D∼W and rewrite βY ∼D|X,W in terms of R-values and σ-values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' standard deviations: βY ∼D|X − βY ∼D|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='W [ii] = −RY ∼D|W σY ∼W σD∼W l � j=1 R2 D∼Wj + l � j=1 RY ∼Wj σY σWj βWj∼D = l � j=1 βWj∼D � RY ∼Wj σY σWj − σD RD∼WjσWj RY ∼D|W σY ∼W σD∼W R2 D∼Wj � In order to extract the factor σY ∼D+W−j/σWj∼D+W−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' we apply rule [iii] – decom- position of unexplained variance – six times and arrive at βY ∼D|X − βY ∼D|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='W [iii] = l � j=1 βWj∼D σY ∼D+W−j σWj∼D+W−j � RY ∼Wj � � � �1 − R2 Wj∼D+W−j 1 − R2 Y ∼D+W−j − RY ∼D|W RD∼Wj � � � �(1 − R2 Wj∼D+W−j)(1 − R2 Y ∼Wj|W−j) (1 − R2 D∼W )(1 − R2 Y ∼D|W−j) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (28) We concentrate on the term in brackets, denoted by Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Invoking rule [v] – reduction of partial correlation – and the (conditional) independence assumption, we infer RY ∼Wj [v] = RY ∼Wj|W−j � 1 − R2 Y ∼W−j, RD∼Wj [v] = RD∼Wj|W−j � 1 − R2 D∼W−j, R2 Wj∼D+W−j = R2 Wj∼D|W−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We insert these relationships into the expression of Tj and simplify it via rule [iii].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' we apply rule [iv] – recursion of partial correlation – on RY ∼D|W and simplify ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='the resulting expression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Tj = RY ∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Wj∼D|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� 1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Y ∼W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Y ∼D+W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='− RY ∼D|W RD∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D∼W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='�(1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Wj∼D|W−j)(1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Y ∼Wj|W−j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='(1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D∼W )(1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Y ∼D|W−j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='[iii] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='= RY ∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='�1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Y ∼D|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='− RY ∼D|W RD∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='�1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Y ∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Y ∼D|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='[iv] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='= RY ∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='�1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Y ∼D|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='−RD∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='RY ∼D|W−j−RY ∼Wj|W−jRD∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Y ∼D|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Sensitivity Analysis with the R2-calculus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='RY ∼Wj|W−j(1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D∼Wj|W−j) − RY ∼D|W−jRD∼Wj|W−j − RY ∼Wj|W−jR2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Y ∼D|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D∼Wj|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='= RY ∼Wj|W−j − RY ∼D|W−j RWj∼D|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Y ∼D|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1 − R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Wj∼D|W−j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='= RY ∼Wj|D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='W−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Returning to equation (28), we plug in Tj = RY ∼Wj|D,W−j and thus finish the proof βY ∼D|X − βY ∼D|X,W = l � j=1 βWj∼D σY ∼D+W−j σWj∼D+W−j RY ∼Wj|D,W−j = l � j=1 βY ∼Wj|D,W−j βWj∼D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Lemma 6 helps us express the bias of OLS and k-class estimands in terms of par- tial R-values which serve as sensitivity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Whether these are intuitive, de- pends on the causal structure of the underlying DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the case of two independent unmeasured variables U1 and U2 which confound or mediate β – the direct effect of D on Y –, the sensitivity parameters (RD∼U1, RD∼U2) and (RY ∼U1|D,U2, RY ∼U2|D,U1) are indeed intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The former tuple targets the dependence between D and U, the latter tuple focuses on the direct effects of U on Y regressing out the remaining variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The following theorem demonstrates how Lemma 6 can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We identify the bias of the k-class estimand in terms of the intuitive sensitivity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Assume the setting of Theorem 1 and let U = (U1, U2) be a two- dimensional random vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Further, suppose U1 ⊥⊥ U2 | X, Z holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Then, βk − β = � fY ∼Z|X,D RD∼Z|X 1 − k + k R2 D∼Z|X + 2 � j=1 Rj fj � 1 − f2 j f2 −j + � R−j � 1−R2 j 1−R2 −j − Rj fj f−j �2 � σY ∼X+Z+D σD∼X+Z , where Rj and fj abbreviate RY ∼Uj|X,Z,D,U−j and fD∼Uj|X,Z, respectively, for j ∈{1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Similarly to the proof of Theorem 1, we expand the difference βk − β as a telescoping sum βk − β = (βk − βY ∼D|X,Z) + (βY ∼D|X,Z − βY ∼D|X,Z,U), Sensitivity Analysis with the R2-calculus 45 which allows us to deal with the two summands separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Following from the same arguments, the first summand equals 1 1 − k(1 − R2 D∼Z|X)(βY ∼D|X − βY ∼D|X,Z) = fY ∼Z|X,D RD∼Z|X 1 − k + k R2 D∼Z|X σY ∼X+Z+D σD∼X+Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' From here onwards, partialing out X and Z is only indicated in the estimands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In order to rewrite and simplify the second summand, we invoke Lemma 6 and the rule on decomposition of unexplained variance βY ∼D|X,Z − βY ∼D|X,Z,U = 2 � j=1 RY ∼Uj|D,U−j σY ∼D+U−j σUj∼D+U−j RD∼Uj σUj σD [iii] = 2 � j=1 RY ∼Uj|D,U−jRD∼Uj � � � � 1 − R2 Y ∼U−j|D 1 − R2 Uj∼D+U−j σY ∼D σD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (29) Due to rule [i] and the conditional independence assumption, RU1∼U2 = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This result can be used to rewrite RUj∼U−j|D via the recursive partial correlation formula [iv];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' moreover, we use the decomposition of unexplained variance [iii] on 1 − R2 Ui∼D+U−i as follows RUj∼U−j|D [iv] = RUj∼U−j − RUj∼D RU−j∼D � 1 − R2 Uj∼D � 1 − R2 U−j∼D = −fD∼Uj fD∼U−j, (30) 1 − R2 Ui∼D+U−i [iii] = (1 − R2 D∼Ui)(1 − R2 Ui∼U−i|D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Inserting these relationships into (29), we find βY ∼D|X,Z − βY ∼D|X,Z,U = 2 � j=1 RY ∼Uj|D,U−j fD∼Uj � � � � 1 − R2 Y ∼U−j|D 1 − f2 D∼U1f2 D∼U2 σY ∼D σD = 2 � j=1 Rj fj � 1 − R2 Y ∼U−j|D 1 − f2 1 f2 2 σY ∼D σD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (31) Lastly, we aim to express � 1 − R2 Y ∼U−j|D in terms of the other sensitivity param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To this end, we use the three-variable identity [vi] with Y ≡ Y , X ≡ U−j, W ≡ Uj and Z ≡ D, where we replace RUj∼U−j|D according to (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' R−j � 1 − R2 −j � 1 − R2 j − f1f2 Rj [vi] = fY ∼U−j|D � 1 − f2 1 f2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' By definition, the identity √ 1 − R2 = 1/ � 1 + f2 holds true for any (partial) R2 Sensitivity Analysis with the R2-calculus 46 and its corresponding f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Thus, we get � 1 − R2 Y ∼U−j|D = � ����1 + � R−j √ 1−R2 −j � 1 − R2 j − f1f2 Rj �2 1 − f2 1 f2 2 � ���� −1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Substituting the � 1 − R2 Y ∼U−j|D term in (31) for the expression above proves the form of the second summand that was required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Derivation of Constraints in Section 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Ordinary Least Squares Specifying a comparative bound on U → Y that partials out D involves two addi- tional sensitivity parameters, RY ∼U|X,Z and RY ∼U| ˜ X, ˙XI,Z,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The former is related to RY ∼U|X,Z,D via equation (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' hence, it remains to find a relationship that con- nects RY ∼U| ˜ X, ˙XI,Z,D to the other sensitivity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To this end, we employ rule [v] – reduction of partial correlation – and the recursive partial correlation formula [iv] for RY ∼U| ˜ X, ˙XI,Z,D and infer RY ∼U|X,Z [v] = RY ∼U| ˜ X, ˙XI,Z � 1 − R2 Y ∼ ˙XIc| ˜ X, ˙XI,Z [iv] = 1 � 1 − R2 Y ∼ ˙XIc| ˜ X, ˙XI,Z � RY ∼D| ˜ X, ˙XI,ZRD∼U| ˜ X, ˙XI,Z + RY ∼U| ˜ X, ˙XI,Z,D � 1 − R2 Y ∼D| ˜ X, ˙XI,Z � 1 − R2 D∼U| ˜ X, ˙XI,Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This equation contains the unknown quantity RD∼U| ˜ X, ˙XI,Z which can be expressed in terms of RD∼U|X,Z via rule [v] RD∼U|X,Z [v] = RD∼U| ˜ X, ˙XI,Z � 1 − R2 D∼ ˙XIc| ˜ X, ˙XI,Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Plugging this relationship into the equation above, we arrive at the constraint RY ∼U|X,Z = 1 � 1 − R2 Y ∼ ˙XIc| ˜ X, ˙XI,Z � RY ∼D| ˜ X, ˙XI,Z � 1 − R2 D∼ ˙XIc| ˜ X, ˙XI,ZRD∼U|X,Z + RY ∼U| ˜ X, ˙XI,Z,D � 1 − R2 Y ∼D| ˜ X, ˙XI,Z � 1 − R2 D∼U|X,Z � 1 − R2 D∼ ˙XIc| ˜ X, ˙XI,Z � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 47 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Two-stage Least Squares The comparative bound on U ↔ Z is in fact equivalent to a bound on the sensitivity parameter RZ∼U|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' First, we relate RZ∼U| ˜ X, ˙X−j to RZ∼U|X via the conditional independence assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Recursion of partial correlation [iv] yields 0 = RU∼ ˙Xj| ˜ XI, ˙X−j,Z [iv] = RU∼ ˙Xj| ˜ X, ˙X−j − RZ∼ ˙Xj| ˜ X, ˙X−jRZ∼U| ˜ X, ˙X−j � 1 − R2 Z∼ ˙Xj| ˜ X, ˙X−j � 1 − R2 Z∼U| ˜ X, ˙X−j ⇔ RU∼ ˙Xj| ˜ X, ˙X−j = RZ∼ ˙Xj| ˜ X, ˙X−jRZ∼U| ˜ X, ˙X−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Employing this relationship and rule [iv] again, we find RZ∼U|X [iv] = RZ∼U| ˜ X, ˙X−j − RZ∼ ˙Xj| ˜ X, ˙X−jRU∼ ˙Xj| ˜ X, ˙X−j � 1 − R2 Z∼ ˙Xj| ˜ X, ˙X−j � 1 − R2 U∼ ˙Xj| ˜ X, ˙X−j = RZ∼U| ˜ X, ˙X−j � � � � 1 − R2 Z∼ ˙Xj| ˜ X, ˙X−j 1 − R2 Z∼ ˙Xj| ˜ X, ˙X−jR2 Z∼U| ˜ X, ˙X−j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' As the right-hand side above is monotone in RZ∼U| ˜ X, ˙Xj, we conclude R2 Z∼U|X ≤ bUZ R2 Z∼ ˙Xj| ˜ X, ˙X−j 1 − R2 Z∼ ˙Xj| ˜ X, ˙X−j 1 − bUZ R4 Z∼ ˙Xj| ˜ X, ˙X−j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' If a practitioner specifies a comparative bound on Z → Y , we need to connect RY ∼ ˙Xj| ˜ X, ˙X−j,U,D to RD∼U|X,Z and RY ∼U|X,Z,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To this end, we employ rule [vi] – the three-variable identity – with Y ≡ Y , X ≡ ˙Xj, W ≡ U and Z ≡ ( ˜X, ˜X−j, D) which yields fY ∼ ˙Xj| ˜ X, ˙X−j,Z,U,D � 1 − R2 Y ∼U|X,Z,D [vi] = fY ∼ ˙Xj| ˜ X, ˙X−j,Z,D � 1 − R2 U∼ ˙Xj| ˜ X, ˙X−j,Z,D − RY ∼U|X,Z,D RU∼ ˙Xj| ˜ X, ˙X−j,Z,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' we use the conditional independence U ⊥⊥ ˙Xj | ˜X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Z both to simplify the following recursive partial correlation formula [iv] and to apply the reduction of partial correlation formula [v] on RD∼U|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z RU∼ ˙Xj| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D [iv] = RU∼ ˙Xj| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z − RD∼ ˙Xj| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z RD∼U| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z � 1 − R2 D∼ ˙Xj| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z � 1 − R2 D∼U| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z = −fD∼ ˙Xj| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z fD∼U| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' RD∼U| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z [v] = RD∼U|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z � 1 − R2 D∼ ˙Xj| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 48 Inserting these two relationships in the three-variable identity above and cancelling some terms, we arrive at fY ∼ ˙Xj| ˜ X, ˙X−j,Z,U,D � 1 − R2 Y ∼U|X,Z,D = � fY ∼ ˙Xj| ˜ X, ˙X−j,Z,D � 1 − R2 D∼U|X,Z + RY ∼U|X,Z,D RD∼ ˙Xj| ˜ X, ˙X−j,Z RD∼U|X,Z ��� 1 − R2 D∼U|X,Z(1 − R2 D∼ ˙Xj| ˜ X, ˙X−j,Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Solving the Optimization Problem Since users can specify any number and kind of bounds on the sensitivity parameters, the resulting constraint set Ψ(ˆθ) is potentially very complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' It may be non-convex and can contain multiple non-linear equality- and inequality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This only leaves few standard optimization algorithms to compute a global solution for (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' These, however, often require careful choice of hyper-parameters and sometimes fail to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For this reason, we propose an adapted grid search algorithm that is more robust and tailored to our specific optimization problem by exploiting the structure of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' First, we characterize the set of potential minimizers and max- imizers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' then, we explain how we can use monotonicity of equality constraints to reduce the number of dimensions of the grid search algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' finally, we give the pseudocode of the algorithm and discuss its computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Characterization of the Solution According to Theorem 1, the objective β is identified in terms of the sensitivity parameters (ψ1, ψ2) = (RD∼U|X,Z, RY ∼U|X,Z,D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Due to its monotonicity in ψ2, the objective β attains its optimal values on a subset of the boundary of Ψ(ˆθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In order to show this, we characterize the feasible set as Ψ(ˆθ) = � ψ1 : Pψ1̸=∅ Pψ1, where Pψ1 = {ψ2 : (ψ1, ψ2) ∈ Ψ(ˆθ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For every fixed ψ1 such that Pψ1 ̸= ∅, the objective β is a linear function in ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This implies that, for any ψ2 ∈ Pψ1, we obtain β(ˆθ, ψ1, min Pψ1) ⋚ β(ˆθ, ψ1, ψ2) ⋚ β(ˆθ, ψ1, max Pψ1), where the direction of the inequalities depends on the sign of ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Therefore, ψ-values that minimize/maximize β are contained in Ψ∗(ˆθ) := � ψ1 : Pψ1̸=∅ {min Pψ1, max Pψ1}, (32) which is a subset of the boundary of Ψ(ˆθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Therefore, it suffices to discretize the set Ψ∗(ˆθ) instead of Ψ(ˆθ) to find an ap- proximate solution to the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 49 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Transfering Bounds via Monotonicity Regular grid search algorithms are highly computationally expensive as their com- plexity grows exponentially in the number of unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Yet, the high computational costs can be significantly reduced by leveraging the monotonicity of many equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We illustrate this with an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Suppose a practitioner specifies the following direct constraint on U → D and comparative constraint on U → Y : RD∼U|X,Z ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5], R2 Y ∼U| ˜ X, ˙X−j,Z ≤ 2R2 Y ∼ ˙Xj| ˜ X, ˙X−j,Z ⇔ R2 Y ∼U|X,Z ≤ 2 f2 Y ∼ ˙Xj| ˜ X, ˙X−j,Z, (33) where the latter equivalence is due to (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In addition, the unknown parame- ters RD∼U|X,Z, RY ∼U|X,Z,D and RY ∼U|X,Z are constrained by the recursive partial correlation formula RY ∼U|X,Z,D [iv] = RY ∼U|X,Z − RY ∼D|X,Z RD∼U|X,Z � 1 − R2 Y ∼D|X,Z � 1 − R2 D∼U|X,Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (34) Note that, for any fixed RD∼U|X,Z value, RY ∼U|X,Z,D is a linear, and hence, mono- tone function of RY ∼U|X,Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In this setting, brute-force grid search creates a three-dimensional grid of points – one dimension per unknown partial R-value – and only keeps those that (approxi- mately) conform with (33) and (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (Partial R- and f-values that only depend on V are estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=') The remaining points are projected onto the (RD∼U|X,Z, RY ∼U|X,Z,D)- plane and, for every fixed RD∼U|X,Z, we can find the smallest/largest value of RY ∼U|X,Z,D to approximate Ψ∗(ˆθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, in this example, the complexity of brute- force grid search is cubic in the number of points per dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Our algorithm, on the other hand, only needs to create a one-dimensional grid of RD∼U|X,Z values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' discretize [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For every such value, we can compute the smallest value for RY ∼U|X,Z,D by plugging RY ∼U|X,Z = − √ 2 | ˆfY ∼ ˙Xj| ˜ X, ˙X−j,Z| into (34) directly and likewise for the largest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Therefore, the complexity only grows linearly in the number of points per dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This principle of using monotonicity of the equality constraints can reduce the dimension of the grid and applies beyond the above example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In fact, when only bounds on U → D and U → Y are given, we solely require a one-dimensional grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, the computational complexity of generating equally spaced points in Ψ∗(ˆθ) grows linearly in the number of grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the most general case, when any (finite) number and kind of bound can be specified, only a three-dimensional grid is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, the worst complexity of the point-generation algorithm is cubic in the number of points per dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Evaluating the objective over Ψ∗(ˆθ) has linear complexity in any case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Adapted Grid-Search Algorithm Our proposed algorithm first constructs a set of equally spaced points that are (approximately) contained in Ψ∗(ˆθ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' then, it evaluates β over this set and takes the Sensitivity Analysis with the R2-calculus 50 Algorithm 1: Grid approximation of Ψ∗(ˆθ) Input: lower and upper bounds given by Al, Au, Bl, Bu, Dl, Du, Eu, El, Ml, Mu, Ol, Ou, bZY Output: vectors A, L and U 1 al ← max{Al};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' au← min{Au} 2 ml← max{Ml};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' mu← min{Mu} 3 Initialize A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' U ∈ RNa 4 for i ∈ [Na] do 5 Ai ← al + (i − 1) (au − al)/(Na − 1) 6 dl ← max{hd(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' El,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Dl} /* Pushing bounds onto b */ 7 du ← min{hd(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Eu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Du} 8 bl ← max{hb(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' dl),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Bl} 9 bu ← min{hb(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' du),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Bu} 10 if bl > bu then 11 Ai ← NA 12 Li ← NA 13 Ui ← NA 14 else 15 fgl← hfg(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Ml) /* Pushing bounds onto g */ 16 fgu← hfg(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Mu) 17 gl ← fgl/ � 1 + f2 gl 18 gu ← fgu/ � 1 + f2gu 19 found ← False /* Finding Li */ 20 for j ∈ [Nb] and not found do 21 Bij ← bl + (j − 1) (bu − bl)/(Nb − 1) 22 fq ← hfq(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Bij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c7) /* Computing bounds on o */ 23 ol ← max{− � bZY · f2q /(1 + f2q ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Ol} 24 ou ← min{ � bZY · f2q /(1 + f2q ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Ou} 25 if ol <= ou then 26 for k ∈ [Ng] and not found do 27 Gik ← gl + (j − 1) (gu − gl)/(Ng − 1) 28 fo ← hfo(Bij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Gik) 29 o ← fo/ � 1 + f2o 30 found ← found ∨(ol ≤ o ∧ o ≤ ou) 31 if found then 32 Li ← Bij 33 if not found then 34 Li ← NA 35 found ← False /* Finding Ui */ 36 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' /* Analogously to Li but the Bij decrease */ 37 return A, L, U Sensitivity Analysis with the R2-calculus 51 minimum/maximum of the obtained β-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The latter step is straightforward whereas the former is complex when multiple interlocking constraints are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In order to keep the the notation short,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' we introduce some abbreviations: a = RD∼U|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' b = RY ∼U|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' d = RY ∼U|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' e = RY ∼U| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙XI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' g = RZ∼U|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' m = RZ∼U|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' o = RY ∼Z|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' q = RY ∼ ˙Xj| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c1 = RY ∼D|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c2 = RY ∼D| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙XI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c3 = RD∼ ˙XIc| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙XI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c4 = RD∼ ˙XIc| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙XI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c5 = RD∼Z|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c6 = RY ∼Z|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c7 = RY ∼ ˙Xj| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' ˙X−j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' hb(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' d) = d − c3 a � 1 − c2 3 √ 1 − a2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' hd(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c4) = 1 � 1 − c2 4 � e � 1 − c2 2 � 1 − a2(1 − c2 3) + c2 � 1 − c2 3 a2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' hfg(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' m) = 1 √ 1 − a2 �� 1 − c2 5 · fm − c5 a � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' hfo(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' g) = 1 √ 1 − b2 �� 1 − g2 · fc6 − b g � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' hfq(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' c7) = √ 1 − a2 · fc7 + c7 a b √ 1 − b2� 1 − a2(1 − c2 7) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' With a slight abuse of notation, the parameter e and the associated constants c2, c3 and c4 as well as q and the associated constant c7 may be scalars or vectors depending on the number of (13)- and (22)-constraints, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The notation fs is a shorthand for the f-transformation of some scalar or vector, that is fs = s/ √ 1 − s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The functions hb and hd are abbreviations for the right-hand sides of the equations (15) and (16), hfo and hfg stem from (17) and (18) and hfq states (23) in the new notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Inserting vectors instead of scalars into the functions is interpreted as componentwise evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In order to compute a set of points that is approximately contained in Ψ∗(ˆθ), we first discretize the interval of all possible a-values and construct the vector A ∈ RNa which contains Na equally spaced points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' (This corresponds to discretizing the interval [min{ψ1 : Pψ1 ̸= ∅}, max{ψ1 : Pψ1 ̸= ∅}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=') Second, we construct the vectors L, U ∈ RNa which approximate the corresponding minima and maxima of β at the respective a-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Thus, we can create the points {(Ai, Li): i ∈ [Na]} ∪ {(Ai, Ui): i ∈ [Na]}, which are (approximately) contained and equally spaced in Ψ∗(ˆθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Evaluating the objective β over this set has complexity O(Na).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In case that only bounds on U → D and U → Y are specified, the computa- tional complexity of generating A, L and U grows linearly in Na and the computed points are actually elements of Ψ∗(ˆθ) instead of merely approximating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The two types of bounds on U → D specify direct constraints on a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, denoting the Sensitivity Analysis with the R2-calculus 52 vectors of upper and lower bounds on a stemming from (9) and (10) Al and Au, we can construct A by equally spacing Na points in the interval [max{Al}, min{Au}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Bounds on U → Y directly constrain b (11), d (14) and e (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Crucially, for every fixed a-value Ai, the functions hd and hb are linear in e and d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, we can transfer bounds on e onto d and, thus, update bounds on d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' likewise, we can then push forward the bounds on d onto b and compute Li and Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In case that at least one bound on U ↔ Z or Z → Y is specified, A can be constructed in the same way as before whereas L and U are more computationally involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We again use the observation that many h-functions are monotone in one argument in order to ”push forward” bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For fixed a-value, we can transfer bounds on m onto g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' for fixed a- and b- value, we can compute bounds on o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' for fixed a- and g-value, we can compute the corresponding o-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We construct Li and Ui by discretizing the range of possible b-values (after successively pushing bounds on e and d onto b) into Nb points and searching for the smallest/largest feasible value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' To test whether a given b-value is feasible, we construct a sequence of Ng values of g and check whether there is at least one value such that the bounds on o are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Therefore, the computational complexity of constructing A, L and U is O(Na · Nb · Ng).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Algorithm 1 contains the pseudocode of the algorithm to generate A, L and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' It concerns the case where at least one bound on U ↔ Z or Z → Y is specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Otherwise, we could directly set Li ← bl and Ui ← bu in line 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' A full implementation of the algorithm will be made available in a public Github repository soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Simulation Study We investigate the empirical coverage of sensitivity intervals computed with the bootstrap in two scenarios: a regression model with one additional covariate and an instrumental variable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In both set-ups, we set the nominal level to 90 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Linear Regression Simulation We generate a sample of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' random vectors (εU, εX, εX, εY )T ∼ N(0, Id) and compute the variables in the model using the following linear structural equations: U := εU, X := εX, D := X + U + εD, Y := D + 2X + U + εY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Based on these structural equations, we derive the covariance matrix of the involved random variables var � � � � � U X D Y � � � � � = � � � � 1 0 1 2 0 1 1 3 1 1 3 6 2 3 6 15 � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 53 It can be used to compute (partial) R-values as well as the bias βY ∼D|X − β = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' If the comparative constraints R2 D∼U ≤ R2 D∼X, R2 Y ∼U ≤ 4 9 R2 Y ∼X are specified, the partially identified region is [1, (3 + √ 3)/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, the true value β = 1 equals the lower end of the PIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The bounds above are sharp in the sense that the lower end of the partially identified range can only be reached when both inequalities are active, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' hold with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In order to construct sensitivity intervals, we generate bootstrap samples of the observed data and solve the corresponding optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Then, we use either percentile or basic bootstrap (Davison and Hinkley, 1997, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 5) to compute the lower and upper end of the sensitivity interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This approach is compared to the heuristic sensitivity intervals of Cinelli and Hazlett (2020) as well as the oracle 90% confidence interval, which could be computed if U was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We simulate data for different sample sizes n and repeat each such experi- ment 1000 times to compute the empirical coverage and length of the sensitiv- ity/confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' More specifically, we evaluate the empirical coverage of β and the PIR for different sensitivity intervals and adapt the notion of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In or- der to account for the fact that the length of typical confidence intervals approaches zero as n → ∞ whereas the length of valid sensitivity intervals is lower bounded by the length of the PIR, we use the distance between the lower end of an interval and 1, when it covers 1, as length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The results of this simulation study are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Percentile boot- strap exhibits coverage of PIR close to the envisaged level of 90%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' its coverage of β is close to 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The latter is expected as the true value of β is the lower end of the PIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' By comparison, the empirical coverage of sensitivity intervals constructed via basic bootstrap is 5 to 10 percentage points below the required level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, we use percentile bootstrap to construct sensitivity intervals in the data example in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Moreover, this simulation study illustrates that Cinelli and Ha- zlett’s heuristic sensitivity intervals do not possess frequentist coverage guarantees: the empirical coverage of the PIR is consistently below 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Finally, we see that sensitivity intervals are substantially longer than the oracle confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We attribute the increased length to the uncertainty stemming from estimating the con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In this simulation study, we did not encounter cases where the estimated constraint set was empty on a bootstrap sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In order to investigate the coverage of bootstrap sensitivity intervals more closely, we consider the distribution of the estimated upper and lower end of the PIR as well as the corresponding bootstrap distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Figure 7 depicts the estimates of these distributions based on 1000 repitions of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' For small sample sizes n, we notice that the bootstrap distribution is both biased and skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Both phenomena diminish as n grows so that the bootstrap distribution approximates the target distribution more closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This is in line with the observation that coverage improves for larger sample sizes, especially for basic bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Sensitivity Analysis with the R2-calculus 54 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Simulation results of the linear regression example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' n Method Coverage Length β PIR Mean Median 200 Percentile bootstrap 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='6% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='748 Basic bootstrap 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='9% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='533 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='310 Heuristic 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='7% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='6% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='213 Oracle 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='124 500 Percentile bootstrap 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='320 Basic bootstrap 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='4% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='197 Heuristic 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='7% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='126 Oracle 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='077 1000 Percentile bootstrap 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='1% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='6% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='207 Basic bootstrap 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='6 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='144 Heuristic 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='089 Oracle 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='053 2000 Percentile bootstrap 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='8% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='135 Basic bootstrap 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='8% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='9% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='117 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='106 Heuristic 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='9% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='063 Oracle 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='037 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Linear Instrumental Variable Simulation We generate 100 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' samples from the distribution (εU, εZ, εD, εY )T ∼ N(0, Id) and compute the variables of the model as follows U := εU, Z := εZ, D := Z + U + εD, Y := D + U + εY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' This data-generating process fulfills the instrumental variable assumptions which renders β = 1 point identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hence, a sensitivity interval where the IV-related sen- sitivity parameters are set to zero ought to be comparable with the confidence inter- val that is based on the asymptotic normality of the TSLS estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In order to use Algorithm 1, we slightly relax the IV assumptions requiring RZ∼U|X, RY ∼Z|X,U,D ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='002] and further set RD∼U|X,Z ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='999, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='999] to bound it away from −1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We compute the empirical coverage and length of sensitivity intervals constructed via percentile and basic bootstrap, the heuristic sensitivity intervals and the oracle confidence intervals over 500 repitions of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Due to the high compu- tational costs, we conduct this simulation study only for sample size n = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The results of this experiment are stated in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We notice that the bootstrap sensitivity intervals are on par with the oracle confidence interval, both in terms Sensitivity Analysis with the R2-calculus 55 n = 1000 n = 2000 n = 200 n = 500 0 1 2 3 0 1 2 3 0 2 4 6 0 2 4 6 PIR lower PIR lower - Boot PIR upper PIR upper - Boot Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Empirical distribution of the lower and upper end of the PIR as well as the corre- sponding bootstrap distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' of coverage and length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' By contrast, the heuristic sensitivity intervals exhibit very high coverage but their length is too long to be informative in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In this simulation study, 24 of the 500 · 500 = 250, 000 constructed bootstrap samples led to an empty constraint set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In these cases, we set the solution of the optimization problem to −∞ and ∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Choice of Hyper-parameters In Table 4, we list the hyper-parameters of Algorithm 1 that were used for different data analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The mesh size of the grid is the same in every dimension, that is Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Simulation results of the instrumental variable example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Method Coverage Length Mean Median Percentile bootstrap 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='338 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='301 Basic bootstrap 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='240 Heuristic 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='581 Oracle 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='290 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='257 Sensitivity Analysis with the R2-calculus 56 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Hyper-parameters for different plots and simulation examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Ngrid Nb-contour Nboot Figure 2 200 500 Figure 3 200 30 Figure 4 150 30 Figure 5 400 Figure 6 300 Table 2 200 500 Table 3 100 500 the numbers of points considered per dimension Na, Nb, and Ng are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We define Ngrid := Na = Nb = Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The number of points per dimension for b-contour plots and the number of bootstrap samples are denoted by Nb-contour and Nboot, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' In the simulation study and data example in this work, we found that the PIR estimates change only marginally for values of Ngrid larger than 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' We recommend to consider at least 100 points per grid dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Ngrid = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The rough struc- ture of the b-contours often becomes apparent for Nb-contour as low as 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' Due to the computational costs of the optimization problem, we choose a relatively low number of bootstrap samples Nboot = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} +page_content=' The simulation studies empirically confirm that percentile bootstrap sensitivity intervals achieve good coverage nonetheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfQPZI/content/2301.00040v1.pdf'} diff --git a/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf b/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b3dab0c4fb0bb9e335837c913b4dcf389a0016e7 --- /dev/null +++ b/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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100644 index 0000000000000000000000000000000000000000..a80afb653328ae04ca1fe84c2f66241af1a26df0 --- /dev/null +++ b/99AyT4oBgHgl3EQfdfcH/content/tmp_files/2301.00301v1.pdf.txt @@ -0,0 +1,1977 @@ +Generalized PTR: User-Friendly Recipes for Data-Adaptive +Algorithms with Differential Privacy +Rachel Redberg, Yuqing Zhu, Yu-Xiang Wang +University of California, Santa Barbara +{rredberg, yuqingzhu, yuxiangw}@ucsb.edu +January 3, 2023 +Abstract +The “Propose-Test-Release” (PTR) framework [Dwork and Lei, 2009] is a classic recipe for +designing differentially private (DP) algorithms that are data-adaptive, i.e. those that add less +noise when the input dataset is “nice”. We extend PTR to a more general setting by privately +testing data-dependent privacy losses rather than local sensitivity, hence making it applicable +beyond the standard noise-adding mechanisms, e.g. to queries with unbounded or undefined +sensitivity. We demonstrate the versatility of generalized PTR using private linear regression +as a case study. Additionally, we apply our algorithm to solve an open problem from “Private +Aggregation of Teacher Ensembles (PATE)” [Papernot et al., 2017, 2018] — privately releasing +the entire model with a delicate data-dependent analysis. +1 +Introduction +The guarantees of differential privacy (DP) [Dwork et al., 2006] are based on worst-case outcomes +across all possible datasets. A common paradigm is therefore to add noise scaled by the global +sensitivity of a query f, i.e. the maximum change in f between any pair of neighboring datasets. +A given dataset X might have a local sensitivity that is much smaller than the global sensitivity, in +which case we can hope to add a smaller amount of noise (calibrated to the local rather than the +global sensitivity) while achieving the same privacy guarantee. However, this must not be undertaken +naïvely – the local sensitivity is a dataset-dependent function and so calibrating noise to the local +sensitivity could leak information about the dataset [Nissim et al., 2007]. +The “Propose-Test-Release” (PTR) framework [Dwork and Lei, 2009] resolves this issue by introducing +a test to privately check whether a proposed bound on the local sensitivity is valid. Only if the +test “passes” is the output released with noise calibrated to the proposed bound on the local +sensitivity. +PTR is a powerful and flexible tool for designing data-adaptive DP algorithms, but it has several +limitations. First, it applies only to noise-adding mechanisms which calibrate noise according to the +sensitivity of a query. Second, the test in “Propose-Test-Release” is computationally expensive for all +but a few simple queries such as privately releasing the median or mode. Third, while some existing +works [Decarolis et al., 2020, Kasiviswanathan et al., 2013, Liu et al., 2021] follow the approach of +testing “nice” properties of a dataset before exploiting these properties in a private release to PTR 1, +1We refer to these as PTR-like methods. +1 +arXiv:2301.00301v1 [cs.LG] 31 Dec 2022 + +there has not been a systematic recipe for discovering which properties should be tested. +In this paper, we propose a generalization of PTR which addresses these limitations. The centerpiece +of our framework is a differentially private test on the data-dependent privacy loss. This test does +not directly consider the local sensitivity of a query and is therefore not limited to additive noise +mechanisms. Moreover, in many cases, the test can be efficiently implemented by privately releasing +a high-probability upper bound, thus avoiding the need to search an exponentially large space of +datasets. Furthermore, the derivation of the test itself often spells out exactly what properties of the +input dataset need to be checked, which streamlines the design of data-adaptive DP algorithms. +Our contributions are summarized as follows: +1. We propose a generalization of PTR which can handle algorithms beyond noise-adding +mechanisms. Generalized PTR allows us to plug in any data-dependent DP analysis to +construct a high-probability DP test that adapts to favorable properties of the input dataset – +without painstakingly designing each test from scratch. +2. We demonstrate that many existing examples of PTR and PTR-like algorithms can be unified +under the generalized PTR framework, sometimes resulting in a tighter analysis (see an +example of report-noisy-max in Sec A.1). +3. We show that one can publish a DP model through privately upper-bounding a one-dimensional +statistic — no matter how complex the output space of the mechanism is. We apply this result +to solve an open problem from PATE [Papernot et al., 2017, 2018]. +4. Our results broaden the applicability of private hyper-parameter tuning [Liu and Talwar, 2019, +Papernot and Steinke, 2021] in enabling joint-parameter selection of DP-specific parameters +(e.g., noise level) and native parameters of the algorithm (e.g., learning rate, regularization +weight), which may jointly affect the data-dependent DP losses. +2 +Related Work +Data-dependent DP algorithms. Privately calibrating noise to the local sensitivity is a well- +studied problem. One approach is to add noise calibrated to the smooth sensitivity [Nissim et al., +2007], an upper bound on the local sensitivity which changes slowly between neighboring datasets. +An alternative to this – and the focus of our work – is Propose-Test-Release (PTR) [Dwork and +Lei, 2009], which works by calculating the distance Dβ(X) to the nearest dataset to X whose local +sensitivity violates a proposed bound β. The PTR algorithm then adds noise to Dβ(X) before +testing whether this privately computed distance is sufficiently large. +PTR spin-offs abound. Notable examples include stability-based methods [Thakurta and Smith, +2013] (stable local sensitivity of 0 near the input data) and privately releasing upper bounds of local +sensitivity [Kasiviswanathan et al., 2013, Liu et al., 2021, Decarolis et al., 2020]. We refer readers to +Chapter 3 of Vadhan [2017] for a concise summary of these classical results. Recent work [Wang +et al., 2022] has provided Rényi DP bounds for PTR and demonstrated its applications to robust +DP-SGD. Our work (see Section 5.2) also considers applications of PTR in data-adaptive private +deep learning: Instead of testing the local sensitivity of each gradient step as in Wang et al. [2022], +our PTR-based PATE algorithm tests the data-dependent privacy loss as a whole. +Liu et al. [2021] proposed a new variant called High-dimensional Propose-Test-Release (HPTR). HPTR +provides a systematic way of solving DP statistical estimation problems by using the exponential +2 + +mechanism (EM) with carefully constructed scores based on certain one-dimensional robust statistics, +which have stable local sensitivity bounds. HPTR focuses on designing data-adaptive DP mechanisms +from scratch; our method, in contrast, converts existing randomized algorithms (including EM and +even some that do not satisfy DP) into those with formal DP guarantees. Interestingly, our proposed +method also depends on a one-dimensional statistic of direct interest: the data-dependent privacy +loss. +Data-dependent DP losses. The flip side of data-dependent DP algorithms is the study of +data-dependent DP losses [Papernot et al., 2018, Soria-Comas et al., 2017, Wang, 2017], which fix +the randomized algorithm but parameterize the resulting privacy loss by the specific input dataset. +For example: In the simple mechanism that adds Laplace noise with parameter b, data-dependent +DP losses are ϵ(X) = ∆LS(X)/b. The data-dependent DP losses are often much smaller than the DP +loss, but they themselves depend on the data and thus may reveal sensitive information; algorithms +satisfying a data-dependent privacy guarantee are not formally DP with guarantees any smaller +than that of the worst-case. Existing work has considered privately publishing these data-dependent +privacy losses [Papernot et al., 2018, Redberg and Wang, 2021], but notice that privately publishing +these losses does not improve the DP parameter of the given algorithm. Part of our contribution is +to resolve this conundrum by showing that a simple post-processing step of the privately released +upper bound of ϵ(Data) gives a formal DP algorithm. +Private hyper-parameter tuning. Our work has a nice connection with private hyper-parameter +tuning. Prior work [Liu and Talwar, 2019, Papernot and Steinke, 2021] requires each candidate +configuration to be released with the same DP (or Rényi DP) parameter set. Another hidden +assumption is that the parameters must not be privacy-correlated (i.e., parameter choice will not +change the privacy guarantee). Otherwise we need to use the largest DP bound across all candidates. +For example, Liu and Talwar [2019] show that if each mechanism (instantiated with one group of +hyper-parameters) is (ϵ, 0)-DP, then running a random number of mechanisms and reporting the best +option satisfies (3ϵ, 0)-DP. Our work directly generalizes the above results by (1) considering a wide +range of hyper-parameters, either privacy-correlated or not; and (2) requiring only that individual +candidates to have a testable data-dependent DP. +3 +Preliminaries +Datasets X, X′ ∈ X are neighbors if they differ by no more than one datapoint – i.e., X ≃ X′ if +d(X, X′) ≤ 1. We will define d(·) to be the number of coordinates that differ between two datasets +of the same size n: d(X, Y ) = #{i ∈ [n] : Xi ̸= Yi}. +We use || · || to denote the radius of the smallest Euclidean ball that contains the input set, e.g. +||X|| = supx∈X ||x||. +The parameter φ denotes the privacy parameters associated with a mechanism (e.g. noise level, +regularization). Mφ is a mechanism parameterized by φ. For mechanisms with continuous output +space, we will take Pr[M(X) = y] to be the probability density function of M(X) at y. +Definition 3.1 (Differential privacy [Dwork et al., 2006]). Fix ϵ, δ ≥ 0. A randomized algorithm +M : X → S satisfies (ϵ, δ)-DP if for all neighboring datasets X ≃ X′ and for all measurable sets +S ⊂ S, +Pr +� +M(X) ∈ S +� +≤ eϵPr +� +M(X′) ∈ S +� ++ δ. +Suppose we wish to privately release the output of a real-valued function f : X → R. We can do so +3 + +by calculating the global sensitivity ∆GS, calibrating the noise scale to the global sensitivity and +then adding sampled noise to the output. +Definition 3.2 (Local / Global sensitivity). The local ℓ∗-sensitivity of a function f is defined as +∆LS(X) = max +X≃X′ ||f(X) − f(X′)||∗ and the global sensitivity of f is ∆GS = supX ∆LS(X). +3.1 +Propose-Test-Release +Calibrating the noise level to the local sensitivity ∆LS(X) of a function would allow us to add less +noise and therefore achieve higher utility for releasing private queries. However, the local sensitivity +is a data-dependent function and naïvely calibrating the noise level to ∆LS(X) will not satisfy +DP. +PTR resolves this issue in a three-step procedure: propose a bound on the local sensitivity, privately +test that the bound is valid (with high probability), and if so calibrate noise according to the bound +and release the output. +PTR privately computes the distance Dβ(X) between the input dataset X and the nearest dataset +X′′ whose local sensitivity exceeds the proposed bound β: +Dβ(X) = min +X′′ {d(X, X′′) : ∆LS(X′′) > β}. +Algorithm 1 Propose-Test-Release [Dwork and Lei, 2009] +1: Input: Dataset X; privacy parameters ϵ, δ; proposed bound β on ∆LS(X); query function +f : X → R. +2: if Dβ(X) + Lap +� 1 +ϵ +� +≤ log(1/δ) +ϵ +then output ⊥, +3: else release f(X) + Lap +� +β +ϵ +� +. +Theorem 3.3. Algorithm 1 satisfies (2ϵ, δ)-DP. [Dwork and Lei, 2009] +Rather than proposing an arbitrary threshold β, one can also privately release an upper bound of +the local sensitivity and calibrate noise according to this upper bound. This was used for node DP +in graph statistics [Kasiviswanathan et al., 2013], and for fitting topic models using spectral methods +[Decarolis et al., 2020]. +4 +Generalized PTR +This section introduces the generalized PTR framework. We first formalize the notion of data- +dependent differential privacy that conditions on an input dataset X. +Definition 4.1 (Data-dependent privacy). Suppose we have δ > 0 and a function ϵ : X → R. We +say that mechanism M satisfies (ϵ(X), δ) data-dependent DP2 for dataset X if for all possible output +sets S and neighboring datasets X′, +Pr +� +M(X) ∈ S +� +≤ eϵ(X)Pr +� +M(X′) ∈ S +� ++ δ, +Pr +� +M(X′) ∈ S +� +≤ eϵ(X)Pr +� +M(X) ∈ S +� ++ δ. +2We will sometimes write that M(X) satisfies ϵ(X) data-dependent DP with respect to δ. +4 + +In generalized PTR, we propose a value φ for the randomized algorithm M, which could be a noise +scale or regularization parameter – or a set including both. For example, φ = (λ, γ) in Example 4.4. +We then say that Mφ is the mechanism M parameterized by φ, and ϵφ(X) its data-dependent +DP. +The following example illustrates how to derive the data-dependent DP for a familiar friend – the +Laplace mechanism. +Example 4.2. ( Data-dependent DP of Laplace Mechanism.) Given a function f : X → R, we will +define +Mφ(X) = f(X) + Lap (φ) . +We then have +log Pr[Mφ(X) = y] +Pr[Mφ(X′) = y] ≤ |f(X) − f(X′)| +φ +. +Maximizing the above calculation over all possible outputs y and using Definition 4.1, +ϵφ(X) = +max +X′:X′≃X +|f(X) − f(X′)| +φ += ∆LS(X) +φ +. +The data-dependent DP ϵφ(X) is a function of both the dataset X and the parameter φ. Maximizing +ϵφ(X) over X recovers the standard DP guarantee of running M with parameter φ. +Algorithm 2 Generalized Propose-Test-Release +1: Input: Dataset X; mechanism Mφ : X → R and its privacy budget ϵ, δ; (ˆϵ, ˆδ)-DP test T ; false +positive rate ≤ δ′; data-dependent DP function ϵφ(·) w.r.t. δ. +2: if not T (X) then output ⊥, +3: else release θ = Mφ(X). +Theorem 4.3 (Privacy guarantee of generalized PTR). Consider a proposal φ and a data-dependent +DP function ϵφ(X) w.r.t. δ. Suppose that we have an (ˆϵ, ˆδ)-DP test T : X → {0, 1} such that when +ϵφ(X) > ϵ, +T (X) = +� +0 with probability 1 − δ′, +1 with probability δ′. +Then Algorithm 2 satisfies (ϵ + ˆϵ, δ + ˆδ + δ′)-DP. +Proof sketch. There are three main cases to consider: +1. We decide not to run Mφ. +2. We decide to run Mφ and ϵφ(X) > ϵ; +3. We decide to run Mφ and ϵφ(X) ≤ ϵ. +5 + +In the first case, the decision to output ⊥ is post-processing of an (ˆϵ, ˆδ)-DP mechanism and inherits +its privacy guarantees. The second case occurs when the (ˆϵ, ˆδ)-DP test "fails" (produces a false +positive) and occurs with probability at most δ′. The third case is a composition of an (ˆϵ, ˆδ)-DP +algorithm and an (ϵ, δ)-DP algorithm. +Generalized PTR is a strict generalization of Propose-Test-Release. For some function f, define Mφ +and T as follows: +Mφ(X) = f(X) + Lap(φ); +T (X) = +� +0 +if Dβ(X) + Lap +� 1 +ϵ +� +> log(1/δ) +ϵ +, +1 +otherwise. +Notice that our choice of parameterization is φ = β +ϵ , where φ is the scale of the Laplace noise. In +other words, we know from Example 4.2 that ϵφ(X) > ϵ exactly when ∆LS(X) > β. +For noise-adding mechanisms such as the Laplace mechanism, the sensitivity is proportional to the +privacy loss (in both the global and local sense, i.e. ∆GS ∝ ϵ and ∆LS ∝ ϵ(X)). Therefore for these +mechanisms the only difference between privately testing the local sensitivity (Algorithm 1) and +privately testing the data-dependent DP (Theorem 4.3) is a change of parameterization. +4.1 +Limitations of local sensitivity +Why do we want to generalize PTR beyond noise-adding mechanisms? Compared to classic PTR, the +generalized PTR framework allows us to be more flexible in both the type of test conducted and also +the type of mechanism whose output we wish to release. For many mechanisms, the local sensitivity +either does not exist or is only defined for specific data-dependent quantities (e.g., the sensitivity of +the score function in the exponential mechanism) rather than the mechanism’s output. +The following example illustrates this issue. +Example 4.4 (Private posterior sampling). Let M : X × Y → Θ be a private posterior sampling +mechanism [Minami et al., 2016, Wang et al., 2015, Gopi et al., 2022] for approximately minimizing +FX(θ). +M samples θ ∼ P(θ) ∝ e−γ(FX(θ)+0.5λ||θ||2) with parameters γ, λ. Note that γ, λ cannot be appro- +priately chosen for this mechanism to satisfy DP without going through a sensitivity calculation of +arg min FX(θ). In fact, the global and local sensitivity of the minimizer is unbounded even in linear +regression problems, i.e when FX(θ) = 1 +2||y − Xθ||2. +Output perturbation algorithms do work for the above problem when we regularize, but they +are known to be suboptimal in theory and in practice [Chaudhuri et al., 2011]. In Section 5.1 +we demonstrate how to apply generalized PTR to achieve a data-adaptive posterior sampling +mechanism. +Even in the cases of noise-adding mechanisms where PTR seems to be applicable, it does not lead to +a tight privacy guarantee. Specifically, by an example of privacy amplification by post-processing +(Example A.1 in the appendix), we demonstrate that the local sensitivity does not capture all +sufficient statistics for data-dependent privacy analysis and thus is loose. +6 + +4.2 +Which φ to propose +The main limitation of generalized PTR is that one needs to “propose” a good guess of parameter φ. +Take the example of φ being the noise level in a noise-adding mechanism. Choosing too small a φ +will result in a useless output ⊥, while choosing too large a φ will add more noise than necessary. +Finding this ’Goldilocks’ φ might require trying out many different possibilities – each of which will +consume privacy budget. +This section introduces a method to jointly tune privacy parameters (e.g., noise scale) along with +parameters related only to the utility of an algorithm (e.g., learning rate or batch size in stochastic +gradient descent) – while avoiding the ⊥ output. +Algorithm 3 takes a list of parameters as input, runs generalized PTR with each of the parameters, +and returns the output with the best utility. We show that the privacy guarantee with respect to ϵ +is independent of the number of φ that we try. +Formally, let φ1, ..., φk be a set of hyper-parameters and ˜θi ∈ {⊥, Range(M)} denotes the output of +running generalized PTR on a private dataset X with φi. Let Xval be a public validation set and +q(˜θi) be the score of evaluating ˜θi with Xval (e.g., validation accuracy). The goal is to select a pair +(˜θi, φi) such that DP model ˜θi maximizes the validation score. +The generalized PTR framework with privacy calibration is described in Algorithm 3. The privacy +guarantee of Algorithm 3 is an application of Liu and Talwar [2019]. +Algorithm 3 PTR with hyper-parameter selection +1: Input: Privacy budget per PTR algorithm (ϵ∗, δ∗), cut-off T, parameters φ1:k, flipping probability +τ and validation score function q(·). +2: Initialize the set S = ∅. +3: Draw G from a geometric distribution Dτ and let ˆT = min(T, G). +4: for i = 1 ,..., ˆT do +5: +pick a random φi from φ1:k. +6: +evaluate φi: (˜θi, q(˜θi)) ← Algorithm 2(φi, (ϵ∗, δ∗)). +7: +S ← S ∪ {˜θi, q(˜θi)}. +8: end for +9: Output the highest scored candidate from S. +Theorem 4.5 ( Theorem 3.4 Liu and Talwar [2019] ). Fix any τ ∈ [0, 1], δ2 > 0 and let T = 1 +τ log 1 +δ2 . +If each oracle access to Algorithm 2 is (ϵ∗, δ∗)-DP, then Algorithm 3 is (3ϵ∗ +3 +√ +2δ∗, +√ +2δ∗T +δ2)-DP. +The theorem implies that one can try a random number of φ while paying a constant ϵ. In practice, +we can roughly set τ = +1 +10k so that the algorithm is likely to test all k parameters. We emphasize +that the privacy and the utility guarantee (stated in the appendix) is not our contribution. But the +idea of applying generalized PTR to enforce a uniform DP guarantee over all choices of parameters +with a data-dependent analysis is new, and in our opinion, significantly broadens the applicability to +generic hyper-parameter tuning machinery from Liu and Talwar [2019]. +4.3 +Construction of the DP test +Classic PTR uses the Laplace mechanism to construct a differentially private upper bound of Dβ(X), +the distance from input dataset X to the closest dataset whose local sensitivity exceeds the proposed +7 + +bound β. The tail bound of the Laplace distribution then ensures that if Dβ(X) = 0 (i.e. if +∆LS(X) > β), then the output will be released with only a small probability δ. +The following theorem shows that we could instead use a differentially private upper bound of the +data-dependent DP ϵφ(X) in order to test whether to run the mechanism Mφ. +Theorem 4.6 (Generalized PTR with private upper bound). Suppose we have a differentially private +upper bound of ϵφ(X) w.r.t. δ such that with probability at least 1 − δ′, ϵP +φ (X) > ϵφ(X). Further +suppose we have an (ˆϵ, ˆδ)-DP test T such that +T(X) = +� +1 +if ϵP +φ (X) < ϵ, +0 +otherwise. +Then Algorithm 2 is (ϵ + ˆϵ, δ + ˆδ + δ′)-DP. +In Section 5.2, we demonstrate that one can upper bound the data-dependent DP through a +modification of the smooth sensitivity framework applied on ϵφ(X). Moreover, in Section 5.1 we +provide a direct application of Theorem 4.6 with private linear regression by making use of the +per-instance DP technique [Wang, 2017]. +The applications in Section 5 are illustrative of two distinct approaches to constructing the DP test +for generalized PTR: +1. Private sufficient statistics release (used in the private linear regression example of Section 5.1) +specifies the data-dependent DP as a function of the dataset and privately releases each +data-dependent component. +2. The second approach (used in the PATE example of Section 5.2) uses the smooth sensitivity +framework to privately release the data-dependent DP as a whole, and then construct a +high-confidence test using the Gaussian mechanism. +These two approaches cover most of the scenarios arising in data-adaptive analysis. For example, +in the appendix we demonstrate the merits of generalized PTR in handling data-adaptive private +generalized linear models (GLMs) using private sufficient statistics release. Moreover, sufficient +statistics release together with our private hyper-parameter tuning (Algorithm 3) can be used to +construct data-adaptive extensions of DP-PCA and Sparse-DP-ERM (see details in the future work +section). +5 +Applications +In this section, we put into action our approaches to construct the DP test and provide applications +in private linear regression and PATE. +5.1 +Private Linear Regression +Theorem 5.1 ([Wang, 2017]). For input data X ∈ X and Y ∈ Y, define the following: +• λmin(X) denotes the smallest eigenvalue of XT X; +• ||θ∗ +λ|| is the magnitude of the solution θ∗ +λ = (XT X + λI)−1XT Y ; +• and L(X, y) := ||X||(||X||||θ∗ +λ|| + ||Y||) is the local Lipschitz constant, denoted L in brief. +8 + +10 +1 +100 +10 +2 +6 × 10 +3 +2 × 10 +2 +3 × 10 +2 +4 × 10 +2 +MSE +UCI Bike dataset (n = 17379, d = 17) +AdaOPS +non-private +OutPert +OPS +OPS with PTR +(a) Bike dataset +10 +1 +100 +2 × 10 +2 +3 × 10 +2 +4 × 10 +2 +6 × 10 +2 +MSE +UCI elevators dataset (n = 8752, d = 18) +AdaOPS +non-private +OutPert +OPS +OPS with PTR +(b) Elevators dataset +Figure 1: Differentially private linear regression algorithms on UCI datasets. y-axis reports the MSE +error with confidence intervals. ϵ is evaluated with δ = 1e − 6. +For brevity, denote λ∗ = λ + λmin(X). The algorithm used in Example 4.4 with parameter φ = (λ, γ) +obeys (ϵφ(Z), δ) data-dependent DP for each dataset Z = (X, Y ) with ϵφ(Z) equal to +� +γL2 log(2/δ) +λ∗ ++ +γL2 +2(λ∗ + ||X||2) + 1 + log(2/δ)||X||2 +2(λ∗) +. +Notice that the data-dependent DP is a function of (λmin, L, ||θ∗ +λ||, λ, γ), where (λmin, L, ||θ∗ +λ||) are +data-dependent quantities. One can apply the generalized PTR framework as in the following +example. +Example 5.2 (OPS with PTR). We demonstrate here how to apply generalized PTR to the one- +posterior sample (OPS) algorithm, a differentially private mechanism which outputs one sample from +the posterior distribution of a Bayesian model with bounded log-likelihood. +• Propose φ = (λ, γ). +• Based on (λ, γ), differentially privately release λmin, ||θ∗ +λ||, L with privacy budget (ϵ, δ/2). +• Condition on a high probability event (with probability at least 1 − δ/2) of λmin, ||θ∗ +λ||, L, test if +ϵP +φ (X) is smaller than the predefined privacy budget (ˆϵ, ˆδ), where ϵP +φ (X) denotes the sanitized +data-dependent DP. +• Based on the outcome of the test, decide whether to release θ ∝ e− γ +2 ||Y −Xθ||2+λ||θ||2. +Theorem 5.3. The algorithm outlined in Example 5.2 satisfies (ϵ + ˆϵ, δ + ˆδ)-DP. +The main idea of the above algorithm boils down to privately releasing all data-dependent quantities +in data-dependent DP, constructing high-probability confidence intervals of these quantities, and +then deciding whether to run the mechanism M with the proposed parameters. We defer the details +of the privacy calibration of data-dependent quantities to the appendix. +One may ask why we cannot directly tune privacy parameters (λ, γ) based on the sanitized data- +dependent DP. This is because, in many scenarios, data-dependent quantities depend on the choice of +privacy parameters, e.g., ||θ∗ +λ|| is a complicated function of λ. Thus, the optimization on λ becomes +9 + +a circular problem — to solve λ, we need to sanitize ||θ∗ +λ||, which needs to choose a λ to begin with. +Alternatively, generalized PTR provides a clear and flexible framework to test the validity of privacy +parameters adapted to the dataset. +Remark 5.4. The above “circular” issue is even more serious for generalized linear models (GLMs) +beyond linear regression. The data-dependent DP there involves a local strong-convexity parameter, +a complex function of the regularizer λ and we only have zeroth-order access to. In the appendix, +we demonstrate how to apply generalized PTR to provide a generic solution to a family of private +GLMs where the link function satisfies a self-concordance assumption. +We next apply Algorithm 3 for Example 5.2 with UCI regression datasets. Standard z-scoring is +applied and each data point is normalize with a Euclidean norm of 1. We consider (60%, 10%, 30%) +splits for training, validation and testing test. +Baselines +• Output Perturbation (Outpert) [Chaudhuri et al., 2011]: θ = (XT X + λI)−1XT y. Release +ˆθ = θ + b with an appropriate λ, where b is a Gaussian random vector. +• Posterior sampling (OPS). Sample ˆθ ∼ P(θ) ∝ e−γ(F(θ)+0.5λ||θ||2) with parameters γ, λ. +• Adaptive posterior sampling (AdaOPS) [Wang, 2018]. Run OPS with (λ, γ) chosen adaptively +according to the dataset. +Outpert and OPS serve as two non-adaptive baselines. In particular, we consider OPS-Balanced [Wang, +2018], which chooses λ to minimize a data-independent upper bound of empirical risk and dominates +other OPS variants. AdaOPS is one state-of-the-art algorithm for adaptive private regression, which +automatically chooses λ by minimizing an upper bound of the data-dependent empirical risk. +We implement OPS-PTR as follows: propose a list of λ through grid search (we choose k = 30 and λ +ranges from [2.5, 2.510] on a logarithmic scale); instantiate Algorithm 3 with τ = 0.1k, T = 1 +τ log(1/δ2) +and δ2 = 1/2δ; calibrate γ to meet the privacy requirement for each λ. sample ˆθ using (λ, γ) and +return the one with the best validation accuracy. Notice that we use a “no ⊥” variant of Algorithm 2 +as the calibration of γ is clear given a fixed λ and privacy budget (see more details in the appendix). +We can propose various combinations of (λ, γ) for more general applications. +Figure 1 demonstrates how the MSE error of the linear regression algorithms varies with the privacy +budget ϵ. OutPert suffers from the large global sensitivity of output θ. OPS performs well but does +not benefit from the data-dependent quantities. AdaOPS is able to adaptively choose (λ, γ) based +on the dataset, but suffers from the estimation error of the data-dependent empirical risk. On the +other hand, OPS-PTR selects a (λ, γ) pair that minimizes the empirical error on the validation set +directly, and the privacy parameter γ adapts to the dataset thus achieving the best result. +5.2 +PATE +In this section, we apply the generalized PTR framework to solve an open problem from the Private +Aggregation of Teacher Ensembles (PATE) [Papernot et al., 2017, 2018] — privately publishing the +entire model through privately releasing data-dependent DP losses. Our algorithm makes use of the +smooth sensitivity framework [Nissim et al., 2007] and the Gaussian mechanism to construct a high- +probability test of the data-dependent DP. The one-dimensional statistical nature of data-dependent +DP enables efficient computations under the smooth sensitivity framework. Thus, this approach is +generally applicable for other private data-adaptive analysis beyond PATE. +10 + +PATE is a knowledge transfer framework for model-agnostic private learning. In this framework, an +ensemble of teacher models is trained on the disjoint private data and uses the teachers’ aggregated +consensus answers to supervise the training of a “student” model agnostic to the underlying machine- +learning algorithms. By publishing only the aggregated answers and by the careful analysis of the +“consensus”, PATE has become a practical technique in recent private model training. +The tight privacy guarantee of PATE heavily relies on a delicate data-dependent DP analysis, for +which the authors of PATE use the smooth sensitivity framework to privately publish the data- +dependent privacy cost. However, it remains an open problem to show that the released model is DP +under data-dependent analysis. Our generalized PTR resolves this gap by carefully testing a private +upper bound of the data-dependent privacy cost. Our algorithm is fully described in Algorithm 4, +where the modification over the original PATE framework is highlighted in blue. +Algorithm 4 takes the input of privacy budget (ϵ′, ˆϵ, δ), unlabeled public data x1:T and K teachers’ +predictions on these data. The parameter ϵ denotes the privacy cost of publishing the data-dependent +DP and ϵ′ is the predefined privacy budget for testing. nj(xi) denotes the the number of teachers +that agree on label j for xi and C denotes the number of classes. The goal is to privately release a +list of plurality outcomes — argmaxj∈[C]nj(xi) for i ∈ [T] — and use these outcomes to supervise +the training of a “student” model in the public domain. The parameter σ1 denotes the noise scale +for the vote count. +In their privacy analysis, Papernot et al. [2018] compute the data-dependent RDPσ1(α, X) of labeling +the entire group of student queries. RDPσ1(α, X) can be orders of magnitude smaller than its data- +independent version if there is a strong agreement among teachers. Note that RDPσ1(α, X) is a +function of the RDP order α and the dataset X, analogous to our Definition 4.1 but subject to +RDP [Mironov, 2017]. +Theorem 5.5 ([Papernot et al., 2018]). If the top three vote counts of xi are n1 > n2 > n3 and +n1 − n2, n2 − n3 ≫ σ1, then the data-dependent RDP of releasing argmaxj{nj + N(0, σ2 +1)} satisfies +(α, exp{−2α/σ2 +1}/α)-RDP and the data-independent RDP (using the Gaussian mechanism) satisfies +(α, α +σ2 +1 )-RDP. +Algorithm 4 PATE with generalized PTR +1: Input: Unlabeled public data x1:T , aggregated teachers prediction n(·), privacy parameter +ˆϵ, ϵ′, δ, noisy parameter σ1. +2: Set α = 2 log(2/δ) +ˆϵ ++ 1, σs = σ2 = +� +3α+2 +ˆϵ +, δ2 = δ/2, smoothness parameter β = 0.2 +α . +3: Compute noisy labels: yip ← argmaxj∈[C]{nj(xi) + N(0, σ2 +1)} for all i ∈ [1 : T]. +4: RDPσ1(α, X) ← data-dependent RDP at the α-th order. +5: SSβ(X) ← the smooth sensitivity of RDPupper +σ1 +(α, X). +6: Privately release µ := log(SSβ(X)) + β · N(0, σ2 +2) + +� +2 log(2/δ2) · σ2 · β +7: RDPupper +σ1 +(α) ← an upper bound of data-dependent RDP through Lemma 5.6. +8: ϵσ1 ← DP guarantee converted from RDPupper +σ1 +(α). +9: If ϵ′ ≥ ϵσ1 return a student model trained using (x1:T ; yp +1:T ). +10: Else return ⊥. +However, RDPσ1(α, X) is data-dependent and thus cannot be revealed. The authors therefore +privately publish the data-dependent RDP using the smooth sensitivity framework [Nissim et al., 2007]. +The smooth sensitivity calculates a smooth upper bound on the local sensitivity of RDPσ1(α, X), +11 + +15 +20 +25 +30 +35 +40 +45 +50 +Noise scale +1 +1 +2 +3 +4 +5 + Gaussian mechanism +PATE-PTR ( + +1) +data-dependent DP (non-private) +(a) High consensus and strong data-dependent DP +15 +20 +25 +30 +35 +40 +45 +50 +Noise scale +1 +1 +2 +3 +4 +5 + Gaussian mechanism +PATE-PTR ( + +1) +data-dependent DP (non-private) +(b) Low consensus and low data-dependent DP +Figure 2: Privacy and utility tradeoffs with PATE. When σ1 is aligned, three algorithms provide the +same utility. y-axis plots the privacy cost of labeling T = 200 public data with δ = 10−5. The left +figure considers the high-consensus case, where the data-adaptive analysis is preferred. +denoted as SSβ(X), such that SSβ(X) ≤ eβSSβ(X′) for any neighboring dataset X and X′. By +adding Gaussian noise scaled by the smooth sensitivity (i.e., release ϵσ1(α, X) + SSβ(X) · N(0, σ2 +s)), +the privacy cost is safely published. +Unlike most noise-adding mechanisms, the standard deviation σs cannot be published since SSβ(X) +is a data-dependent quantity. Moreover, this approach fails to provide a valid privacy guarantee +of the noisy labels obtained through the PATE algorithm, as the published privacy cost could be +smaller than the real privacy cost. Our solution in Algorithm 4 looks like the following: +• Privately release an upper bound of the smooth sensitivity SSβ(X) with eµ. +• Conditioned on a high-probability event of eµ, publish the data-dependent RDP with RDPupper +σ1 +(α). +• Convert RDPupper +σ1 +(α) back to the standard DP guarantee using RDP to DP conversion at δ/2. +• Test if the converted DP is above the predefined budget ϵ′. +The following lemma states that RDPupper +σ1 +(α) is a valid upper bound of the data-dependent +RDP. +Lemma 5.6 (Private upper bound of data-dependent RDP). We are given a RDP function +RDP(α, X) and a β-smooth sensitivity bound SS(·) of RDP(α, X). Let µ (defined in Algorithm 4) +denote the private release of log(SSβ(X)). Let the (β, σs, σ2)-GNSS mechanism be +RDPupper(α):=RDP(α,X)+SSβ(X)·N(0,σ2 +s)+σs +� +2 log( 2 +δ2 )eµ +Then, the release of RDPupper(X) satisfies (α, 3α+2 +2σ2s )-RDP for all 1 < α < +1 +2β; w.p. at least 1 − δ2, +RDPupper(α) is an upper bound of RDP(α, X). +The proof (deferred to the appendix) makes use of the facts that: (1) the log of SSβ(X) has a +bounded global sensitivity β through the definition of smooth sensitivity; (2) releasing RDPσ1(α, X)+ +SSβ(X) · N(0, σ2 +s) is (α, α+1 +σ2s )-RDP (Theorem 23 from Papernot et al. [2018]). +Now, we are ready to state the privacy guarantee of Algorithm 4. +12 + +Theorem 5.7. Algorithm 4 satisfies (ϵ′ + ˆϵ, δ)-DP. +In the proof, the choice of α ensures that the cost of the δ/2 contribution (used in the RDP-to-DP +conversion) is roughly ˆϵ/2. Then the release of RDPupper +σ1 +(α) with σs = +� +2+3α +ˆϵ +accounts for another +cost of (ϵ/2, δ/2)-DP. +Empirical results. We next empirically evaluate Algorithm 4 (PATE-PTR) on the MNIST dataset. +Following the experimental setup from Papernot et al. [2018], we consider the training set to be the +private domain, and the testing set is used as the public domain. We first partition the training set +into 400 disjoint sets and 400 teacher models, each trained individually. Then we select T = 200 +unlabeled data from the public domain, with the goal of privately labeling them. To illustrate the +behaviors of algorithms under various data distributions, we consider two settings of unlabeled +data, high-consensus and low-consensus. In the low-consensus setting, we choose T unlabeled data +such that there is no high agreement among teachers, so the advantage of data-adaptive analysis is +diminished. We provide further details on the distribution of these two settings in the appendix. +Baselines. +We consider the Gaussian mechanism as a data-independent baseline, where the +privacy guarantee is valid but does not take advantage of the properties of the dataset. The data- +dependent DP ( Papernot et al. [2018]) serves as a non-private baseline, which requires further +sanitation. Note that these two baselines provide different privacy analyses of the same algorithm +(see Theorem 5.5). +Figure 2 plots privacy-utility tradeoffs between the three approaches by varying the noise scale σ1. +The purple region denotes a set of privacy budget choices (ˆϵ + ϵ′ used in Algorithm 4) such that the +utility of the three algorithms is aligned under the same σ1. In more detail, the purple region is +lower-bounded by ˆϵ+ϵσ1. We first fix σs = σ2 = 15 such that ˆϵ is fixed. Then we empirically calculate +the average of ϵσ1 (the private upper bound of the data-dependent DP) over 10 trials. Running +Algorithm 4 with any choice of ˆϵ + ϵ′ chosen from the purple region implies ϵ′ > ϵσ1. Therefore, +PATE-PTR will output the same noisy labels (with high probability) as the two baselines. +Observation As σ1 increases, the privacy loss of the Gaussian mechanism decreases, while the +data-dependent DP curve does not change much. This is because the data-dependent DP of each +query is a complex function of both the noise scale and the data and does not monotonically +decrease when σ1 increases (see more details in the appendix). However, the data-dependent DP still +dominates the Gaussian mechanism for a wide range of σ1. Moreover, PATE-PTR nicely interpolates +between the data-independent DP guarantee and the non-private data-adaptive DP guarantee. In the +low-consensus case, the gap between the data-dependent DP and the DP guarantee of the Gaussian +mechanism unsurprisingly decreases. Meanwhile, PATE-PTR (the purple region) performs well +when the noise scale is small but deteriorates when the data-independent approach proves more +advantageous. This example demonstrates that using PTR as a post-processing step to convert +the data-dependent DP to standard DP is effective when the data-adaptive approach dominates +others. +6 +Limitations and Future Work +One weakness of generalized PTR is that it requires a case-specific privacy analysis. Have we simply +exchanged the problem of designing a data-adaptive DP algorithm with the problem of analyzing +the data-dependent privacy loss? We argue that this limitation is inherited from classic PTR. In +situations where classic PTR is not applicable, we’ve outlined several approaches to constructing the +13 + +DP test for our framework (see Sections 4.3 and 5.2). +Furthermore, the data-dependent privacy loss is often more straightforward to compute than local +sensitivity, and often exists in intermediate steps of classic DP analysis already. Most DP analysis +involves providing a high-probability tail bound of the privacy loss random variable. If we stop +before taking the max over the input dataset, then we get a data-dependent DP loss right away (as +in Example 4.2). +There are several exciting directions for applying generalized PTR to more problems. Sufficient +statistics release and our private hyperparameter tuning (Algorithm 3) can be used to construct +data-adaptive extensions of DP-PCA [Dwork et al., 2014] and Sparse-DP-ERM [Kifer et al., 2012]. +For DP-PCA we could use our Algorithm 3 to tune the variance of the noise added to the spectral +gap; for Sparse-DP-ERM we would test the restricted strong convexity parameter (RSC), i.e. not +adding additional regularization if the RSC is already large. +7 +Conclusion +Generalized PTR extends the classic “Propose-Test-Release” framework to a more general setting by +testing the data-dependent privacy loss of an input dataset, rather than its local sensitivity. In this +paper we’ve provided several examples – private linear regression with hyperparameter selection and +PATE – to illustrate how generalized PTR can enhance DP algorithm design via a data-adaptive +approach. +Acknowledgments +The work was partially supported by NSF Award # 2048091 and the Google Research Scholar Award. +Yuqing was supported by the Google PhD Fellowship. +14 + +Contents +1 +Introduction +1 +2 +Related Work +2 +3 +Preliminaries +3 +3.1 +Propose-Test-Release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +4 +Generalized PTR +4 +4.1 +Limitations of local sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +4.2 +Which φ to propose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +4.3 +Construction of the DP test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +5 +Applications +8 +5.1 +Private Linear Regression +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +5.2 +PATE +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +6 +Limitations and Future Work +13 +7 +Conclusion +14 +A Omitted examples in the main body +15 +A.1 Limits of the classic PTR in private binary voting . . . . . . . . . . . . . . . . . . . . +15 +A.2 Self-concordant generalized linear model (GLM) . . . . . . . . . . . . . . . . . . . . . +18 +A.3 Differentially privately release λmin +� +∇2F(θ) +� +. . . . . . . . . . . . . . . . . . . . . . +21 +A.4 Other applications of generalized PTR . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +B Omitted proofs in Section 4 +23 +C Experimental details +23 +C.1 Experimental details in private linear regression . . . . . . . . . . . . . . . . . . . . . +23 +C.2 Details of PATE case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +D Omitted proofs in private GLM +26 +D.1 Per-instance DP of GLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +A +Omitted examples in the main body +In this appendix, we provide more examples to demonstrate the merits of generalized PTR. We +focus on a simple example of post-processed Laplace mechanism in Section A.1 and then an example +on differentially private learning of generalized linear models in Section 4. In both cases, we observe +that generalized PTR provides data-adaptive algorithms with formal DP guarantees, that are simple, +effective and not previously proposed in the literature (to the best of our knowledge). +A.1 +Limits of the classic PTR in private binary voting +The following example demonstrates that classic PTR does not capture sufficient data-dependent +quantities even when the local sensitivity exists and can be efficiently tested. +15 + +Example A.1. Consider a binary class voting problem: n users vote for a binary class {0, 1} and +the goal is to output the class that is supported by the majority. Let ni denote the number of people +who vote for the class i. We consider the report-noisy-max mechanism: +M(X) : argmaxi∈[0,1]ni(X) + Lap(b), +where b = 1/ϵ denotes the scale of Laplace noise. +In the example, we will (1) demonstrate the merit of data-dependent DP; and (2) empirically compare +classic PTR with generalized PTR. +We first explicitly state the data-dependent DP. +Theorem A.2. The data-dependent DP of the above example is +ϵ(X) := max +X′ {| log p +p′ |, | log 1 − p +1 − p′ |}, +where p := Pr[n0(X) + Lap(1/ϵ) > n1(X) + Lap(1/ϵ)] and p′ := Pr[n0(X′) + Lap(1/ϵ) > n1(X′) + +Lap(1/ϵ)]. There are four possible neighboring datasets X′ : n0(X′) = max(n0(X) ± 1, 0), n1(X′) = +n1(X) or n0(X′) = n0(X), n1(X′) = max(n1(X) ± 1, 0). +In Figure 3(a), we empirically compare the above data-dependent DP with the Laplace mechanism +by varying the gap between the two vote counts |n0(X) − n1(X)|. The noise scale is fixed to ϵ = 10. +The data-dependent DP substantially improves over the standard DP if the gap is large. However, +the data-dependent DP is a function of the dataset. We next demonstrate how to apply generalized +PTR to exploit the data-dependent DP. +Notice that the probability n0(X) + Lap(1/ϵ) > n1(X) + Lap(1/ϵ) is equal to the probability that a +random variable Z := X − Y exceeds ϵ(n1(X) − n0(X)), where X, Y are two independent Lap(1) +distributions. We can compute the pdf of Z through the convolution of two Laplace distributions, +which implies fX−Y (z) = 1 + |z| +4e|z| . Let t denote the difference between n1(X) and n0(X), i.e., +t = n1(X) − n0(X). Then we have +p = Pr[Z > ϵ · t] = +2 + ϵ · t +4 exp(ϵ · t) +Similarly, p′ = +2 + ϵ · (t + ℓ) +4 exp(ϵ · (t + ℓ)), where ℓ ∈ [−1, 1] denotes adding or removing one data point to +construct the neighboring dataset X′. Therefore, we can upper bound log(p/p′) by +log p +p′ = +2 + ϵ · t +4 exp(ϵ · t) · 4 exp(ϵ(t + ℓ)) +2 + ϵ · (t + ℓ) +≤ ϵ · log +� +2 + ϵt +2 + ϵ(t + 1) +� += ϵ log +� +1 − +ϵ +2 + ϵ(t + 1) +� +Then we can apply generalized PTR by privately lower-bounding t. +On the other hand, the local sensitivity ∆LS(X) of this noise-adding mechanism is 0 if t > 1. +Specifically, if the gap is larger than one, adding or removing one user will not change the result. To +16 + +0 +5 +10 +15 +20 +25 +30 +35 +40 +The gap t=|n0(X) +n1(X)| +0 +2 +4 +6 +8 +10 + data-dependent DP +Laplace mechanism +(a) data-dependent DP vs Laplace mechanism +10 +28 +10 +23 +10 +18 +10 +13 +10 +8 +10 +3 +102 +Error +10 +2 +10 +1 + + Gen-PTR( +p + ) +classic PTR +Laplace mechanism +(b) Privacy-utility tradeoff between three approaches. +Figure 3: In Figure 3(a), we compare the privacy guarantee by varying the gap. In Figure 3(b) We +fix t = n0(X) − n1(X) = 100 and compare privacy cost when the accuracy is aligned. Gen-PTR with +any choice of privacy budget (˜ϵ + ϵ′) chosen from the purple region would achieve the same utility as +Laplace mechanism but with a smaller privacy cost. The curve of Gen-PTR is always below than +that of the classic PTR, which implies that Gen-PTR can result a tighter privacy analysis when the +utility is aligned. +apply classic PTR, we let γ(X) denote the distance to the nearest dataset X +′′ such that ∆LS > 0 +and test if γ(X) + Lap(1/ϵ) > log(1/δ) +ϵ +. Notice in this example that γ(X) = max(t − 1, 0) can be +computed efficiently. We provide the detailed implementation of these approaches. +1. Gen PTR: lower bound t with tp = t − log(1/δ) +˜ϵ ++ Lap(1/˜ϵ). Calculate an upper bound of +data-dependent DP ϵp using Theorem A.2 with tp. The algorithm then tests if ϵp is within an +predefined privacy budget ϵ′. If the test passes, the algorithm returns argmaxi∈[0,1]ni(X) + +Lap(1/ϵ) satisfies (˜ϵ + ϵ′, δ)-DP. +2. classic PTR: lower bound t with tp = t − log(1/δ) +˜ϵ ++ Lap(1/˜ϵ). If tp > 1, classic PTR outputs +the ground-truth result else returns a random class. This algorithm satisfies (˜ϵ, δ)-DP. +3. Laplace mechanism. M(X) : argmaxi∈[0,1]ni(X) + Lap(1/ϵ). M is (ϵ, δ)-DP. +We argue that though the Gen-PTR and the classic PTR are similar in privately lower-bounding +the data-dependent quantity t, the latter does not capture sufficient information for data-adaptive +analysis. That is to say, only testing the local sensitivity restricts us from learning helpful information +to amplify the privacy guarantee if the test fails. In contrast, our generalized PTR, where privacy +parameters and the local sensitivity parameterize the data-dependent DP, can handle those failure +cases nicely. +To confirm this conjecture, Figure 3(b) plots a privacy-utility trade-off curve between these three +approaches. We consider a voting example with n0(X) = n1(X) + 100 and t = 100, chosen such +that the data-adaptive analysis is favorable. +In Figure 3(b), we vary the noise scale b = 1/ϵ between [0, 0.5]. For each choice of b, we plot the +privacy guarantee of three algorithms when the error rate is aligned. For Gen-PTR, we set ˜ϵ = 1 +2b +and empirically calculate ϵp over 100000 trials. +17 + +In the plot, when ϵ ≪ log(1/δ) +t +, the classic PTR is even worse than the Laplace mechanism. This is +because the classic PTR is likely to return ⊥ while the Laplace mechanism returns argmaxi∈[0,1]ni(X)+ +Lap(1/ϵ), which contains more useful information. Compared to the Laplace mechanism, Gen-PTR +requires an extra privacy allocation ˜ϵ to release the gap t. However, it still achieves an overall smaller +privacy cost when the error rate ≤ 10−5 (the purple region). Meanwhile, Gen-PTR dominates the +classic PTR (i.e., the dashed black curve is always below the blue curve). Note that the classic PTR +and the Gen-PTR utilize the gap information differently: the classic PTR outputs ⊥ if the gap is +not sufficiently large, while the Gen-PTR encodes the gap into the data-dependent DP function +and tests the data-dependent DP in the end. This empirical result suggests that testing the local +sensitivity can be loosely compared to testing the data-dependent DP. Thus, Gen-PTR could provide +a better privacy-utility trade-off. +A.2 +Self-concordant generalized linear model (GLM) +In this section, we demonstrate the effectiveness and flexibility of generalized PTR in handling a +family of GLMs where the link function satisfies a self-concordance assumption. This section is +organized as follows: +• Introduce a family of GLMs with the self-concordance property. +• Introduce a general output perturbation algorithm for private GLMs. +• Analyze the data-dependent DP of GLMs with the self-concordance property. +• Provide an example of applying our generalized PTR framework to logistic regression. +Consider the empirical risk minimization problem of the generalized linear model +θ∗ = argminθ +� +i=1n +li(θ) + r(θ), +where l : R × R → R belongs to a family of convex GLMs: li(θ) = l(y, xT +i θ). Let r : Rd → R be a +regularization function. +We now define the self-concordance property. +Definition A.3 (Generalized self-concordance [Bach, 2010]). A convex and three-times differentiable +function f : Θ → R is R-generalized-self-concordant on an open nonempty convex set Θ∗ ⊂ Θ with +respect to norm ∥ · ∥ if for all u ∈ Θ∗ and all v ∈ Rd, +∇3f(u)[v, v, v] ≤ 2R∥v∥(∇2f(u)[v, v]). +The closer R is to 0, the “nicer” — more self-concordant — the function is. A consequence of (gener- +alized) self-concordance is the spectral (multiplicative) stability of Hessian to small perturbations of +parameters. +Lemma A.4 (Stability of Hessian[Nesterov and Nemirovskii, 1994, Theorem 2.1.1], [Bach, 2010, +Proposition 1]). Let Hθ := ∇2Fs(θ). If Fs is R-self-concordant at θ, then for any v such that +R∥v∥Hθ < 1, we have that +(1 − R∥v∥Hθ)2∇2Fs(θ) ≺ ∇2Fs(θ + v) +≺ +1 +(1 − R∥v∥Hθ)2 ∇2Fs(θ). +18 + +If instead we assume Fs is R-generalized-self-concordant at θ with respect to norm ∥ · ∥, then +e−R∥v∥∇2Fs(θ) ≺ ∇2Fs(θ + v) ≺ eR∥v∥∇2Fs(θ) +The two bounds are almost identical when R∥v∥ and R∥v∥θ are close to 0. In particular, for x ≤ 1/2, +we have that e−2x ≤ 1 − x ≤ e−x. +In particular, the loss function of binary logistic regression is 1-generalized self-concordant. +Example A.5 (Binary logistic regression). Assume ∥x∥2 ≤ 1 for all x ∈ X and y ∈ {−1, 1}. Then +binary logistic regression with datasets in X × Y has a log-likelihood of F(θ) = �n +i=1 log(1 + e−yixT +i θ). +The univariate function l := log(1 + exp(·)) satisfies +|l′′′| = +���� +exp (·)(1 − exp (·)) +(1 + exp (·))3 +���� ≤ +exp (·) +(1 + exp (·))2 := l′′. +We next apply the modified output perturbation algorithm to privately release θ∗. The algorithm is +simply: +1. Solve +θ∗ = argminθ +n +� +i=1 +li(θ) + r(θ). +2. Release +ˆθ = θ∗ + Z, +where γ > 0 is a tuning parameter and Z ∼ N(0, γ−1(�n +i=1 ∇2li(θ) + ∇2r(θ))−1). +The data-dependent DP of the above procedure is stated as follows. +Theorem A.6 (Data-dependent DP of GLM). Denote the smooth part of the loss function Fs = +�n +i=1 l(yi, < xi, · >) + rs(·). Assume the following: +1. The GLM loss function l is convex, three-times continuously differentiable and R-generalized- +self-concordant w.r.t. ∥ · ∥2, +2. Fs is locally α-strongly convex w.r.t. ∥ · ∥2, +3. and in addition, denote L := supθ∈[θ∗,˜θ∗] |l′(y, xT θ)|, β := supθ∈[θ∗,˜θ∗] |l′′(y, xT θ)|. That is, ℓ(·) +is L-Lipschitz and β-smooth. +We then have the data-dependent DP +ϵ(Z) ≤ R(L + β) +α +(1 + log(2/δ)) + γL2 +α ++ +� +γL2 +α +log(2/δ). +The proof follows by taking an upper bound of the per-instance DP loss (Theorem D.1) ϵ(Z, z) over +z = (x, y) ∈ (X, Y). +Notice that the Hessians can be arbitrarily singular and α could be 0, which leads to an infinite +privacy loss without additional assumptions. Thus, we will impose an additional regularization of +form λ +2||θ||2, which ensures that for any dataset FS is λ-strongly convex. +This is not yet DP because it is still about a fixed dataset. We also need a pre-specified privacy +budget (ϵ, δ). We next demonstrate how to apply the generalized PTR to provide a general solution +to the above GLM, using logistic regression as an example. +19 + +Remark A.7 (Logistic regression). For logistic regression, we know L ≤ 1, β ≤ 1/4 and if ∥x∥2 ≤ 1, +it is 1-generalized self-concordant. For any dataset Z = (X, y), the data-dependent DP ϵ(X) w.r.t. +δ can be simplified to: +1.25 +α (1 + log(2/δ)) + γ +α + +�γ +α log(2/δ) +Now, the data-dependent DP is a function of α and γ, where α denotes the local strong convexity at +θ∗ +λ and γ controls the noise scale. We next show how to select these two parameters adapted to the +dataset. +Example A.8. We demonstrate here how we apply generalized PTR to output perturbation of the +logistic regression problem. +1. Take an exponential grid of parameters {λ} and propose each λ. +2. Solve for θ∗ +λ = argminθF(θ) + λ∥θ∥2/2 +3. Calculate the smallest eigenvalue λmin(∇2F(θ∗ +λ)) (e.g., using power method). +4. Differentially privately release λmin with λp +min := max{λmin+ +√ +log(4/δ) +ϵ/2 +·∆GS·Z− +√ +2 log(4/δ)·log(1/δ)∆GS +ϵ/2 +, 0}, +where ∆GS denote the global sensitivity of λmin using Theorem A.11. +5. Let ϵp(·) be instantiated with ϵ(X) w.r.t. δ from Remark A.7, where α = λp +min + λ. Then, +conditioned on a high probability event, ϵp(·) (a function of γ) is a valid DP bound that holds +for all datasets and all parameters γ. +6. Calculate the maximum γ such that ϵp +δ/2(γ) ≤ ϵ/2. +7. Release ˆθ ∼ N(θ∗ +λ, γ−1∇2Fs(θ∗ +λ)−1). +8. Evaluate the utility on the validation set and return the (λ, γ) pair that leads to the highest +utility. +Theorem A.9. For each proposed λ, the algorithm that releases ˆθ ∼ N(θ∗ +λ, γ−1∇2Fs(θ∗ +λ)−1) is +(ϵ, 2δ)-DP. +Proof. The proof follows the recipe of generalized PTR with private upper bound (Example 4.6). First, +the release of λmin(∇2F(θ∗ +λ)) is (ϵ/2, δ/2)-DP. Then, with probability at least 1 − δ, ϵp +δ(·) > ϵδ(X) +holds for all X and γ. Finally, γ is chosen such that the valid upper bound is (ϵ/2, δ/2)-DP. +For the hyper-parameter tuning on λ (Steps 1 and 8), we can use Algorithm 3 to evaluate each λ. +Unlike Example 5.2, the λmin(∇2F(θ∗ +λ)) is a complicated data-dependent function of λ. Thus, we +cannot privately release the data-dependent quantity λmin(∇2F(θ∗ +λ)) without an input λ. The PTR +approach allows us to test a number of different λ and hence get a more favorable privacy-utility +trade-off. +An interesting perspective of this algorithm for logistic regression is that increasing the regularization +α is effectively increasing the number of data points within the soft “margin”3 of separation, hence a +larger contribution to the Hessian from the loss function. +3If we think of logistic regression as a smoothed version of SVM, then increasing α leads to more support vectors. +The “margin” is “softer” in logistic regression, but qualitatively the same. +20 + +Remark A.10. The PTR solution for GLMs follows a similar recipe: propose a regularization +strength λ; construct a lower bound of the strong convexity α at the optimal solution θ∗ +λ; and test +the validity of data-dependent DP using Theorem D.1. +Before moving on to other applications of generalized PTR, we will show how to differentially +privately release λmin according to the requirements of the logistic regression example. +A.3 +Differentially privately release λmin (∇2F(θ)) +To privately release λmin∇2F(θ), we first need to compute its global sensitivity. Once we have that +then we can release it differentially privately using either the Laplace mechanism or the Gaussian +mechanism. +Theorem A.11 (Global sensitivity of the minimum eigenvalue at the optimal solution). Let +F(θ) = �n +i=1 fi(θ) + r(θ) and ˜F(θ) = F(θ) + f(θ) where f1, ..., fn are loss functions corresponding +to a particular datapoint x. Let θ∗ = argminθF(θ) and ˜θ∗ = argminθ ˜F(θ). Assume f is L-Lipschitz +and β-smooth, r(θ) is λ-strongly convex, and F and ˜F are R-self-concordant. If in addition, λ ≥ RL, +then we have +sup +X,x +(λmin(∇2F(θ∗ +λ)) − λmin(∇2 ˜F( ˜θ∗ +λ))) ≤ 2RL + β. +Proof. +λmin(∇2F(θ∗ +λ)) − λmin(∇2 ˜F( ˜θ∗ +λ)) += (λmin(∇2F(θ∗ +λ)) − λmin(∇2 ˜F(θ∗ +λ))) ++ (λmin(∇2 ˜F(θ∗ +λ)) − λmin(∇2 ˜F( ˜θ∗ +λ))). +(1) +We first bound the part on the left. By applying Weyl’s lemma λ(X + E) − λ(X) ≤ ||E||2, we have +sup +x ||∇2F(θ∗ +λ) − ∇2 +˜ +F(θ∗ +λ)||2 = ||∇2f(θ∗ +λ)||2 ≤ β +(2) +In order to bound the part on the right, we apply the semidefinite ordering using self-concordance, +which gives +e−R∥ ˜ +θ∗ +λ−θ∗ +λ∥∇2 ˜F( ˜θ∗ +λ) ≺ ∇2 ˜F(θ∗ +λ) ≺ eR∥ ˜ +θ∗ +λ−θ∗ +λ∥∇2 ˜F( ˜θ∗ +λ). +By the Courant-Fischer Theorem and the monotonicity theorem, we also have that for the smallest +eigenvalue +e−R∥ ˜ +θ∗ +λ−θ∗ +λ∥λmin +� +∇2 ˜F( ˜θ∗ +λ) +� +≤ λmin +� +∇2 ˜F(θ∗ +λ) +� +≤ eR∥ ˜ +θ∗ +λ−θ∗ +λ∥λmin +� +∇2 ˜F( ˜θ∗ +λ) +� +. +(3) +Moreover by Proposition D.2, we have that +∥ ˜θ∗ +λ − θ∗ +λ∥2 ≤ +∥∇f( ˜θ∗λ)∥ +λmin +� +∇2 ˜F( ˜θ∗ +λ) +� ≤ +L +λmin +� +∇2 ˜F( ˜θ∗ +λ) +�. +If λmin +� +∇2 ˜F( ˜θ∗ +λ) +� +≥ RL, then use that ex − 1 ≤ 2x for x ≤ 1. Substituting the above bound to (3) +then to (1) together with (2), we get a data-independent global sensitivity bound of +λmin(∇2F(θ∗ +λ)) − λmin(∇2 ˜F( ˜θ∗ +λ)) ≤ 2RL + β +21 + +as stated. +Proposition A.12. Let ∥ · ∥ be a norm and ∥ · ∥∗ be its dual norm. Let F(θ), f(θ) and ˜F(θ) = +F(θ) + f(θ) be proper convex functions and θ∗ and +˜ +theta +∗ be their minimizers, i.e., 0 ∈ ∂F(θ∗) and +0 ∈ ∂ ˜F( +˜ +theta +∗). If in addition, F, ˜F is α, ˜α-strongly convex with respect to ∥ · ∥ within the restricted +domain θ ∈ {tθ∗ + (1 − t)˜θ∗ | t ∈ [0, 1]}. Then there exists g ∈ ∂f(θ∗) and ˜g ∈ ∂f(˜θ∗) such that +∥θ∗ − ˜θ∗∥ ≤ min +� 1 +α∥˜g∥∗, 1 +˜α∥g∥∗ +� +. +Proof. Apply the first order condition to F restricted to the line segment between ˜θ∗ and θ∗, we get +F(˜θ∗) ≥ F(θ∗) + ⟨∂F(θ∗), ˜θ∗ − θ∗⟩ + α +2 ∥˜θ∗ − θ∗∥2 +(4) +F(θ∗) ≥ F(˜θ∗) + ⟨∂F(˜θ∗), θ∗ − ˜θ∗⟩ + α +2 ∥˜θ∗ − θ∗∥2 +(5) +Note by the convexity of F and f, ∂ ˜F = ∂F + ∂f, where + is the Minkowski Sum. Therefore, +0 ∈ ∂ ˜F(˜θ∗) implies that there exists ˜g such that ˜g ∈ ∂f(˜θ∗) and −˜g ∈ ∂F(˜θ∗). Take −˜g ∈ ∂F(˜θ∗) in +Equation 10 and 0 ∈ ∂F(θ∗) in Equation 9 and add the two inequalities, we obtain +0 ≥ ⟨−˜g, θ∗ − ˜θ∗⟩ + α∥˜θ∗ − θ∗∥2 +≥ −∥˜g∥∗∥θ∗ − ˜θ∗∥ + α∥˜θ∗ − θ∗∥2. +For ∥˜θ∗ − θ∗∥ = 0 the claim is trivially true; otherwise, we can divide both sides of the above +inequality by ∥˜θ∗ − θ∗∥ and get ∥θ∗ − ˜θ∗∥ ≤ 1 +α∥˜g∥∗. +It remains to show that ∥θ∗ − ˜θ∗∥ ≤ 1 +˜α∥g∥∗. This can be obtained by exactly the same arguments +above but applying strong convexity to ˜F instead. Note that we can actually get something slightly +stronger than the statement because the inequality holds for all g ∈ ∂f(θ∗). +A.4 +Other applications of generalized PTR +Besides one-posterior sampling for GLMs, there are plenty of examples that our generalized-PTR +could be applied, e.g., DP-PCA [Dwork et al., 2014] and Sparse-DP-ERM [Kifer et al., 2012] (when +the designed matrix is well-behaved). +[Dwork et al., 2014] provides a PTR style privacy-preserving principle component analysis (PCA). +The key observation of [Dwork et al., 2014] is that the local sensitivity is quite “small” if there is a +large eigengap between the k-th and the k + 1-th eigenvalues. Therefore, their approach (Algorithm +2) chooses to privately release a lower bound of the k-th eigengap (k is fixed as an input) and use +that to construct a high-confidence upper bound of the local sensitivity. +For noise-adding mechanisms, the local sensitivity is proportional to the data-dependent loss and +generalized PTR is applicable. We can formulate the data-dependent DP of DP-PCA as follows: +Theorem A.13. +For a given matrix A ∈ Rm×n, assume each row of A has a bounded ℓ2 norm +being 1. Let Vk denotes the top k eigenvectors of AT A and dk denotes the gap between the k-th +and the k + 1-th eigenvalue. Then releasing VkV T +k + E, where E ∈ Rn×n is a symmetric matrix +with the upper triangle is i.i.d samples from N(0, σ2) satisfies (ϵ(A), δ) data-dependent DP and +ϵ(A) = +2√ +log(1.25/δ) +σ(dk−2) +. +22 + +The proof is based on the local sensitivity result from [Dwork et al., 2014] and the noise calibration +of Gaussian mechanism. +We can combine Theorem A.13 with our Algorithm 3 to instantiate the generalized PTR framework. +The improvement over Dwork et al. [2014] will be to allow joint tuning of the parameter k and the +noise variance (added to the spectral gap dk). +B +Omitted proofs in Section 4 +The utility of Algorithm 3 depends on how many rounds that Algorithm 2 is invoked. We next +provide the utility guarantee of Algorithm 3, which follows a simplification of the result in the +Section A.2 of Papernot and Steinke [2021]. +Theorem B.1. Suppose applying Algorithm 2 with each φi has an equal probability to achieve the +highest validation score. Let ˆT denotes the number of invocation of Algorithm 2, where ˆT follows a +truncated geometric distribution. Then the expected quantile of the highest score candidate is given +by E ˆT +� +1 − +1 +ˆT+1 +� +. +In practice, we can roughly set τ = +1 +10k so that the algorithm is likely to test all k parameters. +Proof. Suppose each oracle access to Q(X) has a probability 1/k of achiving the best validation +accuracy. Let β denote the probability that A (shorthand for Algorithm 3) outputs the best choice +of φi. +β = 1 − Pr[A(X)is not best] += 1 − E ˆT +� +Pr[Q(X)is not best] +ˆT +� += 1 − E ˆT +� +(1 − 1 +k) +ˆT +� +. +Let f(x) = E[x ˆT ]. Applying a first-order approximation on f(1 − 1 +k), we have f(1 − 1 +k) ≈ f(1) − +f′(1) · 1 +k = 1 − E[ ˆT]/k. Then, if k is large and we choose τ = 0.1/k, A can roughly return the best +φi. +C +Experimental details +C.1 +Experimental details in private linear regression +We start with the privacy calibration of the OPS-PTR algorithm. +Algorithm 5 provides the detailed privacy calibration of the private linear regression problem. +Theorem C.1. Algorithm 5 is (ϵ, 2δ)-DP. +Proof. There are three data-dependent quantities in Theorem 5.1: λmin, ||θ∗ +λ|| and L. First, notice +that λmin has a global sensitivity of ||X||2 by Weyl’s lemma. Under the assumption ||X||2 ≤ 1, we +privately release λmin using (ϵ/4, δ/3) in Step 3. Notice that with probability at least 1 − δ/2, ˜λmin +is a lower bound of λmin. +23 + +Algorithm 5 OPS-PTR: One-Posterior Sample with propose-test-release (no-“perp” version) +1: Input: Data X, y. Private budget : ϵ, δ, proposed regularizer λ. +2: Calculate the minimum eigenvalue λmin(XT X). +3: Sample Z ∼ N(0, 1) and privately release ˜λmin = max +� +λmin + +√ +log(6/δ) +ϵ/4 +Z − +√ +2 log(6/δ)·log(2/δ) +ϵ/4 +, 0 +� +4: Calculate ˆθ = (XT X + λI)−1XT y. +5: Sample Z ∼ N(0, 1) and privately release ∆ = log(||Y|| + ||X||||ˆθ||) + log(1+||X||2/(λ+˜λmin)) +ϵ/(4√ +6/δ) +Z + +log(1+||X||2/(λ+˜λmin)) +ϵ/(4√ +2 log(6/δ) log(2/δ)). +6: Set the local Lipschitz ˜L := ||X||e∆. +7: Calibrate γ with Theorem 5.1(δ/3, ϵ/2.) +8: Output ˜θ ∼ p(θ|X, y) ∝ e− γ +2 ||y−Xθ||2+λ||θ||2 +Then, we apply Lemma C.2 from +Wang [2018] to privately release log(||Y|| + ||X||||ˆθ||) using +(ϵ/4, δ/3). Note that both the local Lipschitz constant L and the norm ||θ∗ +λ|| are functions of +log(||Y|| + ||X||||ˆθ||). Thus, we can construct a private upper bound of these by post-processing of +∆. +Then, with probability at least 1 − δ (by a union bound over ˜λmin and ∆), instantiating Theorem 5.1 +with ˜λmin and ˜L provides a valid upper bound of the data-dependent DP. We then tune the parameter +γ using the remaining privacy budget (ϵ/2, δ/3). +Lemma C.2 (Lemma 12 [Wang, 2018]). Let θ∗ +λ be the ridge regression estimate with parameter +λ and the smallest eigenvalue of XT X be λmin, then the function log(||Y + ||X||||θ∗ +λ||) has a local +sensitivity of log(1 + +||X||2 +λmin+λ ). +C.2 +Details of PATE case study +Definition C.3 (Renyi DP [Mironov, 2017]). We say a randomized algorithm M is (α, ϵM(α))-RDP +with order α ≥ 1 if for neighboring datasets X, X′ +Dα(M(X)||M(X′)) := +1 +α − 1 log Eo∼M(X′) +�� Pr[M(X) = o] +Pr[M(X′) = o] +�α� +≤ ϵM(α). +At the limit of α → ∞, RDP reduces to (ϵ, 0)-DP. We now define the data-dependent Renyi DP +that conditioned on an input dataset X. +Definition C.4 (Data-dependent Renyi DP [Papernot et al., 2018]). We say a randomized algorithm +M is (α, ϵM(α, X))-RDP with order α ≥ 1 for dataset X if for neighboring datasets X′ +Dα(M(X)||M(X′)) := +1 +α − 1 log Eo∼M(X′) +�� Pr[M(X) = o] +Pr[M(X′) = o] +�α� +≤ ϵM(α, X). +RDP features two useful properties. +24 + +Lemma C.5 (Adaptive composition). ϵ(M1,M2) = ϵM1(·) + ϵM2(·). +Lemma C.6 (From RDP to DP). If a randomized algorithm M satisfies (α, ϵ(α))-RDP, then M +also satisfies (ϵ(α) + log(1/δ) +α−1 , δ)-DP for any δ ∈ (0, 1). +Definition C.7 (Smooth Sensitivity). Given the smoothness parameter β, a β-smooth sensitivity +of f(X) is defined as +SSβ(X) := max +d≥0 e−βd · +max +˜ +X′:dist(X, ˜ +X′)≤d +∆LS( ˜X′) +Lemma C.8 (Private upper bound of data-dependent RDP, Restatement of Theorem 5.6). ] Given +a RDP function RDP(α, X) and a β-smooth sensitivity bound SS(·) of RDP(α, X). Let µ (defined +in Algorithm 4) denote the private release of log(SSβ(X)). Let (β, σs, σ2)-GNSS mechanism be +RDPupper(α):=RDP(α,X)+SSβ(X)·N(0,σ2 +s)+σs +� +2 log( 2 +δ2 )eµ +Then, the release of RDPupper(X) satisfies (α, 3α+2 +2σ2s )-RDP for all 1 < α < +1 +2β; w.p. at least 1 − δ2, +RDPupper(α) is an upper bound of RDP(α, X). +Proof sketch. We first show that releasing the smooth sensitivity SSβ with eµ satisfies (α, +α +2σ2 +2 )-RDP. +Notice that the log of SSβ(X) has a bounded global sensitivity β (Definition C.7 implies that +| log SSβ(X) − log SSβ(X′)| ≤ β for any neighboring dataset X, X′). By Gaussian mechanism, +scaling noise with βσ2 to log SSβ(X) is (α, +α +2σ2 +2 )-RDP. Therefore, the release of RDP(α, X) is +(α, ϵs(α) + +α +2σ2 +2 )-RDP. Since the release of f(X) + SSβ(X) · N(0, σ2 +s) is (α, α+1 +σ2s )-RDP (Theorem 23 +from Papernot et al. [2018]) for α < +1 +2β, we have ϵs(α) + +α +2σ2 +2 = 3α+2 +2σ2s . +We next prove the second statement. First, notice that with probability at least 1−δ2/2, eµ ≥ SSβ(X) +using the standard Gaussian tail bound. Let E denote the event that eµ ≥ SSβ(X). +Pr +� +RDPupper(α) ≤ RDP(α, X) +� += Pr +� +RDPupper(α) ≤ RDP(α, X)|E +� ++ Pr +� +RDPupper(α) ≤ RDP(α, X)|Ec +� +≤ Pr +� +RDPupper(α) ≤ RDP(α, X)|E +� ++ δ2/2 += Pr +� +N(0, σ2 +s) · SSβ(X) ≥ σs · +� +2 log(2/δ2)eµ|E +� +� +�� +� +denoted by(∗) ++δ2/2 +Condition on the event E, eµ is a valid upper bound of SSβ(X), which implies +(∗) ≤ Pr[N(0, σ2 +s) · SSβ(X) ≥ σs · +� +2 log(2/δ2)SSβ(X)|E] ≤ δ2/2 +Therefore, with probability at least 1 − δ2, RDPupper(α) ≥ RDP(α, X). +Theorem C.9 (Restatement of Theorem 5.7). Algorithm 4 satisfies (ϵ′ + ˆϵ, δ)-DP. +25 + +Proof. The privacy analysis consists of two components — the privacy cost of releasing an upper +bound of data-dependent RDP (ϵupper(α) := ϵs(α)+ +α +2σ2 +2 and the valid upper bound ϵp +σ1(α). First, set +α = 2 log(2/δ) +ϵ ++ 1 and use RDP to DP conversion with δ/2 ensures that the cost of δ/2 contribution +to be roughly ϵ/2 (i.e., log(2/δ) +α−1 += ϵ/2). Second, choosing σs = +� +2+3α +ϵ +gives us another ϵ/2. +Experimental details K = 400 teacher models are trained individually on the disjoint set using +AlexNet model. We set σ2 = σs = 15.0. Our data-dependent RDP calculation and the smooth- +sensitivity calculation follow Papernot et al. [2018]. Specifically, we use the following theorem +(Theorem 6 from Papernot et al. [2018]) to compute the data-dependent RDP of each unlabeled +data x from the public domain. +Theorem C.10 (data-dependent RDP Papernot et al. [2018]). Let ˜q ≥ Pr[M(X) ̸= Argmaxj∈[C]nj(x)], +i.e., an upper bound of the probability that the noisy label does not match the majority label. Assume +α ≤ µ1 and ˜q ≤ e(µ2−1)ϵ2/ +� +µ1 +µ1−1 · +µ2 +µ2−1 +�µ2 +, then we have: +ϵM(α, X) ≤ +1 +α − 1 log +� +(1 − ˜q) · A(˜q, µ2, ϵ2)α−1 + ˜q · B(˜q, µ1, ϵ1)α−1 +� +where A(˜q, µ2, ϵ2) := (1 − ˜q)/ +� +1 − (˜qeϵ2) +µ2−1 +µ2 +� +, B(˜q, µ1, ϵ1) = eϵ1/˜q +1 +µ1−1 , µ2 = σ1 · +� +log(1/˜q), µ1 = +µ2 + 1, ϵ1 = µ1/σ2 +1 and ϵ2 = µ2/σ2 +2. +In the experiments, the non-private data-dependent DP baseline is also based on the above theorem. +Notice that the data-dependent RDP of each query is a function of ˜q, where ˜q denotes an upper +bound of the probability where the plurality output does not match the noisy output. +˜q is a +complex function of both the noisy scale and data and is not monotonically decreasing when σ1 is +increasing. +Simulation of two distributions. The motivation of the experimental design is to compare +three approaches under different data distributions. Notice that there are K = 400 teachers, which +implies the number of the vote count for each class will be bounded by 400. In the simulation of +high-consensus distribution, we choose T = 200 unlabeled public data such that the majority vote +count will be larger than 150 (i.e., maxj∈[C] nj(x) > 150). For the low-consensus distribution, we +choose to select T unlabeled data such that the majority vote count will be smaller than 150. +D +Omitted proofs in private GLM +D.1 +Per-instance DP of GLM +Theorem D.1 (Per-instance differential privacy guarantee). Consider two adjacent data sets Z and +Z′ = [Z, (x, y)], and denote the smooth part of the loss function Fs = �n +i=1 l(yi, ⟨xi, ·⟩) + rs(·) (thus +˜Fs = Fs + l(y, ⟨x, ·⟩). Let the local neighborhood be the line segment between θ∗ and ˜θ∗. Assume +1. the GLM loss function l be convex, three-time continuous differentiable and R-generalized-self- +concordant w.r.t. ∥ · ∥2, +2. Fs is locally α-strongly convex w.r.t. ∥ · ∥2, +3. and in addition, denote L := supθ∈[θ∗,˜θ∗] |l′(y, xT θ)|, β := supθ∈[θ∗,˜θ∗] |l′′(y, xT θ)|. +26 + +Then the algorithm obeys (ϵ, δ)-pDP for Z and z = (x, y) with any 0 < δ < 2/e and +ϵ ≤ ϵ0(1 + log(2/δ)) + e +RL∥x∥2 +α +�γL2∥x∥2 +H−1 +2 ++ +� +γL2∥x∥2 +H−1 log(2/δ) +� +where ϵ0 ≤ e +RL∥x∥2 +α +− 1 + 2β∥x∥2 +H−1 +1 ++ 2β∥x∥2 +˜H−1 +2 . If we instead assume that l is R-self concordant. +Then the same results hold, but with all e +RL∥x∥2 +α +replaced with (1 − RL∥x∥H−1)2. +Under the stronger three-times continuous differentiable assumption, by mean value theorem, there +exists ξ on the line-segment between θ∗ and ˜θ∗ such that +H = +�� 1 +t=0 +∇2Fs((1 − t)θ∗ + t˜θ∗)dt +� += ∇2Fs(ξ). +The two distributions of interests are N(θ∗, [γ∇2Fs(θ∗)]−1) and N(˜θ∗, [γ∇2Fs(˜θ∗)+∇2l(y, xT ˜θ∗)]−1). +Denote [∇2Fs(θ∗)]−1 =: Σ and [∇2Fs(˜θ∗)+∇2l(y, xT ˜θ∗)]−1 =: ˜Σ. Both the means and the covariance +matrices are different, so we cannot use multivariate Gaussian mechanism naively. Instead we will +take the tail bound interpretation of (ϵ, δ)-DP and make use of the per-instance DP framework as +internal steps of the proof. +First, we can write down the privacy loss random variable in analytic form +log |Σ|−1/2e− γ +2 ∥θ−θ∗∥2 +Σ−1 +|˜Σ|−1/2e− γ +2 ∥θ−˜θ∗∥2 +˜Σ−1 += 1 +2 log +�|Σ−1| +|˜Σ−1| +� +� +�� +� +(∗) ++ γ +2 +� +∥θ − θ∗∥2 +Σ−1 − ∥θ − ˜θ∗∥2 +˜Σ−1 +� +� +�� +� +(∗∗) +The general idea of the proof is to simplify the expression above and upper bounding the two terms +separately using self-concordance and matrix inversion lemma, and ultimately show that the privacy +loss random variable is dominated by another random variable having an appropriately scaled shifted +χ-distribution, therefore admits a Gaussian-like tail bound. +To ensure the presentation is readable, we define a few short hands. We will use H and ˜H to denote +the Hessian of Fs and Fs + f respectively and subscript 1 2 indicates whether the Hessian evaluated +at at θ∗ or ˜θ∗. H without any subscript or superscript represents the Hessian of Fs evaluated at ξ as +previously used. +(∗) = 1 +2 log |H1| +|H| +|H| +|H2| +|H2| +| ˜H2| +≤ 1 +2 +� +log |H1| +|H| + log |H| +|H2| + log |H2| +| ˜H2| +� +By the R-generalized self-concordance of Fs, we can apply Lemma D.3, +−∥θ∗ − ξ∥2R ≤ log |H1| +|H| ≤ R∥θ∗ − ξ∥2, +−R∥ξ − ˜θ∗∥2 ≤ log |H| +|H2| ≤ R∥ξ − ˜θ∗∥2. +The generalized linear model ensures that the Hessian of f is rank-1: +∇2f(˜θ∗) = l′′(y, xT ˜θ∗)xxT +and we can apply Lemma ?? in both ways (taking A = H2 and A = ˜H2) and obtain +|H2| +| ˜H2| += +1 +1 + l′′(y, xT ˜θ∗)xT H−1 +2 x += 1 − l′′(y, xT ˜θ∗)xT ˜H2x +27 + +Note that l′′(y, xT ˜θ∗)xT ˜H−1 +2 x is the in-sample leverage-score and l′′(y, xT ˜θ∗)xT H−1 +2 x is the out- +of-sample leverage-score of the locally linearized problem at ˜θ∗. We denote them by µ2 and µ′ +2 +respectively (similarly, for the consistency of notations, we denote the in-sample and out of sample +leverage score at θ∗ by µ1 and µ′ +1 ). +Combine the above arguments we get +(∗) ≤R∥θ∗ − ξ∥2 + R∥ξ − ˜θ∗∥2 + log(1 − µ2) ≤ R∥θ∗ − ˜θ∗∥2 + log(1 − µ2) +(6) +(∗) ≥ − R∥θ∗ − ˜θ∗∥2 − log(1 − µ2). +(7) +We now move on to deal with the second part, where we would like to express everything in terms of +∥θ − θ∗∥H1, which we know from the algorithm is χ-distributed. +(∗∗) = γ +2 +� +∥θ − θ∗∥2 +H1 − ∥θ − θ∗∥2 +H2 + ∥θ − θ∗∥2 +H2 − ∥θ − ˜θ∗∥2 +H2 + ∥θ − ˜θ∗∥2 +H2 − ∥θ − ˜θ∗∥2 +˜H2 +� +By the generalized self-concordance at θ∗ +e−R∥θ∗−˜θ∗∥2∥ · ∥2 +H1 ≤ ∥ · ∥2 +H2 ≤ eR∥θ∗−˜θ∗∥2∥ · ∥2 +H1 +This allows us to convert from ∥ · ∥H2 to ∥ · ∥H1, and as a consequence: +��∥θ − θ∗∥2 +H1 − ∥θ − θ∗∥2 +H2 +�� ≤ [eR∥θ∗−˜θ∗∥2 − 1]∥θ − θ∗∥2 +H1. +Also, +∥θ − θ∗∥2 +H2 − ∥θ − ˜θ∗∥2 +H2 = +� +˜θ∗ − θ∗, 2θ − 2θ∗ + θ∗ − ˜θ∗� +H2 = 2⟨θ − θ∗, ˜θ∗ − θ∗⟩H2 − ∥θ∗ − ˜θ∗∥2 +H2 +Therefore +���∥θ − θ∗∥2 +H2 − ∥θ − ˜θ∗∥2 +H2 +��� ≤ 2∥θ − θ∗∥H2∥θ∗ − ˜θ∗∥H2 + ∥θ∗ − ˜θ∗∥2 +H2 +≤ 2eR∥˜θ∗−θ∗∥2∥θ − θ∗∥H1∥θ∗ − ˜θ∗∥H + eR∥˜θ∗−θ∗∥2∥θ∗ − ˜θ∗∥2 +H. +Then lastly we have +0 ≥ ∥θ − ˜θ∗∥2 +H2 − ∥θ − ˜θ∗∥2 +˜H2 = −l′′(y, xT ˜θ∗) +� +⟨x, θ − θ∗⟩ + ⟨x, θ∗ − ˜θ∗⟩ +�2 +≥ −2β∥x∥2 +H−1 +1 ∥θ − θ∗∥2 +H1 − 2β∥x∥2 +H−1∥θ∗ − ˜θ∗∥2 +H +���∥θ − ˜θ∗∥2 +H2 − ∥θ − ˜θ∗∥2 +˜H2 +��� ≤ 2β∥x∥2 +H−1 +1 ∥θ − θ∗∥2 +H1 + 2β∥x∥2 +H−1∥θ∗ − ˜θ∗∥2 +H +Combine the above derivations, we get +|(∗∗)| ≤ γ +2 +� +a∥θ − θ∗∥2 +H1 + b∥θ − θ∗∥H1 + c +� +(8) +where +a := +� +eR∥θ∗−˜θ∗∥2 − 1 + 2β∥x∥2 +H−1 +1 +� +b :=2eR∥θ∗−˜θ∗∥2∥θ∗ − ˜θ∗∥H +c :=(eR∥θ∗−˜θ∗∥2 + 2β∥x∥2 +H−1)∥θ∗ − ˜θ∗∥2 +H +28 + +Lastly, by (6) and (8), +����log p(θ|Z) +p(θ|Z′) +���� ≤ R∥θ∗ − ˜θ∗∥2 + log(1 − µ2) + γ +2[aW 2 + bW + c]. +where according to the algorithm W := ∥θ − θ∗∥H1 follows a half-normal distribution with σ = +γ−1/2. +By standard Gaussian tail bound, we have for all δ < 2/e. +P(|W| ≤ γ−1/2� +log(2/δ)) ≤ δ. +This implies that a high probability upper bound of the absolute value of the privacy loss random +variable log p(θ|Z) +p(θ|Z′) under p(θ|Z). By the tail bound to privacy conversion lemma (Lemma ??), we +get that for any set S ⊂ Θ P(θ ∈ S|Z) ≤ eϵP(θ ∈ S|Z′) + δ for any 0 < δ < 2/e and +ϵ = R∥θ∗ − ˜θ∗∥2 + log(1 − µ2) + γc +2 + a +2 log(2/δ) + γ1/2b +2 +� +log(2/δ). +Denote v := θ∗ − ˜θ∗, by strong convexity +∥v∥2 ≤ ∥∇l(y, xT θ)[˜θ∗]∥2/α = |l′|∥x∥2/α ≤ L∥x∥2/α +and +∥v∥H ≤ ∥∇l(y, xT θ)[˜θ∗]∥H−1 = |l′|∥x∥H−1 ≤ L∥x∥H−1. +Also use the fact that | log(1 − µ2)| ≤ 2µ2 for µ2 < 0.5 and µ2 ≤ β∥x∥2 +˜H−1 +2 , we can then combine +similar terms and have a more compact representation. +ϵ ≤ ϵ0(1 + log(2/δ)) + e +RL∥x∥2 +α +�γL2∥x∥2 +H−1 +2 ++ +� +γL2∥x∥2 +H−1 log(2/δ) +� +where +ϵ0 ≤ e +RL∥x∥2 +α +− 1 + 2β∥x∥2 +H−1 +1 ++ 2β∥x∥2 +˜H−1 +2 +is the part of the privacy loss that does not get smaller as γ decreases. +Proposition D.2. Let ∥ · ∥ be a norm and ∥ · ∥∗ be its dual norm. Let F(θ), f(θ) and ˜F(θ) = +F(θ) + f(θ) be proper convex functions and θ∗ and +˜ +theta +∗ be their minimizers, i.e., 0 ∈ ∂F(θ∗) and +0 ∈ ∂ ˜F( +˜ +theta +∗). If in addition, F, ˜F is α, ˜α-strongly convex with respect to ∥ · ∥ within the restricted +domain θ ∈ {tθ∗ + (1 − t)˜θ∗ | t ∈ [0, 1]}. Then there exists g ∈ ∂f(θ∗) and ˜g ∈ ∂f(˜θ∗) such that +∥θ∗ − ˜θ∗∥ ≤ min +� 1 +α∥˜g∥∗, 1 +˜α∥g∥∗ +� +. +Proof. Apply the first order condition to F restricted to the line segment between ˜θ∗ and θ∗, there +are we get +F(˜θ∗) ≥ F(θ∗) + ⟨∂F(θ∗), ˜θ∗ − θ∗⟩ + α +2 ∥˜θ∗ − θ∗∥2 +(9) +F(θ∗) ≥ F(˜θ∗) + ⟨∂F(˜θ∗), θ∗ − ˜θ∗⟩ + α +2 ∥˜θ∗ − θ∗∥2 +(10) +29 + +Note by the convexity of F and f, ∂ ˜F = ∂F + ∂f, where + is the Minkowski Sum. Therefore, +0 ∈ ∂ ˜F(˜θ∗) implies that there exists ˜g such that ˜g ∈ ∂f(˜θ∗) and −˜g ∈ ∂F(˜θ∗). Take −˜g ∈ ∂F(˜θ∗) in +Equation 10 and 0 ∈ ∂F(θ∗) in Equation 9 and add the two inequalities, we obtain +0 ≥ ⟨−˜g, θ∗ − ˜θ∗⟩ + α∥˜θ∗ − θ∗∥2 ≥ −∥˜g∥∗∥θ∗ − ˜θ∗∥ + α∥˜θ∗ − θ∗∥2. +For ∥˜θ∗ − θ∗∥ = 0 the claim is trivially true, otherwise, we can divide the both sides of the above +inequality by ∥˜θ∗ − θ∗∥ and get ∥θ∗ − ˜θ∗∥ ≤ 1 +α∥˜g∥∗. +It remains to show that ∥θ∗ − ˜θ∗∥ ≤ 1 +˜α∥g∥∗. This can be obtained by exactly the same arguments +above but applying strong convexity to ˜F instead. Note that we can actually get something slightly +stronger than the statement because the inequality holds for all g ∈ ∂f(θ∗). +A consequence of (generalized) self-concordance is the spectral (multiplicative) stability of Hessian +to small perturbations of parameters. +Lemma D.3 (Stability of Hessian[Nesterov and Nemirovskii, 1994, Theorem 2.1.1], [Bach, 2010, +Proposition 1]). Let Hθ := ∇2Fs(θ). If Fs is R-self-concordant at θ. Then for any v such that +R∥v∥Hθ < 1, we have that +(1 − R∥v∥Hθ)2∇2Fs(θ) ≺ ∇2Fs(θ + v) ≺ +1 +(1 − R∥v∥Hθ)2 ∇2Fs(θ). +If instead we assume Fs is R-generalized-self-concordant at θ with respect to norm ∥ · ∥, then +e−R∥v∥∇2Fs(θ) ≺ ∇2Fs(θ + v) ≺ eR∥v∥∇2Fs(θ) +The two bounds are almost identical when R∥v∥ and R∥v∥θ are close to 0, in particular, for x ≤ 1/2, +e−2x ≤ 1 − x ≤ e−x. +References +Francis Bach. Self-concordant analysis for logistic regression. Electronic Journal of Statistics, 4: +384–414, 2010. +Kamalika Chaudhuri, Claire Monteleoni, and Anand D Sarwate. Differentially private empirical risk +minimization. Journal of Machine Learning Research, 12(3), 2011. +Chris Decarolis, Mukul Ram, Seyed Esmaeili, Yu-Xiang Wang, and Furong Huang. An end-to- +end differentially private latent dirichlet allocation using a spectral algorithm. In International +Conference on Machine Learning, pages 2421–2431. PMLR, 2020. +Cynthia Dwork and Jing Lei. Differential privacy and robust statistics. In ACM symposium on +Theory of computing, pages 371–380, 2009. +Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity +in private data analysis. In Theory of cryptography conference, pages 265–284. Springer, 2006. +Cynthia Dwork, Kunal Talwar, Abhradeep Thakurta, and Li Zhang. Analyze gauss: optimal bounds +for privacy-preserving principal component analysis. In Proceedings of the forty-sixth annual ACM +symposium on Theory of computing, pages 11–20, 2014. +30 + +Sivakanth Gopi, Yin Tat Lee, and Daogao Liu. +Private convex optimization via exponential +mechanism. arXiv preprint arXiv:2203.00263, 2022. +Shiva Prasad Kasiviswanathan, Kobbi Nissim, Sofya Raskhodnikova, and Adam Smith. Analyzing +graphs with node differential privacy. In Theory of Cryptography Conference, pages 457–476. +Springer, 2013. +Daniel Kifer, Adam Smith, and Abhradeep Thakurta. Private convex empirical risk minimization +and high-dimensional regression. In Conference on Learning Theory, pages 25–1. JMLR Workshop +and Conference Proceedings, 2012. +Jingcheng Liu and Kunal Talwar. Private selection from private candidates. In Proceedings of the +51st Annual ACM SIGACT Symposium on Theory of Computing, pages 298–309, 2019. +Xiyang Liu, Weihao Kong, and Sewoong Oh. Differential privacy and robust statistics in high +dimensions. arXiv preprint arXiv:2111.06578, 2021. +Kentaro Minami, HItomi Arai, Issei Sato, and Hiroshi Nakagawa. Differential privacy without +sensitivity. Advances in Neural Information Processing Systems, 29, 2016. +Ilya Mironov. Rényi differential privacy. In 2017 IEEE 30th computer security foundations symposium +(CSF), pages 263–275. IEEE, 2017. +Yurii Nesterov and Arkadii Nemirovskii. Interior-point polynomial algorithms in convex programming. +SIAM, 1994. +Kobbi Nissim, Sofya Raskhodnikova, and Adam Smith. Smooth sensitivity and sampling in private +data analysis. In ACM symposium on Theory of computing (STOC-07), pages 75–84. ACM, 2007. +Nicolas Papernot and Thomas Steinke. Hyperparameter tuning with renyi differential privacy. arXiv +preprint arXiv:2110.03620, 2021. +Nicolas Papernot, Martín Abadi, Úlfar Erlingsson, Ian Goodfellow, and Kunal Talwar. +Semi- +supervised knowledge transfer for deep learning from private training data. In International +Conference on Learning Representations (ICLR-17), 2017. +Nicolas Papernot, Shuang Song, Ilya Mironov, Ananth Raghunathan, Kunal Talwar, and Úlfar +Erlingsson. Scalable private learning with pate. arXiv preprint arXiv:1802.08908, 2018. +Rachel Redberg and Yu-Xiang Wang. Privately publishable per-instance privacy. Advances in Neural +Information Processing Systems, 34, 2021. +Jordi Soria-Comas, Josep Domingo-Ferrer, David Sánchez, and David Megías. Individual differential +privacy: A utility-preserving formulation of differential privacy guarantees. IEEE Transactions on +Information Forensics and Security, 12(6):1418–1429, 2017. +Abhradeep Guha Thakurta and Adam Smith. Differentially private feature selection via stability +arguments, and the robustness of the lasso. In Conference on Learning Theory, pages 819–850. +PMLR, 2013. +Salil Vadhan. The complexity of differential privacy. In Tutorials on the Foundations of Cryptography, +pages 347–450. Springer, 2017. +Jiachen T Wang, Saeed Mahloujifar, Shouda Wang, Ruoxi Jia, and Prateek Mittal. Renyi differential +privacy of propose-test-release and applications to private and robust machine learning. arXiv +preprint arXiv:2209.07716, 2022. +31 + +Yu-Xiang Wang. Per-instance differential privacy and the adaptivity of posterior sampling in linear +and ridge regression. arXiv preprint arXiv:1707.07708, pages 48–71, 2017. +Yu-Xiang Wang. Revisiting differentially private linear regression: optimal and adaptive prediction +& estimation in unbounded domain. arXiv preprint arXiv:1803.02596, 2018. +Yu-Xiang Wang, Stephen Fienberg, and Alex Smola. Privacy for free: Posterior sampling and +stochastic gradient monte carlo. In International Conference on Machine Learning, pages 2493– +2502. PMLR, 2015. +32 + diff --git a/99AyT4oBgHgl3EQfdfcH/content/tmp_files/load_file.txt b/99AyT4oBgHgl3EQfdfcH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..561c577d8d913f6d078cda1ff8aa316c59eb1fb5 --- /dev/null +++ b/99AyT4oBgHgl3EQfdfcH/content/tmp_files/load_file.txt @@ -0,0 +1,1433 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf,len=1432 +page_content='Generalized PTR: User-Friendly Recipes for Data-Adaptive Algorithms with Differential Privacy Rachel Redberg, Yuqing Zhu, Yu-Xiang Wang University of California, Santa Barbara {rredberg, yuqingzhu, yuxiangw}@ucsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='edu January 3, 2023 Abstract The “Propose-Test-Release” (PTR) framework [Dwork and Lei, 2009] is a classic recipe for designing differentially private (DP) algorithms that are data-adaptive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' those that add less noise when the input dataset is “nice”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We extend PTR to a more general setting by privately testing data-dependent privacy losses rather than local sensitivity, hence making it applicable beyond the standard noise-adding mechanisms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' to queries with unbounded or undefined sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We demonstrate the versatility of generalized PTR using private linear regression as a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Additionally, we apply our algorithm to solve an open problem from “Private Aggregation of Teacher Ensembles (PATE)” [Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2017, 2018] — privately releasing the entire model with a delicate data-dependent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 1 Introduction The guarantees of differential privacy (DP) [Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2006] are based on worst-case outcomes across all possible datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' A common paradigm is therefore to add noise scaled by the global sensitivity of a query f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' the maximum change in f between any pair of neighboring datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' A given dataset X might have a local sensitivity that is much smaller than the global sensitivity, in which case we can hope to add a smaller amount of noise (calibrated to the local rather than the global sensitivity) while achieving the same privacy guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' However, this must not be undertaken naïvely – the local sensitivity is a dataset-dependent function and so calibrating noise to the local sensitivity could leak information about the dataset [Nissim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2007].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The “Propose-Test-Release” (PTR) framework [Dwork and Lei, 2009] resolves this issue by introducing a test to privately check whether a proposed bound on the local sensitivity is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Only if the test “passes” is the output released with noise calibrated to the proposed bound on the local sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' PTR is a powerful and flexible tool for designing data-adaptive DP algorithms, but it has several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' First, it applies only to noise-adding mechanisms which calibrate noise according to the sensitivity of a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Second, the test in “Propose-Test-Release” is computationally expensive for all but a few simple queries such as privately releasing the median or mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Third, while some existing works [Decarolis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2020, Kasiviswanathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2013, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2021] follow the approach of testing “nice” properties of a dataset before exploiting these properties in a private release to PTR 1, 1We refer to these as PTR-like methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='00301v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='LG] 31 Dec 2022 there has not been a systematic recipe for discovering which properties should be tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In this paper, we propose a generalization of PTR which addresses these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The centerpiece of our framework is a differentially private test on the data-dependent privacy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This test does not directly consider the local sensitivity of a query and is therefore not limited to additive noise mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Moreover, in many cases, the test can be efficiently implemented by privately releasing a high-probability upper bound, thus avoiding the need to search an exponentially large space of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Furthermore, the derivation of the test itself often spells out exactly what properties of the input dataset need to be checked, which streamlines the design of data-adaptive DP algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Our contributions are summarized as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We propose a generalization of PTR which can handle algorithms beyond noise-adding mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Generalized PTR allows us to plug in any data-dependent DP analysis to construct a high-probability DP test that adapts to favorable properties of the input dataset – without painstakingly designing each test from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We demonstrate that many existing examples of PTR and PTR-like algorithms can be unified under the generalized PTR framework, sometimes resulting in a tighter analysis (see an example of report-noisy-max in Sec A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We show that one can publish a DP model through privately upper-bounding a one-dimensional statistic — no matter how complex the output space of the mechanism is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We apply this result to solve an open problem from PATE [Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2017, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Our results broaden the applicability of private hyper-parameter tuning [Liu and Talwar, 2019, Papernot and Steinke, 2021] in enabling joint-parameter selection of DP-specific parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', noise level) and native parameters of the algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', learning rate, regularization weight), which may jointly affect the data-dependent DP losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2 Related Work Data-dependent DP algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Privately calibrating noise to the local sensitivity is a well- studied problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' One approach is to add noise calibrated to the smooth sensitivity [Nissim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2007], an upper bound on the local sensitivity which changes slowly between neighboring datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' An alternative to this – and the focus of our work – is Propose-Test-Release (PTR) [Dwork and Lei, 2009], which works by calculating the distance Dβ(X) to the nearest dataset to X whose local sensitivity violates a proposed bound β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The PTR algorithm then adds noise to Dβ(X) before testing whether this privately computed distance is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' PTR spin-offs abound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Notable examples include stability-based methods [Thakurta and Smith, 2013] (stable local sensitivity of 0 near the input data) and privately releasing upper bounds of local sensitivity [Kasiviswanathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2013, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2021, Decarolis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We refer readers to Chapter 3 of Vadhan [2017] for a concise summary of these classical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Recent work [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2022] has provided Rényi DP bounds for PTR and demonstrated its applications to robust DP-SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Our work (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2) also considers applications of PTR in data-adaptive private deep learning: Instead of testing the local sensitivity of each gradient step as in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [2022], our PTR-based PATE algorithm tests the data-dependent privacy loss as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [2021] proposed a new variant called High-dimensional Propose-Test-Release (HPTR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' HPTR provides a systematic way of solving DP statistical estimation problems by using the exponential 2 mechanism (EM) with carefully constructed scores based on certain one-dimensional robust statistics, which have stable local sensitivity bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' HPTR focuses on designing data-adaptive DP mechanisms from scratch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' our method, in contrast, converts existing randomized algorithms (including EM and even some that do not satisfy DP) into those with formal DP guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Interestingly, our proposed method also depends on a one-dimensional statistic of direct interest: the data-dependent privacy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Data-dependent DP losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The flip side of data-dependent DP algorithms is the study of data-dependent DP losses [Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2018, Soria-Comas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2017, Wang, 2017], which fix the randomized algorithm but parameterize the resulting privacy loss by the specific input dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For example: In the simple mechanism that adds Laplace noise with parameter b, data-dependent DP losses are ϵ(X) = ∆LS(X)/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The data-dependent DP losses are often much smaller than the DP loss, but they themselves depend on the data and thus may reveal sensitive information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' algorithms satisfying a data-dependent privacy guarantee are not formally DP with guarantees any smaller than that of the worst-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Existing work has considered privately publishing these data-dependent privacy losses [Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2018, Redberg and Wang, 2021], but notice that privately publishing these losses does not improve the DP parameter of the given algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Part of our contribution is to resolve this conundrum by showing that a simple post-processing step of the privately released upper bound of ϵ(Data) gives a formal DP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Private hyper-parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Our work has a nice connection with private hyper-parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Prior work [Liu and Talwar, 2019, Papernot and Steinke, 2021] requires each candidate configuration to be released with the same DP (or Rényi DP) parameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Another hidden assumption is that the parameters must not be privacy-correlated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', parameter choice will not change the privacy guarantee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Otherwise we need to use the largest DP bound across all candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For example, Liu and Talwar [2019] show that if each mechanism (instantiated with one group of hyper-parameters) is (ϵ, 0)-DP, then running a random number of mechanisms and reporting the best option satisfies (3ϵ, 0)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Our work directly generalizes the above results by (1) considering a wide range of hyper-parameters, either privacy-correlated or not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' and (2) requiring only that individual candidates to have a testable data-dependent DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 3 Preliminaries Datasets X, X′ ∈ X are neighbors if they differ by no more than one datapoint – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', X ≃ X′ if d(X, X′) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We will define d(·) to be the number of coordinates that differ between two datasets of the same size n: d(X, Y ) = #{i ∈ [n] : Xi ̸= Yi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We use || · || to denote the radius of the smallest Euclidean ball that contains the input set, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ||X|| = supx∈X ||x||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The parameter φ denotes the privacy parameters associated with a mechanism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' noise level, regularization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Mφ is a mechanism parameterized by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For mechanisms with continuous output space, we will take Pr[M(X) = y] to be the probability density function of M(X) at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 (Differential privacy [Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2006]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Fix ϵ, δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' A randomized algorithm M : X → S satisfies (ϵ, δ)-DP if for all neighboring datasets X ≃ X′ and for all measurable sets S ⊂ S, Pr � M(X) ∈ S � ≤ eϵPr � M(X′) ∈ S � + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Suppose we wish to privately release the output of a real-valued function f : X → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We can do so 3 by calculating the global sensitivity ∆GS, calibrating the noise scale to the global sensitivity and then adding sampled noise to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 (Local / Global sensitivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The local ℓ∗-sensitivity of a function f is defined as ∆LS(X) = max X≃X′ ||f(X) − f(X′)||∗ and the global sensitivity of f is ∆GS = supX ∆LS(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 Propose-Test-Release Calibrating the noise level to the local sensitivity ∆LS(X) of a function would allow us to add less noise and therefore achieve higher utility for releasing private queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' However, the local sensitivity is a data-dependent function and naïvely calibrating the noise level to ∆LS(X) will not satisfy DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' PTR resolves this issue in a three-step procedure: propose a bound on the local sensitivity, privately test that the bound is valid (with high probability), and if so calibrate noise according to the bound and release the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' PTR privately computes the distance Dβ(X) between the input dataset X and the nearest dataset X′′ whose local sensitivity exceeds the proposed bound β: Dβ(X) = min X′′ {d(X, X′′) : ∆LS(X′′) > β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Algorithm 1 Propose-Test-Release [Dwork and Lei, 2009] 1: Input: Dataset X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' privacy parameters ϵ, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' proposed bound β on ∆LS(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' query function f : X → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2: if Dβ(X) + Lap � 1 ϵ � ≤ log(1/δ) ϵ then output ⊥, 3: else release f(X) + Lap � β ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Algorithm 1 satisfies (2ϵ, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [Dwork and Lei, 2009] Rather than proposing an arbitrary threshold β, one can also privately release an upper bound of the local sensitivity and calibrate noise according to this upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This was used for node DP in graph statistics [Kasiviswanathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2013], and for fitting topic models using spectral methods [Decarolis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 4 Generalized PTR This section introduces the generalized PTR framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We first formalize the notion of data- dependent differential privacy that conditions on an input dataset X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 (Data-dependent privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Suppose we have δ > 0 and a function ϵ : X → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We say that mechanism M satisfies (ϵ(X), δ) data-dependent DP2 for dataset X if for all possible output sets S and neighboring datasets X′, Pr � M(X) ∈ S � ≤ eϵ(X)Pr � M(X′) ∈ S � + δ, Pr � M(X′) ∈ S � ≤ eϵ(X)Pr � M(X) ∈ S � + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2We will sometimes write that M(X) satisfies ϵ(X) data-dependent DP with respect to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 4 In generalized PTR, we propose a value φ for the randomized algorithm M, which could be a noise scale or regularization parameter – or a set including both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For example, φ = (λ, γ) in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We then say that Mφ is the mechanism M parameterized by φ, and ϵφ(X) its data-dependent DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The following example illustrates how to derive the data-dependent DP for a familiar friend – the Laplace mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ( Data-dependent DP of Laplace Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=') Given a function f : X → R, we will define Mφ(X) = f(X) + Lap (φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We then have log Pr[Mφ(X) = y] Pr[Mφ(X′) = y] ≤ |f(X) − f(X′)| φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Maximizing the above calculation over all possible outputs y and using Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1, ϵφ(X) = max X′:X′≃X |f(X) − f(X′)| φ = ∆LS(X) φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The data-dependent DP ϵφ(X) is a function of both the dataset X and the parameter φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Maximizing ϵφ(X) over X recovers the standard DP guarantee of running M with parameter φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Algorithm 2 Generalized Propose-Test-Release 1: Input: Dataset X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' mechanism Mφ : X → R and its privacy budget ϵ, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' (ˆϵ, ˆδ)-DP test T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' false positive rate ≤ δ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' data-dependent DP function ϵφ(·) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2: if not T (X) then output ⊥, 3: else release θ = Mφ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='3 (Privacy guarantee of generalized PTR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Consider a proposal φ and a data-dependent DP function ϵφ(X) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Suppose that we have an (ˆϵ, ˆδ)-DP test T : X → {0, 1} such that when ϵφ(X) > ϵ, T (X) = � 0 with probability 1 − δ′, 1 with probability δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then Algorithm 2 satisfies (ϵ + ˆϵ, δ + ˆδ + δ′)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' There are three main cases to consider: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We decide not to run Mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We decide to run Mφ and ϵφ(X) > ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We decide to run Mφ and ϵφ(X) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 5 In the first case, the decision to output ⊥ is post-processing of an (ˆϵ, ˆδ)-DP mechanism and inherits its privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The second case occurs when the (ˆϵ, ˆδ)-DP test "fails" (produces a false positive) and occurs with probability at most δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The third case is a composition of an (ˆϵ, ˆδ)-DP algorithm and an (ϵ, δ)-DP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Generalized PTR is a strict generalization of Propose-Test-Release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For some function f, define Mφ and T as follows: Mφ(X) = f(X) + Lap(φ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' T (X) = � 0 if Dβ(X) + Lap � 1 ϵ � > log(1/δ) ϵ , 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Notice that our choice of parameterization is φ = β ϵ , where φ is the scale of the Laplace noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In other words, we know from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 that ϵφ(X) > ϵ exactly when ∆LS(X) > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For noise-adding mechanisms such as the Laplace mechanism, the sensitivity is proportional to the privacy loss (in both the global and local sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ∆GS ∝ ϵ and ∆LS ∝ ϵ(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Therefore for these mechanisms the only difference between privately testing the local sensitivity (Algorithm 1) and privately testing the data-dependent DP (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='3) is a change of parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 Limitations of local sensitivity Why do we want to generalize PTR beyond noise-adding mechanisms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Compared to classic PTR, the generalized PTR framework allows us to be more flexible in both the type of test conducted and also the type of mechanism whose output we wish to release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For many mechanisms, the local sensitivity either does not exist or is only defined for specific data-dependent quantities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', the sensitivity of the score function in the exponential mechanism) rather than the mechanism’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The following example illustrates this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='4 (Private posterior sampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let M : X × Y → Θ be a private posterior sampling mechanism [Minami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2016, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2015, Gopi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2022] for approximately minimizing FX(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' M samples θ ∼ P(θ) ∝ e−γ(FX(θ)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='5λ||θ||2) with parameters γ, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Note that γ, λ cannot be appro- priately chosen for this mechanism to satisfy DP without going through a sensitivity calculation of arg min FX(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In fact, the global and local sensitivity of the minimizer is unbounded even in linear regression problems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e when FX(θ) = 1 2||y − Xθ||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Output perturbation algorithms do work for the above problem when we regularize, but they are known to be suboptimal in theory and in practice [Chaudhuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 we demonstrate how to apply generalized PTR to achieve a data-adaptive posterior sampling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Even in the cases of noise-adding mechanisms where PTR seems to be applicable, it does not lead to a tight privacy guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Specifically, by an example of privacy amplification by post-processing (Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 in the appendix), we demonstrate that the local sensitivity does not capture all sufficient statistics for data-dependent privacy analysis and thus is loose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 Which φ to propose The main limitation of generalized PTR is that one needs to “propose” a good guess of parameter φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Take the example of φ being the noise level in a noise-adding mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Choosing too small a φ will result in a useless output ⊥, while choosing too large a φ will add more noise than necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Finding this ’Goldilocks’ φ might require trying out many different possibilities – each of which will consume privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This section introduces a method to jointly tune privacy parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', noise scale) along with parameters related only to the utility of an algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', learning rate or batch size in stochastic gradient descent) – while avoiding the ⊥ output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Algorithm 3 takes a list of parameters as input, runs generalized PTR with each of the parameters, and returns the output with the best utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We show that the privacy guarantee with respect to ϵ is independent of the number of φ that we try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Formally, let φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', φk be a set of hyper-parameters and ˜θi ∈ {⊥, Range(M)} denotes the output of running generalized PTR on a private dataset X with φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let Xval be a public validation set and q(˜θi) be the score of evaluating ˜θi with Xval (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', validation accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The goal is to select a pair (˜θi, φi) such that DP model ˜θi maximizes the validation score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The generalized PTR framework with privacy calibration is described in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The privacy guarantee of Algorithm 3 is an application of Liu and Talwar [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Algorithm 3 PTR with hyper-parameter selection 1: Input: Privacy budget per PTR algorithm (ϵ∗, δ∗), cut-off T, parameters φ1:k, flipping probability τ and validation score function q(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2: Initialize the set S = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 3: Draw G from a geometric distribution Dτ and let ˆT = min(T, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 4: for i = 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', ˆT do 5: pick a random φi from φ1:k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 6: evaluate φi: (˜θi, q(˜θi)) ← Algorithm 2(φi, (ϵ∗, δ∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 7: S ← S ∪ {˜θi, q(˜θi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 8: end for 9: Output the highest scored candidate from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='5 ( Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='4 Liu and Talwar [2019] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Fix any τ ∈ [0, 1], δ2 > 0 and let T = 1 τ log 1 δ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If each oracle access to Algorithm 2 is (ϵ∗, δ∗)-DP, then Algorithm 3 is (3ϵ∗ +3 √ 2δ∗, √ 2δ∗T +δ2)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The theorem implies that one can try a random number of φ while paying a constant ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In practice, we can roughly set τ = 1 10k so that the algorithm is likely to test all k parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We emphasize that the privacy and the utility guarantee (stated in the appendix) is not our contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' But the idea of applying generalized PTR to enforce a uniform DP guarantee over all choices of parameters with a data-dependent analysis is new, and in our opinion, significantly broadens the applicability to generic hyper-parameter tuning machinery from Liu and Talwar [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='3 Construction of the DP test Classic PTR uses the Laplace mechanism to construct a differentially private upper bound of Dβ(X), the distance from input dataset X to the closest dataset whose local sensitivity exceeds the proposed 7 bound β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The tail bound of the Laplace distribution then ensures that if Dβ(X) = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' if ∆LS(X) > β), then the output will be released with only a small probability δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The following theorem shows that we could instead use a differentially private upper bound of the data-dependent DP ϵφ(X) in order to test whether to run the mechanism Mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='6 (Generalized PTR with private upper bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Suppose we have a differentially private upper bound of ϵφ(X) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' δ such that with probability at least 1 − δ′, ϵP φ (X) > ϵφ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Further suppose we have an (ˆϵ, ˆδ)-DP test T such that T(X) = � 1 if ϵP φ (X) < ϵ, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then Algorithm 2 is (ϵ + ˆϵ, δ + ˆδ + δ′)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2, we demonstrate that one can upper bound the data-dependent DP through a modification of the smooth sensitivity framework applied on ϵφ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Moreover, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 we provide a direct application of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='6 with private linear regression by making use of the per-instance DP technique [Wang, 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The applications in Section 5 are illustrative of two distinct approaches to constructing the DP test for generalized PTR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Private sufficient statistics release (used in the private linear regression example of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1) specifies the data-dependent DP as a function of the dataset and privately releases each data-dependent component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The second approach (used in the PATE example of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2) uses the smooth sensitivity framework to privately release the data-dependent DP as a whole, and then construct a high-confidence test using the Gaussian mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' These two approaches cover most of the scenarios arising in data-adaptive analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For example, in the appendix we demonstrate the merits of generalized PTR in handling data-adaptive private generalized linear models (GLMs) using private sufficient statistics release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Moreover, sufficient statistics release together with our private hyper-parameter tuning (Algorithm 3) can be used to construct data-adaptive extensions of DP-PCA and Sparse-DP-ERM (see details in the future work section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 5 Applications In this section, we put into action our approaches to construct the DP test and provide applications in private linear regression and PATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 Private Linear Regression Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 ([Wang, 2017]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For input data X ∈ X and Y ∈ Y, define the following: λmin(X) denotes the smallest eigenvalue of XT X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ||θ∗ λ|| is the magnitude of the solution θ∗ λ = (XT X + λI)−1XT Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' and L(X, y) := ||X||(||X||||θ∗ λ|| + ||Y||) is the local Lipschitz constant, denoted L in brief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 8 10 1 100 10 2 6 × 10 3 2 × 10 2 3 × 10 2 4 × 10 2 MSE UCI Bike dataset (n = 17379, d = 17) AdaOPS non-private OutPert OPS OPS with PTR (a) Bike dataset 10 1 100 2 × 10 2 3 × 10 2 4 × 10 2 6 × 10 2 MSE UCI elevators dataset (n = 8752, d = 18) AdaOPS non-private OutPert OPS OPS with PTR (b) Elevators dataset Figure 1: Differentially private linear regression algorithms on UCI datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' y-axis reports the MSE error with confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ϵ is evaluated with δ = 1e − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For brevity, denote λ∗ = λ + λmin(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The algorithm used in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='4 with parameter φ = (λ, γ) obeys (ϵφ(Z), δ) data-dependent DP for each dataset Z = (X, Y ) with ϵφ(Z) equal to � γL2 log(2/δ) λ∗ + γL2 2(λ∗ + ||X||2) + 1 + log(2/δ)||X||2 2(λ∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Notice that the data-dependent DP is a function of (λmin, L, ||θ∗ λ||, λ, γ), where (λmin, L, ||θ∗ λ||) are data-dependent quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' One can apply the generalized PTR framework as in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 (OPS with PTR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We demonstrate here how to apply generalized PTR to the one- posterior sample (OPS) algorithm, a differentially private mechanism which outputs one sample from the posterior distribution of a Bayesian model with bounded log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Propose φ = (λ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Based on (λ, γ), differentially privately release λmin, ||θ∗ λ||, L with privacy budget (ϵ, δ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Condition on a high probability event (with probability at least 1 − δ/2) of λmin, ||θ∗ λ||, L, test if ϵP φ (X) is smaller than the predefined privacy budget (ˆϵ, ˆδ), where ϵP φ (X) denotes the sanitized data-dependent DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Based on the outcome of the test, decide whether to release θ ∝ e− γ 2 ||Y −Xθ||2+λ||θ||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The algorithm outlined in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 satisfies (ϵ + ˆϵ, δ + ˆδ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The main idea of the above algorithm boils down to privately releasing all data-dependent quantities in data-dependent DP, constructing high-probability confidence intervals of these quantities, and then deciding whether to run the mechanism M with the proposed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We defer the details of the privacy calibration of data-dependent quantities to the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' One may ask why we cannot directly tune privacy parameters (λ, γ) based on the sanitized data- dependent DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This is because, in many scenarios, data-dependent quantities depend on the choice of privacy parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', ||θ∗ λ|| is a complicated function of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Thus, the optimization on λ becomes 9 a circular problem — to solve λ, we need to sanitize ||θ∗ λ||, which needs to choose a λ to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Alternatively, generalized PTR provides a clear and flexible framework to test the validity of privacy parameters adapted to the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The above “circular” issue is even more serious for generalized linear models (GLMs) beyond linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The data-dependent DP there involves a local strong-convexity parameter, a complex function of the regularizer λ and we only have zeroth-order access to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In the appendix, we demonstrate how to apply generalized PTR to provide a generic solution to a family of private GLMs where the link function satisfies a self-concordance assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We next apply Algorithm 3 for Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 with UCI regression datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Standard z-scoring is applied and each data point is normalize with a Euclidean norm of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We consider (60%, 10%, 30%) splits for training, validation and testing test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Baselines Output Perturbation (Outpert) [Chaudhuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2011]: θ = (XT X + λI)−1XT y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Release ˆθ = θ + b with an appropriate λ, where b is a Gaussian random vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Posterior sampling (OPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Sample ˆθ ∼ P(θ) ∝ e−γ(F(θ)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='5λ||θ||2) with parameters γ, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Adaptive posterior sampling (AdaOPS) [Wang, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Run OPS with (λ, γ) chosen adaptively according to the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Outpert and OPS serve as two non-adaptive baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In particular, we consider OPS-Balanced [Wang, 2018], which chooses λ to minimize a data-independent upper bound of empirical risk and dominates other OPS variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' AdaOPS is one state-of-the-art algorithm for adaptive private regression, which automatically chooses λ by minimizing an upper bound of the data-dependent empirical risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We implement OPS-PTR as follows: propose a list of λ through grid search (we choose k = 30 and λ ranges from [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='510] on a logarithmic scale);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' instantiate Algorithm 3 with τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1k, T = 1 τ log(1/δ2) and δ2 = 1/2δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' calibrate γ to meet the privacy requirement for each λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' sample ˆθ using (λ, γ) and return the one with the best validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Notice that we use a “no ⊥” variant of Algorithm 2 as the calibration of γ is clear given a fixed λ and privacy budget (see more details in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We can propose various combinations of (λ, γ) for more general applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Figure 1 demonstrates how the MSE error of the linear regression algorithms varies with the privacy budget ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' OutPert suffers from the large global sensitivity of output θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' OPS performs well but does not benefit from the data-dependent quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' AdaOPS is able to adaptively choose (λ, γ) based on the dataset, but suffers from the estimation error of the data-dependent empirical risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' On the other hand, OPS-PTR selects a (λ, γ) pair that minimizes the empirical error on the validation set directly, and the privacy parameter γ adapts to the dataset thus achieving the best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 PATE In this section, we apply the generalized PTR framework to solve an open problem from the Private Aggregation of Teacher Ensembles (PATE) [Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2017, 2018] — privately publishing the entire model through privately releasing data-dependent DP losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Our algorithm makes use of the smooth sensitivity framework [Nissim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2007] and the Gaussian mechanism to construct a high- probability test of the data-dependent DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The one-dimensional statistical nature of data-dependent DP enables efficient computations under the smooth sensitivity framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Thus, this approach is generally applicable for other private data-adaptive analysis beyond PATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 10 PATE is a knowledge transfer framework for model-agnostic private learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In this framework, an ensemble of teacher models is trained on the disjoint private data and uses the teachers’ aggregated consensus answers to supervise the training of a “student” model agnostic to the underlying machine- learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' By publishing only the aggregated answers and by the careful analysis of the “consensus”, PATE has become a practical technique in recent private model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The tight privacy guarantee of PATE heavily relies on a delicate data-dependent DP analysis, for which the authors of PATE use the smooth sensitivity framework to privately publish the data- dependent privacy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' However, it remains an open problem to show that the released model is DP under data-dependent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Our generalized PTR resolves this gap by carefully testing a private upper bound of the data-dependent privacy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Our algorithm is fully described in Algorithm 4, where the modification over the original PATE framework is highlighted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Algorithm 4 takes the input of privacy budget (ϵ′, ˆϵ, δ), unlabeled public data x1:T and K teachers’ predictions on these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The parameter ϵ denotes the privacy cost of publishing the data-dependent DP and ϵ′ is the predefined privacy budget for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' nj(xi) denotes the the number of teachers that agree on label j for xi and C denotes the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The goal is to privately release a list of plurality outcomes — argmaxj∈[C]nj(xi) for i ∈ [T] — and use these outcomes to supervise the training of a “student” model in the public domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The parameter σ1 denotes the noise scale for the vote count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In their privacy analysis, Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [2018] compute the data-dependent RDPσ1(α, X) of labeling the entire group of student queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' RDPσ1(α, X) can be orders of magnitude smaller than its data- independent version if there is a strong agreement among teachers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Note that RDPσ1(α, X) is a function of the RDP order α and the dataset X, analogous to our Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 but subject to RDP [Mironov, 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='5 ([Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2018]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If the top three vote counts of xi are n1 > n2 > n3 and n1 − n2, n2 − n3 ≫ σ1, then the data-dependent RDP of releasing argmaxj{nj + N(0, σ2 1)} satisfies (α, exp{−2α/σ2 1}/α)-RDP and the data-independent RDP (using the Gaussian mechanism) satisfies (α, α σ2 1 )-RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Algorithm 4 PATE with generalized PTR 1: Input: Unlabeled public data x1:T , aggregated teachers prediction n(·), privacy parameter ˆϵ, ϵ′, δ, noisy parameter σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2: Set α = 2 log(2/δ) ˆϵ + 1, σs = σ2 = � 3α+2 ˆϵ , δ2 = δ/2, smoothness parameter β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 3: Compute noisy labels: yip ← argmaxj∈[C]{nj(xi) + N(0, σ2 1)} for all i ∈ [1 : T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 4: RDPσ1(α, X) ← data-dependent RDP at the α-th order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 5: SSβ(X) ← the smooth sensitivity of RDPupper σ1 (α, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 6: Privately release µ := log(SSβ(X)) + β · N(0, σ2 2) + � 2 log(2/δ2) · σ2 · β 7: RDPupper σ1 (α) ← an upper bound of data-dependent RDP through Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 8: ϵσ1 ← DP guarantee converted from RDPupper σ1 (α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 9: If ϵ′ ≥ ϵσ1 return a student model trained using (x1:T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' yp 1:T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 10: Else return ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' However, RDPσ1(α, X) is data-dependent and thus cannot be revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The authors therefore privately publish the data-dependent RDP using the smooth sensitivity framework [Nissim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2007].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The smooth sensitivity calculates a smooth upper bound on the local sensitivity of RDPσ1(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 11 15 20 25 30 35 40 45 50 Noise scale 1 1 2 3 4 5 Gaussian mechanism PATE-PTR ( + 1) data-dependent DP (non-private) (a) High consensus and strong data-dependent DP 15 20 25 30 35 40 45 50 Noise scale 1 1 2 3 4 5 Gaussian mechanism PATE-PTR ( + 1) data-dependent DP (non-private) (b) Low consensus and low data-dependent DP Figure 2: Privacy and utility tradeoffs with PATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' When σ1 is aligned, three algorithms provide the same utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' y-axis plots the privacy cost of labeling T = 200 public data with δ = 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The left figure considers the high-consensus case, where the data-adaptive analysis is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' denoted as SSβ(X), such that SSβ(X) ≤ eβSSβ(X′) for any neighboring dataset X and X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' By adding Gaussian noise scaled by the smooth sensitivity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', release ϵσ1(α, X) + SSβ(X) · N(0, σ2 s)), the privacy cost is safely published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Unlike most noise-adding mechanisms, the standard deviation σs cannot be published since SSβ(X) is a data-dependent quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Moreover, this approach fails to provide a valid privacy guarantee of the noisy labels obtained through the PATE algorithm, as the published privacy cost could be smaller than the real privacy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Our solution in Algorithm 4 looks like the following: Privately release an upper bound of the smooth sensitivity SSβ(X) with eµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Conditioned on a high-probability event of eµ, publish the data-dependent RDP with RDPupper σ1 (α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Convert RDPupper σ1 (α) back to the standard DP guarantee using RDP to DP conversion at δ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Test if the converted DP is above the predefined budget ϵ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The following lemma states that RDPupper σ1 (α) is a valid upper bound of the data-dependent RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='6 (Private upper bound of data-dependent RDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We are given a RDP function RDP(α, X) and a β-smooth sensitivity bound SS(·) of RDP(α, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let µ (defined in Algorithm 4) denote the private release of log(SSβ(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let the (β, σs, σ2)-GNSS mechanism be RDPupper(α):=RDP(α,X)+SSβ(X)·N(0,σ2 s)+σs � 2 log( 2 δ2 )eµ Then, the release of RDPupper(X) satisfies (α, 3α+2 2σ2s )-RDP for all 1 < α < 1 2β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' at least 1 − δ2, RDPupper(α) is an upper bound of RDP(α, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The proof (deferred to the appendix) makes use of the facts that: (1) the log of SSβ(X) has a bounded global sensitivity β through the definition of smooth sensitivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' (2) releasing RDPσ1(α, X)+ SSβ(X) · N(0, σ2 s) is (α, α+1 σ2s )-RDP (Theorem 23 from Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [2018]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Now, we are ready to state the privacy guarantee of Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 12 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Algorithm 4 satisfies (ϵ′ + ˆϵ, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In the proof, the choice of α ensures that the cost of the δ/2 contribution (used in the RDP-to-DP conversion) is roughly ˆϵ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then the release of RDPupper σ1 (α) with σs = � 2+3α ˆϵ accounts for another cost of (ϵ/2, δ/2)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Empirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We next empirically evaluate Algorithm 4 (PATE-PTR) on the MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Following the experimental setup from Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [2018], we consider the training set to be the private domain, and the testing set is used as the public domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We first partition the training set into 400 disjoint sets and 400 teacher models, each trained individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then we select T = 200 unlabeled data from the public domain, with the goal of privately labeling them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' To illustrate the behaviors of algorithms under various data distributions, we consider two settings of unlabeled data, high-consensus and low-consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In the low-consensus setting, we choose T unlabeled data such that there is no high agreement among teachers, so the advantage of data-adaptive analysis is diminished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We provide further details on the distribution of these two settings in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We consider the Gaussian mechanism as a data-independent baseline, where the privacy guarantee is valid but does not take advantage of the properties of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The data- dependent DP ( Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [2018]) serves as a non-private baseline, which requires further sanitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Note that these two baselines provide different privacy analyses of the same algorithm (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Figure 2 plots privacy-utility tradeoffs between the three approaches by varying the noise scale σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The purple region denotes a set of privacy budget choices (ˆϵ + ϵ′ used in Algorithm 4) such that the utility of the three algorithms is aligned under the same σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In more detail, the purple region is lower-bounded by ˆϵ+ϵσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We first fix σs = σ2 = 15 such that ˆϵ is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then we empirically calculate the average of ϵσ1 (the private upper bound of the data-dependent DP) over 10 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Running Algorithm 4 with any choice of ˆϵ + ϵ′ chosen from the purple region implies ϵ′ > ϵσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Therefore, PATE-PTR will output the same noisy labels (with high probability) as the two baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Observation As σ1 increases, the privacy loss of the Gaussian mechanism decreases, while the data-dependent DP curve does not change much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This is because the data-dependent DP of each query is a complex function of both the noise scale and the data and does not monotonically decrease when σ1 increases (see more details in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' However, the data-dependent DP still dominates the Gaussian mechanism for a wide range of σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Moreover, PATE-PTR nicely interpolates between the data-independent DP guarantee and the non-private data-adaptive DP guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In the low-consensus case, the gap between the data-dependent DP and the DP guarantee of the Gaussian mechanism unsurprisingly decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Meanwhile, PATE-PTR (the purple region) performs well when the noise scale is small but deteriorates when the data-independent approach proves more advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This example demonstrates that using PTR as a post-processing step to convert the data-dependent DP to standard DP is effective when the data-adaptive approach dominates others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 6 Limitations and Future Work One weakness of generalized PTR is that it requires a case-specific privacy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Have we simply exchanged the problem of designing a data-adaptive DP algorithm with the problem of analyzing the data-dependent privacy loss?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We argue that this limitation is inherited from classic PTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In situations where classic PTR is not applicable, we’ve outlined several approaches to constructing the 13 DP test for our framework (see Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Furthermore, the data-dependent privacy loss is often more straightforward to compute than local sensitivity, and often exists in intermediate steps of classic DP analysis already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Most DP analysis involves providing a high-probability tail bound of the privacy loss random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If we stop before taking the max over the input dataset, then we get a data-dependent DP loss right away (as in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' There are several exciting directions for applying generalized PTR to more problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Sufficient statistics release and our private hyperparameter tuning (Algorithm 3) can be used to construct data-adaptive extensions of DP-PCA [Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2014] and Sparse-DP-ERM [Kifer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For DP-PCA we could use our Algorithm 3 to tune the variance of the noise added to the spectral gap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' for Sparse-DP-ERM we would test the restricted strong convexity parameter (RSC), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' not adding additional regularization if the RSC is already large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 7 Conclusion Generalized PTR extends the classic “Propose-Test-Release” framework to a more general setting by testing the data-dependent privacy loss of an input dataset, rather than its local sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In this paper we’ve provided several examples – private linear regression with hyperparameter selection and PATE – to illustrate how generalized PTR can enhance DP algorithm design via a data-adaptive approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Acknowledgments The work was partially supported by NSF Award # 2048091 and the Google Research Scholar Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Yuqing was supported by the Google PhD Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 14 Contents 1 Introduction 1 2 Related Work 2 3 Preliminaries 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 Propose-Test-Release .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': 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in the main body 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 Limits of the classic PTR in private binary voting .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='4 Other applications of generalized PTR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 22 B Omitted proofs in Section 4 23 C Experimental details 23 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 Experimental details in private linear regression .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 23 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 Details of PATE case study .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 24 D Omitted proofs in private GLM 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 Per-instance DP of GLM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 26 A Omitted examples in the main body In this appendix, we provide more examples to demonstrate the merits of generalized PTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We focus on a simple example of post-processed Laplace mechanism in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 and then an example on differentially private learning of generalized linear models in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In both cases, we observe that generalized PTR provides data-adaptive algorithms with formal DP guarantees, that are simple, effective and not previously proposed in the literature (to the best of our knowledge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 Limits of the classic PTR in private binary voting The following example demonstrates that classic PTR does not capture sufficient data-dependent quantities even when the local sensitivity exists and can be efficiently tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 15 Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Consider a binary class voting problem: n users vote for a binary class {0, 1} and the goal is to output the class that is supported by the majority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let ni denote the number of people who vote for the class i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We consider the report-noisy-max mechanism: M(X) : argmaxi∈[0,1]ni(X) + Lap(b), where b = 1/ϵ denotes the scale of Laplace noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In the example, we will (1) demonstrate the merit of data-dependent DP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' and (2) empirically compare classic PTR with generalized PTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We first explicitly state the data-dependent DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The data-dependent DP of the above example is ϵ(X) := max X′ {| log p p′ |, | log 1 − p 1 − p′ |}, where p := Pr[n0(X) + Lap(1/ϵ) > n1(X) + Lap(1/ϵ)] and p′ := Pr[n0(X′) + Lap(1/ϵ) > n1(X′) + Lap(1/ϵ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' There are four possible neighboring datasets X′ : n0(X′) = max(n0(X) ± 1, 0), n1(X′) = n1(X) or n0(X′) = n0(X), n1(X′) = max(n1(X) ± 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In Figure 3(a), we empirically compare the above data-dependent DP with the Laplace mechanism by varying the gap between the two vote counts |n0(X) − n1(X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The noise scale is fixed to ϵ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The data-dependent DP substantially improves over the standard DP if the gap is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' However, the data-dependent DP is a function of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We next demonstrate how to apply generalized PTR to exploit the data-dependent DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Notice that the probability n0(X) + Lap(1/ϵ) > n1(X) + Lap(1/ϵ) is equal to the probability that a random variable Z := X − Y exceeds ϵ(n1(X) − n0(X)), where X, Y are two independent Lap(1) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We can compute the pdf of Z through the convolution of two Laplace distributions, which implies fX−Y (z) = 1 + |z| 4e|z| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let t denote the difference between n1(X) and n0(X), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', t = n1(X) − n0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then we have p = Pr[Z > ϵ · t] = 2 + ϵ · t 4 exp(ϵ · t) Similarly, p′ = 2 + ϵ · (t + ℓ) 4 exp(ϵ · (t + ℓ)), where ℓ ∈ [−1, 1] denotes adding or removing one data point to construct the neighboring dataset X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Therefore, we can upper bound log(p/p′) by log p p′ = 2 + ϵ · t 4 exp(ϵ · t) · 4 exp(ϵ(t + ℓ)) 2 + ϵ · (t + ℓ) ≤ ϵ · log � 2 + ϵt 2 + ϵ(t + 1) � = ϵ log � 1 − ϵ 2 + ϵ(t + 1) � Then we can apply generalized PTR by privately lower-bounding t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' On the other hand, the local sensitivity ∆LS(X) of this noise-adding mechanism is 0 if t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Specifically, if the gap is larger than one, adding or removing one user will not change the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' To 16 0 5 10 15 20 25 30 35 40 The gap t=|n0(X) n1(X)| 0 2 4 6 8 10 data-dependent DP Laplace mechanism (a) data-dependent DP vs Laplace mechanism 10 28 10 23 10 18 10 13 10 8 10 3 102 Error 10 2 10 1 Gen-PTR( p + ) classic PTR Laplace mechanism (b) Privacy-utility tradeoff between three approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Figure 3: In Figure 3(a), we compare the privacy guarantee by varying the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In Figure 3(b) We fix t = n0(X) − n1(X) = 100 and compare privacy cost when the accuracy is aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Gen-PTR with any choice of privacy budget (˜ϵ + ϵ′) chosen from the purple region would achieve the same utility as Laplace mechanism but with a smaller privacy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The curve of Gen-PTR is always below than that of the classic PTR, which implies that Gen-PTR can result a tighter privacy analysis when the utility is aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' apply classic PTR, we let γ(X) denote the distance to the nearest dataset X ′′ such that ∆LS > 0 and test if γ(X) + Lap(1/ϵ) > log(1/δ) ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Notice in this example that γ(X) = max(t − 1, 0) can be computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We provide the detailed implementation of these approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Gen PTR: lower bound t with tp = t − log(1/δ) ˜ϵ + Lap(1/˜ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Calculate an upper bound of data-dependent DP ϵp using Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 with tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The algorithm then tests if ϵp is within an predefined privacy budget ϵ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If the test passes, the algorithm returns argmaxi∈[0,1]ni(X) + Lap(1/ϵ) satisfies (˜ϵ + ϵ′, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' classic PTR: lower bound t with tp = t − log(1/δ) ˜ϵ + Lap(1/˜ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If tp > 1, classic PTR outputs the ground-truth result else returns a random class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This algorithm satisfies (˜ϵ, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Laplace mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' M(X) : argmaxi∈[0,1]ni(X) + Lap(1/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' M is (ϵ, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We argue that though the Gen-PTR and the classic PTR are similar in privately lower-bounding the data-dependent quantity t, the latter does not capture sufficient information for data-adaptive analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' That is to say, only testing the local sensitivity restricts us from learning helpful information to amplify the privacy guarantee if the test fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In contrast, our generalized PTR, where privacy parameters and the local sensitivity parameterize the data-dependent DP, can handle those failure cases nicely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' To confirm this conjecture, Figure 3(b) plots a privacy-utility trade-off curve between these three approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We consider a voting example with n0(X) = n1(X) + 100 and t = 100, chosen such that the data-adaptive analysis is favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In Figure 3(b), we vary the noise scale b = 1/ϵ between [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For each choice of b, we plot the privacy guarantee of three algorithms when the error rate is aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For Gen-PTR, we set ˜ϵ = 1 2b and empirically calculate ϵp over 100000 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 17 In the plot, when ϵ ≪ log(1/δ) t , the classic PTR is even worse than the Laplace mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This is because the classic PTR is likely to return ⊥ while the Laplace mechanism returns argmaxi∈[0,1]ni(X)+ Lap(1/ϵ), which contains more useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Compared to the Laplace mechanism, Gen-PTR requires an extra privacy allocation ˜ϵ to release the gap t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' However, it still achieves an overall smaller privacy cost when the error rate ≤ 10−5 (the purple region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Meanwhile, Gen-PTR dominates the classic PTR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', the dashed black curve is always below the blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Note that the classic PTR and the Gen-PTR utilize the gap information differently: the classic PTR outputs ⊥ if the gap is not sufficiently large, while the Gen-PTR encodes the gap into the data-dependent DP function and tests the data-dependent DP in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This empirical result suggests that testing the local sensitivity can be loosely compared to testing the data-dependent DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Thus, Gen-PTR could provide a better privacy-utility trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 Self-concordant generalized linear model (GLM) In this section, we demonstrate the effectiveness and flexibility of generalized PTR in handling a family of GLMs where the link function satisfies a self-concordance assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This section is organized as follows: Introduce a family of GLMs with the self-concordance property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Introduce a general output perturbation algorithm for private GLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Analyze the data-dependent DP of GLMs with the self-concordance property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Provide an example of applying our generalized PTR framework to logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Consider the empirical risk minimization problem of the generalized linear model θ∗ = argminθ � i=1n li(θ) + r(θ), where l : R × R → R belongs to a family of convex GLMs: li(θ) = l(y, xT i θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let r : Rd → R be a regularization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We now define the self-concordance property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='3 (Generalized self-concordance [Bach, 2010]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' A convex and three-times differentiable function f : Θ → R is R-generalized-self-concordant on an open nonempty convex set Θ∗ ⊂ Θ with respect to norm ∥ · ∥ if for all u ∈ Θ∗ and all v ∈ Rd, ∇3f(u)[v, v, v] ≤ 2R∥v∥(∇2f(u)[v, v]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The closer R is to 0, the “nicer” — more self-concordant — the function is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' A consequence of (gener- alized) self-concordance is the spectral (multiplicative) stability of Hessian to small perturbations of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='4 (Stability of Hessian[Nesterov and Nemirovskii, 1994, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1], [Bach, 2010, Proposition 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let Hθ := ∇2Fs(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If Fs is R-self-concordant at θ, then for any v such that R∥v∥Hθ < 1, we have that (1 − R∥v∥Hθ)2∇2Fs(θ) ≺ ∇2Fs(θ + v) ≺ 1 (1 − R∥v∥Hθ)2 ∇2Fs(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 18 If instead we assume Fs is R-generalized-self-concordant at θ with respect to norm ∥ · ∥, then e−R∥v∥∇2Fs(θ) ≺ ∇2Fs(θ + v) ≺ eR∥v∥∇2Fs(θ) The two bounds are almost identical when R∥v∥ and R∥v∥θ are close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In particular, for x ≤ 1/2, we have that e−2x ≤ 1 − x ≤ e−x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In particular, the loss function of binary logistic regression is 1-generalized self-concordant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='5 (Binary logistic regression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Assume ∥x∥2 ≤ 1 for all x ∈ X and y ∈ {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then binary logistic regression with datasets in X × Y has a log-likelihood of F(θ) = �n i=1 log(1 + e−yixT i θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The univariate function l := log(1 + exp(·)) satisfies |l′′′| = ���� exp (·)(1 − exp (·)) (1 + exp (·))3 ���� ≤ exp (·) (1 + exp (·))2 := l′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We next apply the modified output perturbation algorithm to privately release θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The algorithm is simply: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Solve θ∗ = argminθ n � i=1 li(θ) + r(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Release ˆθ = θ∗ + Z, where γ > 0 is a tuning parameter and Z ∼ N(0, γ−1(�n i=1 ∇2li(θ) + ∇2r(θ))−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The data-dependent DP of the above procedure is stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='6 (Data-dependent DP of GLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Denote the smooth part of the loss function Fs = �n i=1 l(yi, < xi, · >) + rs(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Assume the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The GLM loss function l is convex, three-times continuously differentiable and R-generalized- self-concordant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ∥ · ∥2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Fs is locally α-strongly convex w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ∥ · ∥2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' and in addition, denote L := supθ∈[θ∗,˜θ∗] |l′(y, xT θ)|, β := supθ∈[θ∗,˜θ∗] |l′′(y, xT θ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' That is, ℓ(·) is L-Lipschitz and β-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We then have the data-dependent DP ϵ(Z) ≤ R(L + β) α (1 + log(2/δ)) + γL2 α + � γL2 α log(2/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The proof follows by taking an upper bound of the per-instance DP loss (Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1) ϵ(Z, z) over z = (x, y) ∈ (X, Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Notice that the Hessians can be arbitrarily singular and α could be 0, which leads to an infinite privacy loss without additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Thus, we will impose an additional regularization of form λ 2||θ||2, which ensures that for any dataset FS is λ-strongly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This is not yet DP because it is still about a fixed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We also need a pre-specified privacy budget (ϵ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We next demonstrate how to apply the generalized PTR to provide a general solution to the above GLM, using logistic regression as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 19 Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='7 (Logistic regression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For logistic regression, we know L ≤ 1, β ≤ 1/4 and if ∥x∥2 ≤ 1, it is 1-generalized self-concordant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For any dataset Z = (X, y), the data-dependent DP ϵ(X) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' δ can be simplified to: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='25 α (1 + log(2/δ)) + γ α + �γ α log(2/δ) Now, the data-dependent DP is a function of α and γ, where α denotes the local strong convexity at θ∗ λ and γ controls the noise scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We next show how to select these two parameters adapted to the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We demonstrate here how we apply generalized PTR to output perturbation of the logistic regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Take an exponential grid of parameters {λ} and propose each λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Solve for θ∗ λ = argminθF(θ) + λ∥θ∥2/2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Calculate the smallest eigenvalue λmin(∇2F(θ∗ λ)) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', using power method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Differentially privately release λmin with λp min := max{λmin+ √ log(4/δ) ϵ/2 ∆GS·Z− √ 2 log(4/δ)·log(1/δ)∆GS ϵ/2 , 0}, where ∆GS denote the global sensitivity of λmin using Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let ϵp(·) be instantiated with ϵ(X) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' δ from Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='7, where α = λp min + λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then, conditioned on a high probability event, ϵp(·) (a function of γ) is a valid DP bound that holds for all datasets and all parameters γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Calculate the maximum γ such that ϵp δ/2(γ) ≤ ϵ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Release ˆθ ∼ N(θ∗ λ, γ−1∇2Fs(θ∗ λ)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Evaluate the utility on the validation set and return the (λ, γ) pair that leads to the highest utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For each proposed λ, the algorithm that releases ˆθ ∼ N(θ∗ λ, γ−1∇2Fs(θ∗ λ)−1) is (ϵ, 2δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The proof follows the recipe of generalized PTR with private upper bound (Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' First, the release of λmin(∇2F(θ∗ λ)) is (ϵ/2, δ/2)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then, with probability at least 1 − δ, ϵp δ(·) > ϵδ(X) holds for all X and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Finally, γ is chosen such that the valid upper bound is (ϵ/2, δ/2)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For the hyper-parameter tuning on λ (Steps 1 and 8), we can use Algorithm 3 to evaluate each λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Unlike Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2, the λmin(∇2F(θ∗ λ)) is a complicated data-dependent function of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Thus, we cannot privately release the data-dependent quantity λmin(∇2F(θ∗ λ)) without an input λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The PTR approach allows us to test a number of different λ and hence get a more favorable privacy-utility trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' An interesting perspective of this algorithm for logistic regression is that increasing the regularization α is effectively increasing the number of data points within the soft “margin”3 of separation, hence a larger contribution to the Hessian from the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 3If we think of logistic regression as a smoothed version of SVM, then increasing α leads to more support vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The “margin” is “softer” in logistic regression, but qualitatively the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 20 Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The PTR solution for GLMs follows a similar recipe: propose a regularization strength λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' construct a lower bound of the strong convexity α at the optimal solution θ∗ λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' and test the validity of data-dependent DP using Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Before moving on to other applications of generalized PTR, we will show how to differentially privately release λmin according to the requirements of the logistic regression example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='3 Differentially privately release λmin (∇2F(θ)) To privately release λmin∇2F(θ), we first need to compute its global sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Once we have that then we can release it differentially privately using either the Laplace mechanism or the Gaussian mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='11 (Global sensitivity of the minimum eigenvalue at the optimal solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let F(θ) = �n i=1 fi(θ) + r(θ) and ˜F(θ) = F(θ) + f(θ) where f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', fn are loss functions corresponding to a particular datapoint x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let θ∗ = argminθF(θ) and ˜θ∗ = argminθ ˜F(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Assume f is L-Lipschitz and β-smooth, r(θ) is λ-strongly convex, and F and ˜F are R-self-concordant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If in addition, λ ≥ RL, then we have sup X,x (λmin(∇2F(θ∗ λ)) − λmin(∇2 ˜F( ˜θ∗ λ))) ≤ 2RL + β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' λmin(∇2F(θ∗ λ)) − λmin(∇2 ˜F( ˜θ∗ λ)) = (λmin(∇2F(θ∗ λ)) − λmin(∇2 ˜F(θ∗ λ))) + (λmin(∇2 ˜F(θ∗ λ)) − λmin(∇2 ˜F( ˜θ∗ λ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' (1) We first bound the part on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' By applying Weyl’s lemma λ(X + E) − λ(X) ≤ ||E||2, we have sup x ||∇2F(θ∗ λ) − ∇2 ˜ F(θ∗ λ)||2 = ||∇2f(θ∗ λ)||2 ≤ β (2) In order to bound the part on the right, we apply the semidefinite ordering using self-concordance, which gives e−R∥ ˜ θ∗ λ−θ∗ λ∥∇2 ˜F( ˜θ∗ λ) ≺ ∇2 ˜F(θ∗ λ) ≺ eR∥ ˜ θ∗ λ−θ∗ λ∥∇2 ˜F( ˜θ∗ λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' By the Courant-Fischer Theorem and the monotonicity theorem, we also have that for the smallest eigenvalue e−R∥ ˜ θ∗ λ−θ∗ λ∥λmin � ∇2 ˜F( ˜θ∗ λ) � ≤ λmin � ∇2 ˜F(θ∗ λ) � ≤ eR∥ ˜ θ∗ λ−θ∗ λ∥λmin � ∇2 ˜F( ˜θ∗ λ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' (3) Moreover by Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2, we have that ∥ ˜θ∗ λ − θ∗ λ∥2 ≤ ∥∇f( ˜θ∗λ)∥ λmin � ∇2 ˜F( ˜θ∗ λ) � ≤ L λmin � ∇2 ˜F( ˜θ∗ λ) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If λmin � ∇2 ˜F( ˜θ∗ λ) � ≥ RL, then use that ex − 1 ≤ 2x for x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Substituting the above bound to (3) then to (1) together with (2), we get a data-independent global sensitivity bound of λmin(∇2F(θ∗ λ)) − λmin(∇2 ˜F( ˜θ∗ λ)) ≤ 2RL + β 21 as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let ∥ · ∥ be a norm and ∥ · ∥∗ be its dual norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let F(θ), f(θ) and ˜F(θ) = F(θ) + f(θ) be proper convex functions and θ∗ and ˜ theta ∗ be their minimizers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 0 ∈ ∂F(θ∗) and 0 ∈ ∂ ˜F( ˜ theta ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If in addition, F, ˜F is α, ˜α-strongly convex with respect to ∥ · ∥ within the restricted domain θ ∈ {tθ∗ + (1 − t)˜θ∗ | t ∈ [0, 1]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then there exists g ∈ ∂f(θ∗) and ˜g ∈ ∂f(˜θ∗) such that ∥θ∗ − ˜θ∗∥ ≤ min � 1 α∥˜g∥∗, 1 ˜α∥g∥∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Apply the first order condition to F restricted to the line segment between ˜θ∗ and θ∗, we get F(˜θ∗) ≥ F(θ∗) + ⟨∂F(θ∗), ˜θ∗ − θ∗⟩ + α 2 ∥˜θ∗ − θ∗∥2 (4) F(θ∗) ≥ F(˜θ∗) + ⟨∂F(˜θ∗), θ∗ − ˜θ∗⟩ + α 2 ∥˜θ∗ − θ∗∥2 (5) Note by the convexity of F and f, ∂ ˜F = ∂F + ∂f, where + is the Minkowski Sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Therefore, 0 ∈ ∂ ˜F(˜θ∗) implies that there exists ˜g such that ˜g ∈ ∂f(˜θ∗) and −˜g ∈ ∂F(˜θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Take −˜g ∈ ∂F(˜θ∗) in Equation 10 and 0 ∈ ∂F(θ∗) in Equation 9 and add the two inequalities, we obtain 0 ≥ ⟨−˜g, θ∗ − ˜θ∗⟩ + α∥˜θ∗ − θ∗∥2 ≥ −∥˜g∥∗∥θ∗ − ˜θ∗∥ + α∥˜θ∗ − θ∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For ∥˜θ∗ − θ∗∥ = 0 the claim is trivially true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' otherwise, we can divide both sides of the above inequality by ∥˜θ∗ − θ∗∥ and get ∥θ∗ − ˜θ∗∥ ≤ 1 α∥˜g∥∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' It remains to show that ∥θ∗ − ˜θ∗∥ ≤ 1 ˜α∥g∥∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This can be obtained by exactly the same arguments above but applying strong convexity to ˜F instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Note that we can actually get something slightly stronger than the statement because the inequality holds for all g ∈ ∂f(θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='4 Other applications of generalized PTR Besides one-posterior sampling for GLMs, there are plenty of examples that our generalized-PTR could be applied, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', DP-PCA [Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2014] and Sparse-DP-ERM [Kifer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2012] (when the designed matrix is well-behaved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2014] provides a PTR style privacy-preserving principle component analysis (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The key observation of [Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2014] is that the local sensitivity is quite “small” if there is a large eigengap between the k-th and the k + 1-th eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Therefore, their approach (Algorithm 2) chooses to privately release a lower bound of the k-th eigengap (k is fixed as an input) and use that to construct a high-confidence upper bound of the local sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For noise-adding mechanisms, the local sensitivity is proportional to the data-dependent loss and generalized PTR is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We can formulate the data-dependent DP of DP-PCA as follows: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For a given matrix A ∈ Rm×n, assume each row of A has a bounded ℓ2 norm being 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let Vk denotes the top k eigenvectors of AT A and dk denotes the gap between the k-th and the k + 1-th eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then releasing VkV T k + E, where E ∈ Rn×n is a symmetric matrix with the upper triangle is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='d samples from N(0, σ2) satisfies (ϵ(A), δ) data-dependent DP and ϵ(A) = 2√ log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='25/δ) σ(dk−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 22 The proof is based on the local sensitivity result from [Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2014] and the noise calibration of Gaussian mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We can combine Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='13 with our Algorithm 3 to instantiate the generalized PTR framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The improvement over Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [2014] will be to allow joint tuning of the parameter k and the noise variance (added to the spectral gap dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' B Omitted proofs in Section 4 The utility of Algorithm 3 depends on how many rounds that Algorithm 2 is invoked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We next provide the utility guarantee of Algorithm 3, which follows a simplification of the result in the Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 of Papernot and Steinke [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Suppose applying Algorithm 2 with each φi has an equal probability to achieve the highest validation score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let ˆT denotes the number of invocation of Algorithm 2, where ˆT follows a truncated geometric distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then the expected quantile of the highest score candidate is given by E ˆT � 1 − 1 ˆT+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In practice, we can roughly set τ = 1 10k so that the algorithm is likely to test all k parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Suppose each oracle access to Q(X) has a probability 1/k of achiving the best validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let β denote the probability that A (shorthand for Algorithm 3) outputs the best choice of φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' β = 1 − Pr[A(X)is not best] = 1 − E ˆT � Pr[Q(X)is not best] ˆT � = 1 − E ˆT � (1 − 1 k) ˆT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let f(x) = E[x ˆT ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Applying a first-order approximation on f(1 − 1 k), we have f(1 − 1 k) ≈ f(1) − f′(1) · 1 k = 1 − E[ ˆT]/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then, if k is large and we choose τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1/k, A can roughly return the best φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' C Experimental details C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 Experimental details in private linear regression We start with the privacy calibration of the OPS-PTR algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Algorithm 5 provides the detailed privacy calibration of the private linear regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Algorithm 5 is (ϵ, 2δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' There are three data-dependent quantities in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1: λmin, ||θ∗ λ|| and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' First, notice that λmin has a global sensitivity of ||X||2 by Weyl’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Under the assumption ||X||2 ≤ 1, we privately release λmin using (ϵ/4, δ/3) in Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Notice that with probability at least 1 − δ/2, ˜λmin is a lower bound of λmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 23 Algorithm 5 OPS-PTR: One-Posterior Sample with propose-test-release (no-“perp” version) 1: Input: Data X, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Private budget : ϵ, δ, proposed regularizer λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 2: Calculate the minimum eigenvalue λmin(XT X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 3: Sample Z ∼ N(0, 1) and privately release ˜λmin = max � λmin + √ log(6/δ) ϵ/4 Z − √ 2 log(6/δ)·log(2/δ) ϵ/4 , 0 � 4: Calculate ˆθ = (XT X + λI)−1XT y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 5: Sample Z ∼ N(0, 1) and privately release ∆ = log(||Y|| + ||X||||ˆθ||) + log(1+||X||2/(λ+˜λmin)) ϵ/(4√ 6/δ) Z + log(1+||X||2/(λ+˜λmin)) ϵ/(4√ 2 log(6/δ) log(2/δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 6: Set the local Lipschitz ˜L := ||X||e∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 7: Calibrate γ with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1(δ/3, ϵ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=') 8: Output ˜θ ∼ p(θ|X, y) ∝ e− γ 2 ||y−Xθ||2+λ||θ||2 Then, we apply Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 from Wang [2018] to privately release log(||Y|| + ||X||||ˆθ||) using (ϵ/4, δ/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Note that both the local Lipschitz constant L and the norm ||θ∗ λ|| are functions of log(||Y|| + ||X||||ˆθ||).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Thus, we can construct a private upper bound of these by post-processing of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then, with probability at least 1 − δ (by a union bound over ˜λmin and ∆), instantiating Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 with ˜λmin and ˜L provides a valid upper bound of the data-dependent DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We then tune the parameter γ using the remaining privacy budget (ϵ/2, δ/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 (Lemma 12 [Wang, 2018]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let θ∗ λ be the ridge regression estimate with parameter λ and the smallest eigenvalue of XT X be λmin, then the function log(||Y + ||X||||θ∗ λ||) has a local sensitivity of log(1 + ||X||2 λmin+λ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2 Details of PATE case study Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='3 (Renyi DP [Mironov, 2017]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We say a randomized algorithm M is (α, ϵM(α))-RDP with order α ≥ 1 if for neighboring datasets X, X′ Dα(M(X)||M(X′)) := 1 α − 1 log Eo∼M(X′) �� Pr[M(X) = o] Pr[M(X′) = o] �α� ≤ ϵM(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' At the limit of α → ∞, RDP reduces to (ϵ, 0)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We now define the data-dependent Renyi DP that conditioned on an input dataset X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='4 (Data-dependent Renyi DP [Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 2018]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We say a randomized algorithm M is (α, ϵM(α, X))-RDP with order α ≥ 1 for dataset X if for neighboring datasets X′ Dα(M(X)||M(X′)) := 1 α − 1 log Eo∼M(X′) �� Pr[M(X) = o] Pr[M(X′) = o] �α� ≤ ϵM(α, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' RDP features two useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 24 Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='5 (Adaptive composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ϵ(M1,M2) = ϵM1(·) + ϵM2(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='6 (From RDP to DP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If a randomized algorithm M satisfies (α, ϵ(α))-RDP, then M also satisfies (ϵ(α) + log(1/δ) α−1 , δ)-DP for any δ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='7 (Smooth Sensitivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Given the smoothness parameter β, a β-smooth sensitivity of f(X) is defined as SSβ(X) := max d≥0 e−βd · max ˜ X′:dist(X, ˜ X′)≤d ∆LS( ˜X′) Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='8 (Private upper bound of data-dependent RDP, Restatement of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ] Given a RDP function RDP(α, X) and a β-smooth sensitivity bound SS(·) of RDP(α, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let µ (defined in Algorithm 4) denote the private release of log(SSβ(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let (β, σs, σ2)-GNSS mechanism be RDPupper(α):=RDP(α,X)+SSβ(X)·N(0,σ2 s)+σs � 2 log( 2 δ2 )eµ Then, the release of RDPupper(X) satisfies (α, 3α+2 2σ2s )-RDP for all 1 < α < 1 2β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' at least 1 − δ2, RDPupper(α) is an upper bound of RDP(α, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We first show that releasing the smooth sensitivity SSβ with eµ satisfies (α, α 2σ2 2 )-RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Notice that the log of SSβ(X) has a bounded global sensitivity β (Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='7 implies that | log SSβ(X) − log SSβ(X′)| ≤ β for any neighboring dataset X, X′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' By Gaussian mechanism, scaling noise with βσ2 to log SSβ(X) is (α, α 2σ2 2 )-RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Therefore, the release of RDP(α, X) is (α, ϵs(α) + α 2σ2 2 )-RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Since the release of f(X) + SSβ(X) · N(0, σ2 s) is (α, α+1 σ2s )-RDP (Theorem 23 from Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [2018]) for α < 1 2β, we have ϵs(α) + α 2σ2 2 = 3α+2 2σ2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We next prove the second statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' First, notice that with probability at least 1−δ2/2, eµ ≥ SSβ(X) using the standard Gaussian tail bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let E denote the event that eµ ≥ SSβ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Pr � RDPupper(α) ≤ RDP(α, X) � = Pr � RDPupper(α) ≤ RDP(α, X)|E � + Pr � RDPupper(α) ≤ RDP(α, X)|Ec � ≤ Pr � RDPupper(α) ≤ RDP(α, X)|E � + δ2/2 = Pr � N(0, σ2 s) · SSβ(X) ≥ σs · � 2 log(2/δ2)eµ|E � � �� � denoted by(∗) +δ2/2 Condition on the event E, eµ is a valid upper bound of SSβ(X), which implies (∗) ≤ Pr[N(0, σ2 s) · SSβ(X) ≥ σs · � 2 log(2/δ2)SSβ(X)|E] ≤ δ2/2 Therefore, with probability at least 1 − δ2, RDPupper(α) ≥ RDP(α, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='9 (Restatement of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Algorithm 4 satisfies (ϵ′ + ˆϵ, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The privacy analysis consists of two components — the privacy cost of releasing an upper bound of data-dependent RDP (ϵupper(α) := ϵs(α)+ α 2σ2 2 and the valid upper bound ϵp σ1(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' First, set α = 2 log(2/δ) ϵ + 1 and use RDP to DP conversion with δ/2 ensures that the cost of δ/2 contribution to be roughly ϵ/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', log(2/δ) α−1 = ϵ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Second, choosing σs = � 2+3α ϵ gives us another ϵ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Experimental details K = 400 teacher models are trained individually on the disjoint set using AlexNet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We set σ2 = σs = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Our data-dependent RDP calculation and the smooth- sensitivity calculation follow Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Specifically, we use the following theorem (Theorem 6 from Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [2018]) to compute the data-dependent RDP of each unlabeled data x from the public domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='10 (data-dependent RDP Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' [2018]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let ˜q ≥ Pr[M(X) ̸= Argmaxj∈[C]nj(x)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', an upper bound of the probability that the noisy label does not match the majority label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Assume α ≤ µ1 and ˜q ≤ e(µ2−1)ϵ2/ � µ1 µ1−1 · µ2 µ2−1 �µ2 , then we have: ϵM(α, X) ≤ 1 α − 1 log � (1 − ˜q) · A(˜q, µ2, ϵ2)α−1 + ˜q · B(˜q, µ1, ϵ1)α−1 � where A(˜q, µ2, ϵ2) := (1 − ˜q)/ � 1 − (˜qeϵ2) µ2−1 µ2 � , B(˜q, µ1, ϵ1) = eϵ1/˜q 1 µ1−1 , µ2 = σ1 · � log(1/˜q), µ1 = µ2 + 1, ϵ1 = µ1/σ2 1 and ϵ2 = µ2/σ2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In the experiments, the non-private data-dependent DP baseline is also based on the above theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Notice that the data-dependent RDP of each query is a function of ˜q, where ˜q denotes an upper bound of the probability where the plurality output does not match the noisy output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ˜q is a complex function of both the noisy scale and data and is not monotonically decreasing when σ1 is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Simulation of two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The motivation of the experimental design is to compare three approaches under different data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Notice that there are K = 400 teachers, which implies the number of the vote count for each class will be bounded by 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In the simulation of high-consensus distribution, we choose T = 200 unlabeled public data such that the majority vote count will be larger than 150 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', maxj∈[C] nj(x) > 150).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For the low-consensus distribution, we choose to select T unlabeled data such that the majority vote count will be smaller than 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' D Omitted proofs in private GLM D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 Per-instance DP of GLM Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1 (Per-instance differential privacy guarantee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Consider two adjacent data sets Z and Z′ = [Z, (x, y)], and denote the smooth part of the loss function Fs = �n i=1 l(yi, ⟨xi, ·⟩) + rs(·) (thus ˜Fs = Fs + l(y, ⟨x, ·⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let the local neighborhood be the line segment between θ∗ and ˜θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Assume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' the GLM loss function l be convex, three-time continuous differentiable and R-generalized-self- concordant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ∥ · ∥2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Fs is locally α-strongly convex w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ∥ · ∥2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' and in addition, denote L := supθ∈[θ∗,˜θ∗] |l′(y, xT θ)|, β := supθ∈[θ∗,˜θ∗] |l′′(y, xT θ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 26 Then the algorithm obeys (ϵ, δ)-pDP for Z and z = (x, y) with any 0 < δ < 2/e and ϵ ≤ ϵ0(1 + log(2/δ)) + e RL∥x∥2 α �γL2∥x∥2 H−1 2 + � γL2∥x∥2 H−1 log(2/δ) � where ϵ0 ≤ e RL∥x∥2 α − 1 + 2β∥x∥2 H−1 1 + 2β∥x∥2 ˜H−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If we instead assume that l is R-self concordant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then the same results hold, but with all e RL∥x∥2 α replaced with (1 − RL∥x∥H−1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Under the stronger three-times continuous differentiable assumption, by mean value theorem, there exists ξ on the line-segment between θ∗ and ˜θ∗ such that H = �� 1 t=0 ∇2Fs((1 − t)θ∗ + t˜θ∗)dt � = ∇2Fs(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The two distributions of interests are N(θ∗, [γ∇2Fs(θ∗)]−1) and N(˜θ∗, [γ∇2Fs(˜θ∗)+∇2l(y, xT ˜θ∗)]−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Denote [∇2Fs(θ∗)]−1 =: Σ and [∇2Fs(˜θ∗)+∇2l(y, xT ˜θ∗)]−1 =: ˜Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Both the means and the covariance matrices are different, so we cannot use multivariate Gaussian mechanism naively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Instead we will take the tail bound interpretation of (ϵ, δ)-DP and make use of the per-instance DP framework as internal steps of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' we can write down the privacy loss random variable in analytic form log |Σ|−1/2e− γ 2 ∥θ−θ∗∥2 Σ−1 |˜Σ|−1/2e− γ 2 ∥θ−˜θ∗∥2 ˜Σ−1 = 1 2 log �|Σ−1| |˜Σ−1| � � �� � (∗) + γ 2 � ∥θ − θ∗∥2 Σ−1 − ∥θ − ˜θ∗∥2 ˜Σ−1 � � �� � (∗∗) The general idea of the proof is to simplify the expression above and upper bounding the two terms separately using self-concordance and matrix inversion lemma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' and ultimately show that the privacy loss random variable is dominated by another random variable having an appropriately scaled shifted χ-distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' therefore admits a Gaussian-like tail bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' To ensure the presentation is readable, we define a few short hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We will use H and ˜H to denote the Hessian of Fs and Fs + f respectively and subscript 1 2 indicates whether the Hessian evaluated at at θ∗ or ˜θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' H without any subscript or superscript represents the Hessian of Fs evaluated at ξ as previously used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' (∗) = 1 2 log |H1| |H| |H| |H2| |H2| | ˜H2| ≤ 1 2 � log |H1| |H| + log |H| |H2| + log |H2| | ˜H2| � By the R-generalized self-concordance of Fs, we can apply Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='3, −∥θ∗ − ξ∥2R ≤ log |H1| |H| ≤ R∥θ∗ − ξ∥2, −R∥ξ − ˜θ∗∥2 ≤ log |H| |H2| ≤ R∥ξ − ˜θ∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The generalized linear model ensures that the Hessian of f is rank-1: ∇2f(˜θ∗) = l′′(y, xT ˜θ∗)xxT and we can apply Lemma ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' in both ways (taking A = H2 and A = ˜H2) and obtain |H2| | ˜H2| = 1 1 + l′′(y, xT ˜θ∗)xT H−1 2 x = 1 − l′′(y, xT ˜θ∗)xT ˜H2x 27 Note that l′′(y, xT ˜θ∗)xT ˜H−1 2 x is the in-sample leverage-score and l′′(y, xT ˜θ∗)xT H−1 2 x is the out- of-sample leverage-score of the locally linearized problem at ˜θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' We denote them by µ2 and µ′ 2 respectively (similarly, for the consistency of notations, we denote the in-sample and out of sample leverage score at θ∗ by µ1 and µ′ 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Combine the above arguments we get (∗) ≤R∥θ∗ − ξ∥2 + R∥ξ − ˜θ∗∥2 + log(1 − µ2) ≤ R∥θ∗ − ˜θ∗∥2 + log(1 − µ2) (6) (∗) ≥ − R∥θ∗ − ˜θ∗∥2 − log(1 − µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' (7) We now move on to deal with the second part, where we would like to express everything in terms of ∥θ − θ∗∥H1, which we know from the algorithm is χ-distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' (∗∗) = γ 2 � ∥θ − θ∗∥2 H1 − ∥θ − θ∗∥2 H2 + ∥θ − θ∗∥2 H2 − ∥θ − ˜θ∗∥2 H2 + ∥θ − ˜θ∗∥2 H2 − ∥θ − ˜θ∗∥2 ˜H2 � By the generalized self-concordance at θ∗ e−R∥θ∗−˜θ∗∥2∥ · ∥2 H1 ≤ ∥ · ∥2 H2 ≤ eR∥θ∗−˜θ∗∥2∥ · ∥2 H1 This allows us to convert from ∥ · ∥H2 to ∥ · ∥H1, and as a consequence: ��∥θ − θ∗∥2 H1 − ∥θ − θ∗∥2 H2 �� ≤ [eR∥θ∗−˜θ∗∥2 − 1]∥θ − θ∗∥2 H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Also, ∥θ − θ∗∥2 H2 − ∥θ − ˜θ∗∥2 H2 = � ˜θ∗ − θ∗, 2θ − 2θ∗ + θ∗ − ˜θ∗� H2 = 2⟨θ − θ∗, ˜θ∗ − θ∗⟩H2 − ∥θ∗ − ˜θ∗∥2 H2 Therefore ���∥θ − θ∗∥2 H2 − ∥θ − ˜θ∗∥2 H2 ��� ≤ 2∥θ − θ∗∥H2∥θ∗ − ˜θ∗∥H2 + ∥θ∗ − ˜θ∗∥2 H2 ≤ 2eR∥˜θ∗−θ∗∥2∥θ − θ∗∥H1∥θ∗ − ˜θ∗∥H + eR∥˜θ∗−θ∗∥2∥θ∗ − ˜θ∗∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then lastly we have 0 ≥ ∥θ − ˜θ∗∥2 H2 − ∥θ − ˜θ∗∥2 ˜H2 = −l′′(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' xT ˜θ∗) � ⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' θ − θ∗⟩ + ⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' θ∗ − ˜θ∗⟩ �2 ≥ −2β∥x∥2 H−1 1 ∥θ − θ∗∥2 H1 − 2β∥x∥2 H−1∥θ∗ − ˜θ∗∥2 H ���∥θ − ˜θ∗∥2 H2 − ∥θ − ˜θ∗∥2 ˜H2 ��� ≤ 2β∥x∥2 H−1 1 ∥θ − θ∗∥2 H1 + 2β∥x∥2 H−1∥θ∗ − ˜θ∗∥2 H Combine the above derivations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' we get |(∗∗)| ≤ γ 2 � a∥θ − θ∗∥2 H1 + b∥θ − θ∗∥H1 + c � (8) where a := � eR∥θ∗−˜θ∗∥2 − 1 + 2β∥x∥2 H−1 1 � b :=2eR∥θ∗−˜θ∗∥2∥θ∗ − ˜θ∗∥H c :=(eR∥θ∗−˜θ∗∥2 + 2β∥x∥2 H−1)∥θ∗ − ˜θ∗∥2 H 28 Lastly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' by (6) and (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ����log p(θ|Z) p(θ|Z′) ���� ≤ R∥θ∗ − ˜θ∗∥2 + log(1 − µ2) + γ 2[aW 2 + bW + c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' where according to the algorithm W := ∥θ − θ∗∥H1 follows a half-normal distribution with σ = γ−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' By standard Gaussian tail bound, we have for all δ < 2/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' P(|W| ≤ γ−1/2� log(2/δ)) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This implies that a high probability upper bound of the absolute value of the privacy loss random variable log p(θ|Z) p(θ|Z′) under p(θ|Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' By the tail bound to privacy conversion lemma (Lemma ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ), we get that for any set S ⊂ Θ P(θ ∈ S|Z) ≤ eϵP(θ ∈ S|Z′) + δ for any 0 < δ < 2/e and ϵ = R∥θ∗ − ˜θ∗∥2 + log(1 − µ2) + γc 2 + a 2 log(2/δ) + γ1/2b 2 � log(2/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Denote v := θ∗ − ˜θ∗, by strong convexity ∥v∥2 ≤ ∥∇l(y, xT θ)[˜θ∗]∥2/α = |l′|∥x∥2/α ≤ L∥x∥2/α and ∥v∥H ≤ ∥∇l(y, xT θ)[˜θ∗]∥H−1 = |l′|∥x∥H−1 ≤ L∥x∥H−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Also use the fact that | log(1 − µ2)| ≤ 2µ2 for µ2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='5 and µ2 ≤ β∥x∥2 ˜H−1 2 , we can then combine similar terms and have a more compact representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ϵ ≤ ϵ0(1 + log(2/δ)) + e RL∥x∥2 α �γL2∥x∥2 H−1 2 + � γL2∥x∥2 H−1 log(2/δ) � where ϵ0 ≤ e RL∥x∥2 α − 1 + 2β∥x∥2 H−1 1 + 2β∥x∥2 ˜H−1 2 is the part of the privacy loss that does not get smaller as γ decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let ∥ · ∥ be a norm and ∥ · ∥∗ be its dual norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let F(θ), f(θ) and ˜F(θ) = F(θ) + f(θ) be proper convex functions and θ∗ and ˜ theta ∗ be their minimizers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=', 0 ∈ ∂F(θ∗) and 0 ∈ ∂ ˜F( ˜ theta ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If in addition, F, ˜F is α, ˜α-strongly convex with respect to ∥ · ∥ within the restricted domain θ ∈ {tθ∗ + (1 − t)˜θ∗ | t ∈ [0, 1]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then there exists g ∈ ∂f(θ∗) and ˜g ∈ ∂f(˜θ∗) such that ∥θ∗ − ˜θ∗∥ ≤ min � 1 α∥˜g∥∗, 1 ˜α∥g∥∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Apply the first order condition to F restricted to the line segment between ˜θ∗ and θ∗, there are we get F(˜θ∗) ≥ F(θ∗) + ⟨∂F(θ∗), ˜θ∗ − θ∗⟩ + α 2 ∥˜θ∗ − θ∗∥2 (9) F(θ∗) ≥ F(˜θ∗) + ⟨∂F(˜θ∗), θ∗ − ˜θ∗⟩ + α 2 ∥˜θ∗ − θ∗∥2 (10) 29 Note by the convexity of F and f, ∂ ˜F = ∂F + ∂f, where + is the Minkowski Sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Therefore, 0 ∈ ∂ ˜F(˜θ∗) implies that there exists ˜g such that ˜g ∈ ∂f(˜θ∗) and −˜g ∈ ∂F(˜θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Take −˜g ∈ ∂F(˜θ∗) in Equation 10 and 0 ∈ ∂F(θ∗) in Equation 9 and add the two inequalities, we obtain 0 ≥ ⟨−˜g, θ∗ − ˜θ∗⟩ + α∥˜θ∗ − θ∗∥2 ≥ −∥˜g∥∗∥θ∗ − ˜θ∗∥ + α∥˜θ∗ − θ∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' For ∥˜θ∗ − θ∗∥ = 0 the claim is trivially true, otherwise, we can divide the both sides of the above inequality by ∥˜θ∗ − θ∗∥ and get ∥θ∗ − ˜θ∗∥ ≤ 1 α∥˜g∥∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' It remains to show that ∥θ∗ − ˜θ∗∥ ≤ 1 ˜α∥g∥∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' This can be obtained by exactly the same arguments above but applying strong convexity to ˜F instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Note that we can actually get something slightly stronger than the statement because the inequality holds for all g ∈ ∂f(θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' A consequence of (generalized) self-concordance is the spectral (multiplicative) stability of Hessian to small perturbations of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='3 (Stability of Hessian[Nesterov and Nemirovskii, 1994, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='1], [Bach, 2010, Proposition 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Let Hθ := ∇2Fs(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If Fs is R-self-concordant at θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Then for any v such that R∥v∥Hθ < 1, we have that (1 − R∥v∥Hθ)2∇2Fs(θ) ≺ ∇2Fs(θ + v) ≺ 1 (1 − R∥v∥Hθ)2 ∇2Fs(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' If instead we assume Fs is R-generalized-self-concordant at θ with respect to norm ∥ · ∥, then e−R∥v∥∇2Fs(θ) ≺ ∇2Fs(θ + v) ≺ eR∥v∥∇2Fs(θ) The two bounds are almost identical when R∥v∥ and R∥v∥θ are close to 0, in particular, for x ≤ 1/2, e−2x ≤ 1 − x ≤ e−x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' References Francis Bach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Self-concordant analysis for logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Electronic Journal of Statistics, 4: 384–414, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Kamalika Chaudhuri, Claire Monteleoni, and Anand D Sarwate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Differentially private empirical risk minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Journal of Machine Learning Research, 12(3), 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Chris Decarolis, Mukul Ram, Seyed Esmaeili, Yu-Xiang Wang, and Furong Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' An end-to- end differentially private latent dirichlet allocation using a spectral algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 2421–2431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Cynthia Dwork and Jing Lei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Differential privacy and robust statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In ACM symposium on Theory of computing, pages 371–380, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Calibrating noise to sensitivity in private data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In Theory of cryptography conference, pages 265–284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Springer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Cynthia Dwork, Kunal Talwar, Abhradeep Thakurta, and Li Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Analyze gauss: optimal bounds for privacy-preserving principal component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 11–20, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 30 Sivakanth Gopi, Yin Tat Lee, and Daogao Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Private convex optimization via exponential mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='00263, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Shiva Prasad Kasiviswanathan, Kobbi Nissim, Sofya Raskhodnikova, and Adam Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Analyzing graphs with node differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In Theory of Cryptography Conference, pages 457–476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Springer, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Daniel Kifer, Adam Smith, and Abhradeep Thakurta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Private convex empirical risk minimization and high-dimensional regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In Conference on Learning Theory, pages 25–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' JMLR Workshop and Conference Proceedings, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Jingcheng Liu and Kunal Talwar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Private selection from private candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 298–309, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Xiyang Liu, Weihao Kong, and Sewoong Oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Differential privacy and robust statistics in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='06578, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Kentaro Minami, HItomi Arai, Issei Sato, and Hiroshi Nakagawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Differential privacy without sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Ilya Mironov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Rényi differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In 2017 IEEE 30th computer security foundations symposium (CSF), pages 263–275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' IEEE, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Yurii Nesterov and Arkadii Nemirovskii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Interior-point polynomial algorithms in convex programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' SIAM, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Kobbi Nissim, Sofya Raskhodnikova, and Adam Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Smooth sensitivity and sampling in private data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In ACM symposium on Theory of computing (STOC-07), pages 75–84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' ACM, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Nicolas Papernot and Thomas Steinke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Hyperparameter tuning with renyi differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='03620, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Nicolas Papernot, Martín Abadi, Úlfar Erlingsson, Ian Goodfellow, and Kunal Talwar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Semi- supervised knowledge transfer for deep learning from private training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In International Conference on Learning Representations (ICLR-17), 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Nicolas Papernot, Shuang Song, Ilya Mironov, Ananth Raghunathan, Kunal Talwar, and Úlfar Erlingsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Scalable private learning with pate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' arXiv preprint arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='08908, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Rachel Redberg and Yu-Xiang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Privately publishable per-instance privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Jordi Soria-Comas, Josep Domingo-Ferrer, David Sánchez, and David Megías.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Individual differential privacy: A utility-preserving formulation of differential privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' IEEE Transactions on Information Forensics and Security, 12(6):1418–1429, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Abhradeep Guha Thakurta and Adam Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Differentially private feature selection via stability arguments, and the robustness of the lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In Conference on Learning Theory, pages 819–850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' PMLR, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Salil Vadhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' The complexity of differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In Tutorials on the Foundations of Cryptography, pages 347–450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Jiachen T Wang, Saeed Mahloujifar, Shouda Wang, Ruoxi Jia, and Prateek Mittal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Renyi differential privacy of propose-test-release and applications to private and robust machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='07716, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 31 Yu-Xiang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Per-instance differential privacy and the adaptivity of posterior sampling in linear and ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' arXiv preprint arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='07708, pages 48–71, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Yu-Xiang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Revisiting differentially private linear regression: optimal and adaptive prediction & estimation in unbounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' arXiv preprint arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content='02596, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Yu-Xiang Wang, Stephen Fienberg, and Alex Smola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' Privacy for free: Posterior sampling and stochastic gradient monte carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 2493– 2502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' PMLR, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} +page_content=' 32' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfdfcH/content/2301.00301v1.pdf'} diff --git a/99FPT4oBgHgl3EQfZDSN/content/tmp_files/2301.13076v1.pdf.txt b/99FPT4oBgHgl3EQfZDSN/content/tmp_files/2301.13076v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0486650d565a3d675bce9001b83f8667413a2849 --- /dev/null +++ b/99FPT4oBgHgl3EQfZDSN/content/tmp_files/2301.13076v1.pdf.txt @@ -0,0 +1,1286 @@ +EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE +MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG +Abstract. We generate anti-self-polar polytopes via a numerical implementation of the +gradient flow induced by the diameter functional on the space of all finite subsets of the +sphere, and prove related results on the critical points of the diameter functional as well as +results about the combinatorics of such polytopes. We also discuss potential connections to +Borsuk’s conjecture. +Contents +1. +Introduction +1 +2. +Pointwise extremal sets +3 +2.1. +The pyramid construction +4 +2.2. +Construction of k-stacks +5 +3. +Minimal sets on S2 with diameter below the first accumulation critical value +7 +3.1. +Configuration space +8 +3.2. +Finiteness results +8 +3.3. +A labeling strategy for the points in Bk +9 +4. +Anti-self-polar polytopes +10 +4.1. +ASP polytopes +11 +4.2. +Borsuk’s conjecture +12 +4.3. +Proof of Lovasz’s theorem +13 +4.4. +4-dimensional polytopes +15 +5. +Implementation of the diameter gradient flow +16 +6. +Computational results +17 +6.1. +Pointwise extremal configurations on S2 +18 +6.2. +Pointwise extremal configurations on S3 +20 +Appendix A. +Semi-algebraic sets +21 +References +21 +1. Introduction +Let (X, dX) be a metric space. The Kuratowski embedding x �→ dX(x, ·) is an embedding +of X into L∞(X), the space of all bounded real-valued functions on X with the uniform +norm. When X is the unit sphere with its geodesic distance, the homotopy types of the +r-neighborhoods Br(X, L∞(X)) in the Kuratowski embedding of X were studied by Katz +in [Kat91]. The values at which the homotopy type changes are closely related to the critical +configurations of the diameter functional diam of X which maps a finite subset A of X to +diam(A) := maxa,a′∈A dX(a, a′). When X is the unit circle, such critical values turn out to +be exactly one-half of the diameter values of odd regular polygons inscribed in S1. Note that +1 +arXiv:2301.13076v1 [math.CO] 30 Jan 2023 + +2 +MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG +the vertex sets of odd regular polygons are exactly the configurations that are local minima +of the diameter functional on the space of all finite subsets of S1 equipped with Hausdorff +distance. In [Kat89], Katz studied the diameter-extremal configurations on S2 and S3. The +latter provide candidates for testing Borsuk’s conjecture in R4 (see below). +Recently, Lim, M´emoli, and Okutan [LMO22, Theorem 5] proved that the homotopy types +of neighborhoods of the Kuratowski embedding of X are naturally homotopy equivalent to +the so-called Vietoris–Rips complexes of X, a central object in the field of applied algebraic +topology. Therefore, the study of diameter-extremal configurations is also of interest for +understanding the properties of the Vietoris-Rips complex of spheres [AA17, AAF18]. +In this paper, we extend the investigation of diameter-extremal configurations on spheres +started in [Kat89]. +In the S1 case, the critical values of the diameter functional form a +convergent sequence with the only accumulation point being π. It is natural to wonder to +what extent a similar behavior is true on S2. We consider two canonical families of diameter- +extremal configurations on S2 which we call pyramids Ak that contains 2k + 2 points (see +Section 2.1) and stacked-triangles Bk that contains 3k + 1 points (see Section 2.2). Both +families contain infinitely many members with diameters monotonically approaching 2π +3 . We +prove in Theorem 3.10 that 2π +3 is in fact the first accumulation point of the set of critical +values of the diameter functional. In Proposition 3.17, we prove that the two families Ak and +Bk do not exhaust all the possible configurations with similar diameter bounds, and in fact +there are infinitely many additional diameter-extremal configurations. Diameter-extremal +configuration with 3k points can be found by performing diameter gradient flow on a certain +subset of Bk. When k is odd, by a parity argument, the resulting configuration cannot be +an instance of Ak or Bk. +We next devise and implement a computational algorithm (see Algorithm 1) that attempts +to produce diameter extremal configurations. We use this algorithm to find new configu- +rations not in Ak or Bk. Furthermore, we found configurations not isometric to the ones +produced in the course of proving Proposition 3.17. See Table 1 for a complete list of all the +configurations we found in this way with up to 10 points. The list contains 10 previously +unknown configurations where 8 of those exhibit Z2 symmetry and the remaining two are +asymmetric; see Figures 7 and 8. +The convex hulls of certain diameter-extremal configurations give rise to anti-self-polar +polytopes (ASP), for example, the regular tetrahedron and any Ak or Bk. ASPs are polytopes +P characterized by the property that the polar of P equals −P (see Definition 4.3). ASPs +have been studied by Lov´asz in the context of answering a question by Erd¨os and Graham +[Lov83] and were also considered in [Kat89, Section 5] in the context of Borsuk’s conjecture. +Borsuk’s conjecture (see Section 4.2) for a finite point set X in Rn is equivalent to the +property that the chromatic number of the diameter graph (see Definition 4.7) of X is +bounded above by n + 1. We continue to explore the suggestion in [Kat89] to use diameter- +extremal configurations on S3 to test Borsuk’s conjecture in R4 (a case that is still open). +As shown by Lovasz [Lov83], the chromatic number of the diameter graph associated to +any ASP in Rn is at least n + 1. An ASP for which the inequality is strict would disprove +Borsuk’s conjecture. +It was conjectured in [Kat89] that the number of edges in the diameter graph of an ASP +4-polytope with v vertices is at least 3v − 5. We use Kalai’s inequality from [Kalai94, Sec- +tion 4.3] to prove such a bound in Theorem 4.21 below. We then formulate conjectures + +EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE +3 +about the number of edges in the diameter graph for more general subsets on S3, see Conjec- +tures 4.22 and 4.23. A calculation based on these two conjectures suggests that the maximum +possible chromatic number of the diameter graph of a finite subset X ⊆ R4 is 6 instead 5, +the number predicted by Borsuk’s conjecture. +We perform experiments attempting to identify diameter-extremal configurations on the +three-dimensional sphere. The interest in these experiments is twofold. On the one hand, +it is naturally interesting to obtain an understanding of critical configurations beyond the +case of S1 and S2. On the other hand, whereas Borsuk’s conjecture is known to be true +in dimensions 2 and 3 but false in dimensions 64 and higher, its status for dimension 4 is +unknown. Hence, by the above, it is tempting to seek a diameter-extremal configuration X of +S3 whose convex hull is an ASP such that its diameter graph has chromatic number at least +6. We discovered 65 new configurations on S3 not obtained by the pyramid construction +(see 2.1) on a previously known configuration on S2; see Theorem 6.1. However, all the +diameter graphs of these configurations have a chromatic number precisely 5. +Acknowledgements. This work was partially supported by BSF #2020124, NSF CCF +#1740761, and NSF IIS #1901360. +2. Pointwise extremal sets +Let Sn ⊆ Rn+1 be the unit sphere with its geodesic distance. For a subset Y ⊆ Sn, its +diameter diam(Y ) is computed with respect to the geodesic distance on the sphere. +Definition 2.1 (Taut sets in Sn). A finite subset Y ⊂ Sn is taut if one of the following +equivalent conditions is satisfied: +(1) the convex hull of Y contains the origin; +(2) there are non-negative real numbers {ay}y∈Y , not all zero, satisfying +� +y∈Y +ay y = 0, +where y denotes the position vector of the point y ∈ Rn+1. +Jung’s theorem immediately gives the following result. +Proposition 2.2. If Y ⊂ Sn is taut, then diam(Y ) ≥ arccos +� −1 +n+1 +� +. +The following observation will be useful in the sequel. +Corollary 2.3. Let Y ⊂ Sn be a taut set such that |Y | = n+2 and diam(Y ) < arccos +� +− 1 +n +� +. +Then the dimension of the vector space spanned by Y is equal to n + 1. +In particular, if {a1, . . . , an+2} is any set of non-negative coefficients such that +n+2 +� +i=1 +aiyi = 0, +then all ai must be positive. +Proof. Suppose the vector space spanned by all points in Y is of dimension at most n. Then, +the set {y1, y2, . . . , yn+2} must lie on some great sphere Sn−1 ⊆ Sn and it must be taut in +Sn−1. Then, by Proposition 2.2, the set Y must have diameter at least arccos +� +− 1 +n +� +which +contradicts the assumptions on Y . This concludes the first part of the proof. +For the second part, without loss of generality, we assume that a1 = 0, then the set of vectors + +4 +MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG +{y2, y3 . . . , yn+2} is linearly dependent and hence dim(span{y2, y3, . . . , yn+2}) < n + 1. The +contradiction with the first part establishes the result. +□ +Let Y be a subset of a metric space (X, dX). For any two points y, y′ ∈ Y , we say that y +and y′ are comaximal in Y if dX(y, y′) = diam(Y ). In such a case, y is called a comaximal +point with y′. We use the notation comaxY (y) to denote the set of all points in Y which are +comaximal with y. +For two points x, x′ ∈ Sn with distance less than π, there is a unique arclength-parametrized +geodesic γx,x′ connecting x to x′ such that γx,x′(0) = x. Consider the unit tangent vector +˙γx,x′(0) in the tangent space TxSn. +We recall the notion of pointwise extremal subsets in Sn as in [Kat89]. +Definition 2.4 ([Kat89]). Let Y ⊆ Sn be a finite subset with no antipodal pairs. We say +that y ∈ Y is held (in place) by Y if the set of vectors ˙γy,y′(0) as y′ runs over comaxY (y) is +a taut set. We say that Y is pointwise extremal if every point y ∈ Y is held by Y . +When n = 1, it is not difficult to see that, for all integers k ≥ 1, the vertex set of an +inscribed regular (2k + 1)-gon is pointwise extremal. The following proposition shows the +converse. +Proposition 2.5. Let Y ⊆ S1 be a pointwise extremal set containing no pair of antipodal +points. Then Y is the vertex set of an odd regular polygon inscribed in S1. +Proof. Let y ∈ Y and let D = diam(Y ). Let RD be the clockwise rotation on S1 by angle D. +As y is held by Y ⊆ S1, the set Y must contain both points in S1 at distance D from y. In +particular, the set Y is invariant under the rotation RD. As Y is a finite subset, the quotient +D +2π must be rational. Let m +n be the representation of +D +2π in lowest terms. Then the orbit of +y under the rotation RD forms the vertex set of an inscribed regular n-gon Y ′ ⊆ S1. As Y +does not contain any antipodal pairs, n is necessarily odd. Therefore, Y contains the vertex +set of a odd regular n-gon Y ′ of the same diameter as Y . Then Y must coincide with Y ′ as +adding any additional point to the set Y ′ would strictly increase the diameter. +□ +2.1. The pyramid construction. In this section, we describe a class of pointwise extremal +subsets of Sn called pyramids in [Kat89]. For any pointwise extremal subset Y ⊂ Sn−1, the +pyramid construction provides a corresponding pointwise extremal subset in Sn that consists +of a rescaled copy of Y together with one extra point. Let Sn ⊆ Rn+1 be the unit sphere. +Let Z = (0, . . . , 0, 1) denote the “north pole”. Let xn+1 be the last coordinate of Rn+1. Then +for each plane {xn+1 = a}, a ∈ R that meets Sn at more than one point, the intersection is +a rescaled copy of Sn−1 which we call a horizontal section. Each horizontal section contains +a suitable rescaled copy of Y which is isometrically embedded into it. +Definition 2.6. The pyramid over Y is the subset of Sn consisting of the north pole Z +together with a rescaled copy Y ′ of Y inside some horizontal section such that the diameter +of Y ′ equals the distance from Z to the horizontal section. Denote by Pyr(Y ) the pyramid +over a pointwise extremal subset Y . +Let x, y ∈ Y be points with dSn−1(x, y) = diam(Y ). Let x′, y′ ∈ Pyr(Y ) be points corre- +sponding to x, y. Then the triple Z, x′, y′ is the vertex set of a spherical equilateral triangle, +with spherical angle ∢ x′Zy′ = diam(Y ). Applying the spherical theorem of cosines to the +geodesic triangle △ x′Zy′, we obtain the following relationship: +diam(Pyr(Y )) = arcsec +� +sec +� +diam(Y ) +� +− 1 +� +. + +EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE +5 +Example 2.7 (The Ak family in S2). Let k ≥ 1. We apply the pyramid construction to the +regular (2k + 1)-gon on S1 to obtain a pointwise extremal configuration Ak ⊆ S2, consisting +of the north pole of S2 together with a suitably rescaled copy of the regular (2k + 1)-gon, so +that diam(Ak) = arcsec +� +sec +� 2kπ +2k+1 +� +− 1 +� +; see Figure 1. In particular, the diameter diam(Ak) +tends to 2π +3 as k goes to infinity. +Figure 1. The configuration A2 consists of the north pole and the vertices +of a regular pentagon. +2.2. Construction of k-stacks. Following [Kat89], let a β-digon be the convex region on +S2 bounded by two meridians (great semicircles joining the north and south poles), with +angle β between the two meridians. +Given a β-digon, we now introduce a procedure that will be used to produce a certain +type of pointwise extremal set Y ⊆ Sn called a k-stack. The digon procedure is a “walking +process” on the digon that takes as input an odd integer 2k + 1 ≥ 3 and outputs a suitable +step length d1 > β. +We start walking with equal steps from the north pole on alternating sides of the digon, +with step length d1 calibrated so as to get exactly to the south pole after 2k + 1 steps; see +Figure 2. +Let Z ∈ Sn be the north pole. A regular n-simplex inscribed in the equator Sn−1 ⊂ Sn +defines n + 1 meridians passing through the vertices of the simplex. +Let ℓ ∈ (0, π). The set of points on Sn which are at distance ℓ away from the north pole Z +is a rescaled (n−1)-sphere Sn−1 +ℓ +, namely a horizontal section of Sn. The intersection between +Sn−1 +ℓ +and the set of n + 1 meridians is the vertex set of an inscribed n-simplex in Sn−1 +ℓ +. +Let k ≥ 1. A k-stacked configuration Y (see Figure 2) consists of the north pole Z together +with the union of the vertex sets of k stacked n-simplices each obtained as the intersection of +a horizontal (n − 1)-sphere Sn−1 +ℓi +with the n + 1 meridians. The distances ℓ1, . . . , ℓk between +the horizontal sections and the north pole are determined by the digon procedure as follows. +Let d1 be the step length that comes from the digon procedure with input 2k + 1. Consider +the sequence of numbers {dj}2k+1 +j=0 +where dj is the distance to the north pole of the point +obtained after walking j steps in via the digon procedure. Then, the sequence of numbers +{ℓi}1≤i≤k is defined in terms of {di}2k+1 +i=0 +by setting +ℓi = d2i +for +1 ≤ i ≤ k. +Note that d2k = diam(Y ) and d1 = π − d2k. +Given an odd integer 2k + 1 ≥ 3, the following system of equations summarizes the +computation of di for 1 ≤ i ≤ 2k + 1. + +6 +MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG +Figure 2. Each value di in Equation (2.1) is the distance between the point +pi shown in this figure and the north pole. For 1 ≤ i ≤ 2k + 1, the distance +between pi and pi+1 is d1. The two conditions in Equation (2.1) are obtained by +requiring p0 to be the north pole and p2k+1 to be the south pole. The conditions +in the second line of Equation (2.1) are obtained by applying the theorem of +cosines for the geodesic spherical triangles with vertices {Z, pi, pi+1}, for each +1 ≤ i ≤ 2k + 1. +The third line Equations (2.1) is obtained by symmetry +considerations. +Let βn = arccos( 1 +n). The values {di}0≤i≤2k+1 are determined by n and k via the following +equations (see Figure 2): +(2.1) +� +� +� +� +� +d0 = 0, d2k+1 = π +cos(di) cos(di+1) + sin(di) sin(di+1) cos(βn) = cos(d1), 1 ≤ i ≤ 2k +di + d2k+1−i = π, 0 ≤ i ≤ 2k + 1. +Remark 2.8. Let d1 be the output of the digon procedure with input 2k + 1 on a digon of +angle β. If we perform the “walking process” on a digon of angle π − β with complementary +step length π − d1, we will eventually get close to the south pole (but will not reach it) and +then will start walking back to the north pole and reach it after 2k + 1 steps. If we add an +edge between the points that we traveled during the “walking process”, we obtain the diameter +graph (see Definition 4.7) of a regular 2k + 1-gon. +Example 2.9 (The Bk family in S2). When n = 2, for each k, we denote the stacks that +result from the digon procedure by Bk, which consists of the vertices of k stacked triangles +(2-simplices) together with the north pole. Note that B1 coincides with the configuration +A1 from Example 2.7. By construction, diam(Bk) = π − d1 < π − arccos( 1 +2) = +2π +3 and +limk→∞ diam(Bk) = 2π +3 . + +EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE +7 +Figure 3. The configuration B2 that consists of the north pole and vertices +of two stacked triangles. The green dash lines are meridians; the red dot is the +north pole, and points of the same color are of the same distance to the north +pole. +Example 2.10 (The Tk family in S3). Let n = 3. For each k, we denote the stacks that +result from the digon procedure by Tk, which consists of the vertices of k stacked tetrahedra +together with the north pole. +3. Minimal sets on S2 with diameter below the first accumulation critical +value +Let d > 0 and let D(S2, d) be the set of all finite subsets Y ⊂ S2 with diam(Y ) < d. +As each finite subset on S2 is closed, the Hausdorff distance dH is a metric on D(S2, d). +Definition 3.1 (Diameter-extremal sets in D(S2, d) [Kat89]). A subset Y ∈ D(S2, d) is called +diameter-extremal for the diameter functional if there is a little-o function such that +diam(Y ) ≤ diam(Y ′) + o( dH(Y, Y ′)) +for all Y ′ ⊂ S2. In other words, we have +lim +dH(Y ′,Y )→0 +diam(Y ′) − diam(Y ) +dH(Y ′, Y ) +≥ 0. +Remark 3.2. An n-point set Y is diameter-extremal if and only if at the corresponding +point in the configuration space (S2)×n, the gradients of the distances between pairs of points +at maximal distance form a taut set (see further in Section 3.1). +Lemma 3.3 ([Kat89, Corollary 3.4]). A diameter-extremal set Y ∈ D(S2, 2π +3 ) is necessarily +pointwise extremal. +Definition 3.4 (Minimal set in D(S2, d) [Kat89]). A subset Y ∈ D(S2, d) is called a minimal +set if there is some δ > 0 such that diam(Y ) ≤ diam(Y ′) for all finite subsets Y ′ with +dH(Y, Y ′) ≤ δ. +Clearly, every minimal set is diameter-extremal. In fact, there is a converse. +Theorem 3.5 ( [Kat89, Theorem 2]). Every diameter-extremal set in D(S2, 2π +3 ) is a minimal +set on S2. +By a mountain-pass argument, one obtains the following consequence. +Lemma 3.6 ([Kat89, Corollary 2]). There is exactly one (up to congruence) minimal set in +each connected component of D(S2, 2π +3 ). + +8 +MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG +3.1. Configuration space. We will now estimate the number of such connected compo- +nents. We use the notation +k� +diam≤d +S2 to denote the set of all tuples (y1, . . . , yk) in �k S2 such +that the diameter of its associated set {y1, . . . , yk} is less than or equal to d. Note that, for +any ϵ > 0, we have a natural continuous map +k +� +diam≤d +S2 −→ D(S2, d + ϵ). +By realizing +k� +diam≤d +S2 as a closed semi-algebraic set, we obtain the following upper bound on +the number of connected components in +k� +diam≤d +S2. +Lemma 3.7. Let k ≥ 0. We set sk = 2k+ k(k+1) +2 +. Then, for every d > 0, the number b0(k, d) +of connected components of +k� +diam≤d +S2 satisfies +b0(k, d) ≤ 2sk(4sk − 1)3k−1. +Proof. We will first describe the set +k� +diam≤d +S2 as a closed basic semi-algebraic set in R3k. Let +xi,j, where 1 ≤ i ≤ k and 1 ≤ j ≤ 3, denote the standard coordinates in R3k. Then the set +k� +diam≤d +S2 is characterized by the following conditions: +� +x2 +i,1 + x2 +i,2 + x2 +i,3 = 1 +for all 1 ≤ i ≤ k, +(xi,1 − xi′,1)2 + (xi,2 − xi′,2)2 + (xi,3 − xi′,3)2 ≤ d2 +for all 1 ≤ i < i′ ≤ k. +Therefore the set +k� +diam≤d +S2 is a basic semi-algebraic set given by sk = 2k + k(k+1) +2 +non-strict +inequalities. Then Theorem A.5 implies that b0(k, d) ≤ 1 +2(2sk + 2)(2sk + 1)3k−1. +□ +3.2. Finiteness results. Lemma 4.1 and Lemma 4.3 in [Kat89] imply the following result. +Lemma 3.8 ([Kat89]). Let 0 < d < +2π +3 . +Let Y +∈ D(S2, d) be a pointwise extremal +set. +Then for any pair of distinct points y, y′ in Y , the distance dS2(y, y′) is at least +arccos +� +2 cos2(d) +cos2(d/2) − 1 +� +. +By a packing argument on the sphere, we obtain the following result. +Corollary 3.9. For each ϵ > 0, there is a positive integer N(ϵ) such that every pointwise +extremal subset Y of diameter less than 2π +3 − ϵ contains fewer than N(ϵ) points. +Theorem 3.10. For each 0 < ϵ < 2π +3 , there are only finitely many diameter-extremal sets +in D(S2, 2π +3 − ϵ). +In particular, +2π +3 is the first accumulation point of the critical values of the diameter +functional of S2. + +EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE +9 +Proof. Let dϵ = +2π +3 − ϵ. By Theorem 3.5, it suffices to show that there are only finitely +many minimal sets in D(S2, dϵ). By Corollary 3.9, there is some N such that every pointwise +extremal set in D(S2, dϵ) contains no more than N points. +Therefore the image of the +continuous map φ +φ : +N +� +diam≤dϵ +S2 −→ D(S2, 2π +3 ) +contains all pointwise extremal configurations with diameter less than or equal to dϵ. By +Lemma 3.3, the image of φ (in particular) contains all minimal sets with diameter not +exceeding dϵ. +Let Cϵ be the number of connected components which contain a minimal set with diameter +no more than dϵ. By Lemma 3.6, the number of minimal sets in D(S2, dϵ) is at most Cϵ. As +the image of φ contains all minimal sets with diameter no more than dϵ, the number Cϵ is +bounded by the rank of the map +φ∗ : H0 +� +N +� +diam≤dϵ +S2 +� +−→ H0 +� +D(S2, 2π +3 ) +� +. +The claim now follows by invoking the upper bound on the dimension of H0 +� +N� +diam≤dϵ +S2 +� +from Lemma 3.7. +□ +3.3. A labeling strategy for the points in Bk. Recall that Bk ⊆ S2 consists of the north +pole and the vertices of k stacked triangles, and that the vertices of the stacked triangles are +distributed along three meridians. +We label the north pole as Z, then label the vertices of the i-th triangle (counting from the +north pole) by Pi, Qi, Ri in such a way that all the Pi, 1 ≤ i ≤ k are on a common longitude +and similarly for all Qi, 1 ≤ i ≤ k and Ri, 1 ≤ i ≤ k. +Definition 3.11. The subset dEBk is obtained by removing the points with indexes in E +from Bk. +Definition 3.12. A set Y ⊆ S2 is separable if for each pair of points x, y ∈ Y there are two +other points z, w ∈ Y such that the 4-tuple {x, y, z, w} is taut. +Lemma 3.13 ([Kat89, Lemma 4.1]). A pointwise extremal subset Y ⊂ S2 with diam(Y ) < 2π +3 +is necessarily separable. +The proof of the above lemma in [Kat89] gives the following stronger result. +Lemma 3.14. Let Y ⊂ S2 be a subset with diam(Y ) < 2π +3 . Suppose x ∈ Y is held by Y . +Then for any other point y ∈ Y , there exist z, w ∈ Y such that the four-point set {x, y, z, w} +is taut. +We will now analyze variations of subsets which are continuous with respect to the Haus- +dorff distance. +Lemma 3.15. Let {Yt, t ∈ [0, 1]} be a continuous family of subsets of S2 with at most 4 +points. Suppose the following two conditions hold: +• the set Y0 is taut, +• Yt ∈ D(S2, 2π +3 ) for every t ∈ [0, 1]. + +10 +MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG +Then Yt is taut for each t ∈ [0, 1]. +Proof. As the set Y0 is taut and diam(Y ) < 2π +3 , Corollary 2.3 implies that the convex hull +H0 of Y0 is a tetrahedron and that the origin 0 is in interior of H0. For each t ∈ [0, 1] let Ht +be the convex hull of the set Yt. To show that each set Yt is taut, it suffices to show that the +origin 0 stays in the interior of Ht for all t ∈ [0, 1]. +Suppose the contrary. Let t0 be the supremum of t such that 0 is in the interior of Ht +for all smaller values of t. Either Ht0 is nondegenerate and then 0 must belong to one of its +(triangular) faces, or it is degenerate, i.e., lies in a plane through the origin. In either case, +we obtain a taut subset of the circle given by the intersection of the plane with the sphere, +and can apply Jung’s theorem. +Namely, by Proposition 2.2 we obtain diam(Yt0) ≥ 2π +3 , contradicting the hypothesis Yt0 ∈ +D(S2, 2π +3 ) and proving the lemma. +□ +Corollary 3.16. Let Yt, t ∈ [0, 1] be a path in D(S2, 2π +3 ). If a certain 4-tuple in Y0 is taut, +then it continues to be taut for all t ∈ [0, 1]. +Proposition 3.17. There exist infinitely many (up to congruence) pointwise extremal sets +in D(S2, 2π +3 ) that are not contained in the family Ak or Bk. +Proof. Since each connected component contains a (unique) minimal set, it suffices to show +that for each k, the configuration dPkBk is separable. +By Lemma 3.14, we can separate most pairs of points from dPkBk except for a pair of +points from the triple of points at maximal distance from Pk, namely the points Z, Q1, and +R1. Let us check that such pairs don’t coalesce, either. This is immediate from the fact that +if we remove all layers except the first and the k-th, the remaining configuration is in the +connected component in D(S2, 2π +3 ) of the 7-point minimal set B2. Thus, by Corollary 3.16, +it suffices to check that if we remove P2 from B2, no remaining points coalesce. This can be +checked directly, and also follows from the fact that the diameter flow applied to the 6-point +configuration dP2B2 produces the 6-point minimal set A2 (see Section 5). +□ +4. Anti-self-polar polytopes +In this paper, we adopt the following restricted definition of a polytope: a (convex) polytope +will be the convex hull of any finite set of points in Rn. +Definition 4.1. The affine hull aff(S) of a set S ⊆ Rn is +aff(S) = +� k +� +i=1 +αixi +����� k > 0, xi ∈ S, αi ∈ R, +k +� +i=1 +αi = 1 +� +. +We now give the formal definition of a face of a polytope following [Zie12]. +Definition 4.2 ([Zie12, Definition 2.1]). Let P ⊆ Rd be a convex polytope. A linear inequality +⟨c, x⟩ ≤ c0 is valid for P if it is satisfied for all points x ∈ P. A face of P is any set of the +form +F = P ∩ +� +x ∈ Rd : ⟨c, x⟩ = c0 +� +where ⟨c, x⟩ ≤ c0 is a valid inequality for P. + +EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE +11 +The dimension of a polytope P is defined to be the dimension of its affine hull aff(P) +(regarded as an affine space). A 3-dimensional polytope is a polyhedron. The codimension- +one faces of a polytope P are called facets; the codimension-two faces are called ridges. If +each face of P is a simplex, then P is called a simplicial polytope. We will use fi(P) to +denote the number of i-faces of the polytope P. When there is no risk of confusion, we will +denote fi(P) by just fi. For a n-dimensional polytope, the vector (f0, f1, . . . , fn−1) is called +the f-vector of P. +4.1. ASP polytopes. In [Lov83], Lov´asz introduced the following type of polytopes which +we will refer to as anti-self-polar (ASP) polytopes.1 +Our terminology will be justified in +Remark 4.4. +Definition 4.3 (Anti-self-polar polytopes). Let P ⊆ Rn be a n-dimensional polytope. We +say that P is anti-self-polar (ASP) if the following three conditions hold: +(1) P is inscribed in the unit sphere Sn−1 ⊆ Rn. +(2) P is circumscribed around a sphere centered at the origin with radius s for some +0 < s < 1. +(3) There is a bijection σ between vertices and facets of P such that if v is any vertex +then the facet σ(v) is orthogonal to the vector v. +Remark 4.4. Let P ⊂ Rn be a polytope containing the origin 0. Let Sn−1 +r +(0) be the sphere +centered at 0 ∈ Rn with radius r > 0. The polar body of P with respect to the sphere Sn−1 +r +(0) +is defined to be the set +polarr(P) = {x ∈ Rn| ⟨x, y⟩ ≤ r2 for all y ∈ P}. +As shown in [Hor21], the condition for an ASP polytope in Rn can be restated using the +terminology of polar bodies. In terms of our definition of polarity, if P is an ASP polytope, +then there exists some r such that the following relation holds; see [Hor21, Lemma 1]. +polarr(P) = −P. +The polar body description shows that for each 0 ≤ i ≤ n − 1, the bijection σ in condition +(3) can be extended to a bijection between the set of i-dimensional faces and the set of +(n − i − 1)-dimensional faces; see [Hor21, Lemma 2]. +Proposition 4.5 ([Kat89, Remark after Theorem 1]). Let Y ⊂ S2 be a pointwise extremal +subset with diam(Y ) < 2π +3 . Then the convex hull of Y is an ASP polyhedron. +Remark 4.6. The result above no longer holds if the restriction on the diameter is removed. +A counterexample is given by an 8-point configuration Y ⊆ S2 consisting of the vertices of +an antiprism over a square (see Figure 4). If the diameter of Y is exactly attained by the +diagonals of the two squares and by the pairs that consist of a vertex of one square and one +of the two farthest vertices of the other square, then Y is pointwise extremal. However, the +convex hull of Y is not ASP. Indeed, note that the top square is a facet of the convex hull +of Y . If the convex hull of Y were ASP, then there would be a vertex y0 ∈ Y such that the +distance from y0 to each vertex of the top square would equal diam(Y ). But, our construction +of Y does not satisfy this. +1Lov´asz [Lov83] and Horv`ath[Hor21] use the terminology “strongly self-dual polytopes”. + +12 +MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG +Figure 4. The antiprism on a square. +Definition 4.7. Let Y ⊆ Rn be a finite subset. The diameter graph G(Y ) of Y is defined +to be the graph with vertex set V (G) = Y and two vertices y, y′ in G are connected if and +only if y and y′ are comaximal in Y . +Given a polytope P, we will refer to the diameter graph of the vertex set of P simply as +the diameter graph of P. We denote the diameter graph of P by G(P). +Definition 4.8. The chromatic number χ(G) of a graph G is the smallest number of colors +needed to color the vertices so that no two adjacent vertices share the same color. +The following property of the diameter graph G(P) of an ASP polytope P follows from +[Lov83, Lemma 2 and Lemma 3]. Recall that σ denotes the bijection between the vertex set +and the set of facets of P. In [Lov83, Lemma 1], it is shown that for any two vertices v, v′ +of P, the condition v ∈ σ(v′) is equivalent to v′ ∈ σ(v). +Proposition 4.9. Let P be an ASP polytope. Two vertices v, v′ in G(P) are connected by +an edge in G(P) if and only if v ∈ σ(v′), when viewed as vertices in P. +Theorem 4.10 ([Lov83, Theorem 2]). The diameter graph G(P) of an n-dimensional ASP +polytope P ⊆ Rn satisfies χ(G(P)) ≥ n + 1. +The proof of the theorem is discussed in Section 4.3. The chromatic number of a diameter +graph G(Y ) of a subset Y ⊂ Rn is closely related to the following conjecture of Borsuk. +4.2. Borsuk’s conjecture. +Conjecture 4.11 (Borsuk’s conjecture). Let Y be a bounded subset of Rn. Then there is a +partition of Y into n + 1 sets each of which has a smaller diameter than Y . +For finite subsets, Borsuk’s conjecture has the following equivalent form in terms of diam- +eter graphs: +For every finite bounded subset Y ⊆ Rn, the chromatic number of the diam- +eter graph G(Y ) of Y is no greater than n + 1. +To see the above equivalence, a partition {Y1, . . . , Yk} of Y is equivalent to a coloring of +Y by requiring that two points are of the same color if and only if they both belong to + +EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE +13 +some Yi, 1 ≤ i ≤ k. Therefore, since Y is a finite set, the condition that the diameter of each +subset Yi is less than the diameter of Y is equivalent to requiring that the coloring associated +to the partition {Y1, . . . , Yk} has the property that no two adjacent vertices in the diameter +graph G(Y ) share the same color. +Borsuk’s conjecture holds when n = 2 (Borsuk [Bor33]) and n = 3 (Perkal [Per47]). The +general conjecture was disproved by Khan and Kalai [KK93]. The lowest dimensional coun- +terexample currently known was constructed by Jenrich and Brouwer (and based on a con- +struction by Bondarenko) in dimension 64 [JB14]. For additional information on the histori- +cal developments on the construction of counterexamples to Borsuk’s conjecture, see [Rai13, +Section 2]. +Remark 4.12. Let Y ⊂ Sn−1 be a finite subset. Given a regular geodesic n + 1-simplex +∆geodesic +n+1 +, Sn−1 can be partitioned into n + 1 connected parts {X1, X2, . . . , Xn+1} where each +Xi contains the interior of one of the faces of ∆geodesic +n+1 +. Therefore, by coloring points of +Y according to which partition set Xi the point belongs to, we obtain a proper coloring of +the diameter graph of Y provided that the diameter diam(Y ) diameter of Y is greater than +ηn−1, the diameter of a face of ∆geodesic +n+1 +. The above coloring strategy was first described in +[Lov83, Section 0]. Though notice that [Lov83] made a mistake in computing the exact value +of ηn−1 [Rai12, Rai13]. The correct values of ηn−1 first appeared in [San46] and reproduced +in the context of ASP polytopes in [Hor21]. +By Theorem 4.10 and the fact that Borsuk’s conjecture is true for n = 3, the chromatic +number χ(G(P)) of an ASP polyhedron P ⊆ R3 equals 4. In Figures 7 and 8, we display +4-colorings of the diameter graphs of all the ASP polyhedra in Tables 5 and 6. +Remark 4.13. Borsuk’s conjecture is still open for 4 ≤ n ≤ 63. Theorem 4.10 suggests +that ASP polytopes are a natural source of potential counterexamples to Borsuk’s conjecture. +Additionally, by Proposition 4.5, pointwise extremal configurations are closely related to ASP +polytopes. In Section 6.2, we present some pointwise extremal subsets on S3 obtained through +computer experiments. However, the pointwise extremal subsets that we have found so far +all have chromatic number 5; cf. Section 6.2. +4.3. Proof of Lovasz’s theorem. Theorem 4.10 was proved in [Lov83] by analyzing the +neighborhood complex of the diameter graph of ASP polytopes. +Definition 4.14 (Neighborhood complex). Let G be a finite graph. +The neighborhood +complex N(G) is the simplicial complex with vertex set V (G) such that a subset A ⊆ V (G) +forms a simplex if and only if the points of A have a neighbor in common. +In [Lov78], Lov´asz shows the following lower bound of the chromatic number of a graph +with respect to the connectivity of its neighborhood complex. Recall a topological space X +is k-connected if its homotopy groups are trivial up to degree k. +Theorem 4.15 ([Lov78]). Let G be a graph and suppose that N(G) is k-connected (k ≥ 0). +Then χ(G) ≥ k + 3. +Lemma 4.16 ([Lov83, Lemma 4]). Let P be an ASP polytope and G(P) be its diameter +graph. Then N(G(P)) is homotopy equivalent to the boundary of P. +Proof of Theorem 4.10. By Lemma 4.16, N(G) is homotopy equivalent to the boundary of +P. + +14 +MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG +As P is a (convex) polytope, the boundary of P is homeomorphic to Sn−1. Hence N(G) +is homotopy equivalent to Sn−1. Therefore, N(G) is (n − 2) connected. By Theorem 4.15, +χ(G) ≥ n + 1. +□ +Let d ≥ 2 and n ≥ 1 be integers and let e(d, n) be the maximum possible number of edges +in the diameter graph of a subset of Rd with n points. When d = 2, it is shown in [HP34] +that e(2, n) = n. This fact leads to one proof of Borsuk’s conjecture for finite subsets Y of +R2. When d = 3, it was conjectured by V´azsonyi that e(3, n) = 2n − 2; see [Erd46]. The +V´azsonyi’s conjecture was proved independently by Gr¨unbaum [Gr¨u56], Heppes [Hep56] and +Straszewicz [Str57]. As mentioned in Heppes [Hep56], V´azsonyi’s conjecture implies that +Borsuk’s conjecture is true for finite subsets in R3. We have already seen in Theorem 4.10 +that the diameter graph of an ASP polytope has high chromatic number, suggesting a +possible approach to seeking higher-dimensional counterexamples. +We now introduce a set of enumerative invariants fij(P) of a polytope P which will be +used below. Informally, for i < j, fij(P) counts the number of pairs “i-face contained in a +j-face” in the polytope P. Precisely, +fij(P) := ♯{(φi, φj) | φi is a i-face of P, φj is a j-face of P, and φi ⊆ φj.} +When there is no risk of confusion, we will simply use fij to denote fij(P). Thus f01 is the +number of pairs “vertex contained in an edge”, namely just twice the number f1 of edges +in P. +Lemma 4.17. Let P be an anti-self-polar polytope of dimension d + 1. Let e(G(P)) be the +number of edges in the graph G(P). Then f0d(P) = 2e(G(P)). +Proof. Let V be the set of vertices of P and let W be the set of faces in P. Recall that σ +denotes the bijection between V and the set of facets of P. By Proposition 4.9, we have +2e(G(P)) = +� +v∈V +f0(σ(v)) = +� +φd⊂W +f0(φd) = f0d. +The second equality above follows from the definition of σ. +□ +Proposition 4.18. Every ASP polyhedron P ⊆ R3 satisfies e(G(P)) = 2f0 − 2. +Proof. By Lemma 4.17, we have 2e(G(P)) = f02. Furthermore by duality we have f01 = f12. +This enables us to give a possibly generalizable proof as follows. +Note that, each face has as many vertices as edges and therefore f02 = f12. By duality, +f12 = f01 which is twice the number of edges, namely 2f1. Thus the number of maximal +distances is the same as the number of edges. +Meanwhile by the formula for the Euler +characteristic, for an anti-self-polar polyhedron we have f1 = 2f0 − 2. Altogether, we have +f02 = f12 = f01 = 2f1 = 2(2f0 − 2). +Thus the number of maximal distances is also 2f0 − 2. +□ +In fact, it is shown in [Kat89] that every pointwise extremal set in S2 with diameter less +than 2π +3 exhibits the maximum number of possible edges. +Theorem 4.19 ([Kat89, Theorem 1]). Suppose Y ⊂ S2 is a pointwise extremal set with +N = |Y | and diam(Y ) < 2π +3 . Then the number of edges in the diameter graph G(Y ) equals +2N − 2. + +EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE +15 +As noted in [Kat89, page 118], the example of the antiprism on a square constructed +in Remark 4.13 shows that the above result is no longer true if we remove the diameter +constraint: the diameter graph of the antiprism on a square has 8 vertices but only 12 edges. +4.4. 4-dimensional polytopes. Consider the V´azsonyi’s problem in R4, that is, for a fixed +n, determine the maximal possible number of edges e(4, n) amongst the diameter graphs of +all possible n point sets in R4. +Example 4.20. Let m be a positive integer and let Y := A ∪ B ⊂ S3 be a subset consisting +of 2m points constructed as follows. The set A consists of m points on an arc of length less +than π +2 on a great circle whereas the (disjoint) set B consists of m points also on an arc of +length less than π +2 on an orthogonal great circle. +Then each pair of points a ∈ A, b ∈ B is comaximal in Y . Thus e(4, n) is at least quadratic +in n. It is shown in [Erd67] that e(4, n) exactly has quadratic growth rate in n. +For an anti-self-polar polytope P ⊆ R4, we prove the following lower bound on the number +of edges in the diameter graph G(P), originally conjectured in [Kat89, Section 5]. +Theorem 4.21. Let P ⊆ R4 be a 4-dimensional anti-self-polar polytope. Then the number +of edges e(G(P)) in the diameter graph G(P) is at least 3f0(P) − 5. +Proof. By Lemma 4.17, the assertion is equivalent to the bound f03(P) ≥ 6f0(P) − 10. For +each facet φ of P, let aj +φ be the number of j-gons occurring as faces of φ, and let aj denote +the total number of j-gons occurring as faces of P. Kalai [Kalai94, Section 4.3] proved that +every 4-dimensional polytope satisfies g2 ≥ 0 or equivalently +a4 + 2a5 + · · · ≥ 4f0(P) − f1(P) − 10. +Let φ run through all the facets of P. By Euler’s formula, we have +f03(P) = +� +φ +f0(φ) += +� +φ +2 + f1(φ) − f2(φ) += +� +φ +2 + 1 +2(3a3 +φ + 4a4 +φ + 5a5 +φ + · · · ) − f2(φ) += +� +φ +2 + 1 +2f2(φ) + 1 +2(a4 +φ + 2a5 +φ + · · · ) += 2f3(P) + f2(P) + 1 +2 +� +φ +(a4 +φ + 2a5 +φ + · · · ) += 2f3(P) + f2(P) + (a4 + 2a5 + · · · ) +≥ 2f3(P) + f2(P) + 4f0(P) − f1(P) − 10 += (2f3(P) + 4f0(P)) + (f2(P) − f1(P)) − 10 += 6f0(P) − 10 +by duality, as required. +□ +The above results suggest formulating the following conjectures. + +16 +MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG +Conjecture 4.22. Every ASP polytope P ⊆ R4 satisfies e(G(P)) = 3f0(P) − 5. +In Section 6.2, we report 65 configurations that we generate through numerical experi- +ments. Each of those configurations confirms the above conjecture. +Conjecture 4.23. Every subset X ⊆ S3 with diam(X) > π +2 satisfies e(G(X)) ≤ 3|X| − 5. +Assuming these conjectures and by an argument similar to the case of the S2 discussed +on page 14, one can show that the chromatic number of the diameter graph of any set X +in S3 with its diameter greater than π +2 would be at most 6. Indeed, Conjecture 4.23 implies +that one can always choose a point x0 ∈ X comaximal with at most 5 other points, by the +pigeonhole principle. Thus, if X − {x0} can be colored with 6 colors, then X can be so +colored, also, by using the color not used up by any of its 5 (or fewer) comaximal points, and +we conclude by induction. The fact that this calculation produces the number 6 instead of +5 would provide weak evidence toward the possibility that the Borsuk number of R4 might +be the former rather than the latter. +5. Implementation of the diameter gradient flow +This section describes the implementation of the diameter gradient flow on spheres. Given +a finite subset Y of Sn, we first test whether every point in Y is held. If there is a point y +that is not held by Y , we then move y in the direction that points toward the center of the +minimum bounding sphere of the tangent vectors determined by points in comaxY (y). We +continue this process until every point in Y is held. In other words the point y is updated to +a point yt = +y0+tv0 +||y0+tv0|| where t > 0 is a parameter value determined through the Armijo rule +[Arm66], and v0 is the unit tangent vector at y that points toward the center of the minimum + +EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE +17 +bounding sphere of the set {˙γy,y′ | y′ ∈ comaxyY }. The pseudocode of the algorithm is shown +below. +Algorithm 1: DiameterGradientFlow +Input: An initial finite subset Y on unit sphere Sn. +Parameters: β, η ∈ (0, 1) for determing Armijo Rule stepsa. +Output: The extremal configurations obtained under the diameter gradient flow +with initial condition Y . +1 Function IsHeld(y, Y ): +2 +E ← comaxY (y) +3 +Ty(E) ← {˙γy,y′ for y′ ∈ comaxY (y)} +4 +if 0 in the convex hull of Ty(E) then +5 +return True +6 +else +7 +return False +8 +end if +9 +10 Function Main(Y , β, η): +11 +/* Initialize convergence tag +*/ +12 +tag = False +13 +while tag == False do +14 +for y0 ∈ Y do +15 +if IsHeld(y0, Y ) then +16 +tag == True +17 +else +18 +E ← comaxY (y) +19 +Ty0(E) ← {˙γy,y′} for y′ ∈ E} +20 +v0 ← center of the minimum bounding sphere of Ty(E). +21 +/* Determine the step size tk > 0 using Armijo Rule +*/ +22 +tk = maxl∈N0 βl +s.t. +diam +� +Y \{y0} ∪ +� +y0+tkv0 +||y0+tkv0|| +�� +≤ diam(Y ) − βlη +23 +Y ← Y \{y0} ∪ +� +y0+tkv0 +||y0+tkv0|| +� +24 +tag == False +25 +break +26 +end if +27 +end for +28 +end while +29 return +asee [Arm66] +6. Computational results +In this section, we describe our computational results regarding pointwise extremal config- +urations on S2 and S3 using the Algorithm 1. In most of our experiments, we set parameters +β = 0.5, η = 0.001 and use the Python package MINIBALL([Dev21]) for finding the optimal +direction for decreasing the diameter by moving a single point. + +18 +MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG +6.1. Pointwise extremal configurations on S2. In this section, we present the compu- +tational results from running the diameter gradient flow Algorithm 1 with initial sets, which +are obtained by removing up to six points from Bk (Example 2.9) with k ≤ 5. +In total, we obtain 54 configurations. We present in Table 1 the configurations with up to +10 points2 that we found upon convergence of the gradient flow. +Shape +v +f +r +t +Diameter +Symmetry Group +Initial Set +A1(= B1) +4 +3 +4 +4 +1.91064 +S3 +dZB2 +A2 +6 +5 +1 +5 +2.03446 +D5 +dP2B2 +B2 +7 +4 +3 +4 +2.07654 +S3 +dZB3 +C1 +8 +5 +1 +4 +2.08707 +Z2 +d{Q1,P3}B3 +A3 +8 +7 +1 +7 +2.06459 +D7 +d{P1,P3}B3 +C2 +9 +5 +1 +3 +2.09335 +Z2 +d{P1,R1,Q3,Q4}B4 +C3 +9 +5 +1 +4 +2.09079 +Z2 +dP3B3 +C4 +9 +6 +1 +5 +2.09016 +Z2 +dP1B3 +B3 +10 +4 +6 +4 +2.09303 +S3 +dZB4 +D1 +10 +5 +1 +3 +2.09409 +{e} +d{P1,Q1,P2,Q4,R4,R5}B5 +C5 +10 +5 +1 +4 +2.09317 +Z2 +d{P1,R3,Q4}B4 +C6 +10 +5 +2 +4 +2.09356 +Z2 +d{P1,Q1}B4 +C7 +10 +5 +3 +4 +2.09240 +Z2 +d{P1,R3,P4}B4 +D2 +10 +6 +1 +4 +2.09360 +{e} +d{P1,R3,R4}B4 +C8 +10 +7 +1 +6 +2.09174 +Z2 +d{P1,P3,Q4}B4 +A4 +10 +9 +1 +9 +2.07654 +D9 +d{P1,P3,P4}B4 +Table 1. +Pointwise extremal configurations on S2 with up to v = 10 vertices, +sorted first by v, then by f (maximal number of edges in a face), then by r +(number of faces with a maximal number of edges), then by t (number of +triangles in the configuration’s diameter graph). For each of the 10 pointwise +extremal configurations that we found, in the last column we list one initial +set which leads to that configuration under the diameter gradient flow (a given +pointwise extremal configuration may be reached from different initial sets). +2An interactive visualization of the table can be found through the link: +https://ndag.github.io/ +anti-self-dual-polyhedra/ + +EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE +19 +(a) C1 +(b) C2 +(c) C3 +(d) C4 +(e) C5 +(f) C6 +(g) C7 +(h) C8 +Figure 5. Eight Z2 symmetric pointwise extremal configurations with at +most 10 points. +(a) D1 +(b) D2 +Figure 6. Two asymmetric pointwise extremal configurations with 10 points. + +20 +MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG +(a) C1 +(b) C2 +(c) C3 +(d) C4 +(e) C5 +(f) C6 +(g) C7 +(h) C8 +Figure 7. Diameter graphs of Z2 symmetric pointwise extremal configura- +tions with less than 10 with a minimal coloring. Note that all diameter graphs +above can be colored with four colors. +(a) D1 +(b) D2 +Figure 8. The diameter graph of the two asymmetric pointwise extremal +configurations D1 and D2. +6.2. Pointwise extremal configurations on S3. In this section we present some compu- +tational results on pointwise extremal configurations on S3. Recall that Tk ⊆ S3 denotes the +k-stack; cf. Example 2.10. The Tk consists of the north pole and the vertices of k stacked +3-simplices, for a total of 4k + 1 points. +We use similar indexing for the points in Tk, that is, the north pole is denoted Z, then +we label the verticees of i-th tetrahedron (counting from the north pole) by Pi, Qi, Ri, Si +in such a way that all the Pi, 1 ≤ i ≤ k are on a common longitude and similarly for all +Qi, 1 ≤ i ≤ k, Ri, 1 ≤ i ≤ k, and Si, 1 ≤ i ≤ k. +Theorem 6.1. Applying the diameter gradient flow to the initial sets of the diameter gradient +flow be the subsets of T1, T2, T3, T4 with at most four points removed, one obtains at least 65 +distinct pointwise-extremal configurations which are not pyramids. 3 +3A comprehensive table containing statistics for the 65 configurations, similar to Table 1, can be accessed +through the following link: https://ndag.github.io/anti-self-dual-polyhedra/table.html + +EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE +21 +Through exact calculation via the Python package NetworkX([HSS]), we find that the +diameter graph of each of these 65 configurations has chromatic number equal to 5 and also +satisfies e = 3v − 5 where e and v are the number of edges and the number of vertices in the +diameter graph, respectively. +Appendix A. Semi-algebraic sets +Let k ≥ 1. Let R[x1, . . . , xk] be the k-dimensional ring of polynomials with real coefficients. +We now introduce the notion of semi-algebraic subset following [BCR13]. +Definition A.1 ([BCR13, Definition 2.1.4]). Let {ri}s +i=1 be a set of positive integers. A +semi-algebraic subset of Rn is a subset of the form +s� +i=1 +ri +� +j=1 +{x ∈ Rn | fi,j ∗i,j 0} , +where fi,j ∈ R [X1, . . . , Xn] and the operation ∗i,j is either < or =, for i = 1, , . . . , s and +j = 1, . . . , ri. +Definition A.2. A collection A of subsets of a set X is called an algebra of sets if A +contains the empty set and is closed under finite union, finite intersection and under taking +complements. +Remark A.3. Semi-algebraic subsets of Rn form the smallest algebra of sets that contains +all sets of the form +{x ∈ Rn | f(x) > 0} , where f ∈ R [X1, . . . , Xn] . +Definition A.4 ([BCR13, Definition 2.7.1]). A basic open semi-algebraic subset of Rn is a +set of the form +{x ∈ Rn | f1(x) > 0, . . . , fs(x) > 0} +where f1, . . . , fs ∈ R [X1, . . . , Xn]. A basic closed semi-algebraic subset of Rn is a set of the +form +{x ∈ Rn | f1(x) ≥ 0, . . . , fs(x) ≥ 0} +where f1, . . . , fs ∈ R [X1, . . . , Xn] +By applying Morse theory, Milnor [Mil64] obtained the following bound on the number of +Betti numbers of a closed basic semi-algebraic set. +Theorem A.5 ([Mil64, Theorem 3]). If X ⊂ Rn is a basic closed semi-algebraic subset +defined by p polynomial inequalities f1 ≥ 0, . . . , fp ≥ 0 of degree ≤ d, then the sum of the +Betti numbers of X is at most 1 +2(dp + 2)(dp + 1)n−1. +References +[AA17] Michal Adamaszek and Henry Adams, The Vietoris–Rips complexes of a circle, Pacific Journal of +Mathematics 290.1 (2017) 1–40. +[AAF18] Michal Adamaszek, Henry Adams and Florian Frick, Metric reconstruction via optimal transport, +SIAM Journal on Applied Algebra and Geometry 2.4 (2018) 597–619. +[Arm66] Larry Armijo, Minimization of functions having lipschitz continuous first partial derivatives, Pacific +Journal of mathematics 16 (1966), no. 1, 1–3. +[BCR13] Jacek Bochnak, Michel Coste, and Marie-Fran¸coise Roy, Real algebraic geometry, vol. 36, Springer +Science & Business Media, 2013. + +22 +MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG +[BHMH18] Logan Beal, Daniel Hill, R Martin, and John Hedengren, Gekko optimization suite, Processes 6 +(2018), no. 8, 106. +[Bor33] Karol Borsuk, Drei s¨atze ¨uber die n-dimensionale euklidische sph¨are, Fundamenta Mathematicae 20 +(1933), no. 1, 177–190. +[Bro12] Arne Brondsted, An introduction to convex polytopes, vol. 90, Springer Science & Business Media, +2012. +[Dev21] Alexandre Devert, Miniball, https://github.com/marmakoide/miniball, 2021. +[Erd46] Paul Erd¨os, On sets of distances of n points, The American Mathematical Monthly 53 (1946), no. 5, +248–250. +[Erd67] P Erd¨os, On some applications of graph theory to geometry, Canadian Journal of Mathematics 19 +(1967), 968–971. +[Gr¨u56] B Gr¨unbaum, A proof of v´azsonyi’s conjecture, Bull. Res. Council Israel, Sect. A 6 (1956), 77–78. +[Hep56] A Heppes, Beweis einer vermutung von a. V´azsonyi, Acta Mathematica Hungarica 7 (1956), no. 3-4, +463–466. +[Hor21] ´Akos. G Horv´ath, Strongly self-dual polytopes and distance graphs in the unit sphere, Acta Mathe- +matica Hungarica 163 (2021), no. 2, 640–651. +[HP34] Heinz Hopf and Erika Pannwitz, Aufgabe nr. 167, Jahresbericht Deutsch. Math.-Verein 43 (1934), +114. +[HSS] +Aric A Hagberg, Daniel A Schult, and Pieter J Swart, Exploring Network Structure, Dynamics, +and Function using NetworkX, Proceedings of the 7th Python in Science Conference (SciPy 2008), +11–16. +[JB14] +Thomas Jenrich and Andries E Brouwer, A 64-dimensional counterexample to borsuk’s conjecture, +The Electronic Journal of Combinatorics (2014), P4–29. +[Kat89] Mikhail Katz, Diameter-extremal subsets of spheres, Discrete & Computational Geometry 4 (1989), +no. 2, 117–137. +[Kat91] +, On neighborhoods of the kuratowski imbedding beyond the first extremum of the diameter +functional, Fundamenta Mathematicae 137 (1991), no. 3, 161–175. +[KK93] Jeff Kahn and Gil Kalai, A counterexample to Borsuk’s conjecture, Bulletin of the American Math- +ematical Society 29 (1993), no. 1, 60–62. +[LMO22] Sunhyuk Lim, Facundo M´emoli, and Osman B Okutan, Vietoris–Rips persistent homology, injec- +tive metric spaces, and the filling radius, Accepted to appear in Algebraic & Geometric Topology +(2022). +[Kalai94] Gil Kalai. Some aspects of the combinatorial theory of convex polytopes. Polytopes: abstract, +convex and computational (Scarborough, ON, 1993), 205–229, NATO Adv. Sci. Inst. Ser. C: Math. +Phys. Sci., 440, Kluwer, Dordrecht, 1994. +[Lov78] L´aszl´o Lov´asz, Kneser’s conjecture, chromatic number, and homotopy, Journal of Combinatorial +Theory, Series A 25 (1978), no. 3, 319–324. +[Lov83] L´asl´o Lov´asz, Self-dual polytopes and the chromatic number of distance graphs on the sphere, Acta +Sci. Math.(Szeged) 45 (1983), no. 1-4, 317–323. +[Mil64] John Milnor, On the Betti numbers of real varieties, Proceedings of the American Mathematical +Society 15 (1964), no. 2, 275–280. +[MS71] Peter McMullen and Geoffrey Colin Shephard, Convex polytopes and the upper bound conjecture, +Cambridge University Press, 1971. +[Per47] Julian Perkal, Sur la subdivision des ensembles en parties de diam`etre inf´erieur, Colloq. Math, vol. 1, +1947, p. 45. +[Rai12] Andrei M Raigorodskii, On the chromatic numbers of spheres in Rn, Combinatorica 32 (2012), +no. 1, 111–123. +[Rai13] Andrei M Raigorodskii, Cliques and cycles in distance graphs and graphs of diameters., Discrete +geometry and algebraic combinatorics, 2013, pp. 93–109. +[San46] LA Santal´o, Convex regions on the n-dimensional spherical surface, Annals of Mathematics (1946), +448–459. +[Str57] +Stefan Straszewicz, Sur un probleme g´eom´etrique de P. Erd¨os, Bull. Acad. Polon. Sci. Cl. III 5 +(1957), 39–40. + +EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE +23 +[Wal70] David W Walkup, The lower bound conjecture for 3-and 4-manifolds, Acta Mathematica 125 (1970), +75–107. +[Zie12] +G¨unter M Ziegler, Lectures on polytopes, vol. 152, Springer Science & Business Media, 2012. +Bar Ilan University. +Email address: katzmik@math.biu.ac.il +The Ohio State University. +Email address: facundo.memoli@gmail.com +University of Utah. +Email address: qswang@math.utah.edu + diff --git a/99FPT4oBgHgl3EQfZDSN/content/tmp_files/load_file.txt b/99FPT4oBgHgl3EQfZDSN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e0177c16f1b1d0f6ba82c9037a728dd6d7c597d --- /dev/null +++ b/99FPT4oBgHgl3EQfZDSN/content/tmp_files/load_file.txt @@ -0,0 +1,859 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf,len=858 +page_content='EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We generate anti-self-polar polytopes via a numerical implementation of the gradient flow induced by the diameter functional on the space of all finite subsets of the sphere, and prove related results on the critical points of the diameter functional as well as results about the combinatorics of such polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We also discuss potential connections to Borsuk’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal sets 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The pyramid construction 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Construction of k-stacks 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Minimal sets on S2 with diameter below the first accumulation critical value 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Configuration space 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Finiteness results 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A labeling strategy for the points in Bk 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Anti-self-polar polytopes 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' ASP polytopes 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Borsuk’s conjecture 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof of Lovasz’s theorem 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 4-dimensional polytopes 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Implementation of the diameter gradient flow 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Computational results 17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal configurations on S2 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal configurations on S3 20 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Semi-algebraic sets 21 References 21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Introduction Let (X, dX) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The Kuratowski embedding x �→ dX(x, ·) is an embedding of X into L∞(X), the space of all bounded real-valued functions on X with the uniform norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When X is the unit sphere with its geodesic distance, the homotopy types of the r-neighborhoods Br(X, L∞(X)) in the Kuratowski embedding of X were studied by Katz in [Kat91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The values at which the homotopy type changes are closely related to the critical configurations of the diameter functional diam of X which maps a finite subset A of X to diam(A) := maxa,a′∈A dX(a, a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When X is the unit circle, such critical values turn out to be exactly one-half of the diameter values of odd regular polygons inscribed in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Note that 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='13076v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='CO] 30 Jan 2023 2 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG the vertex sets of odd regular polygons are exactly the configurations that are local minima of the diameter functional on the space of all finite subsets of S1 equipped with Hausdorff distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In [Kat89], Katz studied the diameter-extremal configurations on S2 and S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The latter provide candidates for testing Borsuk’s conjecture in R4 (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Recently, Lim, M´emoli, and Okutan [LMO22, Theorem 5] proved that the homotopy types of neighborhoods of the Kuratowski embedding of X are naturally homotopy equivalent to the so-called Vietoris–Rips complexes of X, a central object in the field of applied algebraic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore, the study of diameter-extremal configurations is also of interest for understanding the properties of the Vietoris-Rips complex of spheres [AA17, AAF18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In this paper, we extend the investigation of diameter-extremal configurations on spheres started in [Kat89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In the S1 case, the critical values of the diameter functional form a convergent sequence with the only accumulation point being π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' It is natural to wonder to what extent a similar behavior is true on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We consider two canonical families of diameter- extremal configurations on S2 which we call pyramids Ak that contains 2k + 2 points (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) and stacked-triangles Bk that contains 3k + 1 points (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Both families contain infinitely many members with diameters monotonically approaching 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We prove in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 that 2π 3 is in fact the first accumulation point of the set of critical values of the diameter functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='17, we prove that the two families Ak and Bk do not exhaust all the possible configurations with similar diameter bounds, and in fact there are infinitely many additional diameter-extremal configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Diameter-extremal configuration with 3k points can be found by performing diameter gradient flow on a certain subset of Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When k is odd, by a parity argument, the resulting configuration cannot be an instance of Ak or Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We next devise and implement a computational algorithm (see Algorithm 1) that attempts to produce diameter extremal configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We use this algorithm to find new configu- rations not in Ak or Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Furthermore, we found configurations not isometric to the ones produced in the course of proving Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' See Table 1 for a complete list of all the configurations we found in this way with up to 10 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The list contains 10 previously unknown configurations where 8 of those exhibit Z2 symmetry and the remaining two are asymmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see Figures 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The convex hulls of certain diameter-extremal configurations give rise to anti-self-polar polytopes (ASP), for example, the regular tetrahedron and any Ak or Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' ASPs are polytopes P characterized by the property that the polar of P equals −P (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' ASPs have been studied by Lov´asz in the context of answering a question by Erd¨os and Graham [Lov83] and were also considered in [Kat89, Section 5] in the context of Borsuk’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Borsuk’s conjecture (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2) for a finite point set X in Rn is equivalent to the property that the chromatic number of the diameter graph (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7) of X is bounded above by n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We continue to explore the suggestion in [Kat89] to use diameter- extremal configurations on S3 to test Borsuk’s conjecture in R4 (a case that is still open).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As shown by Lovasz [Lov83], the chromatic number of the diameter graph associated to any ASP in Rn is at least n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' An ASP for which the inequality is strict would disprove Borsuk’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' It was conjectured in [Kat89] that the number of edges in the diameter graph of an ASP 4-polytope with v vertices is at least 3v − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We use Kalai’s inequality from [Kalai94, Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3] to prove such a bound in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='21 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We then formulate conjectures EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 3 about the number of edges in the diameter graph for more general subsets on S3, see Conjec- tures 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='22 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A calculation based on these two conjectures suggests that the maximum possible chromatic number of the diameter graph of a finite subset X ⊆ R4 is 6 instead 5, the number predicted by Borsuk’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We perform experiments attempting to identify diameter-extremal configurations on the three-dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The interest in these experiments is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' On the one hand, it is naturally interesting to obtain an understanding of critical configurations beyond the case of S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' On the other hand, whereas Borsuk’s conjecture is known to be true in dimensions 2 and 3 but false in dimensions 64 and higher, its status for dimension 4 is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Hence, by the above, it is tempting to seek a diameter-extremal configuration X of S3 whose convex hull is an ASP such that its diameter graph has chromatic number at least 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We discovered 65 new configurations on S3 not obtained by the pyramid construction (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) on a previously known configuration on S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' However, all the diameter graphs of these configurations have a chromatic number precisely 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' This work was partially supported by BSF #2020124, NSF CCF #1740761, and NSF IIS #1901360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal sets Let Sn ⊆ Rn+1 be the unit sphere with its geodesic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For a subset Y ⊆ Sn, its diameter diam(Y ) is computed with respect to the geodesic distance on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1 (Taut sets in Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A finite subset Y ⊂ Sn is taut if one of the following equivalent conditions is satisfied: (1) the convex hull of Y contains the origin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' (2) there are non-negative real numbers {ay}y∈Y , not all zero, satisfying � y∈Y ay y = 0, where y denotes the position vector of the point y ∈ Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Jung’s theorem immediately gives the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If Y ⊂ Sn is taut, then diam(Y ) ≥ arccos � −1 n+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The following observation will be useful in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊂ Sn be a taut set such that |Y | = n+2 and diam(Y ) < arccos � − 1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the dimension of the vector space spanned by Y is equal to n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In particular, if {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , an+2} is any set of non-negative coefficients such that n+2 � i=1 aiyi = 0, then all ai must be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Suppose the vector space spanned by all points in Y is of dimension at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then, the set {y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , yn+2} must lie on some great sphere Sn−1 ⊆ Sn and it must be taut in Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2, the set Y must have diameter at least arccos � − 1 n � which contradicts the assumptions on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' This concludes the first part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For the second part, without loss of generality, we assume that a1 = 0, then the set of vectors 4 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG {y2, y3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , yn+2} is linearly dependent and hence dim(span{y2, y3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , yn+2}) < n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The contradiction with the first part establishes the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ Let Y be a subset of a metric space (X, dX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For any two points y, y′ ∈ Y , we say that y and y′ are comaximal in Y if dX(y, y′) = diam(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In such a case, y is called a comaximal point with y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We use the notation comaxY (y) to denote the set of all points in Y which are comaximal with y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For two points x, x′ ∈ Sn with distance less than π, there is a unique arclength-parametrized geodesic γx,x′ connecting x to x′ such that γx,x′(0) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Consider the unit tangent vector ˙γx,x′(0) in the tangent space TxSn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We recall the notion of pointwise extremal subsets in Sn as in [Kat89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4 ([Kat89]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊆ Sn be a finite subset with no antipodal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We say that y ∈ Y is held (in place) by Y if the set of vectors ˙γy,y′(0) as y′ runs over comaxY (y) is a taut set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We say that Y is pointwise extremal if every point y ∈ Y is held by Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When n = 1, it is not difficult to see that, for all integers k ≥ 1, the vertex set of an inscribed regular (2k + 1)-gon is pointwise extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The following proposition shows the converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊆ S1 be a pointwise extremal set containing no pair of antipodal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then Y is the vertex set of an odd regular polygon inscribed in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let y ∈ Y and let D = diam(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let RD be the clockwise rotation on S1 by angle D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As y is held by Y ⊆ S1, the set Y must contain both points in S1 at distance D from y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In particular, the set Y is invariant under the rotation RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As Y is a finite subset, the quotient D 2π must be rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let m n be the representation of D 2π in lowest terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the orbit of y under the rotation RD forms the vertex set of an inscribed regular n-gon Y ′ ⊆ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As Y does not contain any antipodal pairs, n is necessarily odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore, Y contains the vertex set of a odd regular n-gon Y ′ of the same diameter as Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then Y must coincide with Y ′ as adding any additional point to the set Y ′ would strictly increase the diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The pyramid construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In this section, we describe a class of pointwise extremal subsets of Sn called pyramids in [Kat89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For any pointwise extremal subset Y ⊂ Sn−1, the pyramid construction provides a corresponding pointwise extremal subset in Sn that consists of a rescaled copy of Y together with one extra point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Sn ⊆ Rn+1 be the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Z = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , 0, 1) denote the “north pole”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let xn+1 be the last coordinate of Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then for each plane {xn+1 = a}, a ∈ R that meets Sn at more than one point, the intersection is a rescaled copy of Sn−1 which we call a horizontal section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Each horizontal section contains a suitable rescaled copy of Y which is isometrically embedded into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The pyramid over Y is the subset of Sn consisting of the north pole Z together with a rescaled copy Y ′ of Y inside some horizontal section such that the diameter of Y ′ equals the distance from Z to the horizontal section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Denote by Pyr(Y ) the pyramid over a pointwise extremal subset Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let x, y ∈ Y be points with dSn−1(x, y) = diam(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let x′, y′ ∈ Pyr(Y ) be points corre- sponding to x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the triple Z, x′, y′ is the vertex set of a spherical equilateral triangle, with spherical angle ∢ x′Zy′ = diam(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Applying the spherical theorem of cosines to the geodesic triangle △ x′Zy′, we obtain the following relationship: diam(Pyr(Y )) = arcsec � sec � diam(Y ) � − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 5 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7 (The Ak family in S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We apply the pyramid construction to the regular (2k + 1)-gon on S1 to obtain a pointwise extremal configuration Ak ⊆ S2, consisting of the north pole of S2 together with a suitably rescaled copy of the regular (2k + 1)-gon, so that diam(Ak) = arcsec � sec � 2kπ 2k+1 � − 1 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In particular, the diameter diam(Ak) tends to 2π 3 as k goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The configuration A2 consists of the north pole and the vertices of a regular pentagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Construction of k-stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Following [Kat89], let a β-digon be the convex region on S2 bounded by two meridians (great semicircles joining the north and south poles), with angle β between the two meridians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Given a β-digon, we now introduce a procedure that will be used to produce a certain type of pointwise extremal set Y ⊆ Sn called a k-stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The digon procedure is a “walking process” on the digon that takes as input an odd integer 2k + 1 ≥ 3 and outputs a suitable step length d1 > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We start walking with equal steps from the north pole on alternating sides of the digon, with step length d1 calibrated so as to get exactly to the south pole after 2k + 1 steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Z ∈ Sn be the north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A regular n-simplex inscribed in the equator Sn−1 ⊂ Sn defines n + 1 meridians passing through the vertices of the simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let ℓ ∈ (0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The set of points on Sn which are at distance ℓ away from the north pole Z is a rescaled (n−1)-sphere Sn−1 ℓ , namely a horizontal section of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The intersection between Sn−1 ℓ and the set of n + 1 meridians is the vertex set of an inscribed n-simplex in Sn−1 ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A k-stacked configuration Y (see Figure 2) consists of the north pole Z together with the union of the vertex sets of k stacked n-simplices each obtained as the intersection of a horizontal (n − 1)-sphere Sn−1 ℓi with the n + 1 meridians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The distances ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , ℓk between the horizontal sections and the north pole are determined by the digon procedure as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let d1 be the step length that comes from the digon procedure with input 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Consider the sequence of numbers {dj}2k+1 j=0 where dj is the distance to the north pole of the point obtained after walking j steps in via the digon procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then, the sequence of numbers {ℓi}1≤i≤k is defined in terms of {di}2k+1 i=0 by setting ℓi = d2i for 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Note that d2k = diam(Y ) and d1 = π − d2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Given an odd integer 2k + 1 ≥ 3, the following system of equations summarizes the computation of di for 1 ≤ i ≤ 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 6 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Each value di in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) is the distance between the point pi shown in this figure and the north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For 1 ≤ i ≤ 2k + 1, the distance between pi and pi+1 is d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The two conditions in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) are obtained by requiring p0 to be the north pole and p2k+1 to be the south pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The conditions in the second line of Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) are obtained by applying the theorem of cosines for the geodesic spherical triangles with vertices {Z, pi, pi+1}, for each 1 ≤ i ≤ 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The third line Equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) is obtained by symmetry considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let βn = arccos( 1 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The values {di}0≤i≤2k+1 are determined by n and k via the following equations (see Figure 2): (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) � � � � � d0 = 0, d2k+1 = π cos(di) cos(di+1) + sin(di) sin(di+1) cos(βn) = cos(d1), 1 ≤ i ≤ 2k di + d2k+1−i = π, 0 ≤ i ≤ 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let d1 be the output of the digon procedure with input 2k + 1 on a digon of angle β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If we perform the “walking process” on a digon of angle π − β with complementary step length π − d1, we will eventually get close to the south pole (but will not reach it) and then will start walking back to the north pole and reach it after 2k + 1 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If we add an edge between the points that we traveled during the “walking process”, we obtain the diameter graph (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7) of a regular 2k + 1-gon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='9 (The Bk family in S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When n = 2, for each k, we denote the stacks that result from the digon procedure by Bk, which consists of the vertices of k stacked triangles (2-simplices) together with the north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Note that B1 coincides with the configuration A1 from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By construction, diam(Bk) = π − d1 < π − arccos( 1 2) = 2π 3 and limk→∞ diam(Bk) = 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The configuration B2 that consists of the north pole and vertices of two stacked triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The green dash lines are meridians;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' the red dot is the north pole, and points of the same color are of the same distance to the north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 (The Tk family in S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For each k, we denote the stacks that result from the digon procedure by Tk, which consists of the vertices of k stacked tetrahedra together with the north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Minimal sets on S2 with diameter below the first accumulation critical value Let d > 0 and let D(S2, d) be the set of all finite subsets Y ⊂ S2 with diam(Y ) < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As each finite subset on S2 is closed, the Hausdorff distance dH is a metric on D(S2, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1 (Diameter-extremal sets in D(S2, d) [Kat89]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A subset Y ∈ D(S2, d) is called diameter-extremal for the diameter functional if there is a little-o function such that diam(Y ) ≤ diam(Y ′) + o( dH(Y, Y ′)) for all Y ′ ⊂ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In other words, we have lim dH(Y ′,Y )→0 diam(Y ′) − diam(Y ) dH(Y ′, Y ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' An n-point set Y is diameter-extremal if and only if at the corresponding point in the configuration space (S2)×n, the gradients of the distances between pairs of points at maximal distance form a taut set (see further in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3 ([Kat89, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A diameter-extremal set Y ∈ D(S2, 2π 3 ) is necessarily pointwise extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4 (Minimal set in D(S2, d) [Kat89]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A subset Y ∈ D(S2, d) is called a minimal set if there is some δ > 0 such that diam(Y ) ≤ diam(Y ′) for all finite subsets Y ′ with dH(Y, Y ′) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Clearly, every minimal set is diameter-extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In fact, there is a converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5 ( [Kat89, Theorem 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Every diameter-extremal set in D(S2, 2π 3 ) is a minimal set on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By a mountain-pass argument, one obtains the following consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='6 ([Kat89, Corollary 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' There is exactly one (up to congruence) minimal set in each connected component of D(S2, 2π 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 8 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We will now estimate the number of such connected compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We use the notation k� diam≤d S2 to denote the set of all tuples (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , yk) in �k S2 such that the diameter of its associated set {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , yk} is less than or equal to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Note that, for any ϵ > 0, we have a natural continuous map k � diam≤d S2 −→ D(S2, d + ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By realizing k� diam≤d S2 as a closed semi-algebraic set, we obtain the following upper bound on the number of connected components in k� diam≤d S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We set sk = 2k+ k(k+1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then, for every d > 0, the number b0(k, d) of connected components of k� diam≤d S2 satisfies b0(k, d) ≤ 2sk(4sk − 1)3k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We will first describe the set k� diam≤d S2 as a closed basic semi-algebraic set in R3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let xi,j, where 1 ≤ i ≤ k and 1 ≤ j ≤ 3, denote the standard coordinates in R3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the set k� diam≤d S2 is characterized by the following conditions: � x2 i,1 + x2 i,2 + x2 i,3 = 1 for all 1 ≤ i ≤ k, (xi,1 − xi′,1)2 + (xi,2 − xi′,2)2 + (xi,3 − xi′,3)2 ≤ d2 for all 1 ≤ i < i′ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore the set k� diam≤d S2 is a basic semi-algebraic set given by sk = 2k + k(k+1) 2 non-strict inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5 implies that b0(k, d) ≤ 1 2(2sk + 2)(2sk + 1)3k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Finiteness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3 in [Kat89] imply the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='8 ([Kat89]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let 0 < d < 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ∈ D(S2, d) be a pointwise extremal set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then for any pair of distinct points y, y′ in Y , the distance dS2(y, y′) is at least arccos � 2 cos2(d) cos2(d/2) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By a packing argument on the sphere, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For each ϵ > 0, there is a positive integer N(ϵ) such that every pointwise extremal subset Y of diameter less than 2π 3 − ϵ contains fewer than N(ϵ) points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For each 0 < ϵ < 2π 3 , there are only finitely many diameter-extremal sets in D(S2, 2π 3 − ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In particular, 2π 3 is the first accumulation point of the critical values of the diameter functional of S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let dϵ = 2π 3 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5, it suffices to show that there are only finitely many minimal sets in D(S2, dϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='9, there is some N such that every pointwise extremal set in D(S2, dϵ) contains no more than N points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore the image of the continuous map φ φ : N � diam≤dϵ S2 −→ D(S2, 2π 3 ) contains all pointwise extremal configurations with diameter less than or equal to dϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3, the image of φ (in particular) contains all minimal sets with diameter not exceeding dϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Cϵ be the number of connected components which contain a minimal set with diameter no more than dϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='6, the number of minimal sets in D(S2, dϵ) is at most Cϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As the image of φ contains all minimal sets with diameter no more than dϵ, the number Cϵ is bounded by the rank of the map φ∗ : H0 � N � diam≤dϵ S2 � −→ H0 � D(S2, 2π 3 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The claim now follows by invoking the upper bound on the dimension of H0 � N� diam≤dϵ S2 � from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A labeling strategy for the points in Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Recall that Bk ⊆ S2 consists of the north pole and the vertices of k stacked triangles, and that the vertices of the stacked triangles are distributed along three meridians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We label the north pole as Z, then label the vertices of the i-th triangle (counting from the north pole) by Pi, Qi, Ri in such a way that all the Pi, 1 ≤ i ≤ k are on a common longitude and similarly for all Qi, 1 ≤ i ≤ k and Ri, 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The subset dEBk is obtained by removing the points with indexes in E from Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A set Y ⊆ S2 is separable if for each pair of points x, y ∈ Y there are two other points z, w ∈ Y such that the 4-tuple {x, y, z, w} is taut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='13 ([Kat89, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A pointwise extremal subset Y ⊂ S2 with diam(Y ) < 2π 3 is necessarily separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The proof of the above lemma in [Kat89] gives the following stronger result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊂ S2 be a subset with diam(Y ) < 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Suppose x ∈ Y is held by Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then for any other point y ∈ Y , there exist z, w ∈ Y such that the four-point set {x, y, z, w} is taut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We will now analyze variations of subsets which are continuous with respect to the Haus- dorff distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let {Yt, t ∈ [0, 1]} be a continuous family of subsets of S2 with at most 4 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Suppose the following two conditions hold: the set Y0 is taut, Yt ∈ D(S2, 2π 3 ) for every t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 10 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG Then Yt is taut for each t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As the set Y0 is taut and diam(Y ) < 2π 3 , Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3 implies that the convex hull H0 of Y0 is a tetrahedron and that the origin 0 is in interior of H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For each t ∈ [0, 1] let Ht be the convex hull of the set Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' To show that each set Yt is taut, it suffices to show that the origin 0 stays in the interior of Ht for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Suppose the contrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let t0 be the supremum of t such that 0 is in the interior of Ht for all smaller values of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Either Ht0 is nondegenerate and then 0 must belong to one of its (triangular) faces, or it is degenerate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=', lies in a plane through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In either case, we obtain a taut subset of the circle given by the intersection of the plane with the sphere, and can apply Jung’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Namely, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2 we obtain diam(Yt0) ≥ 2π 3 , contradicting the hypothesis Yt0 ∈ D(S2, 2π 3 ) and proving the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Yt, t ∈ [0, 1] be a path in D(S2, 2π 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If a certain 4-tuple in Y0 is taut, then it continues to be taut for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' There exist infinitely many (up to congruence) pointwise extremal sets in D(S2, 2π 3 ) that are not contained in the family Ak or Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Since each connected component contains a (unique) minimal set, it suffices to show that for each k, the configuration dPkBk is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='14, we can separate most pairs of points from dPkBk except for a pair of points from the triple of points at maximal distance from Pk, namely the points Z, Q1, and R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let us check that such pairs don’t coalesce, either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' This is immediate from the fact that if we remove all layers except the first and the k-th, the remaining configuration is in the connected component in D(S2, 2π 3 ) of the 7-point minimal set B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Thus, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='16, it suffices to check that if we remove P2 from B2, no remaining points coalesce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' This can be checked directly, and also follows from the fact that the diameter flow applied to the 6-point configuration dP2B2 produces the 6-point minimal set A2 (see Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Anti-self-polar polytopes In this paper, we adopt the following restricted definition of a polytope: a (convex) polytope will be the convex hull of any finite set of points in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The affine hull aff(S) of a set S ⊆ Rn is aff(S) = � k � i=1 αixi ����� k > 0, xi ∈ S, αi ∈ R, k � i=1 αi = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We now give the formal definition of a face of a polytope following [Zie12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2 ([Zie12, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P ⊆ Rd be a convex polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A linear inequality ⟨c, x⟩ ≤ c0 is valid for P if it is satisfied for all points x ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A face of P is any set of the form F = P ∩ � x ∈ Rd : ⟨c, x⟩ = c0 � where ⟨c, x⟩ ≤ c0 is a valid inequality for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 11 The dimension of a polytope P is defined to be the dimension of its affine hull aff(P) (regarded as an affine space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A 3-dimensional polytope is a polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The codimension- one faces of a polytope P are called facets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' the codimension-two faces are called ridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If each face of P is a simplex, then P is called a simplicial polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We will use fi(P) to denote the number of i-faces of the polytope P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When there is no risk of confusion, we will denote fi(P) by just fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For a n-dimensional polytope, the vector (f0, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , fn−1) is called the f-vector of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' ASP polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In [Lov83], Lov´asz introduced the following type of polytopes which we will refer to as anti-self-polar (ASP) polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1 Our terminology will be justified in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3 (Anti-self-polar polytopes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P ⊆ Rn be a n-dimensional polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We say that P is anti-self-polar (ASP) if the following three conditions hold: (1) P is inscribed in the unit sphere Sn−1 ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' (2) P is circumscribed around a sphere centered at the origin with radius s for some 0 < s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' (3) There is a bijection σ between vertices and facets of P such that if v is any vertex then the facet σ(v) is orthogonal to the vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P ⊂ Rn be a polytope containing the origin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Sn−1 r (0) be the sphere centered at 0 ∈ Rn with radius r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The polar body of P with respect to the sphere Sn−1 r (0) is defined to be the set polarr(P) = {x ∈ Rn| ⟨x, y⟩ ≤ r2 for all y ∈ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As shown in [Hor21], the condition for an ASP polytope in Rn can be restated using the terminology of polar bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In terms of our definition of polarity, if P is an ASP polytope, then there exists some r such that the following relation holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see [Hor21, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' polarr(P) = −P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The polar body description shows that for each 0 ≤ i ≤ n − 1, the bijection σ in condition (3) can be extended to a bijection between the set of i-dimensional faces and the set of (n − i − 1)-dimensional faces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see [Hor21, Lemma 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5 ([Kat89, Remark after Theorem 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊂ S2 be a pointwise extremal subset with diam(Y ) < 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the convex hull of Y is an ASP polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The result above no longer holds if the restriction on the diameter is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A counterexample is given by an 8-point configuration Y ⊆ S2 consisting of the vertices of an antiprism over a square (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If the diameter of Y is exactly attained by the diagonals of the two squares and by the pairs that consist of a vertex of one square and one of the two farthest vertices of the other square, then Y is pointwise extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' However, the convex hull of Y is not ASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Indeed, note that the top square is a facet of the convex hull of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If the convex hull of Y were ASP, then there would be a vertex y0 ∈ Y such that the distance from y0 to each vertex of the top square would equal diam(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' But, our construction of Y does not satisfy this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 1Lov´asz [Lov83] and Horv`ath[Hor21] use the terminology “strongly self-dual polytopes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 12 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The antiprism on a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊆ Rn be a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The diameter graph G(Y ) of Y is defined to be the graph with vertex set V (G) = Y and two vertices y, y′ in G are connected if and only if y and y′ are comaximal in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Given a polytope P, we will refer to the diameter graph of the vertex set of P simply as the diameter graph of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We denote the diameter graph of P by G(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The chromatic number χ(G) of a graph G is the smallest number of colors needed to color the vertices so that no two adjacent vertices share the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The following property of the diameter graph G(P) of an ASP polytope P follows from [Lov83, Lemma 2 and Lemma 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Recall that σ denotes the bijection between the vertex set and the set of facets of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In [Lov83, Lemma 1], it is shown that for any two vertices v, v′ of P, the condition v ∈ σ(v′) is equivalent to v′ ∈ σ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P be an ASP polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Two vertices v, v′ in G(P) are connected by an edge in G(P) if and only if v ∈ σ(v′), when viewed as vertices in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 ([Lov83, Theorem 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The diameter graph G(P) of an n-dimensional ASP polytope P ⊆ Rn satisfies χ(G(P)) ≥ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The proof of the theorem is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The chromatic number of a diameter graph G(Y ) of a subset Y ⊂ Rn is closely related to the following conjecture of Borsuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Borsuk’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='11 (Borsuk’s conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y be a bounded subset of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then there is a partition of Y into n + 1 sets each of which has a smaller diameter than Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For finite subsets, Borsuk’s conjecture has the following equivalent form in terms of diam- eter graphs: For every finite bounded subset Y ⊆ Rn, the chromatic number of the diam- eter graph G(Y ) of Y is no greater than n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' To see the above equivalence, a partition {Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Yk} of Y is equivalent to a coloring of Y by requiring that two points are of the same color if and only if they both belong to EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 13 some Yi, 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore, since Y is a finite set, the condition that the diameter of each subset Yi is less than the diameter of Y is equivalent to requiring that the coloring associated to the partition {Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Yk} has the property that no two adjacent vertices in the diameter graph G(Y ) share the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Borsuk’s conjecture holds when n = 2 (Borsuk [Bor33]) and n = 3 (Perkal [Per47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The general conjecture was disproved by Khan and Kalai [KK93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The lowest dimensional coun- terexample currently known was constructed by Jenrich and Brouwer (and based on a con- struction by Bondarenko) in dimension 64 [JB14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For additional information on the histori- cal developments on the construction of counterexamples to Borsuk’s conjecture, see [Rai13, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊂ Sn−1 be a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Given a regular geodesic n + 1-simplex ∆geodesic n+1 , Sn−1 can be partitioned into n + 1 connected parts {X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Xn+1} where each Xi contains the interior of one of the faces of ∆geodesic n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore, by coloring points of Y according to which partition set Xi the point belongs to, we obtain a proper coloring of the diameter graph of Y provided that the diameter diam(Y ) diameter of Y is greater than ηn−1, the diameter of a face of ∆geodesic n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The above coloring strategy was first described in [Lov83, Section 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Though notice that [Lov83] made a mistake in computing the exact value of ηn−1 [Rai12, Rai13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The correct values of ηn−1 first appeared in [San46] and reproduced in the context of ASP polytopes in [Hor21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 and the fact that Borsuk’s conjecture is true for n = 3, the chromatic number χ(G(P)) of an ASP polyhedron P ⊆ R3 equals 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In Figures 7 and 8, we display 4-colorings of the diameter graphs of all the ASP polyhedra in Tables 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Borsuk’s conjecture is still open for 4 ≤ n ≤ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 suggests that ASP polytopes are a natural source of potential counterexamples to Borsuk’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Additionally, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5, pointwise extremal configurations are closely related to ASP polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2, we present some pointwise extremal subsets on S3 obtained through computer experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' However, the pointwise extremal subsets that we have found so far all have chromatic number 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof of Lovasz’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 was proved in [Lov83] by analyzing the neighborhood complex of the diameter graph of ASP polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='14 (Neighborhood complex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let G be a finite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The neighborhood complex N(G) is the simplicial complex with vertex set V (G) such that a subset A ⊆ V (G) forms a simplex if and only if the points of A have a neighbor in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In [Lov78], Lov´asz shows the following lower bound of the chromatic number of a graph with respect to the connectivity of its neighborhood complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Recall a topological space X is k-connected if its homotopy groups are trivial up to degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='15 ([Lov78]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let G be a graph and suppose that N(G) is k-connected (k ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then χ(G) ≥ k + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='16 ([Lov83, Lemma 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P be an ASP polytope and G(P) be its diameter graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then N(G(P)) is homotopy equivalent to the boundary of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='16, N(G) is homotopy equivalent to the boundary of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 14 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG As P is a (convex) polytope, the boundary of P is homeomorphic to Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Hence N(G) is homotopy equivalent to Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore, N(G) is (n − 2) connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='15, χ(G) ≥ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ Let d ≥ 2 and n ≥ 1 be integers and let e(d, n) be the maximum possible number of edges in the diameter graph of a subset of Rd with n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When d = 2, it is shown in [HP34] that e(2, n) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' This fact leads to one proof of Borsuk’s conjecture for finite subsets Y of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When d = 3, it was conjectured by V´azsonyi that e(3, n) = 2n − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see [Erd46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The V´azsonyi’s conjecture was proved independently by Gr¨unbaum [Gr¨u56], Heppes [Hep56] and Straszewicz [Str57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As mentioned in Heppes [Hep56], V´azsonyi’s conjecture implies that Borsuk’s conjecture is true for finite subsets in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We have already seen in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 that the diameter graph of an ASP polytope has high chromatic number, suggesting a possible approach to seeking higher-dimensional counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We now introduce a set of enumerative invariants fij(P) of a polytope P which will be used below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Informally, for i < j, fij(P) counts the number of pairs “i-face contained in a j-face” in the polytope P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Precisely, fij(P) := ♯{(φi, φj) | φi is a i-face of P, φj is a j-face of P, and φi ⊆ φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='} When there is no risk of confusion, we will simply use fij to denote fij(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Thus f01 is the number of pairs “vertex contained in an edge”, namely just twice the number f1 of edges in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P be an anti-self-polar polytope of dimension d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let e(G(P)) be the number of edges in the graph G(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then f0d(P) = 2e(G(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let V be the set of vertices of P and let W be the set of faces in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Recall that σ denotes the bijection between V and the set of facets of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='9, we have 2e(G(P)) = � v∈V f0(σ(v)) = � φd⊂W f0(φd) = f0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The second equality above follows from the definition of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Every ASP polyhedron P ⊆ R3 satisfies e(G(P)) = 2f0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='17, we have 2e(G(P)) = f02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Furthermore by duality we have f01 = f12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' This enables us to give a possibly generalizable proof as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Note that, each face has as many vertices as edges and therefore f02 = f12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By duality, f12 = f01 which is twice the number of edges, namely 2f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Thus the number of maximal distances is the same as the number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Meanwhile by the formula for the Euler characteristic, for an anti-self-polar polyhedron we have f1 = 2f0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Altogether, we have f02 = f12 = f01 = 2f1 = 2(2f0 − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Thus the number of maximal distances is also 2f0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ In fact, it is shown in [Kat89] that every pointwise extremal set in S2 with diameter less than 2π 3 exhibits the maximum number of possible edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='19 ([Kat89, Theorem 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Suppose Y ⊂ S2 is a pointwise extremal set with N = |Y | and diam(Y ) < 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the number of edges in the diameter graph G(Y ) equals 2N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 15 As noted in [Kat89, page 118], the example of the antiprism on a square constructed in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='13 shows that the above result is no longer true if we remove the diameter constraint: the diameter graph of the antiprism on a square has 8 vertices but only 12 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 4-dimensional polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Consider the V´azsonyi’s problem in R4, that is, for a fixed n, determine the maximal possible number of edges e(4, n) amongst the diameter graphs of all possible n point sets in R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let m be a positive integer and let Y := A ∪ B ⊂ S3 be a subset consisting of 2m points constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The set A consists of m points on an arc of length less than π 2 on a great circle whereas the (disjoint) set B consists of m points also on an arc of length less than π 2 on an orthogonal great circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then each pair of points a ∈ A, b ∈ B is comaximal in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Thus e(4, n) is at least quadratic in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' It is shown in [Erd67] that e(4, n) exactly has quadratic growth rate in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For an anti-self-polar polytope P ⊆ R4, we prove the following lower bound on the number of edges in the diameter graph G(P), originally conjectured in [Kat89, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P ⊆ R4 be a 4-dimensional anti-self-polar polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the number of edges e(G(P)) in the diameter graph G(P) is at least 3f0(P) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='17, the assertion is equivalent to the bound f03(P) ≥ 6f0(P) − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For each facet φ of P, let aj φ be the number of j-gons occurring as faces of φ, and let aj denote the total number of j-gons occurring as faces of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Kalai [Kalai94, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3] proved that every 4-dimensional polytope satisfies g2 ≥ 0 or equivalently a4 + 2a5 + · · · ≥ 4f0(P) − f1(P) − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let φ run through all the facets of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Euler’s formula, we have f03(P) = � φ f0(φ) = � φ 2 + f1(φ) − f2(φ) = � φ 2 + 1 2(3a3 φ + 4a4 φ + 5a5 φ + · · · ) − f2(φ) = � φ 2 + 1 2f2(φ) + 1 2(a4 φ + 2a5 φ + · · · ) = 2f3(P) + f2(P) + 1 2 � φ (a4 φ + 2a5 φ + · · · ) = 2f3(P) + f2(P) + (a4 + 2a5 + · · · ) ≥ 2f3(P) + f2(P) + 4f0(P) − f1(P) − 10 = (2f3(P) + 4f0(P)) + (f2(P) − f1(P)) − 10 = 6f0(P) − 10 by duality, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ The above results suggest formulating the following conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 16 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Every ASP polytope P ⊆ R4 satisfies e(G(P)) = 3f0(P) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2, we report 65 configurations that we generate through numerical experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Each of those configurations confirms the above conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Every subset X ⊆ S3 with diam(X) > π 2 satisfies e(G(X)) ≤ 3|X| − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Assuming these conjectures and by an argument similar to the case of the S2 discussed on page 14, one can show that the chromatic number of the diameter graph of any set X in S3 with its diameter greater than π 2 would be at most 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Indeed, Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='23 implies that one can always choose a point x0 ∈ X comaximal with at most 5 other points, by the pigeonhole principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Thus, if X − {x0} can be colored with 6 colors, then X can be so colored, also, by using the color not used up by any of its 5 (or fewer) comaximal points, and we conclude by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The fact that this calculation produces the number 6 instead of 5 would provide weak evidence toward the possibility that the Borsuk number of R4 might be the former rather than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Implementation of the diameter gradient flow This section describes the implementation of the diameter gradient flow on spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Given a finite subset Y of Sn, we first test whether every point in Y is held.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If there is a point y that is not held by Y , we then move y in the direction that points toward the center of the minimum bounding sphere of the tangent vectors determined by points in comaxY (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We continue this process until every point in Y is held.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In other words the point y is updated to a point yt = y0+tv0 ||y0+tv0|| where t > 0 is a parameter value determined through the Armijo rule [Arm66], and v0 is the unit tangent vector at y that points toward the center of the minimum EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 17 bounding sphere of the set {˙γy,y′ | y′ ∈ comaxyY }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The pseudocode of the algorithm is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Algorithm 1: DiameterGradientFlow Input: An initial finite subset Y on unit sphere Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Parameters: β, η ∈ (0, 1) for determing Armijo Rule stepsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Output: The extremal configurations obtained under the diameter gradient flow with initial condition Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 1 Function IsHeld(y, Y ): 2 E ← comaxY (y) 3 Ty(E) ← {˙γy,y′ for y′ ∈ comaxY (y)} 4 if 0 in the convex hull of Ty(E) then 5 return True 6 else 7 return False 8 end if 9 10 Function Main(Y , β, η): 11 /* Initialize convergence tag / 12 tag = False 13 while tag == False do 14 for y0 ∈ Y do 15 if IsHeld(y0, Y ) then 16 tag == True 17 else 18 E ← comaxY (y) 19 Ty0(E) ← {˙γy,y′} for y′ ∈ E} 20 v0 ← center of the minimum bounding sphere of Ty(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 21 /* Determine the step size tk > 0 using Armijo Rule / 22 tk = maxl∈N0 βl s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' diam � Y \\{y0} ∪ � y0+tkv0 ||y0+tkv0|| �� ≤ diam(Y ) − βlη 23 Y ← Y \\{y0} ∪ � y0+tkv0 ||y0+tkv0|| � 24 tag == False 25 break 26 end if 27 end for 28 end while 29 return asee [Arm66] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Computational results In this section, we describe our computational results regarding pointwise extremal config- urations on S2 and S3 using the Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In most of our experiments, we set parameters β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='001 and use the Python package MINIBALL([Dev21]) for finding the optimal direction for decreasing the diameter by moving a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 18 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal configurations on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In this section, we present the compu- tational results from running the diameter gradient flow Algorithm 1 with initial sets, which are obtained by removing up to six points from Bk (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='9) with k ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In total, we obtain 54 configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We present in Table 1 the configurations with up to 10 points2 that we found upon convergence of the gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Shape v f r t Diameter Symmetry Group Initial Set A1(= B1) 4 3 4 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='91064 S3 dZB2 A2 6 5 1 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='03446 D5 dP2B2 B2 7 4 3 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='07654 S3 dZB3 C1 8 5 1 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='08707 Z2 d{Q1,P3}B3 A3 8 7 1 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='06459 D7 d{P1,P3}B3 C2 9 5 1 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09335 Z2 d{P1,R1,Q3,Q4}B4 C3 9 5 1 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09079 Z2 dP3B3 C4 9 6 1 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09016 Z2 dP1B3 B3 10 4 6 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09303 S3 dZB4 D1 10 5 1 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09409 {e} d{P1,Q1,P2,Q4,R4,R5}B5 C5 10 5 1 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09317 Z2 d{P1,R3,Q4}B4 C6 10 5 2 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09356 Z2 d{P1,Q1}B4 C7 10 5 3 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09240 Z2 d{P1,R3,P4}B4 D2 10 6 1 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09360 {e} d{P1,R3,R4}B4 C8 10 7 1 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09174 Z2 d{P1,P3,Q4}B4 A4 10 9 1 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='07654 D9 d{P1,P3,P4}B4 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal configurations on S2 with up to v = 10 vertices, sorted first by v, then by f (maximal number of edges in a face), then by r (number of faces with a maximal number of edges), then by t (number of triangles in the configuration’s diameter graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For each of the 10 pointwise extremal configurations that we found, in the last column we list one initial set which leads to that configuration under the diameter gradient flow (a given pointwise extremal configuration may be reached from different initial sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 2An interactive visualization of the table can be found through the link: https://ndag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='io/ anti-self-dual-polyhedra/ EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 19 (a) C1 (b) C2 (c) C3 (d) C4 (e) C5 (f) C6 (g) C7 (h) C8 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Eight Z2 symmetric pointwise extremal configurations with at most 10 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' (a) D1 (b) D2 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Two asymmetric pointwise extremal configurations with 10 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 20 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG (a) C1 (b) C2 (c) C3 (d) C4 (e) C5 (f) C6 (g) C7 (h) C8 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Diameter graphs of Z2 symmetric pointwise extremal configura- tions with less than 10 with a minimal coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Note that all diameter graphs above can be colored with four colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' (a) D1 (b) D2 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The diameter graph of the two asymmetric pointwise extremal configurations D1 and D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal configurations on S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In this section we present some compu- tational results on pointwise extremal configurations on S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Recall that Tk ⊆ S3 denotes the k-stack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The Tk consists of the north pole and the vertices of k stacked 3-simplices, for a total of 4k + 1 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We use similar indexing for the points in Tk, that is, the north pole is denoted Z, then we label the verticees of i-th tetrahedron (counting from the north pole) by Pi, Qi, Ri, Si in such a way that all the Pi, 1 ≤ i ≤ k are on a common longitude and similarly for all Qi, 1 ≤ i ≤ k, Ri, 1 ≤ i ≤ k, and Si, 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Applying the diameter gradient flow to the initial sets of the diameter gradient flow be the subsets of T1, T2, T3, T4 with at most four points removed, one obtains at least 65 distinct pointwise-extremal configurations which are not pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 3 3A comprehensive table containing statistics for the 65 configurations, similar to Table 1, can be accessed through the following link: https://ndag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='io/anti-self-dual-polyhedra/table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='html EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 21 Through exact calculation via the Python package NetworkX([HSS]), we find that the diameter graph of each of these 65 configurations has chromatic number equal to 5 and also satisfies e = 3v − 5 where e and v are the number of edges and the number of vertices in the diameter graph, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Semi-algebraic sets Let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let R[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , xk] be the k-dimensional ring of polynomials with real coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We now introduce the notion of semi-algebraic subset following [BCR13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1 ([BCR13, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let {ri}s i=1 be a set of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A semi-algebraic subset of Rn is a subset of the form s� i=1 ri � j=1 {x ∈ Rn | fi,j ∗i,j 0} , where fi,j ∈ R [X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Xn] and the operation ∗i,j is either < or =, for i = 1, , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , s and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A collection A of subsets of a set X is called an algebra of sets if A contains the empty set and is closed under finite union, finite intersection and under taking complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Semi-algebraic subsets of Rn form the smallest algebra of sets that contains all sets of the form {x ∈ Rn | f(x) > 0} , where f ∈ R [X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Xn] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4 ([BCR13, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A basic open semi-algebraic subset of Rn is a set of the form {x ∈ Rn | f1(x) > 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , fs(x) > 0} where f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , fs ∈ R [X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Xn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A basic closed semi-algebraic subset of Rn is a set of the form {x ∈ Rn | f1(x) ≥ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , fs(x) ≥ 0} where f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , fs ∈ R [X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Xn] By applying Morse theory, Milnor [Mil64] obtained the following bound on the number of Betti numbers of a closed basic semi-algebraic set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5 ([Mil64, Theorem 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If X ⊂ Rn is a basic closed semi-algebraic subset defined by p polynomial inequalities f1 ≥ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , fp ≥ 0 of degree ≤ d, then the sum of the Betti numbers of X is at most 1 2(dp + 2)(dp + 1)n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' References [AA17] Michal Adamaszek and Henry Adams, The Vietoris–Rips complexes of a circle, Pacific Journal of Mathematics 290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1 (2017) 1–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [AAF18] Michal Adamaszek, Henry Adams and Florian Frick, Metric reconstruction via optimal transport, SIAM Journal on Applied Algebra and Geometry 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4 (2018) 597–619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Arm66] Larry Armijo, Minimization of functions having lipschitz continuous first partial derivatives, Pacific Journal of mathematics 16 (1966), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 1, 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [BCR13] Jacek Bochnak, Michel Coste, and Marie-Fran¸coise Roy, Real algebraic geometry, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 36, Springer Science & Business Media, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 22 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG [BHMH18] Logan Beal, Daniel Hill, R Martin, and John Hedengren, Gekko optimization suite, Processes 6 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 8, 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Bor33] Karol Borsuk, Drei s¨atze ¨uber die n-dimensionale euklidische sph¨are, Fundamenta Mathematicae 20 (1933), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 1, 177–190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Bro12] Arne Brondsted, An introduction to convex polytopes, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 90, Springer Science & Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Dev21] Alexandre Devert, Miniball, https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='com/marmakoide/miniball, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Erd46] Paul Erd¨os, On sets of distances of n points, The American Mathematical Monthly 53 (1946), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 5, 248–250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Erd67] P Erd¨os, On some applications of graph theory to geometry, Canadian Journal of Mathematics 19 (1967), 968–971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Gr¨u56] B Gr¨unbaum, A proof of v´azsonyi’s conjecture, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Council Israel, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A 6 (1956), 77–78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Hep56] A Heppes, Beweis einer vermutung von a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' V´azsonyi, Acta Mathematica Hungarica 7 (1956), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 3-4, 463–466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Hor21] ´Akos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' G Horv´ath, Strongly self-dual polytopes and distance graphs in the unit sphere, Acta Mathe- matica Hungarica 163 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 2, 640–651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [HP34] Heinz Hopf and Erika Pannwitz, Aufgabe nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 167, Jahresbericht Deutsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='-Verein 43 (1934), 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [HSS] Aric A Hagberg, Daniel A Schult, and Pieter J Swart, Exploring Network Structure, Dynamics, and Function using NetworkX, Proceedings of the 7th Python in Science Conference (SciPy 2008), 11–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [JB14] Thomas Jenrich and Andries E Brouwer, A 64-dimensional counterexample to borsuk’s conjecture, The Electronic Journal of Combinatorics (2014), P4–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Kat89] Mikhail Katz, Diameter-extremal subsets of spheres, Discrete & Computational Geometry 4 (1989), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 2, 117–137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Kat91] , On neighborhoods of the kuratowski imbedding beyond the first extremum of the diameter functional, Fundamenta Mathematicae 137 (1991), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 3, 161–175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [KK93] Jeff Kahn and Gil Kalai, A counterexample to Borsuk’s conjecture, Bulletin of the American Math- ematical Society 29 (1993), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 1, 60–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [LMO22] Sunhyuk Lim, Facundo M´emoli, and Osman B Okutan, Vietoris–Rips persistent homology, injec- tive metric spaces, and the filling radius, Accepted to appear in Algebraic & Geometric Topology (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Kalai94] Gil Kalai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Some aspects of the combinatorial theory of convex polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Polytopes: abstract, convex and computational (Scarborough, ON, 1993), 205–229, NATO Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' C: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=', 440, Kluwer, Dordrecht, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Lov78] L´aszl´o Lov´asz, Kneser’s conjecture, chromatic number, and homotopy, Journal of Combinatorial Theory, Series A 25 (1978), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 3, 319–324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Lov83] L´asl´o Lov´asz, Self-dual polytopes and the chromatic number of distance graphs on the sphere, Acta Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' (Szeged) 45 (1983), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 1-4, 317–323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Mil64] John Milnor, On the Betti numbers of real varieties, Proceedings of the American Mathematical Society 15 (1964), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 2, 275–280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [MS71] Peter McMullen and Geoffrey Colin Shephard, Convex polytopes and the upper bound conjecture, Cambridge University Press, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Per47] Julian Perkal, Sur la subdivision des ensembles en parties de diam`etre inf´erieur, Colloq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Math, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 1, 1947, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Rai12] Andrei M Raigorodskii, On the chromatic numbers of spheres in Rn, Combinatorica 32 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 1, 111–123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Rai13] Andrei M Raigorodskii, Cliques and cycles in distance graphs and graphs of diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=', Discrete geometry and algebraic combinatorics, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 93–109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [San46] LA Santal´o, Convex regions on the n-dimensional spherical surface, Annals of Mathematics (1946), 448–459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Str57] Stefan Straszewicz, Sur un probleme g´eom´etrique de P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Erd¨os, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Polon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' III 5 (1957), 39–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 23 [Wal70] David W Walkup, The lower bound conjecture for 3-and 4-manifolds, Acta Mathematica 125 (1970), 75–107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Zie12] G¨unter M Ziegler, Lectures on polytopes, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 152, Springer Science & Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Bar Ilan University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Email address: katzmik@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='biu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='il The Ohio State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Email address: facundo.' metadata={'source': 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transition region +Gabriel Pelouze1, 2, Tom Van Doorsselaere2 , Konstantinos Karampelas2, 3 , Julia M. Riedl2 , and Timothy Duckenfield2 +1 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405, Orsay, France +2 Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, +Belgium. +e-mail: tom.vandoorsselaere@kuleuven.be +3 Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK +Received 23 September 2022 / Accepted 3 January 2023 +ABSTRACT +Context. Transverse oscillations are ubiquitously observed in the solar corona, both in coronal loops and open magnetic flux tubes. +Numerical simulations suggest that their dissipation could heat coronal loops, counterbalancing radiative losses. These models rely +on a continuous driver at the footpoint of the loops. However, analytical works predict that transverse waves are subject to a cut-off in +the transition region. It is thus unclear whether they can reach the corona, and indeed heat coronal loops. +Aims. Our aims are to determine how the cut-off of kink waves affects their propagation into the corona, and to characterize the +variation of the cut-off frequency with altitude. +Methods. Using 3D magnetohydrodynamic simulations, we modelled the propagation of kink waves in a magnetic flux tube, embed- +ded in a realistic atmosphere with thermal conduction, that starts in the chromosphere and extends into the corona. We drove kink +waves at four different frequencies, and determined whether they experienced a cut-off. We then calculated the altitude at which the +waves were cut-off, and compared it to the prediction of several analytical models. +Results. We show that kink waves indeed experience a cut-off in the transition region, and we identified the analytical model that +gives the best predictions. In addition, we show that waves with periods shorter than approximately 500 s can still reach the corona by +tunnelling through the transition region, with little to no attenuation of their amplitude. This means that such waves can still propagate +from the footpoints of loop, and result in heating in the corona. +Key words. Sun: atmosphere – Sun: oscillations – magnetohydrodynamics (MHD) – waves – methods: numerical +1. Introduction +Recent advances in observations and modelling have shown +that magnetohydrodynamic (MHD) waves could significantly +contribute to the heating of the solar corona (see review by +Van Doorsselaere et al. 2020). In particular, transverse waves are +ubiquitously observed, and they come in several kinds. The type +that was first discovered are the transverse waves that are impul- +sively excited after a flare (Nakariakov et al. 1999). However, +these transverse waves are only sporadically excited and do not +play an important role in the energy budget of the solar corona +(Terradas & Arregui 2018). Later on, it was discovered that the +corona is filled by small-amplitude transverse waves (Tomczyk +et al. 2007; Tomczyk & McIntosh 2009; McIntosh et al. 2011; +Tian et al. 2012). These were observed in coronal loops as prop- +agating (Tiwari et al. 2019) or standing waves (Anfinogentov +et al. 2015). These low-amplitude transverse waves were also +observed as propagating waves in open-field regions (Thurgood +et al. 2014; Morton et al. 2015). These low-amplitude waves +show little-to-no decay (Morton et al. 2021) and are thus named +“decayless”. +Because the flare-excited standing waves are rapidly decay- +ing (Goddard et al. 2016; Nechaeva et al. 2019) due to reso- +nant absorption (Goossens et al. 2002) and non-linear Kelvin- +Helmholtz instability (KHI) damping (Terradas et al. 2008; An- +tolin et al. 2014; Van Doorsselaere et al. 2021; Arregui 2021), +it is generally thought that the decayless waves must be con- +tinuously supplied with energy to counteract its strong damp- +ing. Several mechanisms for excitation have been proposed: slip- +stick driving with steady flows (Nakariakov et al. 2016; Karam- +pelas & Van Doorsselaere 2020), vortex shedding (Nakariakov +et al. 2009; Karampelas & Van Doorsselaere 2021) or footpoint +driving (Nisticò et al. 2013; Karampelas et al. 2017) through p- +modes (Morton et al. 2019) or convective shuffling. The latter +option of footpoint driving has had some success in generating +standing mode decayless waves (Afanasyev et al. 2020), which +counterbalance the non-linear damping through the KHI (Guo +et al. 2019) and lead to heating of loops (Shi et al. 2021). +However, for the driving of decayless waves through their +footpoints, it is not well understood how the transverse waves +propagate through the complicated structure of the chromo- +sphere and transition region. The simulations of transverse-wave +induced KHI heating (e.g. Karampelas et al. 2019) only take into +account the coronal part of the loop, that is imposing a driver at +the top of the transition region. To properly model the whole loop +evolution due to the wave heating, it is essential to also model the +wave driver in the photosphere, and accurately capture its influ- +ence on the coronal loop dynamics. +In plane-parallel atmospheres, the propagation of fast and +slow waves has been well studied. It was found that these modes +couple efficiently to Alfvén waves through resonant absorption +(Hansen & Cally 2009; Cally & Andries 2010; Khomenko & +Cally 2012). Currently, investigations are ongoing to what hap- +pens if the cross-field structuring is included into the wave prop- +agation model (Cally & Khomenko 2019; Riedl et al. 2019, +2021). Another crucial ingredient is the wave’s behaviour in +Article number, page 1 of 8 +arXiv:2301.03100v1 [astro-ph.SR] 8 Jan 2023 + +A&A proofs: manuscript no. kink_cutoff +strong (i.e. non-WKB) stratification. It is well-known that slow +waves experience a cut-off while propagating through a stratified +medium (Bel & Leroy 1977). This has been verified observation- +ally (Jess et al. 2013) and numerically (Felipe et al. 2018). Still, +up to now, it is unknown if a similar cut-off exists for transverse +waves in structured media. For the driving of the observed decay- +less waves in the corona, this is a crucial property to understand. +Several analytical works predict that transverse waves are +cut-off in the transition below a given frequency. The first for- +mula was derived by Spruit (1981): +ω2 +Sp81 = g +8H +1 +2β + 1, +(1) +where g is the gravity projected along the loop, H the pressure +scale height, and β the ratio between the gas and magnetic pres- +sures. For a typical isothermal atmosphere, this corresponds to +a cut-off period of 700 s (Spruit 1981). However, Lopin et al. +(2014) showed that this classical cut-off is suppressed when the +radial component of the magnetic field is taken into account. +Lopin & Nagorny (2017) later showed that transverse waves can +still be cut-off, provided a non-isothermal atmosphere. They pre- +dict the following cut-off frequency: +ω2 +LN17 = +c2 +k0 +4H0H(z) +� +δ2 +B +dH(z) +dz ++ H2(z) +z2 +� +, +(2) +where z is the altitude, ck0 is the kink speed at the base of atmo- +sphere (z = z0), H is the pressure scale height, H0 = H(z0), and +δ2 +B = +� +B2 +0i − B2 +0e +� +/ +� +B2 +0i + B2 +0e +� +is the relative difference between +the magnetic field inside (B0,i) and outside (B0,e) the flux tube, at +z = z0. Finally, an alternative formula was derived by Snow et al. +(2017): +ω2 +Sn17 = v2 +A(z) +4z2 , +(3) +where z is the altitude, and vA is the Alfvén speed. +In this article, we modelled the propagation of kink waves +in an open magnetic flux tube, embedded in a non-isothermal +atmosphere. The atmosphere extends from the chromosphere to +the corona, and includes gravitational stratification and thermal +conduction (Sect. 2). We drove kink waves at different periods, +and determined whether they experienced a cut-off (Sect. 3). +We compare these results to the three analytical formulas given +above in Sect. 4, and summarize our conclusions in Sect. 5. +2. Numerical model: magnetic flux tube through the +transition region +We modelled a vertical magnetic flux tube of radius R = 1 Mm +embedded in a stratified atmosphere, starting in the chromo- +sphere (altitude z = 0 Mm) and extending through the transi- +tion region (z ≈ 4 Mm) into the corona. Kink waves were ex- +cited in the flux tube by applying a monoperiodic driver at the +bottom of the domain (z = 0 Mm). In the upper half of the do- +main (z > 50 Mm), we implemented a “velocity rewrite layer” +to absorb the kink waves. The driver and the velocity rewrite +layer are described in Sect. 2.1. A sketch of the domain is shown +on Fig. 1. We solved the 3D MHD evolution of this tube using +the PLUTO code (Mignone et al. 2007), version 4.3. This code +solves the conservative MHD equations (mass continuity, mo- +mentum conservation, energy conservation, and induction equa- +tion). We used the corner transport upwind finite volume scheme, +x [Mm] +z [Mm] +Driver +Transition +region +Velocity +rewrite +layer +Corona +Magnetic tube +Kink +wave +2 Mm +Chromosphere +0 +−8 +8 +0 +100 +50 +0 +3 +−3 +y [Mm] +Fig. 1. Sketch of the simulation domain, showing the magnetic flux +tube, the location of the kink wave driver (bottom boundary), chromo- +sphere, transition region, corona, and velocity rewrite layer. +where characteristic tracing is used for the time stepping, and a +linear spatial reconstruction with a monotonized central differ- +ence limiter is performed. The magnetic field divergence was +kept small using the extended divergence cleaning method (gen- +eralized Lagrange multiplier, or GLM), and flux was computed +with the linearized Roe Riemann solver. We did not include ex- +plicit viscosity, resistivity, or cooling. However, numerical dis- +sipation results in higher effective viscosity and resistivity than +what is expected for the solar corona, as discussed by Karam- +pelas et al. (2019). We included a modified thermal conduction, +as described below. +The transition region between the chromosphere and the +corona is characterized by a very sharp temperature gradient. +Resolving such gradient requires a very high resolution along +the tube (∼ 1 km in the transition region). In order to keep com- +putational costs reasonable, we artificially broadened the tran- +sition region (thus reducing the temperature gradient). To that +end, we modified the thermal conductivity using the method de- +veloped by Linker et al. (2001); Lionello et al. (2009); Miki´c +et al. (2013). Below the cut-off temperature Tc = 2.5 · 105 K, +the parallel thermal conductivity was set to κ∥ = C0T 5/2 +c +with +C0 = 9 · 10−12 Wm−1K−7/2. Above Tc, κ∥ = C0T 5/2. This al- +lowed us to use a resolution of 98 km along the tube. This grid +allows to fully resolve the broadened transition region, which +has a minimum temperature scale length of 1.6 Mm (see John- +ston & Bradshaw 2019). The dimensions of the domain were +(Lx, Ly, Lz) = (16, 6, 100) Mm. We used a uniform grid of +400 × 150 × 1024 cells, with a size of 40 km in the x and y direc- +tions, and 98 km in the z direction. Furthermore, we verified that +the results did not change significantly when using a resolution +of 40 km in the z direction. To that end, we ran a separate sim- +ulation and verified that the resulting cut-off altitude and com- +parison to the analytical formulas (see Sect. 4) were not strongly +modified. We note that such resolution is too costly in terms of +compute time to be used for all simulations in this work. +The strong stratification in the transition region makes it +challenging to obtain a relaxed initial state for the model. We +first initialized the domain with a field-aligned hydrostatic equi- +librium (Sect. 2.2). We then let the simulation relax in 2D for +47 ks (Sect. 2.3). Finally, we filled the 3D domain with this re- +Article number, page 2 of 8 + +G. Pelouze et al.: Cut-off of transverse waves +0 +20 +40 +60 +80 +100 +Altitude [Mm] +0.9995 +0.9996 +0.9997 +0.9998 +0.9999 +1.0000 +Velocity rewrite coefficient αv +αv(t≤15.7 ks) +αv(t=18.8 ks) +αv(t=21.9 ks) +αv(t=25.1 ks) +αv(t=28.2 ks) +αv(t≥31.3 ks)=αv,3D +Fig. 2. Velocity-rewrite coefficient αv, applied to the velocity above +50 Mm so that upper-propagating waves are not reflected back into the +domain. αv is shown for different times of the 2D relaxation run. The +last profile (t ≥ 31.3 ks) is also applied in the 3D driven simulations. +laxed state through cylindrical symmetry, where we drove kink +waves of different periods for a duration up to 2.7 ks (Sect. 2.4). +2.1. Boundary conditions and driver +We first describe the boundary conditions used for the relaxation +(2D) and kink wave (3D) simulations. +Bottom boundary +At the bottom boundary (base of the chro- +mosphere, z = 0), the density and pressure were extrapolated +using the hydrostatic equilibrium equation. The magnetic field +was extrapolated using the zero normal-gradient condition de- +scribed by Karampelas et al. (2019, section 2.4). For vz, we ei- +ther imposed a reflective boundary condition (2D relaxation, see +Sect. 2.3), or imposed vz = 0 (in 3D, see Sect. 2.4). We verified +that both boundary conditions give the same results in 3D sim- +ulations. The parallel velocity components vx and vy were set to +obey either a zero-gradient boundary condition (2D relaxation), +or to follow a driver that excites kink waves (in 3D). We used +a monoperiodic, dipole-like, driver developed by Pascoe et al. +(2010) and updated by Karampelas et al. (2017). Inside the tube, +the driver imposes: +� +vx(x, y, t), vy(x, y, t) +� += {v(t), 0} , +(4) +where v(t) = v0 cos (2πt/P0), with v0 the driver amplitude, set +to 2 km s−1. The driver period, P0, was set to different values +in order to test the cut-off of kink waves. Outside the tube, the +driver imposes: +� +vx(x, y, t), vy(x, y, t) +� += v(t)R2 +� +(x − x0(t))2 − y2, 2 (x − x0(t)) y +� +� +(x − x0(t))2 + y2�2 +, +(5) +where x0(t) = v0P0/(2π) · sin (2πt/P0) is the centre of the tube’s +footpoint at time t. This driver generates a kink wave polarized +in the x direction. +Upper boundary At the upper boundary (top of the corona, z = +100 Mm), the magnetic field was kept symmetric. All other vari- +ables obeyed a reflective boundary condition. In order to absorb +the upwards waves excited by the driver, we artificially modified +the velocity in the upper half of the domain (z > 50 Mm). At +each time step, after solving the MHD equations, we decreased +each component of the velocity vi by multiplying it by a quantity +αv ≲ 1: +v′ +i = αv(t, z)vi. +(6) +In the driven 3D simulations αv was kept constant in time, and +varied linearly along the loop, from 1 at z = zv = 50 Mm, to +αv,min = 0.9995 at z = L = 100 Mm: +αv,3D(z) = +������� +1 +if z ≤ zv, +1 − �1 − αv,min +� � z−zv +L−zv +� +else. +(7) +In the 2D relaxation run, the first third of the simulation (t1/3 = +15.7 ks) was run without modifying the velocity (i.e. αv = 1). +During the second third, αv was linearly ramped down in time to +match the profile αv,3D(z) described above. Finally, the last third +of the simulation was run with the constant αv,3D(z): +αv,2D(z, t) = +����������� +1 +if t ≤ t1/3, +1 − �1 − αv,3D(z)� � t−t1/3 +t1/3 +� +if t1/3 < t ≤ 2t1/3, +αv,3D(z) +else. +(8) +The evolution of αv is shown in Fig. 2. This “velocity rewrite +layer” can successfully absorb the kink waves that are excited +by the driver at the bottom of the chromosphere. As a result, +these waves are not reflected at the upper boundary, and do not +propagate downwards back into the domain. We stress that the +solution obtained inside the velocity rewrite layer (i.e. above z = +50 Mm) is not physical, and that this layer should be considered +as a part of the upper boundary. +Side boundaries At the side boundaries (x and y axes), all vari- +ables obeyed a zero-gradient boundary condition. In the 2D re- +laxation run, we only simulated half of the tube radius (x > 0). +For these simulations, we imposed a reflective boundary condi- +tion on all variables at the centre of the tube (x = 0). +2.2. Initial conditions: field-aligned hydrostatic equilibrium +The simulation was initialized with a uniform vertical magnetic +field of magnitude B0 = 42 G. Along the tube, we imposed +the following temperature profile, derived from Aschwanden & +Schrijver (2002): +T(x, y, z) = +��������� +Tch +if z ≤ ∆ch, +Tch + (Tcor(x, y) − Tch) +� +1 − +� L−z +L−∆ch +�2�0.3 +else, +(9) +where z is the altitude, L is the height of the computational +domain, ∆ch = 4 Mm is thickness of the chromosphere, and +Tch = 20 000 K is the temperature in the chromosphere. We de- +fined the transverse temperature profile at the top of the domain, +Tcor(x, y), as: +Tcor(x, y) = Tcor,ext + (Tcor,int − Tcor,ext)ζ(x, y), +(10) +where Tcor,int = 1.2 MK is the temperature inside the tube, and +Tcor,ext = 3.6 MK is the temperature outside the tube. The shape +of the profile was set by ζ(x, y): +ζ(x, y) = 1 +2 +� +1 − tanh +�� � +x2 + y2/R − 1 +� +b +�� +, +(11) +Article number, page 3 of 8 + +A&A proofs: manuscript no. kink_cutoff +0.1 +1 +10 +100 +Altitude [Mm] +10−2 +10−1 +100 +Temperature [MK] +Tint +Text +10−13 +10−12 +10−11 +10−10 +10−9 +10−8 +Density [kg m⁻³] +ρint +ρext +39 +40 +41 +42 +43 +44 +Magnetic field [G] +Bint +Bext +(a) Field-aligned hydrostatic equilibrium +0.1 +1 +10 +100 +Altitude [Mm] +10−2 +10−1 +100 +Temperature [MK] +Tint +Text +10−13 +10−12 +10−11 +10−10 +10−9 +10−8 +Density [kg m⁻³] +ρint +ρext +9 +10 +11 +12 +13 +14 +Magnetic field [G] +Bint +Bext +(b) 2D magnetohydrodynamic relaxation +Fig. 3. Temperature (black), density (red), and magnetic field magnitude (blue) profiles inside (r = 0 Mm; solid lines) and outside (r = 8 Mm; +dashed lines) the flux tube. (a) After solving the field-aligned hydrostatic equilibrium. (b) After the 2D magnetohydrodynamic relaxation. +where R = 1 Mm is the tube radius, and b = 5 is a dimensionless +number setting the width of the inhomogeneous layer between +the interior and exterior of the tube (l ≈ 6R/b). ζ(x, y) is close to +1 inside the tube, and to 0 outside. +We also set the density at the bottom of the chromosphere +(z = 0) to: +ρch(x, y, z = 0) = ρch,ext + (ρch,int − ρch,ext)ζ(x, y), +(12) +where ρch,int = 3.51 · 10−8 kg m−3 is the density inside the tube, +and ρch,ext = 1.17 · 10−8 kg m−3 is the density outside. We then +integrated the field-aligned hydrostatic equilibrium equation nu- +merically using a Crank-Nicholson scheme. The profiles of the +imposed temperature and of the density resulting from the inte- +gration are shown in Fig. 3 (a). The temperature contrast (interior +temperature divided by exterior temperature) is 1 in the chromo- +sphere, and decreases to 1/3 in the corona. The density contrast +is 3 in the chromosphere, increases to around 7 in the transition +region, and decreases again to about 4 in the upper corona. The +pressure contrast is 3 in the chromosphere, and slowly decreases +to reach 1.2 in the upper corona. +However, this initial state is not in magnetohydrostatic +(MHS) equilibrium, because the pressure varies across the flux +tube, while the magnetic field does not. To fix this, we let the +tube relax by running a 2D magnetohydrodynamic simulation +(Sect. 2.3). We then used this relaxed state to initialize the 3D +simulation of kink waves (Sect. 2.4). +2.3. Flux tube relaxation (2D) +In order to obtain a flux tube in MHS equilibrium, we first +run a 2D simulation, initialized with the initial state described +in Sect. 2.2. The MHD equations were solved in a longitudi- +nal plane at y = 0 (see Fig. 1), with x ∈ [0, 8.56] Mm, and +z ∈ [0, 100] Mm. We used a uniform grid of 64 × 2048 cells with +a size of 134 km×49 km. The resolution along z is higher than in +the 3D runs in order to resolve the sharper gradients in the tran- +sition region (see Fig. 3). We verified that a resolution of 40 km +in the x direction yielded the same results, by running a separate +2D simulation followed by a 3D driven simulation (P0 = 200 s), +and verifying that the cut-off altitude and comparison to the an- +alytical formulas (Sect. 4) were not significantly modified. +We let the system evolve for 47 ks, during which the velocity +rewrite parameter αv varied as described in Eq. (8). As a result +of the relaxation, periodic longitudinal flows with a velocity of +about 15 km s−1 develop along the tube. They are damped during +the later stages of the simulation, as the velocity rewrite layer is +gradually introduced. At the end of the relaxation run, residual +velocities are lower than 0.5 km s−1 everywhere in the domain. +The resulting temperature, density, and magnetic field profiles +are shown on Fig. 3 (b). Compared to the initial state (Fig. 3 a), +the transition region is significantly broadened, with a thickness +of about 7 Mm. This is the direct result of the modified thermal +conductivity used in this setup, and allows for a coarser resolu- +tion along the loop in the 3D simulations. In addition, the tem- +perature and density decrease, both inside and outside the tube. +Overall, the density contrast (ρint/ρext) decreases: it reaches 1 +in the chromosphere, 1.2 in the transition region, and 1.8 in the +corona. The temperature contrast also changes to about 1.3 in +the transition, and about 0.8 in the corona. Finally, the magnetic +field amplitude contrast remains very close to 1 everywhere in +the domain (0.97 in the chromosphere and 1 in the corona), with +a magnitude of about 11 G everywhere in the domain. Compared +to the initial uniform magnetic field, the magnitude is divided by +about four, while the contrast remains close to 1. The final tem- +perature and density profile significantly differ from the initial +conditions of 2D relaxation run. However, this is not an issue, as +the goal of this study is to investigate how the analytical formulas +we consider (Spruit 1981; Lopin & Nagorny 2017; Snow et al. +2017) predict the cut-off frequency for a given temperature and +density profile. By using the relaxed profiles as an input to these +analytical formulas, we obtained predictions for the relaxed sys- +tem. +This relaxed 2D simulation was then mapped onto the 3D +domain through cylindrical symmetry. We used a rotation about +the line x = 0 (i.e. the centre of the loop), and a trilinear interpo- +lation to project onto the 3D Cartesian grid. +2.4. Kink waves propagation (3D) +In order to simulate the propagation of kink waves from the chro- +mosphere to the corona, we drove the 3D simulations with the +monoperiodic, dipole-like, driver described in Eqs. (4) and (5). +We ran four simulations, with different driver periods P0: 200 s, +Article number, page 4 of 8 + +G. Pelouze et al.: Cut-off of transverse waves +0 +200 +400 +Time [s] +0 +10 +20 +30 +40 +50 +Altitude [Mm] +(a) P0 =200 s +−15 +−10 +−5 +0 +5 +10 +15 +Velocity [km s⁻¹] +0 +250 +500 +750 +1000 +Time [s] +(b) P0 =335 s +−6 +−4 +−2 +0 +2 +4 +6 +Velocity [km s⁻¹] +0 +500 +1000 +1500 +2000 +Time [s] +(c) P0 =700 s +−3 +−2 +−1 +0 +1 +2 +3 +Velocity [km s⁻¹] +0 +1000 +2000 +Time [s] +(d) P0 =2000 s +−2 +−1 +0 +1 +2 +Velocity [km s⁻¹] +Fig. 4. Kink waves transverse velocity (vx) at the loop centre (x = y = 0), as a function of altitude and time. The velocity is shown for four 3D +simulations with different driver periods P0, after an initial settling time of 2P0 (for P0 = 200 s, 335 s and 700 s), or 0.42P0 (for P0 = 2000 s). The +dashed black lines represent a propagation at the kink speed (see Eq. (13)), and are independent of the driver period. +335 s, 700 s, and 2000 s. The propagating kink waves generated +by the driver are absorbed by the velocity rewrite layer at the top +of the domain, and are thus not reflected downwards. The first +three simulations were run for a duration of 5P0. The last simula- +tion was run for 1.75P0. At the beginning of the simulations, the +system goes through an initial transitory phase before the propa- +gating kink wave is fully established (i.e. its amplitude does not +change with time). We waited for 2P0 (0.42P0 for P0 = 2000 s) +for the kink wave to enter a stable sinusoidal regime. After this +duration, we saved high-cadence snapshots at the centre of the +loop (line x = y = 0). For all further analysis, we used the snap- +shots saved after the transitory phase. The transverse velocity vx +at the loop centre is shown in Fig. 4. As can be seen on this +figure, the amplitude of the kink wave decreases as the period +increases. For the two longer driver periods (700 and 2000 s), +the amplitude of the kink wave is small enough for some pertur- +bations to become visible. They travel at the Alfvén speed, and +appear to be triggered by the flows remaining after the relaxation +(see Sect. 2.3). These perturbations have amplitudes smaller than +0.2 km s−1, and should thus have no effect on the wave. +3. Results: cut-off and tunnelling of transverse +waves +In order to determine whether the kink waves driven in the 3D +simulations are experiencing a cut-off, we looked at the evolution +of the velocity amplitude (Sect. 3.1), as well as the phase speed +(Sect. 3.2) as a function of altitude. The analysis of these profiles +allows us to establish that the transverse waves are subject to a +low-frequency cut-off in the transition region. +3.1. Wave amplitude increases with frequency +In order to compute the velocity amplitude of the kink wave, we +fitted the function Ax(z) sin (ω(z)t + φ(z)) to the transverse ve- +locity vx(z, t), at each altitude (z). Ax(z) is the velocity amplitude, +ω(z) is the kink wave frequency, and φ(z) is the phase. The fre- +quency varies by less than 1 % with altitude, confirming theoret- +ical understanding. The velocity amplitude is shown in Fig. 5. +In all simulations, the wave amplitude increases with altitude, +because of the density decreases with altitude and energy con- +servation. Across simulations, the amplitude at a given altitude +increases with the frequency of the wave. This means that kink +waves with higher frequencies propagate better from the chro- +0.1 +1 +10 +Altitude [Mm] +2 +4 +6 +8 +10 +12 +14 +16 +Velocity amplitude [km s⁻¹] +P0 =200 s +P0 =335 s +P0 =700 s +P0 =2000 s +0.1 +1 +10 +2.0 +2.2 +2.4 +2.6 +0.05 +50 +Fig. 5. Velocity amplitude of kink waves, as a function of altitude. The +velocity is shown for four different driver periods (P0). The inset has the +same axes as the main figure, with a zoom-in on the vertical axis. +mosphere to the corona. This would be consistent with the low- +frequency cut-off predicted by analytical models (see Sect. 1). +3.2. Evanescent waves in the transition region +To determine the altitude at which the waves are cut-off, we +compared their phase speed vp(z) to the kink speed of the +flux tube ck(z). The inverse phase speed is equivalent to the +phase difference ∆φ(z) between two altitudes separated by ∆z: +1/vp(z) = ∆φ(z)/(ω∆z). The phase difference has been success- +fully used to determine the cut-off frequency of acoustic and +slow-magnetosonic waves in observations (Centeno et al. 2006; +Felipe et al. 2010; Krishna Prasad et al. 2017; Felipe et al. 2018), +and in simulations (Felipe & Sangeetha 2020). In these articles, +the authors determine the phase speed for a wide range of fre- +quencies, but at a limited number of altitude positions. In the +present study however, we could only examine four frequencies, +because of the high computational cost of a simulation. However, +we computed the phase difference at all altitudes of the simula- +tion domain. This allows us to determine the altitude at which +the wave is cut-off. +Article number, page 5 of 8 + +A&A proofs: manuscript no. kink_cutoff +The phase speed at a given altitude z was computed from the +transverse velocity in the cells above and below, that is vx(t, z + +∆z/2) and vx(t, z − ∆z/2), where ∆z = 98 km is the cell size. +We apodized these velocity time series with a Hann window, +and computed the cross-correlation C(τ, z) = vx(t, z + ∆z/2) ⋆ +vx(t, z−∆z/2). We then determined the time delay ∆τ(z), by find- +ing the maximum of C(τ, z). To that end, we fitted the function +A + B cos (ω(τ − ∆τ)/δ) to C(τ, z), with τ ∈ [−P0/4, +P0/4]. Fi- +nally, the phase difference was given by ∆φ(z) = ω∆τ(z), and the +inverse phase speed by 1/vp(z) = ∆τ(z)/∆z. The inverse phase +speed is shown on Fig. 6, alongside the inverse kink speed for +the simulated flux tube. The kink speed ck is calculated using: +c2 +k(z) = ρi(z)v2 +A i(z) + ρe(z)v2 +A e(z) +ρi(z) + ρe(z) +, +(13) +where ρ(z) is the density, vA(z) = B(z)/ +� +µ0ρ(z) is the Alfvén +speed, B(z) is the magnetic field amplitude, and µ0 is the mag- +netic permittivity of vacuum. The indices i and e correspond, +respectively, to internal and external quantities relatively to the +flux tube, and are taken at x = 0 and x = 8 Mm. +In simulations with short driver periods, the inverse phase +speed is somewhat smaller than the inverse kink speed in the +chromosphere and transition region (vp/ck ≈ 2 for P0 = 200 s, +and 5 for P0 = 335 s), and equals the inverse kink speed in the +corona. On the other hand, in simulations with longer periods, +the inverse phase speeds are much lower than the inverse kink +speed below a given altitude. For P0 = 700 s, 1/vp is about 250 +times smaller than 1/ck below z = 1 Mm. For P0 = 2000 s, a +similar drop occurs below z = 20 Mm. +For a propagating kink wave, the inverse phase speed is ex- +pected to be equal to the inverse kink speed. Conversely, stand- +ing and evanescent (i.e. cut-off) waves have inverse phase speeds +smaller than the inverse kink speed. Thus, the decreased inverse +phase speed for higher periods indicates that the waves are cut- +off in at least some regions. +To distinguish between the standing and evanescent cases, +we have also looked at the wave amplitude (Fig. 5). In the +absence of vertical stratification, the amplitude of evanescent +waves decreases with altitude. However, in a stratified atmo- +sphere (our case), the amplitude increases with altitude because +of the density decrease, even for evanescent waves. On Fig. 5, the +amplitude of waves with longer periods (for which 1/vp ≪ 1/ck) +increases less with altitude compared to waves with shorter pe- +riods (for which 1/vp ≲ 1/ck). We thus conclude that the waves +with longer periods are evanescent in parts of the low atmo- +sphere, where their inverse phase speed is much lower than the +inverse kink speed. This means that these long-period waves are +cut-off in the transition region. +3.3. Wave tunnelling at higher frequencies +Waves with shorter periods (P0 = 200 and 335 s) also show signs +of cut-off at low altitudes. Below z = 3 Mm, the inverse phase +speed 1/vp is lower than the inverse kink speed 1/ck (Fig. 6), +and the amplitude increase with altitude is smaller for P0 = 335 s +than for P0 = 200 s (Fig. 5). However, this cut-off is significantly +weaker than in the long-period case. This is explained by the fact +that the cut-off region (where 1/vp < 1/ck) is narrower for short +periods (∼ 1 Mm) than for long periods (∼ 10 Mm). As a result, +short-period waves can tunnel through the cut-off region, and +propagate into the corona. Furthermore, the weak attenuation in +the cut-off region (1/vp ≲ 1/ck) results further reduces the effect +of the cut-off. +0.1 +1 +10 +Altitude [Mm] +10−1 +100 +101 +102 +1/v [s Mm⁻¹] +1/ck +1/vp (P0 =200 s) +1/vp (P0 =335 s) +1/vp (P0 =700 s) +1/vp (P0 =2000 s) +50 +Fig. 6. Inverse phase speed of the kink wave (1/vp), and inverse kink +speed of the flux tube (1/ck), as a function of altitude. The phase speed +is given for four different driver periods (P0). +0.1 +1 +10 +Altitude [Mm] +10−3 +10−2 +10−1 +ωc [s⁻¹] +Models +Sp81 +Sn17 +LN17 (z0 = 24 km) +LN17 (z0 = 659 km) +LN17 (z0 = 1343 km) +LN17 (z0 = 1978 km) +Simulations +tr =0.2 +tr =0.3 +tr =0.4 +tr =0.5 +50 +Fig. 7. Kink wave cut-off frequency as a function of altitude, from an- +alytical models (left column of the legend), and from our numerical +simulations (right column of the legend). We show the analytical pre- +dictions of Spruit (1981, SP81), Snow et al. (2017, Sn17), and of Lopin +& Nagorny (2017, LN17) (coloured lines). For the last model, we com- +puted the cut-off frequency for different values of z0, the “base of the +atmosphere”. We show the cut-off altitude (zc) for the four simulations +that we ran with different driver frequencies (black markers). The cut- +off altitudes are computed with different thresholds tr, indicated on the +legend and described in the text. +4. Discussion: comparison to analytical formulas +In order to compare our simulations to the analytical models, +we quantified the cut-off frequency as a function of altitude. We +define zc, the altitude at which ck/vp goes above a given threshold +tr. This corresponds to the altitude where the wave leaves the cut- +off regime and enters the propagating regime. That is, the cut- +off altitude. We computed zc for four values of tr between 0.2 +and 0.5. Considering the four simulations with different driver +frequencies ω, we obtained the cut-off altitude as a function of +the frequency, zc(ω). We compare this to the cut-off frequency as +a function of altitude, ωc(z), predicted by the analytical models +presented in Sect. 1. +Article number, page 6 of 8 + +G. Pelouze et al.: Cut-off of transverse waves +On Fig. 7, we show the cut-off frequency and altitude com- +puted in our simulations, for different values of tr (black points). +On the same figure, we show the predictions of the analytical +formulas of Spruit (1981, Eq. (1)), Lopin & Nagorny (2017, +Eq. (2)), and Snow et al. (2017, Eq. (3)) (coloured lines), com- +puted for the temperature and density profiles used in our simu- +lations. We implement the formula of Lopin & Nagorny (2017) +for different values of z0, defined by the authors as “the base of +the atmosphere”, with no further details. Because this quantity +is not accurately defined, we used four values of z0 in the range +of 24 km (bottom cell of our simulation domain), to 1978 km. +This loosely defined parameter broadens the range for the cut- +off frequencies predicted by this formula. While the match is +rather loose, the cut-off altitude zc(ω) measured in our simula- +tions matches the overall variation the cut-off frequency ωc(z) +predicted by the Lopin & Nagorny (2017) formula. In particular, +the shape of the profiles are in good agreement. On the contrary, +the Snow et al. (2017) model correctly predicts the cut-off fre- +quency only in the lower transition region, but fails to do so in +the upper transition region and corona. In particular, their model +predicts a slower decrease of the cut-off frequency above 20 Mm, +while the simulations and the Lopin & Nagorny (2017) show a +continued decrease. Finally, the Spruit (1981) predictions are off +by almost an order of magnitude at all altitudes. Thus, the for- +mula of Lopin & Nagorny (2017) best predicts the cut-off fre- +quency of transverse waves at different altitudes. +While the broadened transition region in our simulations +could affect the altitude-dependence of the cut-off frequency, this +should have little impact on the validation of the analytical for- +mulas. Indeed, these formulas include the atmospheric stratifi- +cation through altitude-dependent profiles of either the pressure +scale height or the Alfvén speed (see Sect. 1). Because they make +no hypothesis on these profiles, they should be valid regardless +of the atmosphere considered. As such, the agreement with the +simulations should not depend on the broadening of the transi- +tion region, provided the appropriate profile is fed into the for- +mulas. After validating the Lopin & Nagorny (2017) formula by +comparing it to our simulations, it should be applicable to other +stratification profiles. +We note that while analytical formulas can predict the kink +cut-off frequency, this is not sufficient to know whether a kink +wave with a given frequency will propagate into the corona. To +that end, the thickness of the cut-off region and the strength of +the attenuation have to be taken into account. As shown by our +simulations, kink waves with higher frequencies (≥ 3 mHz) can +propagate into the corona by tunnelling through a region where +they are cut-off (Sect. 3.3). Furthermore, these waves only expe- +rience a weak attenuation, because their frequency is close to the +cut-off frequency. In fact, the cut-off frequency does not consti- +tute a clear-cut boundary between oscillatory and non-oscillatory +solutions. This was also reported for sound waves by Felipe & +Sangeetha (2020). Although the question of whether a solution +is oscillating is well-defined mathematically, this is not straight- +forward to translate into a single cut-off frequency (Schmitz & +Fleck 1998). For this reason, there exist several canonical def- +initions for cut-off frequencies, set within the continuous vari- +ation between the oscillating and non-oscillating regimes (see +e.g. Schmitz & Fleck 1998 for sound waves in the solar atmo- +sphere). As a result, cut-off frequencies are bound to be mere +indications, rather than strong constraints, on the physical be- +haviour of a wave (Chae & Litvinenko 2018). +5. Conclusions +Transverse waves are a candidate mechanism for heating the so- +lar corona. However, several analytical models predicted that +they are cut-off in the transition region. In order to assess +whether transverse waves can indeed heat the corona, it is thus +crucial to determine whether they can propagate through the +transition region. To that end, we have simulated the propagation +of transverse kink waves in an open magnetic flux tube, embed- +ded in an atmosphere extending from the chromosphere to the +corona. We found that transverse waves are indeed cut-off in the +lower solar atmosphere. However, only waves with low frequen- +cies (ν ≲ 2 mHz) are significantly affected. At higher frequen- +cies, the cut-off occurs in a very thin layer (∼ 1 Mm), and results +in a weak attenuation. In this case, waves can tunnel through +the cut-off layer, experiencing little to no amplitude attenuation. +This means that transverse waves with high frequencies are able +to transport energy from the chromosphere to the corona, where +it can be dissipated and result in heating. +Furthermore, we compared our simulations to several ana- +lytical models that predict the cut-off frequency of transverse +waves. We conclude that the formula proposed by Lopin & +Nagorny (2017) gives the best prediction. While our simulations +use a broadened transition, we expect it to have little impact on +the validation of analytical formulas. As such, the formula by +Lopin & Nagorny (2017) should be able to predict the cut-off +frequency for any atmospheric stratification profile. We note that +while the cut-off frequency is a good first indicator of whether a +wave can propagate into the corona, it cannot alone predict the +whole behaviour of the wave. In particular, waves with frequen- +cies just below the cut-off frequency (that should thus be cut-off) +can still reach the corona, thanks to a combination of tunnelling, +and weak attenuation. +Acknowledgements. This project has received funding from the European Re- +search Council (ERC) under the European Union’s Horizon 2020 research and +innovation program (grant agreement No. 724326). GP was supported by a +CNES postdoctoral allocation. TVD was supported by the European Research +Council (ERC) under the European Union’s Horizon 2020 research and inno- +vation programme (grant agreement No 724326) and the C1 grant TRACEs- +pace of Internal Funds KU Leuven. K.K. recognises support from a postdoctoral +mandate from KU Leuven Internal Funds (PDM/2019), from a UK Science and +Technology Facilities Council (STFC) grant ST/T000384/1, and from a FWO +(Fonds voor Wetenschappelijk Onderzoek – Vlaanderen) postdoctoral fellowship +(1273221N). The results received support from the FWO senior research project +with number G088021N. Software: Astropy (Astropy Collaboration et al. 2013, +2018), +References +Afanasyev, A. N., Van Doorsselaere, T., & Nakariakov, V. M. 2020, A&A, 633, +L8 +Anfinogentov, S. A., Nakariakov, V. M., & Nisticò, G. 2015, A&A, 583, A136 +Antolin, P., Yokoyama, T., & Van Doorsselaere, T. 2014, ApJ, 787, L22 +Arregui, I. 2021, ApJ, 915, L25 +Aschwanden, M. J. & Schrijver, C. J. 2002, ApJS, 142, 269 +Astropy Collaboration, Price-Whelan, A. M., Sip˝ocz, B. M., et al. 2018, AJ, 156, +123 +Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, +A33 +Bel, N. & Leroy, B. 1977, A&A, 55, 239 +Cally, P. S. & Andries, J. 2010, Sol. Phys., 266, 17 +Cally, P. S. & Khomenko, E. 2019, ApJ, 885, 58 +Centeno, R., Collados, M., & Trujillo Bueno, J. 2006, ApJ, 640, 1153 +Chae, J. & Litvinenko, Y. E. 2018, ApJ, 869, 36 +Felipe, T., Khomenko, E., Collados, M., & Beck, C. 2010, ApJ, 722, 131 +Felipe, T., Kuckein, C., & Thaler, I. 2018, A&A, 617, A39 +Felipe, T. & Sangeetha, C. R. 2020, A&A, 640, A4 +Goddard, C. R., Nisticò, G., Nakariakov, V. M., & Zimovets, I. V. 2016, A&A, +585, A137 +Article number, page 7 of 8 + +A&A proofs: manuscript no. kink_cutoff +Goossens, M., Andries, J., & Aschwanden, M. J. 2002, A&A, 394, L39 +Guo, M., Van Doorsselaere, T., Karampelas, K., & Li, B. 2019, ApJ, 883, 20 +Hansen, S. C. & Cally, P. S. 2009, Sol. Phys., 255, 193 +Jess, D. B., Reznikova, V. E., Van Doorsselaere, T., Keys, P. H., & Mackay, D. H. +2013, ApJ, 779, 168 +Johnston, C. D. & Bradshaw, S. J. 2019, ApJ, 873, L22 +Karampelas, K. & Van Doorsselaere, T. 2020, ApJ, 897, L35 +Karampelas, K. & Van Doorsselaere, T. 2021, ApJ, 908, L7 +Karampelas, K., Van Doorsselaere, T., & Antolin, P. 2017, A&A, 604, A130 +Karampelas, K., Van Doorsselaere, T., & Guo, M. 2019, A&A, 623, A53 +Khomenko, E. & Cally, P. S. 2012, ApJ, 746, 68 +Krishna Prasad, S., Jess, D. B., Van Doorsselaere, T., et al. 2017, ApJ, 847, 5 +Linker, J. A., Lionello, R., Miki´c, Z., & Amari, T. 2001, J. Geophys. Res., 106, +25165 +Lionello, R., Linker, J. A., & Miki´c, Z. 2009, ApJ, 690, 902 +Lopin, I. & Nagorny, I. 2017, ApJ, 154, 141 +Lopin, I. P., Nagorny, I. G., & Nippolainen, E. 2014, Sol. Phys., 289, 3033 +McIntosh, S. W., de Pontieu, B., Carlsson, M., et al. 2011, Nature, 475, 477 +Mignone, A., Bodo, G., Massaglia, S., et al. 2007, ApJS, 170, 228 +Miki´c, Z., Lionello, R., Mok, Y., Linker, J. A., & Winebarger, A. R. 2013, ApJ, +773, 94 +Morton, R. J., Tiwari, A. K., Van Doorsselaere, T., & McLaughlin, J. A. 2021, +ApJ, 923, 225 +Morton, R. J., Tomczyk, S., & Pinto, R. 2015, Nat. Com., 6, 7813 +Morton, R. J., Weberg, M. J., & McLaughlin, J. A. 2019, Nat. Astron., 3, 223 +Nakariakov, V. M., Anfinogentov, S. A., Nisticò, G., & Lee, D.-H. 2016, A&A, +591, L5 +Nakariakov, V. M., Aschwanden, M. J., & van Doorsselaere, T. 2009, A&A, 502, +661 +Nakariakov, V. M., Ofman, L., Deluca, E. E., Roberts, B., & Davila, J. M. 1999, +Sci, 285, 862 +Nechaeva, A., Zimovets, I. V., Nakariakov, V. M., & Goddard, C. R. 2019, ApJS, +241, 31 +Nisticò, G., Nakariakov, V. M., & Verwichte, E. 2013, A&A, 552, A57 +Pascoe, D. J., Wright, A. N., & De Moortel, I. 2010, ApJ, 711, 990 +Riedl, J. M., Doorsselaere, T. V., Reale, F., et al. 2021, ApJ, 922, 225 +Riedl, J. M., Van Doorsselaere, T., & Santamaria, I. C. 2019, A&A, 625, A144 +Schmitz, F. & Fleck, B. 1998, A&A, 337, 487 +Shi, M., Van Doorsselaere, T., Guo, M., et al. 2021, ApJ, 908, 233 +Snow, B., Fedun, V., Verth, G., & Erdelyi, R. 2017, New Insights into Kink Wave +Cut-off Frequency Due to Longitudinal Stratification, UK National Astron- +omy Meeting, 2017 +Spruit, H. C. 1981, A&A, 98, 155 +Terradas, J., Andries, J., Goossens, M., et al. 2008, ApJ, 687, L115 +Terradas, J. & Arregui, I. 2018, Research Notes of the American Astronomical +Society, 2, 196 +Thurgood, J. O., Morton, R. J., & McLaughlin, J. A. 2014, ApJ, 790, L2 +Tian, H., McIntosh, S. W., Wang, T., et al. 2012, ApJ, 759, 144 +Tiwari, A. K., Morton, R. J., Régnier, S., & McLaughlin, J. A. 2019, ApJ, 876, +106 +Tomczyk, S. & McIntosh, S. W. 2009, ApJ, 697, 1384 +Tomczyk, S., McIntosh, S. W., Keil, S. L., et al. 2007, Sci, 317, 1192 +Van Doorsselaere, T., Goossens, M., Magyar, N., Ruderman, M. S., & Ismayilli, +R. 2021, ApJ, 910, 58 +Van Doorsselaere, T., Srivastava, A. K., Antolin, P., et al. 2020, Space Sci. Rev., +216, 140 +Article number, page 8 of 8 + diff --git a/AdE1T4oBgHgl3EQfVQSI/content/tmp_files/load_file.txt b/AdE1T4oBgHgl3EQfVQSI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..75715798f7e89402163e221bf63244241ed80e5a --- /dev/null +++ b/AdE1T4oBgHgl3EQfVQSI/content/tmp_files/load_file.txt @@ -0,0 +1,792 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf,len=791 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' kink_cutoff ©ESO 2023 January 10, 2023 Cut-off of transverse waves through the solar transition region Gabriel Pelouze1, 2, Tom Van Doorsselaere2 , Konstantinos Karampelas2, 3 , Julia M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Riedl2 , and Timothy Duckenfield2 1 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405, Orsay, France 2 Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' e-mail: tom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='vandoorsselaere@kuleuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='be 3 Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK Received 23 September 2022 / Accepted 3 January 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Transverse oscillations are ubiquitously observed in the solar corona, both in coronal loops and open magnetic flux tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Numerical simulations suggest that their dissipation could heat coronal loops, counterbalancing radiative losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' These models rely on a continuous driver at the footpoint of the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' However, analytical works predict that transverse waves are subject to a cut-off in the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' It is thus unclear whether they can reach the corona, and indeed heat coronal loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Our aims are to determine how the cut-off of kink waves affects their propagation into the corona, and to characterize the variation of the cut-off frequency with altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Using 3D magnetohydrodynamic simulations, we modelled the propagation of kink waves in a magnetic flux tube, embed- ded in a realistic atmosphere with thermal conduction, that starts in the chromosphere and extends into the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We drove kink waves at four different frequencies, and determined whether they experienced a cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We then calculated the altitude at which the waves were cut-off, and compared it to the prediction of several analytical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We show that kink waves indeed experience a cut-off in the transition region, and we identified the analytical model that gives the best predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In addition, we show that waves with periods shorter than approximately 500 s can still reach the corona by tunnelling through the transition region, with little to no attenuation of their amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This means that such waves can still propagate from the footpoints of loop, and result in heating in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Sun: atmosphere – Sun: oscillations – magnetohydrodynamics (MHD) – waves – methods: numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Introduction Recent advances in observations and modelling have shown that magnetohydrodynamic (MHD) waves could significantly contribute to the heating of the solar corona (see review by Van Doorsselaere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In particular, transverse waves are ubiquitously observed, and they come in several kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The type that was first discovered are the transverse waves that are impul- sively excited after a flare (Nakariakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' However, these transverse waves are only sporadically excited and do not play an important role in the energy budget of the solar corona (Terradas & Arregui 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Later on, it was discovered that the corona is filled by small-amplitude transverse waves (Tomczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Tomczyk & McIntosh 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' McIntosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' These were observed in coronal loops as prop- agating (Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019) or standing waves (Anfinogentov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' These low-amplitude transverse waves were also observed as propagating waves in open-field regions (Thurgood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Morton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' These low-amplitude waves show little-to-no decay (Morton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2021) and are thus named “decayless”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Because the flare-excited standing waves are rapidly decay- ing (Goddard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Nechaeva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019) due to reso- nant absorption (Goossens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2002) and non-linear Kelvin- Helmholtz instability (KHI) damping (Terradas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' An- tolin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Van Doorsselaere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Arregui 2021), it is generally thought that the decayless waves must be con- tinuously supplied with energy to counteract its strong damp- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Several mechanisms for excitation have been proposed: slip- stick driving with steady flows (Nakariakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Karam- pelas & Van Doorsselaere 2020), vortex shedding (Nakariakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Karampelas & Van Doorsselaere 2021) or footpoint driving (Nisticò et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2017) through p- modes (Morton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019) or convective shuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The latter option of footpoint driving has had some success in generating standing mode decayless waves (Afanasyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2020), which counterbalance the non-linear damping through the KHI (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019) and lead to heating of loops (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' However, for the driving of decayless waves through their footpoints, it is not well understood how the transverse waves propagate through the complicated structure of the chromo- sphere and transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The simulations of transverse-wave induced KHI heating (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019) only take into account the coronal part of the loop, that is imposing a driver at the top of the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' To properly model the whole loop evolution due to the wave heating, it is essential to also model the wave driver in the photosphere, and accurately capture its influ- ence on the coronal loop dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In plane-parallel atmospheres, the propagation of fast and slow waves has been well studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' It was found that these modes couple efficiently to Alfvén waves through resonant absorption (Hansen & Cally 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Cally & Andries 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Khomenko & Cally 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Currently, investigations are ongoing to what hap- pens if the cross-field structuring is included into the wave prop- agation model (Cally & Khomenko 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Riedl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Another crucial ingredient is the wave’s behaviour in Article number, page 1 of 8 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='03100v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='SR] 8 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' kink_cutoff strong (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' non-WKB) stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' It is well-known that slow waves experience a cut-off while propagating through a stratified medium (Bel & Leroy 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This has been verified observation- ally (Jess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2013) and numerically (Felipe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Still, up to now, it is unknown if a similar cut-off exists for transverse waves in structured media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' For the driving of the observed decay- less waves in the corona, this is a crucial property to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Several analytical works predict that transverse waves are cut-off in the transition below a given frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The first for- mula was derived by Spruit (1981): ω2 Sp81 = g 8H 1 2β + 1, (1) where g is the gravity projected along the loop, H the pressure scale height, and β the ratio between the gas and magnetic pres- sures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' For a typical isothermal atmosphere, this corresponds to a cut-off period of 700 s (Spruit 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' However, Lopin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2014) showed that this classical cut-off is suppressed when the radial component of the magnetic field is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Lopin & Nagorny (2017) later showed that transverse waves can still be cut-off, provided a non-isothermal atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' They pre- dict the following cut-off frequency: ω2 LN17 = c2 k0 4H0H(z) � δ2 B dH(z) dz + H2(z) z2 � , (2) where z is the altitude, ck0 is the kink speed at the base of atmo- sphere (z = z0), H is the pressure scale height, H0 = H(z0), and δ2 B = � B2 0i − B2 0e � / � B2 0i + B2 0e � is the relative difference between the magnetic field inside (B0,i) and outside (B0,e) the flux tube, at z = z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Finally, an alternative formula was derived by Snow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2017): ω2 Sn17 = v2 A(z) 4z2 , (3) where z is the altitude, and vA is the Alfvén speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In this article, we modelled the propagation of kink waves in an open magnetic flux tube, embedded in a non-isothermal atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The atmosphere extends from the chromosphere to the corona, and includes gravitational stratification and thermal conduction (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We drove kink waves at different periods, and determined whether they experienced a cut-off (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We compare these results to the three analytical formulas given above in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 4, and summarize our conclusions in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Numerical model: magnetic flux tube through the transition region We modelled a vertical magnetic flux tube of radius R = 1 Mm embedded in a stratified atmosphere, starting in the chromo- sphere (altitude z = 0 Mm) and extending through the transi- tion region (z ≈ 4 Mm) into the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Kink waves were ex- cited in the flux tube by applying a monoperiodic driver at the bottom of the domain (z = 0 Mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In the upper half of the do- main (z > 50 Mm), we implemented a “velocity rewrite layer” to absorb the kink waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The driver and the velocity rewrite layer are described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' A sketch of the domain is shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We solved the 3D MHD evolution of this tube using the PLUTO code (Mignone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2007), version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This code solves the conservative MHD equations (mass continuity, mo- mentum conservation, energy conservation, and induction equa- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We used the corner transport upwind finite volume scheme, x [Mm] z [Mm] Driver Transition region Velocity rewrite layer Corona Magnetic tube Kink wave 2 Mm Chromosphere 0 −8 8 0 100 50 0 3 −3 y [Mm] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Sketch of the simulation domain, showing the magnetic flux tube, the location of the kink wave driver (bottom boundary), chromo- sphere, transition region, corona, and velocity rewrite layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' where characteristic tracing is used for the time stepping, and a linear spatial reconstruction with a monotonized central differ- ence limiter is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The magnetic field divergence was kept small using the extended divergence cleaning method (gen- eralized Lagrange multiplier, or GLM), and flux was computed with the linearized Roe Riemann solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We did not include ex- plicit viscosity, resistivity, or cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' However, numerical dis- sipation results in higher effective viscosity and resistivity than what is expected for the solar corona, as discussed by Karam- pelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We included a modified thermal conduction, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The transition region between the chromosphere and the corona is characterized by a very sharp temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Resolving such gradient requires a very high resolution along the tube (∼ 1 km in the transition region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In order to keep com- putational costs reasonable, we artificially broadened the tran- sition region (thus reducing the temperature gradient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' To that end, we modified the thermal conductivity using the method de- veloped by Linker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Lionello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Miki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Below the cut-off temperature Tc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='5 · 105 K, the parallel thermal conductivity was set to κ∥ = C0T 5/2 c with C0 = 9 · 10−12 Wm−1K−7/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Above Tc, κ∥ = C0T 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This al- lowed us to use a resolution of 98 km along the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This grid allows to fully resolve the broadened transition region, which has a minimum temperature scale length of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='6 Mm (see John- ston & Bradshaw 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The dimensions of the domain were (Lx, Ly, Lz) = (16, 6, 100) Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We used a uniform grid of 400 × 150 × 1024 cells, with a size of 40 km in the x and y direc- tions, and 98 km in the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Furthermore, we verified that the results did not change significantly when using a resolution of 40 km in the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' To that end, we ran a separate sim- ulation and verified that the resulting cut-off altitude and com- parison to the analytical formulas (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 4) were not strongly modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We note that such resolution is too costly in terms of compute time to be used for all simulations in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The strong stratification in the transition region makes it challenging to obtain a relaxed initial state for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We first initialized the domain with a field-aligned hydrostatic equi- librium (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We then let the simulation relax in 2D for 47 ks (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Finally, we filled the 3D domain with this re- Article number, page 2 of 8 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Pelouze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' : Cut-off of transverse waves 0 20 40 60 80 100 Altitude [Mm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='9995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='9996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='9997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='9998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='9999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='0000 Velocity rewrite coefficient αv αv(t≤15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='7 ks) αv(t=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='8 ks) αv(t=21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='9 ks) αv(t=25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='1 ks) αv(t=28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2 ks) αv(t≥31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3 ks)=αv,3D Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Velocity-rewrite coefficient αv, applied to the velocity above 50 Mm so that upper-propagating waves are not reflected back into the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' αv is shown for different times of the 2D relaxation run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The last profile (t ≥ 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3 ks) is also applied in the 3D driven simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' laxed state through cylindrical symmetry, where we drove kink waves of different periods for a duration up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='7 ks (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Boundary conditions and driver We first describe the boundary conditions used for the relaxation (2D) and kink wave (3D) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Bottom boundary At the bottom boundary (base of the chro- mosphere, z = 0), the density and pressure were extrapolated using the hydrostatic equilibrium equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The magnetic field was extrapolated using the zero normal-gradient condition de- scribed by Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2019, section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' For vz, we ei- ther imposed a reflective boundary condition (2D relaxation, see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3), or imposed vz = 0 (in 3D, see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We verified that both boundary conditions give the same results in 3D sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The parallel velocity components vx and vy were set to obey either a zero-gradient boundary condition (2D relaxation), or to follow a driver that excites kink waves (in 3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We used a monoperiodic, dipole-like, driver developed by Pascoe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2010) and updated by Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Inside the tube, the driver imposes: � vx(x, y, t), vy(x, y, t) � = {v(t), 0} , (4) where v(t) = v0 cos (2πt/P0), with v0 the driver amplitude, set to 2 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The driver period, P0, was set to different values in order to test the cut-off of kink waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Outside the tube, the driver imposes: � vx(x, y, t), vy(x, y, t) � = v(t)R2 � (x − x0(t))2 − y2, 2 (x − x0(t)) y � � (x − x0(t))2 + y2�2 , (5) where x0(t) = v0P0/(2π) · sin (2πt/P0) is the centre of the tube’s footpoint at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This driver generates a kink wave polarized in the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Upper boundary At the upper boundary (top of the corona, z = 100 Mm), the magnetic field was kept symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' All other vari- ables obeyed a reflective boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In order to absorb the upwards waves excited by the driver, we artificially modified the velocity in the upper half of the domain (z > 50 Mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' At each time step, after solving the MHD equations, we decreased each component of the velocity vi by multiplying it by a quantity αv ≲ 1: v′ i = αv(t, z)vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (6) In the driven 3D simulations αv was kept constant in time, and varied linearly along the loop, from 1 at z = zv = 50 Mm, to αv,min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='9995 at z = L = 100 Mm: αv,3D(z) = ������� 1 if z ≤ zv, 1 − �1 − αv,min � � z−zv L−zv � else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (7) In the 2D relaxation run, the first third of the simulation (t1/3 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='7 ks) was run without modifying the velocity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' αv = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' During the second third, αv was linearly ramped down in time to match the profile αv,3D(z) described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Finally, the last third of the simulation was run with the constant αv,3D(z): αv,2D(z, t) = ����������� 1 if t ≤ t1/3, 1 − �1 − αv,3D(z)� � t−t1/3 t1/3 � if t1/3 < t ≤ 2t1/3, αv,3D(z) else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (8) The evolution of αv is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This “velocity rewrite layer” can successfully absorb the kink waves that are excited by the driver at the bottom of the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' As a result, these waves are not reflected at the upper boundary, and do not propagate downwards back into the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We stress that the solution obtained inside the velocity rewrite layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' above z = 50 Mm) is not physical, and that this layer should be considered as a part of the upper boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Side boundaries At the side boundaries (x and y axes), all vari- ables obeyed a zero-gradient boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In the 2D re- laxation run, we only simulated half of the tube radius (x > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' For these simulations, we imposed a reflective boundary condi- tion on all variables at the centre of the tube (x = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Initial conditions: field-aligned hydrostatic equilibrium The simulation was initialized with a uniform vertical magnetic field of magnitude B0 = 42 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Along the tube, we imposed the following temperature profile, derived from Aschwanden & Schrijver (2002): T(x, y, z) = ��������� Tch if z ≤ ∆ch, Tch + (Tcor(x, y) − Tch) � 1 − � L−z L−∆ch �2�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3 else, (9) where z is the altitude, L is the height of the computational domain, ∆ch = 4 Mm is thickness of the chromosphere, and Tch = 20 000 K is the temperature in the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We de- fined the transverse temperature profile at the top of the domain, Tcor(x, y), as: Tcor(x, y) = Tcor,ext + (Tcor,int − Tcor,ext)ζ(x, y), (10) where Tcor,int = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2 MK is the temperature inside the tube, and Tcor,ext = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='6 MK is the temperature outside the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The shape of the profile was set by ζ(x, y): ζ(x, y) = 1 2 � 1 − tanh �� � x2 + y2/R − 1 � b �� , (11) Article number, page 3 of 8 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' kink_cutoff 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='1 1 10 100 Altitude [Mm] 10−2 10−1 100 Temperature [MK] Tint Text 10−13 10−12 10−11 10−10 10−9 10−8 Density [kg m⁻³] ρint ρext 39 40 41 42 43 44 Magnetic field [G] Bint Bext (a) Field-aligned hydrostatic equilibrium 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='1 1 10 100 Altitude [Mm] 10−2 10−1 100 Temperature [MK] Tint Text 10−13 10−12 10−11 10−10 10−9 10−8 Density [kg m⁻³] ρint ρext 9 10 11 12 13 14 Magnetic field [G] Bint Bext (b) 2D magnetohydrodynamic relaxation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Temperature (black), density (red), and magnetic field magnitude (blue) profiles inside (r = 0 Mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' solid lines) and outside (r = 8 Mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' dashed lines) the flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (a) After solving the field-aligned hydrostatic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (b) After the 2D magnetohydrodynamic relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' where R = 1 Mm is the tube radius, and b = 5 is a dimensionless number setting the width of the inhomogeneous layer between the interior and exterior of the tube (l ≈ 6R/b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' ζ(x, y) is close to 1 inside the tube, and to 0 outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We also set the density at the bottom of the chromosphere (z = 0) to: ρch(x, y, z = 0) = ρch,ext + (ρch,int − ρch,ext)ζ(x, y), (12) where ρch,int = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='51 · 10−8 kg m−3 is the density inside the tube, and ρch,ext = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='17 · 10−8 kg m−3 is the density outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We then integrated the field-aligned hydrostatic equilibrium equation nu- merically using a Crank-Nicholson scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The profiles of the imposed temperature and of the density resulting from the inte- gration are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The temperature contrast (interior temperature divided by exterior temperature) is 1 in the chromo- sphere, and decreases to 1/3 in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The density contrast is 3 in the chromosphere, increases to around 7 in the transition region, and decreases again to about 4 in the upper corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The pressure contrast is 3 in the chromosphere, and slowly decreases to reach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2 in the upper corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' However, this initial state is not in magnetohydrostatic (MHS) equilibrium, because the pressure varies across the flux tube, while the magnetic field does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' To fix this, we let the tube relax by running a 2D magnetohydrodynamic simulation (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We then used this relaxed state to initialize the 3D simulation of kink waves (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Flux tube relaxation (2D) In order to obtain a flux tube in MHS equilibrium, we first run a 2D simulation, initialized with the initial state described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The MHD equations were solved in a longitudi- nal plane at y = 0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 1), with x ∈ [0, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='56] Mm, and z ∈ [0, 100] Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We used a uniform grid of 64 × 2048 cells with a size of 134 km×49 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The resolution along z is higher than in the 3D runs in order to resolve the sharper gradients in the tran- sition region (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We verified that a resolution of 40 km in the x direction yielded the same results, by running a separate 2D simulation followed by a 3D driven simulation (P0 = 200 s), and verifying that the cut-off altitude and comparison to the an- alytical formulas (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 4) were not significantly modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We let the system evolve for 47 ks, during which the velocity rewrite parameter αv varied as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' As a result of the relaxation, periodic longitudinal flows with a velocity of about 15 km s−1 develop along the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' They are damped during the later stages of the simulation, as the velocity rewrite layer is gradually introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' At the end of the relaxation run, residual velocities are lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='5 km s−1 everywhere in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The resulting temperature, density, and magnetic field profiles are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Compared to the initial state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3 a), the transition region is significantly broadened, with a thickness of about 7 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This is the direct result of the modified thermal conductivity used in this setup, and allows for a coarser resolu- tion along the loop in the 3D simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In addition, the tem- perature and density decrease, both inside and outside the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Overall, the density contrast (ρint/ρext) decreases: it reaches 1 in the chromosphere, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2 in the transition region, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='8 in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The temperature contrast also changes to about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3 in the transition, and about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='8 in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Finally, the magnetic field amplitude contrast remains very close to 1 everywhere in the domain (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='97 in the chromosphere and 1 in the corona), with a magnitude of about 11 G everywhere in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Compared to the initial uniform magnetic field, the magnitude is divided by about four, while the contrast remains close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The final tem- perature and density profile significantly differ from the initial conditions of 2D relaxation run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' However, this is not an issue, as the goal of this study is to investigate how the analytical formulas we consider (Spruit 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Lopin & Nagorny 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Snow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2017) predict the cut-off frequency for a given temperature and density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' By using the relaxed profiles as an input to these analytical formulas, we obtained predictions for the relaxed sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This relaxed 2D simulation was then mapped onto the 3D domain through cylindrical symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We used a rotation about the line x = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' the centre of the loop), and a trilinear interpo- lation to project onto the 3D Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Kink waves propagation (3D) In order to simulate the propagation of kink waves from the chro- mosphere to the corona, we drove the 3D simulations with the monoperiodic, dipole-like, driver described in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We ran four simulations, with different driver periods P0: 200 s, Article number, page 4 of 8 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Pelouze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' : Cut-off of transverse waves 0 200 400 Time [s] 0 10 20 30 40 50 Altitude [Mm] (a) P0 =200 s −15 −10 −5 0 5 10 15 Velocity [km s⁻¹] 0 250 500 750 1000 Time [s] (b) P0 =335 s −6 −4 −2 0 2 4 6 Velocity [km s⁻¹] 0 500 1000 1500 2000 Time [s] (c) P0 =700 s −3 −2 −1 0 1 2 3 Velocity [km s⁻¹] 0 1000 2000 Time [s] (d) P0 =2000 s −2 −1 0 1 2 Velocity [km s⁻¹] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Kink waves transverse velocity (vx) at the loop centre (x = y = 0), as a function of altitude and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The velocity is shown for four 3D simulations with different driver periods P0, after an initial settling time of 2P0 (for P0 = 200 s, 335 s and 700 s), or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='42P0 (for P0 = 2000 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The dashed black lines represent a propagation at the kink speed (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (13)), and are independent of the driver period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 335 s, 700 s, and 2000 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The propagating kink waves generated by the driver are absorbed by the velocity rewrite layer at the top of the domain, and are thus not reflected downwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The first three simulations were run for a duration of 5P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The last simula- tion was run for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='75P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' At the beginning of the simulations, the system goes through an initial transitory phase before the propa- gating kink wave is fully established (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' its amplitude does not change with time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We waited for 2P0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='42P0 for P0 = 2000 s) for the kink wave to enter a stable sinusoidal regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' After this duration, we saved high-cadence snapshots at the centre of the loop (line x = y = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' For all further analysis, we used the snap- shots saved after the transitory phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The transverse velocity vx at the loop centre is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' As can be seen on this figure, the amplitude of the kink wave decreases as the period increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' For the two longer driver periods (700 and 2000 s), the amplitude of the kink wave is small enough for some pertur- bations to become visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' They travel at the Alfvén speed, and appear to be triggered by the flows remaining after the relaxation (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' These perturbations have amplitudes smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2 km s−1, and should thus have no effect on the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Results: cut-off and tunnelling of transverse waves In order to determine whether the kink waves driven in the 3D simulations are experiencing a cut-off, we looked at the evolution of the velocity amplitude (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='1), as well as the phase speed (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2) as a function of altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The analysis of these profiles allows us to establish that the transverse waves are subject to a low-frequency cut-off in the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Wave amplitude increases with frequency In order to compute the velocity amplitude of the kink wave, we fitted the function Ax(z) sin (ω(z)t + φ(z)) to the transverse ve- locity vx(z, t), at each altitude (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Ax(z) is the velocity amplitude, ω(z) is the kink wave frequency, and φ(z) is the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The fre- quency varies by less than 1 % with altitude, confirming theoret- ical understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The velocity amplitude is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In all simulations, the wave amplitude increases with altitude, because of the density decreases with altitude and energy con- servation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Across simulations, the amplitude at a given altitude increases with the frequency of the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This means that kink waves with higher frequencies propagate better from the chro- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='1 1 10 Altitude [Mm] 2 4 6 8 10 12 14 16 Velocity amplitude [km s⁻¹] P0 =200 s P0 =335 s P0 =700 s P0 =2000 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='1 1 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='05 50 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Velocity amplitude of kink waves, as a function of altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The velocity is shown for four different driver periods (P0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The inset has the same axes as the main figure, with a zoom-in on the vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' mosphere to the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This would be consistent with the low- frequency cut-off predicted by analytical models (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Evanescent waves in the transition region To determine the altitude at which the waves are cut-off, we compared their phase speed vp(z) to the kink speed of the flux tube ck(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The inverse phase speed is equivalent to the phase difference ∆φ(z) between two altitudes separated by ∆z: 1/vp(z) = ∆φ(z)/(ω∆z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The phase difference has been success- fully used to determine the cut-off frequency of acoustic and slow-magnetosonic waves in observations (Centeno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Felipe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Krishna Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Felipe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2018), and in simulations (Felipe & Sangeetha 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In these articles, the authors determine the phase speed for a wide range of fre- quencies, but at a limited number of altitude positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In the present study however, we could only examine four frequencies, because of the high computational cost of a simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' However, we computed the phase difference at all altitudes of the simula- tion domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This allows us to determine the altitude at which the wave is cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Article number, page 5 of 8 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' kink_cutoff The phase speed at a given altitude z was computed from the transverse velocity in the cells above and below, that is vx(t, z + ∆z/2) and vx(t, z − ∆z/2), where ∆z = 98 km is the cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We apodized these velocity time series with a Hann window, and computed the cross-correlation C(τ, z) = vx(t, z + ∆z/2) ⋆ vx(t, z−∆z/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We then determined the time delay ∆τ(z), by find- ing the maximum of C(τ, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' To that end, we fitted the function A + B cos (ω(τ − ∆τ)/δ) to C(τ, z), with τ ∈ [−P0/4, +P0/4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Fi- nally, the phase difference was given by ∆φ(z) = ω∆τ(z), and the inverse phase speed by 1/vp(z) = ∆τ(z)/∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The inverse phase speed is shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 6, alongside the inverse kink speed for the simulated flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The kink speed ck is calculated using: c2 k(z) = ρi(z)v2 A i(z) + ρe(z)v2 A e(z) ρi(z) + ρe(z) , (13) where ρ(z) is the density, vA(z) = B(z)/ � µ0ρ(z) is the Alfvén speed, B(z) is the magnetic field amplitude, and µ0 is the mag- netic permittivity of vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The indices i and e correspond, respectively, to internal and external quantities relatively to the flux tube, and are taken at x = 0 and x = 8 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In simulations with short driver periods, the inverse phase speed is somewhat smaller than the inverse kink speed in the chromosphere and transition region (vp/ck ≈ 2 for P0 = 200 s, and 5 for P0 = 335 s), and equals the inverse kink speed in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' On the other hand, in simulations with longer periods, the inverse phase speeds are much lower than the inverse kink speed below a given altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' For P0 = 700 s, 1/vp is about 250 times smaller than 1/ck below z = 1 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' For P0 = 2000 s, a similar drop occurs below z = 20 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' For a propagating kink wave, the inverse phase speed is ex- pected to be equal to the inverse kink speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Conversely, stand- ing and evanescent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' cut-off) waves have inverse phase speeds smaller than the inverse kink speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Thus, the decreased inverse phase speed for higher periods indicates that the waves are cut- off in at least some regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' To distinguish between the standing and evanescent cases, we have also looked at the wave amplitude (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In the absence of vertical stratification, the amplitude of evanescent waves decreases with altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' However, in a stratified atmo- sphere (our case), the amplitude increases with altitude because of the density decrease, even for evanescent waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' On Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 5, the amplitude of waves with longer periods (for which 1/vp ≪ 1/ck) increases less with altitude compared to waves with shorter pe- riods (for which 1/vp ≲ 1/ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We thus conclude that the waves with longer periods are evanescent in parts of the low atmo- sphere, where their inverse phase speed is much lower than the inverse kink speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This means that these long-period waves are cut-off in the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Wave tunnelling at higher frequencies Waves with shorter periods (P0 = 200 and 335 s) also show signs of cut-off at low altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Below z = 3 Mm, the inverse phase speed 1/vp is lower than the inverse kink speed 1/ck (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 6), and the amplitude increase with altitude is smaller for P0 = 335 s than for P0 = 200 s (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' However, this cut-off is significantly weaker than in the long-period case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This is explained by the fact that the cut-off region (where 1/vp < 1/ck) is narrower for short periods (∼ 1 Mm) than for long periods (∼ 10 Mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' As a result, short-period waves can tunnel through the cut-off region, and propagate into the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Furthermore, the weak attenuation in the cut-off region (1/vp ≲ 1/ck) results further reduces the effect of the cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='1 1 10 Altitude [Mm] 10−1 100 101 102 1/v [s Mm⁻¹] 1/ck 1/vp (P0 =200 s) 1/vp (P0 =335 s) 1/vp (P0 =700 s) 1/vp (P0 =2000 s) 50 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Inverse phase speed of the kink wave (1/vp), and inverse kink speed of the flux tube (1/ck), as a function of altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The phase speed is given for four different driver periods (P0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='1 1 10 Altitude [Mm] 10−3 10−2 10−1 ωc [s⁻¹] Models Sp81 Sn17 LN17 (z0 = 24 km) LN17 (z0 = 659 km) LN17 (z0 = 1343 km) LN17 (z0 = 1978 km) Simulations tr =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2 tr =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3 tr =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='4 tr =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='5 50 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Kink wave cut-off frequency as a function of altitude, from an- alytical models (left column of the legend), and from our numerical simulations (right column of the legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We show the analytical pre- dictions of Spruit (1981, SP81), Snow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2017, Sn17), and of Lopin & Nagorny (2017, LN17) (coloured lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' For the last model, we com- puted the cut-off frequency for different values of z0, the “base of the atmosphere”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We show the cut-off altitude (zc) for the four simulations that we ran with different driver frequencies (black markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The cut- off altitudes are computed with different thresholds tr, indicated on the legend and described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Discussion: comparison to analytical formulas In order to compare our simulations to the analytical models, we quantified the cut-off frequency as a function of altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We define zc, the altitude at which ck/vp goes above a given threshold tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This corresponds to the altitude where the wave leaves the cut- off regime and enters the propagating regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' That is, the cut- off altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We computed zc for four values of tr between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Considering the four simulations with different driver frequencies ω, we obtained the cut-off altitude as a function of the frequency, zc(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We compare this to the cut-off frequency as a function of altitude, ωc(z), predicted by the analytical models presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Article number, page 6 of 8 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Pelouze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' : Cut-off of transverse waves On Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 7, we show the cut-off frequency and altitude com- puted in our simulations, for different values of tr (black points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' On the same figure, we show the predictions of the analytical formulas of Spruit (1981, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (1)), Lopin & Nagorny (2017, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2)), and Snow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2017, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (3)) (coloured lines), com- puted for the temperature and density profiles used in our simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We implement the formula of Lopin & Nagorny (2017) for different values of z0, defined by the authors as “the base of the atmosphere”, with no further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Because this quantity is not accurately defined, we used four values of z0 in the range of 24 km (bottom cell of our simulation domain), to 1978 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This loosely defined parameter broadens the range for the cut- off frequencies predicted by this formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' While the match is rather loose, the cut-off altitude zc(ω) measured in our simula- tions matches the overall variation the cut-off frequency ωc(z) predicted by the Lopin & Nagorny (2017) formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In particular, the shape of the profiles are in good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' On the contrary, the Snow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' (2017) model correctly predicts the cut-off fre- quency only in the lower transition region, but fails to do so in the upper transition region and corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In particular, their model predicts a slower decrease of the cut-off frequency above 20 Mm, while the simulations and the Lopin & Nagorny (2017) show a continued decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Finally, the Spruit (1981) predictions are off by almost an order of magnitude at all altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Thus, the for- mula of Lopin & Nagorny (2017) best predicts the cut-off fre- quency of transverse waves at different altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' While the broadened transition region in our simulations could affect the altitude-dependence of the cut-off frequency, this should have little impact on the validation of the analytical for- mulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Indeed, these formulas include the atmospheric stratifi- cation through altitude-dependent profiles of either the pressure scale height or the Alfvén speed (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Because they make no hypothesis on these profiles, they should be valid regardless of the atmosphere considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' As such, the agreement with the simulations should not depend on the broadening of the transi- tion region, provided the appropriate profile is fed into the for- mulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' After validating the Lopin & Nagorny (2017) formula by comparing it to our simulations, it should be applicable to other stratification profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We note that while analytical formulas can predict the kink cut-off frequency, this is not sufficient to know whether a kink wave with a given frequency will propagate into the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' To that end, the thickness of the cut-off region and the strength of the attenuation have to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' As shown by our simulations, kink waves with higher frequencies (≥ 3 mHz) can propagate into the corona by tunnelling through a region where they are cut-off (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Furthermore, these waves only expe- rience a weak attenuation, because their frequency is close to the cut-off frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In fact, the cut-off frequency does not consti- tute a clear-cut boundary between oscillatory and non-oscillatory solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This was also reported for sound waves by Felipe & Sangeetha (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Although the question of whether a solution is oscillating is well-defined mathematically, this is not straight- forward to translate into a single cut-off frequency (Schmitz & Fleck 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' For this reason, there exist several canonical def- initions for cut-off frequencies, set within the continuous vari- ation between the oscillating and non-oscillating regimes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Schmitz & Fleck 1998 for sound waves in the solar atmo- sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' As a result, cut-off frequencies are bound to be mere indications, rather than strong constraints, on the physical be- haviour of a wave (Chae & Litvinenko 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Conclusions Transverse waves are a candidate mechanism for heating the so- lar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' However, several analytical models predicted that they are cut-off in the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In order to assess whether transverse waves can indeed heat the corona, it is thus crucial to determine whether they can propagate through the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' To that end, we have simulated the propagation of transverse kink waves in an open magnetic flux tube, embed- ded in an atmosphere extending from the chromosphere to the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We found that transverse waves are indeed cut-off in the lower solar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' However, only waves with low frequen- cies (ν ≲ 2 mHz) are significantly affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' At higher frequen- cies, the cut-off occurs in a very thin layer (∼ 1 Mm), and results in a weak attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In this case, waves can tunnel through the cut-off layer, experiencing little to no amplitude attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This means that transverse waves with high frequencies are able to transport energy from the chromosphere to the corona, where it can be dissipated and result in heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Furthermore, we compared our simulations to several ana- lytical models that predict the cut-off frequency of transverse waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We conclude that the formula proposed by Lopin & Nagorny (2017) gives the best prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' While our simulations use a broadened transition, we expect it to have little impact on the validation of analytical formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' As such, the formula by Lopin & Nagorny (2017) should be able to predict the cut-off frequency for any atmospheric stratification profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' We note that while the cut-off frequency is a good first indicator of whether a wave can propagate into the corona, it cannot alone predict the whole behaviour of the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' In particular, waves with frequen- cies just below the cut-off frequency (that should thus be cut-off) can still reach the corona, thanks to a combination of tunnelling, and weak attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' This project has received funding from the European Re- search Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 724326).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' GP was supported by a CNES postdoctoral allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' TVD was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and inno- vation programme (grant agreement No 724326) and the C1 grant TRACEs- pace of Internal Funds KU Leuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' recognises support from a postdoctoral mandate from KU Leuven Internal Funds (PDM/2019), from a UK Science and Technology Facilities Council (STFC) grant ST/T000384/1, and from a FWO (Fonds voor Wetenschappelijk Onderzoek – Vlaanderen) postdoctoral fellowship (1273221N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' The results received support from the FWO senior research project with number G088021N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Software: Astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2013, 2018), References Afanasyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Nakariakov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2020, A&A, 633, L8 Anfinogentov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Nakariakov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Nisticò, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2015, A&A, 583, A136 Antolin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Yokoyama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2014, ApJ, 787, L22 Arregui, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2021, ApJ, 915, L25 Aschwanden, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Schrijver, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2002, ApJS, 142, 269 Astropy Collaboration, Price-Whelan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Sip˝ocz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2018, AJ, 156, 123 Astropy Collaboration, Robitaille, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Tollerud, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2013, A&A, 558, A33 Bel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Leroy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 1977, A&A, 55, 239 Cally, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Andries, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2010, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', 266, 17 Cally, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Khomenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019, ApJ, 885, 58 Centeno, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Collados, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Trujillo Bueno, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2006, ApJ, 640, 1153 Chae, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Litvinenko, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2018, ApJ, 869, 36 Felipe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Khomenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Collados, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Beck, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2010, ApJ, 722, 131 Felipe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Kuckein, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Thaler, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2018, A&A, 617, A39 Felipe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Sangeetha, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2020, A&A, 640, A4 Goddard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Nisticò, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Nakariakov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Zimovets, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2016, A&A, 585, A137 Article number, page 7 of 8 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' kink_cutoff Goossens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Andries, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Aschwanden, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2002, A&A, 394, L39 Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Karampelas, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019, ApJ, 883, 20 Hansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Cally, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2009, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', 255, 193 Jess, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Reznikova, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Keys, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Mackay, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2013, ApJ, 779, 168 Johnston, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Bradshaw, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019, ApJ, 873, L22 Karampelas, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2020, ApJ, 897, L35 Karampelas, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2021, ApJ, 908, L7 Karampelas, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Antolin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2017, A&A, 604, A130 Karampelas, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019, A&A, 623, A53 Khomenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Cally, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2012, ApJ, 746, 68 Krishna Prasad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Jess, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2017, ApJ, 847, 5 Linker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Lionello, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Miki´c, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Amari, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2001, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', 106, 25165 Lionello, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Linker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Miki´c, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2009, ApJ, 690, 902 Lopin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Nagorny, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2017, ApJ, 154, 141 Lopin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Nagorny, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Nippolainen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2014, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', 289, 3033 McIntosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', de Pontieu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Carlsson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2011, Nature, 475, 477 Mignone, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Bodo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Massaglia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2007, ApJS, 170, 228 Miki´c, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Lionello, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Mok, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Linker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Winebarger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2013, ApJ, 773, 94 Morton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Tiwari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & McLaughlin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2021, ApJ, 923, 225 Morton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Tomczyk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Pinto, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2015, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', 6, 7813 Morton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Weberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & McLaughlin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', 3, 223 Nakariakov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Anfinogentov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Nisticò, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2016, A&A, 591, L5 Nakariakov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Aschwanden, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2009, A&A, 502, 661 Nakariakov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Ofman, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Deluca, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Roberts, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Davila, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 1999, Sci, 285, 862 Nechaeva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Zimovets, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Nakariakov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Goddard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019, ApJS, 241, 31 Nisticò, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Nakariakov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Verwichte, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2013, A&A, 552, A57 Pascoe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Wright, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & De Moortel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2010, ApJ, 711, 990 Riedl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Reale, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2021, ApJ, 922, 225 Riedl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Santamaria, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019, A&A, 625, A144 Schmitz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Fleck, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 1998, A&A, 337, 487 Shi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2021, ApJ, 908, 233 Snow, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Fedun, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Verth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Erdelyi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2017, New Insights into Kink Wave Cut-off Frequency Due to Longitudinal Stratification, UK National Astron- omy Meeting, 2017 Spruit, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 1981, A&A, 98, 155 Terradas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Andries, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Goossens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2008, ApJ, 687, L115 Terradas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & Arregui, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2018, Research Notes of the American Astronomical Society, 2, 196 Thurgood, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Morton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & McLaughlin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2014, ApJ, 790, L2 Tian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', McIntosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2012, ApJ, 759, 144 Tiwari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Morton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Régnier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & McLaughlin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2019, ApJ, 876, 106 Tomczyk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' & McIntosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2009, ApJ, 697, 1384 Tomczyk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', McIntosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Keil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2007, Sci, 317, 1192 Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Goossens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Magyar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Ruderman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', & Ismayilli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2021, ApJ, 910, 58 Van Doorsselaere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Srivastava, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', Antolin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' 2020, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfVQSI/content/2301.03100v1.pdf'} +page_content=', 216, 140 Article 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7733293 diff --git a/DNE2T4oBgHgl3EQfSAfe/content/tmp_files/2301.03789v1.pdf.txt b/DNE2T4oBgHgl3EQfSAfe/content/tmp_files/2301.03789v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..79b11535dbe76dca35eede831a57772a02c85a7b --- /dev/null +++ b/DNE2T4oBgHgl3EQfSAfe/content/tmp_files/2301.03789v1.pdf.txt @@ -0,0 +1,3874 @@ +1 + +Determination of the Zak phase of one-dimensional photonic +systems via far-field diffraction + +C. Liu*, H.R. Wang*, and H.C. Onga) + +Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s +Republic of China + +Bloch waves in 1D periodic systems carry Zak phase, which plays a key role in determining +the band topology. In general, for systems that possess inversion symmetry, the Zak phase of +an isolated band is quantized as 0 or  and is associated with the spatial field symmetries at the +Brillouin zone center and boundary. The phase is  if the field symmetries are different but is +0 when they are the same. Since the radiation losses from leaky systems are strongly associated +with the Bloch waves, one may probe the far-field continuum to determine the Zak phases. +Here, we formulate the diffractions from photonic systems at the zone center and boundary and +find their spectral profiles reveal the Bloch wave symmetries and thereby the corresponding +Zak phase. The field symmetries also generalize the occurrence of bound states in the +continuum at high symmetry points. For verification, we have studied the Zak phases of one- +dimensional TM plasmonic and TE photonic crystals by electrodynamic simulations and +measuring the optical properties of plasmonic crystals using Fourier space diffraction +spectroscopy and common path interferometry. In addition, a topological protected interface +state is demonstrated when two 0 and  systems are joined together. The results prove our +method provides a simple way for characterizing the band topology of non-Hermitian systems +via far-fields. + +* These authors equally contributed to this work +a) hcong@phy.cuhk.edu.hk + + +2 + +I. +INTRODUCTION +Topological physics has attracted a widespread of interest not only in condensed matter +physics [1-3] but also in other branches such as ultracold atom [4,5], electromagnetism [6-8], +mechanics [9], acoustics [10,11], and oceanography [12]. Much attention in this field is focused +on realizing the so-called topologically protected states, which support robust wave +propagation against perturbation and disorder [1-12]. To produce such states, two systems that +are topologically trivial and nontrivial are brought together to facilitate the occurrence of +topological phase transition at the interface. As most of the matters are topologically trivial, +the identification and the growth of different classes of topological systems are currently under +intensive investigation [13,14]. Likewise, developing methods to characterize the topological +properties of the systems is equally important. +In analogy to the Su-Schrieffer-Heeger (SSH) model, the band topology of one- +dimensional (1D) periodic systems is determined by Zak phase, , which is a geometric phase +[15,16]. For a particular th isolated band,  emerges when the Bloch wave travels in +momentum space adiabatically along the band across the first Brillouin zone from k = -/P to +/P, where P is the period of the system. [16]. If systems possess inversion symmetry,  is +quantized as either 0 or  [16].  defines the topological invariant of two band systems. For +systems that support higher order bands, the topology of the band gap of interest is the +summation of all  below that gap, giving rise to a  summation that is either even or odd +multiple of  for indicating whether the system is topologically trivial or nontrivial [17,18]. A +zero-dimensional interface state is then formed between two odd and even  systems. +One notable feature that comes with  is the distinctive spatial wave symmetries at the +Brillouin zone center and boundary of the band [16-18]. The field symmetries, typically even + +3 + +and odd with respect to the unit cell center, are the same for  = 0 system but different when + =  [18]. The association between n and the field symmetry can be understood from the +standpoint of Wannier function, which sums the Bloch waves carrying all k along a band [19]. +Consider the Bloch waves at the zone center and boundary that have the same field symmetry, +the Wannier function has either the +( +) +( ) +W +x +W +x +− += + or +( +) +( ) +W +x +W +x +− += − + spatial dependence, +leading to  = +( ) +2 +2 +x W +x +dx +P + + +− + = 0 [16]. On the other hand, for the waves that exhibit +different spatial symmetries at two high symmetry points, the Wannier function now shows +( +) +( ) +W +x +P +W +x +− + += + or +( +) +( ) +W +x +P +W +x +− + += − + dependence, which gives  =  [16]. +Therefore, instead of tracing the Bloch waves one by one along the band to determine n, one +can simply examine the field symmetries. However, how to measure the Bloch wave symmetry +remains challenging. +To date, there have been only a few studies focusing on measuring the geometric phase, +either Zak or Berry phase. Demler and Bloch are among the first to combine Bloch oscillation +and interferometry in a dimerized cold atom system to mobilize the Bloch wave across the +Brillouin zone and subsequently measure  [20,21]. They prove  =  evolves when the +intercell interaction is stronger than that of intracell. Cardano et al have demonstrated the use +of mean displacement method to determine  in a chiral Floquet system [22]. Such method is +then extended to other more generalized SSH systems where the next nearest neighbor +interaction is strong enough to break the chiral symmetry [23]. While most of them trace the +Bloch waves, Gorlach et al adopt an alternative approach by probing the spectral positions of +the dipolar (bright) and quadrupolar (dark) characteristics of far-field radiations, which scale +with the topological invariant of the system as deduced by using temporal coupled mode theory + +4 + +(CMT) [24]. When the bright and dark radiation bands are at longer and shorter wavelengths, +the system is trivial, but becomes nontrivial upon switching places. However, their method is +limited to the lowest band gap at the zone center. Recently, Chan and his coworkers have +formulated that the sign of the reflection phase for wavelengths within the th band gap can +resolves the trivial and nontrivial  [17,18]. The determination of  via measuring the +reflection phase of the band gap is then demonstrated in several photonic and acoustic systems +[25-28]. +Here, we further extend the CMT to formulate the diffractions arising from 1D leaky +optical systems and show the mirror symmetric diffraction orders taken at the zone center and +boundary directly reveal the near-field symmetries and thereby the corresponding . It is found +the odd and even near-field symmetries dictate the far-field interferences, shaping the overall +radiation profiles including the bound states in the continuum (BICs) [29-35] and Fano +resonances [36]. We find destructive interference always occurs between the diffraction orders +of the first band gap at the zone center, resulting in a symmetry-protected quasi-BIC [34]. To +verify the CMT, we first conduct finite-difference time-domain (FDTD) simulations on 1D Au +plasmonic and SiO2/Au photonic crystals which respectively support TM- and TE-polarized +surface waves and the results agree very well with the theory. We then fabricate plasmonic +crystals (PmCs) with different geometries and measure their polarization- and angle-resolved +diffraction and phase profiles by Fourier space spectroscopy and common path interferometry +to study . Changing the groove width of PmCs leads to band inversion and thus effectively +varies the band topology. Finally, a topological protected interface state is demonstrated by +joining two topological trivial and nontrivial PmCs together. +II. +TEMPORAL COUPLED MODE THEORY + +5 + +At high symmetry points in 1D Brillouin zone, two degenerate but counter propagating +Bloch modes interact with each other to yield two coupled modes separated by an energy gap +[37,38]. Such interaction can be described within the framework of CMT [37-40]. As shown +in Fig. 1(a), for an optically thick system that possesses inversion symmetry, the dynamics of +two mode amplitudes, a1 and a2, taken under TM or TE polarization can be written as: +  +1 +1 +2 +2 +o +c +T +c +o +a +a +d +i +K +s +a +a +dt + + + + ++ + + + + + + += ++ + + + + + + + + + + + + +, + + +(1) +where +o + and +c + are the complex frequency and coupling constant, which are expressed as +( +) 2 +o +o +a +r +i + + += ++ + + + and +c +i + + + += ++ +, where o is the resonant angular frequency, a and +r are the absorption and radiative decay rates, and  and  are the real and imaginary parts of +the coupling constant. For a given polarization, the discrete incoming power amplitude vector +is   +0 +T +N , +, +N , +s +s +s +s ++ +− ++ ++ ++ +=  + + + , where the subscript N is an integer  0. +0, +s + is denoted +as the surface normal power and +N , +s ++ are two mirror symmetric powers defined obliquely with +respect to the surface normal. +1 +0 1 +1 +2 +0 2 +2 +N , +, +N , +T +N , +, +N , +K + + + + + + +− +− + + +=  + + + +, where +1 +N, + + and +2 +N, + + are the +complex in-coupling constants for inputting energy from the continuum to a1 and a2. N +depends on the number of available ports, which is governed by the diffraction equation as +( +) +m +m +P sin +sin + + + += +− +, where m is the diffraction order,  is the incident polar angle, and m +is the diffraction angle [41]. For example, as shown in Fig. 1(b), for the lowest band gap at the +zone center,  = 0o, such that only one m = 0th propagating order exists in free space. For the +second band gap at the zone boundary where +2P sin + + += +, two m = 0th and 1st orders are +present at +m + + +=  . In general, zone center supports an odd number of ports including +0 + +whereas an even number of ports is found at zone boundary where +0 + is always zero [41]. + +6 + +To see how the field symmetry is revealed, we solve the eigenvalues and eigenvectors of +the homogeneous part of Eq. (1) by diagonalization. The complex frequencies of the coupled +modes as: +( +) +( +) +( +) +2 +o +a +r +i + + + + + = + ++ + + + +, indicating their spectral positions and decay +rates depend on  and . For the real part, we see the spectral positions of the coupled modes +are determined by the magnitude and sign of  and they are separated by an energy gap = 2. +On the other hand, for the imaginary part, one mode has larger decay rate whereas another one +has lower, featuring the bright (dipolar) and dark (quadrupolar) modes [42]. In particular, if +2 +0 +r + + − += +, one coupled mode exhibits zero radiation damping, resulting in a quasi-BIC [34]. +The unit eigenvectors are +1 +2 +1 +2 +1 +2 +a +a +a +a +a +a ++ +− ++ + + + + += + + + + +− + + + + +, which are orthogonal and carry odd and even +symmetries with respect to the unit cell center. As a result, for an isolated energy band,  = 0 +if both the eigenvectors at the zone center and boundary are either a+ or a− but =  if they are +different. +We study the spatial field symmetries of a for TM and TE polarized waves. Leaky +evanescent waves are considered here as an example. For TM modes such as Bloch-like +surface plasmon polaritons (SPPs) propagating in the x-direction, the magnetic fields of a+ are +( )( +) +1 +2 +x +x +z +ik x +ik x +k z +k +ˆ +H +H +Ae +u +x +e +e +y +− +− ++ += +− +, where A is a constant, kx and kz are the propagation +constants in the x- and z-directions, and +( ) +ku +x is the periodic function [43]. +( ) +ku +x is assumed +to be an even function for simplicity as its symmetry does not affect the Zak phase results. The +corresponding +electric +fields +are +( +) +( ) +( +) +( +) +( +) +1 +2 +2 +zk z +k +z +x +x +x +H +H +A +ˆ +ˆ +E +e +u +x +k sin k x x +k cos k x z +i +− + ++ +− += += ++ +−  + +, revealing the in-plane x- +and out-of-plane z-components are odd and even in the x-direction, or +( ) +( +) +x +x +E +x +E +x += − +− + and + +7 + +( ) +( +) +z +z +E +x +E +x += +− +. Likewise, for a− , we have even +( ) +( +) +x +x +E +x +E +x += +− + and odd +( ) +( +) +z +z +E +x +E +x += − +− +. Conversely, for TE modes such as waveguide modes, the in-plane electric +fields of a+ and a− are +( ) +( +) +2 +zk z +k +x +ˆ +iAe +u +x sin k x y +− + and +( ) +( +) +2 +zk z +x +ˆ +Ae +u x cos k x y +− +, giving rise to +odd +( ) +( +) +y +y +E +x +E +x += − +− + and even +( ) +( +) +y +y +E +x +E +x += +− +, respectively. Therefore, for the in-plane +components, the TM and TE polarized a+ and a− are odd and even in the x-direction. +Once the field symmetries of a are known, their spectral positions will then be deduced +via far-field. By using conservation of energy and time reversal symmetry, the outgoing ports +are expressed as   +  +1 +2 +a +s +C s +K a +− ++ + + += ++ + + + + +, where   +0 +T +N , +, +N , +s +s +s +s +− +− +− +− +− +=  + + + and C is the +nonresonant scattering matrix [38]. We find the transformation matrix to be +1 +1 +1 +1 +1 +2 +T +T + + += + + +− + + + +so that the outgoing fields can now be rewritten as: +  +  +  +1 +2 +1 +2 +0 1 +0 2 +0 1 +0 2 +1 +2 +1 +2 +1 +1 +2 +2 +N , +N , +N , +N , +T +, +, +, +, +N , +N , +N , +N , +a +s +C s +T K +C s +a +a +a + + + + + + + + + + + + +− +− +− +− ++ +− ++ ++ ++ +− +− ++ +− + + + + + + + + + + + + + + + + + + ++ +− += ++ += ++ ++ + + + + + + + + + + + + + + + + ++ +− + + + + +. +(2) +Eq. (2) can be further simplified by using the relationships between +n,i + − + and +n,i + +, where i = 1 +or 2 and n  N is the diffraction order. As provided in the Supplementary Information [44], +given the fact that both far- and near-fields should follow the same spatial symmetry, the +radiation patterns of TM a arising from the interferences between the decay ports should +preserve the same +( ) +( +) +F +F +x +x +E +x +E +x += − +− + and +( ) +( +) +F +F +x +x +E +x +E +x += +− + dependences, where the +superscript +F +denotes +the +far-fields, +leading +to +( +) +1 +2 +1 +2 +n, +n, +n, +n, + + + + +− +− ++ += − ++ + and +1 +2 +1 +2 +n, +n, +n, +n, + + + + +− +− +− += +− + for a+ and a− . Likewise, for TE a , +( ) +( +) +F +F +y +y +E +x +E +x += − +− + and + +8 + +( ) +( +) +F +F +y +y +E +x +E +x += +− + also give +( +) +1 +2 +1 +2 +n, +n, +n, +n, + + + + +− +− ++ += − ++ + and +1 +2 +1 +2 +n, +n, +n, +n, + + + + +− +− +− += +− +. More +importantly, both polarizations indicate +,1 +,2 +n +n + +− += − + and +,1 +,2 +n +n + + +− += − +, which agree with the +fact that the system should fulfill the inversion symmetry requirement. However, +1 +n, + − + ( +2 +n, + − +) +is not necessarily equal to +1 +n, + +( +2 +n, + +). In addition, for a+ , +( +) +0 1 +0 2 +0 1 +0 2 +, +, +, +, + + + + ++ += − ++ + implies the +normal diffraction order is always missing, resulting in an even number of decay ports at both +the zone center and boundary. Therefore, at the zone center for TM and TE polarizations, Eq. +(2) can be reduced as: +  +( +) +0 +0 +1 +1 +0 +2 +2 +2 +N +N +N , +N +N +, +N +N +N , +N +N +s +s +C s +a +a +s + + + + + + + + + +− +− +− +− +− ++ ++ +− +− +− +− +− ++ + + + + + + + + + + + + + + + + + + + + + + + + += ++ ++ + + + + + + + + + + + + + + + + + + +− +− ++ + + + + + + +, + +(3) +where the 1,2 subscripts are now dropped. On the other hand, at the zone boundary, the +outgoing fields carry the same analytical form as Eq. (3) except +0, +s − = 0 since +0 +0 + = +. +Eq. (3) reveals additional information about the occurrence of quasi-BIC at high +symmetry points. In general, quasi-BIC occurs when all the decay ports are zero. Therefore, +at the zone center, unless +0 + = 0, quasi-BIC can only be observed from a+ . Particularly, for +the lowest zone center band gap where only the N = 0 port is present, an a+ quasi-BIC is always +present, making it symmetry protected [34]. However, for higher order band gaps, while the +normal N = 0 port is still zero, other N > 0 ports are not necessary. Quasi-BIC can still be +found if +n +n + + +− = +. In other words, if all the mirror symmetric decay ports of the uncoupled +mode are identical and in-phase, destructive interferences occur everywhere across all +diffraction orders, resulting in quasi-BIC. Such special condition can only be met for certain +tailored system geometry. If +n +n + + +−  +, a+ appears as bright or dark mode depending on the + +9 + +sign of . On the other hand, at the zone boundary where +0, +s −is always zero, a+ or a− can be +quasi-BIC if +n +n + +− += + or +n +n + +− += − + is fulfilled. +We then explicitly formulate the diffraction orders. By considering only one single +incidence port q such that   +0 +0 +T +q, +s +s ++ ++ + + +=  + , the coupled mode amplitudes are +( +) +( +) +, +1 +2 +q +q +qs +a +i +− ++ ++ ++ +− += +− + + + + + and +( +) +( +) +, +1 +2 +q +q +qs +a +i + + + + +− ++ +− +− ++ += +− +. Two mirror symmetric n  N diffraction +orders thus are: +( +)( +) +( +) +( +)( +) +( +) +( +)( +) +( +) +( +)( +) +( +) +, +, +, +, +1 +1 +, +2 +2 +1 +1 +, +2 +2 +n +n +q +n +n +n +n +q +q +n +n +q +q +n +n +q +q +n +n +q +q +q +s +c +s +s +s +i +i +i +c +i +− +− +− ++ ++ +− ++ +− +− +− +− ++ +− +− +− +− +− ++ +− +− +− ++ ++ ++ +− +− +− +− += +− ++ +− += ++ ++ ++ +− + + + + + + + + + + + + + + + + + + + + + + + + + + +(4) +where +n +c are the complex nonresonant scattering coefficients. We see from Eq. (4) that the +radiations from a+ and a− have odd and even symmetries [37]. While a− gives two in phase +diffraction orders, those from a+ are  out of phase. Therefore, by fitting the magnitude and +phase, +2 +, +, +n +q +s +s + +− ++ and +( +) +, +, +arg +n +q +s +s + +− ++ , spectra of any pair of oblique mirror diffraction orders +at the zone center and boundary with Eq. (4) to determine their relative phase, the spectral +positions +( +) +Re + + + can be deduced to find out whether a+ or a− is associated with the energy +band of interest. +III. +FINITE-DIFFERENCE TIME DOMAIN SIMULATION +We verify the CMT model by FDTD simulations. Two types of optical systems are +considered, and they are 1D Au plasmonic and SiO2/Au photonic crystals. While the plasmonic +crystals (PmCs) support TM-polarized Bloch-like SPPs [45], the photonic crystals (PhCs) +excite TE waveguide modes [46]. We will present the results of PmCs here and those of the + +10 + +PhCs are provided in the Supplementary Information [44]. For the PmCs, the unit cell is shown +in Fig. 2(a), with the period P and groove height H are set at 900 nm and 50 nm, respectively, +and the groove width W is varied from 100 and 700 nm with a step size of 150 nm. The +corresponding TM-polarized k- and wavelength-resolved total reflectivity, which sums all the +diffraction orders, mappings are calculated along the -X direction in Fig 2(b) – (f), showing +the presence of the dispersive ±1 and -2 Bloch-like SPP bands, which follow the phase +matching equation given as +2 +2 +1 +1 +2 +Au +SP +Au +n +k +P + + + + + + + + = ++ + + + + ++  + + + +, where +Au + + is the dielectric constant +of Au and nSP is the SPP band, as illustrated by the dash lines in Fig 2(b) [37,45]. More +importantly, one sees ±1 SPPs cross at k = 0 m-1 and +1 and -2 SPPs cross at k = /P m-1, +yielding two band gaps at  = 925 and 650 nm for the zone center and boundary. In agreement +with the CMT model, the coupled modes exhibit dark and bright radiation characteristics. +We attempt to determine the Zak phase of the +1 SPP band. At the zone center for all +PmCs, the dark mode is quasi-BIC and located at the +1 band for W = 100 – 400 nm but flips +to the -1 band when W increases further. The corresponding reflectivity spectra are plotted in +Fig. 3(a) for illustration, clearly showing only one single reflectivity dip as the bright mode. +As a result, we conclude a+ locates at the +1 band for W = 100 – 400 nm but flips to the -1 +band for wider W. On the other hand, at the zone boundary, we can no longer differentiate the +spectral positions of a simply by examining the total reflectivity spectra because two dark +and bright modes are present. Since only a pair of mirror symmetric m = 0th and 1st, or n = 1, +diffraction orders is available, Fig. 3(b) & (c) show the simulated +2 +1, +1, +s +s + − ++ + and +( +) +1, +1, +arg s +s + − ++ spectra and we fit them by by Eq. (4) to determine the relative phases between +the diffraction pairs of two modes. The best fits are displayed as the solid lines. The +corresponding +( +) +Re  of all PmCs are summarized in Table 1, in which the highlights are the + +11 + +coupled modes sitting on the +1 band at the zone center (high energy mode) and boundary (low +energy mode). If the highlights at two regions are either a+ or a− , the Zak phase is 0, but  +when they are different. As a result, by comparing the modes at the zone center and boundary +of the +1 band,  =  for W = 100, 250, 550 nm but  = 0 for 400 and 700 nm. +To confirm our findings, we have simulated the near-field intensity profiles at the zone +center and boundary of the +1 band by FDTD in Fig. 4(a) & (b) for different W. At the zone +center, we see the profiles are even with respect to the groove center for W = 100 – 400 nm but +change to odd afterwards [18,47]. On the other hand, the profiles at the zone boundary are odd +for W = 100, 250, and 700 nm but are even for 400 and 550 nm. As a result, the field +symmetries indicate  =  for W = 100, 250 and 550 nm but 0 for 400 and 700 nm, in consistent +with the far-field simulations. In addition, we have calculated the near-field patterns across the +first Brillouin zone for all PmCs in the Supplementary Information [44] and then employ the +Wilson loop method to directly determine  given as +( ) +P +P +X +k dk + + +− +, where +( ) +X +k is the Berry +connection given as +( ) ( ) +( ) ( ) +* +,k +k +unit cell +* +k +,k +unit cell +u +( x ) +i +u +x +x +dx +k +u +x +x u +( x )dx + + + + + + + [47,48]. The evolutions of the individal +phase difference, which is +( ) +X +k +k + , of the +1 band as a fucntion of k with k = 0.04π/P are +plotted in Fig. 4(c). The integrated areas yield the  phases that once again support our results. +IV. +EXPERIMENTAL VERIFICATION +A series of 1D periodic Au rectangular groove PmCs has been fabricated by focused ion +beam (FIB) and their scanning electron microscopy (SEM) images are shown in the insets of +Fig. 5(a) – (e), showing they have P = 900 nm, H = 50 nm, and W varying from 100 to 700 nm +[47]. After the sample preparation, the PmCs are then transferred to a homebuilt Fourier space + +12 + +optical microscope described in the Supplementary Information for angle- and wavelength- +resolved diffraction measurements [44]. Briefly, a supercontinuum generation laser is +illuminated on the sample at a well-defined incident angle  via the microscope objective lens +and the signals from the sample are collected by the same objective lens in which the diffraction +orders are projected onto the momentum space [49,50]. By using an aperture to filter out the +desired diffraction order, a spectrometer-based CCD detector and a common path +interferometer are used for measuring the magnitude and phase spectra [51,52]. +By varying  sequentially and at the same time measuring the total reflection spectra, we +contour plot the TM-polarized reflectivity mappings in Fig. 5(a) – (e) for different W along the +-X direction. They show ±1 and -2 SPP bands are present, and the bands are consistent with +the phase-matching equation as illustrated by the dash lines. From the mappings, we see at +normal incidence, or the zone center, BIC-like mode is always observed near the band gap. +The +1 band has a+ for W = 100 – 400 nm but a− for wider W. On the other hand, at the zone +boundary where +1 and -2 SPPs cross at  ~ 20.5o, we see the dark and bright modes are found +and their positions depend on W. To estimate the spectral positions of a , we measure the +corresponding m = 0th and 1st, or n = ±1, reflectivity and TM-TE phase difference spectra in +Fig. 6(a) & (b) and fit them by Eq. (4) to determine +( +) +Re  in Table 1, which shows the +1 +band is a− for W = 100, 250 and 700 nm is a+ for 400 and 550 nm. Therefore,  =  for W = +100, 250 and 550 nm but = 0 for 400 and 700 nm. +Finally, we demonstrate a topologically protected state is formed at the interface between +two topological trivial and nontrivial PmCs [47]. We construct a heterostructure by joining +two W = 100 and 400 nm PmCs together. In prior to joining, we have examined by FDTD the +field symmetries at the zone center and boundary of two PmCs and determine the  of the 0, - +1, and +1 SPP bands to be  ,  and  for W = 100 nm and  ,  and 0 for W = 400 nm. + +13 + +Therefore, the sums of  give  and 0 for W = 100 and 400 nm PmCs, indicating the -2/+1 +energy gaps at the zone boundary are topological trivial and nontrivial. We then simulate the +heterostructure supercell as shown in Fig. 7(a) that consists of 14 unit cells of W = 100 and 400 +nm PmCs on the right- and left-handed sides [47]. Fig. 7(b) shows the TM-polarized k- and +wavelength-resolved reflectivity mapping at the zone boundary along the -X direction, clearly +demonstrating a localized mode is located at k = 0.5/P or θ = 20.5o, and  ~ 640 nm in the +mid of the band gap. We also have simulated the wavelength-dependent near-field mapping +of the heterostructure. For different wavelengths, the near-field intensities at 20 nm above the +surface is simulated across the heterostructure and then contour plotted in Fig. 7(c), showing +the interface is located at x = 0 m and the trivial and nontrivial regions are at x > 0 m and < +0 m, respectively. One sees two strong fields are visible at ~ 620 and 670 nm in the PmC +bulk regions away from the interface due to the excitations of the upper and lower coupled +modes. However, the strongest field strength is observed at the interface, x = 0 µm, at 640 nm, +and it decays rapidly into the bulk regions, signifying the presence of a topologically protected +interface state [47]. We have prepared the heterostructure by FIB and its SEM image is shown +in Fig. 7(d) with W = 100 and 400 nm PmCs on the right- and left-hand sides. The TM- +polarized k- and wavelength-resolved reflectivity mapping of the sample is illustrated in Fig. +7(e), clearly showing an interface state is found at  = 20.5o and  ~ 625 nm in the +1/-2 band +gap at the zone boundary. +V. +CONCLUSION +In summary, we have formulated an analytical model based on temporal CMT to +determine the Zak phase of an isolated band in leaky photonic systems. At the Brillouin zone +center and boundary, as the far- and near-fields of the systems share the same spatial symmetry, +the mirror symmetric diffractions are either in or  out of phase depending on the Bloch wave + +14 + +symmetry. Therefore, the near-field symmetries can be probed by studying the diffraction +profiles. In addition, our model generalizes the occurrence of quasi-BIC at the high symmetry +points. The interplay between the in-coupling constants of different ports plays a decisive role +in manifesting quasi-BICs. For verification, we have studied 1D PmCs and PhCs that support +TM- and TE-polarized SPP and waveguide modes by FDTD and the results agree very well +with the theory. We also have prepared 1D PmCs by FIB and examined their diffractions by +using Fourier space diffraction spectroscopy and common path interferometry for determining +the Zak phases. In the end, a topological protected interface state is demonstrated by joining +two topological trivial and nontrivial PmCs together. +VI. +ACKNOWLEDGMENT +This research was supported by the Chinese University of Hong Kong through Area of +Excellence (AoE/P-02/12) and Innovative Technology Fund Guangdong-Hong Kong +Technology Cooperation Funding Scheme (GHP/077/20GD). + + + +15 + +Reference +1. X.L. Qi and S.C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, +1057 (2011). +2. M.Z. Hasan and C.L. Kane, Topological insulators, Rev. Mod. Phys. 82, 3045 (2010). +3. J.E. Moore, The birth of topological insulators, Nature 464, 194 (2010). +4. N.R. Cooper, J. Dalibard, and I.B. Spielman, Topological bands for ultracold atoms, Rev. +Mod. Phys. 91, 015005 (2019). +5. D.W. Zhang, Y.Q. Zhu, Y.X. Zhao, H. Yan, S.L. Zhu, Topological quantum matter with +cold atoms, Adv. Phys. 67, 253 (2019). +6. T. Ozawa, H.M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M.C. Rechtsman, D. +Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. +Phys. 91, 015006 (2019). +7. L. Lu, J.D. Joannopoulos and M. Soljačić, Topological photonics, Nat. Photon. 8, 821 +(2018). +8. A.B. Khanikaev and G. Shvets, Two-dimensional topological photonics, Nat. Photon. 11, +763 (2017). +9. S.D. Huber, Topological mechanics, Nat. Phys. 12, 621 (2016). +10. Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y.D Chong, and B. Zhang, Topological Acoustics, +Phys. Rev. Lett. 114, 114301 (2015). +11. G. Ma, M. Xiao and C.T. Chan, Topological phases in acoustic and mechanical systems, +Nat. Rev. Phys. 1, 281 (2019). +12. P. Delplace, J.B. Marston, and A. Venaille, Topological origin of equatorial waves, +Science 358, 1075 (2017). +13. F. Tang, H.C. Po, A. Vishwanath, and X. Wan, Comprehensive search for topological +materials using symmetry indicators, Nature 566, 486 (2019). + +16 + +14. X. Chen, X. Ma, K. He, J.F. Jia, and Q.-K. Xue, Molecular beam epitaxial growth of +topological insulators, Adv. Mater. 23, 1162 (2011). +15. W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in Polyacetylene, Phys. Rev. Lett. 42, +1698 (1979). +16. J. Zak, Berry’s Phase for Energy Bands in Solids, Phys. Rev. Lett. 62, 2747 (1989). +17. M. Xiao, Z.Q. Zhang, and C.T. Chan, Surface impedance and bulk band geometric phases +in one-dimensional systems, Phys. Rev. X 4, 021017 (2014). +18. M. Xiao, G. Ma, Z. Yang, P. Sheng, Z.Q. Zhang, and C.T. Chan, Geometric phase and +band inversion in periodic acoustic systems, Nat. Phys. 11, 240 (2015). +19. J.K. Asbóth, L. Oroszlány, and A. Pályi, A Short Course on Topological Insulators +(Springer, New York, 2016). +20. D.A. Abanin, T. Kitagawa, I. Bloch, and E. Demler, Interferometric approach to measuring +band topology in 2D optical lattices, Phys. Rev. Lett. 110, 165304 (2013). +21. M. Atala, M. Aidelsburger, J.T. Barreiro, D. Abanin, T. Kitagawa, E. Demler and I. Bloch, +Direct measurement of the Zak phase in topological Bloch bands, Nat. Phys. 9, 795 (2013). +22. F. Cardano, A. D’Errico1, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, +V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, Detection of +Zak phases and topological invariants in a chiral quantum walk of twisted photons, Nat. +Comm. 8, 15516 (2017). +23. Z. Jiao, S. Longhi, X. Wang, J. Gao, W. Zhou, Y. Wang, Y. Fu, L. Wang, R. Ren, L.-F. +Qiao, and X.-M. Jin, Experimentally detecting quantized Zak phases without chiral symmetry in +photonic lattices, Phys. Rev. Lett. 127, 147401 (2021). +24. M.A. Gorlach, X. Ni, D.A. Smirnova, D. Korobkin, D. Zhirihin, A.P. Slobozhanyuk, P.A. +Belov, A. Alù, and A.B. Khanikaev, Far-field probing of leaky topological states in all +dielectric metasurfaces, Nat. Comm. 9, 909 (2018). + +17 + +25. W. Gao, M. Xiao, C.T. Chan, and W.Y. Tam, Determination of Zak phase by reflection +phase in 1D photonic crystals, Opt. Lett. 40, 5259 (2015). +26. Q. Wang, M. Xiao, H. Liu, S. Zhu, and C.T. Chan, Measurement of the Zak phase of photonic +bands through the interface states of a metasurface/photonic crystal, Phys. Rev. B 93, 041415(R) +(2016). +27. L. Fan, W. Yu, S. Zhang, H. Zhang, and J. Ding, Zak phases and band properties in acoustic +metamaterials with negative modulus or negative density, Phys. Rev. B 94, 174307 (2016). +28. W. Zhu, Y. Ding, J. Ren, Y. Sun, Y. Li, H. Jiang, and H. Chen, Zak phase and band +inversion in dimerized one-dimensional locally resonant metamaterials, Phys. Rev. B 97, +195307 (2018). +29. K. Koshelev, A. Bogdanov, and Y. Kivshar, Engineering with bound states in the +continuum, Opt. Photon. News 31, 38 (2020). +30. B. Zhen, C.W. Hsu, L. Lu, A.D. Stone, and M. Soljacic, Topological nature of optical +bound states in the continuum, Phys. Rev. Lett. 113, 257401 (2014). +31. Y. Zhang, A. Chen, W.H. Liu, C.W. Hsu, B. Wang, F. Guan, X. Liu, L. Shi, L. Lu, and J. +Zi, Observation of polarization vortices in momentum space, Phys. Rev. Lett. 120, 186103 +(2018). +32. W. Ye, Y. Gao, and J. Liu, Singular points of polarizations in the momentum space of +photonic crystal slabs, Phys. Rev. Lett. 124, 153904 (2020). +33. T. Yoda and M. Notomi, Generation and annihilation of topologically protected bound +states in the continuum and circularly polarized states by symmetry breaking, Phys. Rev. +Lett. 125, 053902 (2020). +34. C.W. Hsu, B. Zhen, A.D. Stone, J.D. Joannopoulos, and M. Soljacic, Bound states in the +continuum, Nat. Rev. Mater. 1, 16048 (2016). + +18 + +35. H.M. Doeleman, F. Monticone, W. den Hollander, A. Alu, and A.F. Koenderink, +Experimental observation of a polarization vortex at an optical bound state in the +continuum, Nat. Photon. 12 397 (2018). +36. M.F. Limonov, M.V. Rybin, A.N. Poddubny, and Y.S. Kivshar, Fano resonances in +photonics, Nat. Photon. 11, 543 (2017) +37. Z. L. Cao and H. C. Ong, Determination of the absorption and radiative decay rates of dark +and bright plasmonic modes, Opt. Exp. 22, 16112 (2014). +38. L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. H. Fan, From electromagnetically +induced transparency to superscattering with a single structure: A coupled-mode theory +for doubly resonant structures, Phys. Rev. Lett. 108, 083902 (2012). +39. H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, New Jersey, 1984). +40. S. Fan, in Optical Fiber Telecommunications V, edited by I. P. Kaminow, T. Li, and A. E. +Willner (Academic Press, Burlington, 2008). +41. E.G. Loewen and E. Popov, Diffraction gratings and applications (CRC Press, New York, +1997). +42. C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, +Femtosecond light transmission and subradiant damping in plasmonic crystals, Phys. Rev. +Lett. 94 113901 (2005). +43. C. Billaudeau, S. Collin, C. Sauvan, N. Bardou, F. Pardo, and J.-L. Pelouard, Angle- +resolved transmission measurements through anisotropic two-dimensional plasmonic +crystals, Opt. Lett. 33, 165 (2008). +44. See Supplementary Materials for the connection between the far- and near-fields from one- +dimensional periodic optical system, simulated near-field patterns of the +1 surface +plasmon polariton (SPP) band of 1D PmCs across the first Brillouin zone, FDTD +simulation results of 1D SiO2/Au photonic crystals (PhCs), and the Fourier space optical + +19 + +microscope for angle- and wavelength resolved diffraction mapping and common path +interferometry +45. X. Guo, C. Liu, and H. C. Ong, Generalization of the circular dichroism from metallic +arrays that support Bloch-like surface plasmon polaritons, Phys. Rev. Appl. 15, 024048 +(2021). +46. A. Christ, S.G. Tikhodeev, N.A. Gippius, J. Kuhl, and H. Giessen, Waveguide-plasmon +polaritons: Strong coupling of photonic and electronic resonances in a metallic photonic +crystal slab, Phys. Rev. Lett. 91, 183901 (2003). +47. C. Liu and H. C. Ong, Realization of topological superlattices and the associated interface +states in one-dimensional plasmonic crystals, Phys. Rev. B 106, 045401 (2022). +48. H. Wang, G.-Y. Guo, and J.-H. Jiang, Band topology in classical waves: Wilson-loop +approach to topological numbers and fragile topology, New. J. Phys. 21, 093029 (2019). +49. B. Huang, F. Yu, and R. N. Zare, Surface plasmon resonance imaging using a high +numerical aperture microscope objective, Anal. Chem. 79, 2979 (2007). +50. F. Bleckmann, Z. Cherpakova, S. Linden, and A. Alberti, Spectral imaging of topological +edge states in plasmonic waveguide arrays, Phys. Rev. B 96, 045417 (2017). +51. Z.L. Cao, S.L. Wong, S.Y. Wu, H.P. Ho, and H.C. Ong, High performing phase-based +surface plasmon resonance sensing from metallic nanohole arrays, Appl. Phys. Lett. 104, +171116 (2014). +52. S.L. Wong and H.C. Ong, Phase difference mapping of two-dimensional metallic nanohole +arrays, Appl. Phys. Lett. 100, 233102 (2012). + + + +20 + + +Fig. 1. (a) The schematic shows at the Brillouin zone center and boundary in 1D leaky optical +system, two Bloch-like modes a1,2 counter propagate in opposite directions with each supports +discrete in-coupling channels +1 2 +0 1 2 +1 2 +N, , +, , +N, , + + +− +. They interact with each other to form two +coupled a at higher and lower energies separated by an energy band gap. (b) +0 1 2 +0 +, , + + + at +the zone center but +0 1 2 +0 +, , + += + at the zone boundary. + + + +Second zone + boundary +band gap +Lowest zone +center band gap +Lowest zone +boundary +band gap21 + + +Fig. 2. (a) The unit cell of 1D PmC for FDTD simulations. The simulated TM-polarized k- +and wavelength-resolved total reflectivity mappings of PmCs with W = (b) 100, (c) 250, (d) +400, (e) 550, and (f) 700 nm taken along the -X direction. The white dash lines are calculated +by using the phase-matching equation, indicating ±1 and -2 Bloch-like SPPs are excited. At +the zone center and boundary where k = 0 and 0.5, two energy band gaps are formed, featuring +two dark and bright modes are located above or below the gap. Particularly, at k = 0, a quasi- +BIC is observed at either above or below the gap. + + +(a) +Air +(b) +-2 SPP +p +↑H ++1: +SPP +W +Au +、-1SPP +C) +(d) +(f) +e22 + + +Fig. 3. The TM-polarized total reflectivity spectra of PmCs taken at the zone center for +different W, exhibiting only one single reflectivity dip as the bright mode. The red dash line +is the band gap center, indicating the quasi-BIC occurs at shorter wavelength for W = 100, 250 +and 400 nm but longer wavelength for W = 550 and 700 nm. At the zone boundary, two TM- +polarized mirror symmetric n = -1 (black square) and 1 (red circle) (b) reflectivity and (c) phase +spectra for W = 100 (top) to 700 (bottom). The green and blue solid lines are the best fits +determined by CMT. + +XX +X23 + + +Fig. 4. The FDTD simulated near-field patterns of the PmCs for different W taken at the +Brillouin zone (a) center and (b) boundary, showing their field symmetries are the same for W += 400 and 500 nm but different for W = 100, 250, and 700 nm. (c) The individual phase profiles +determined by the Wilson loop method. The integration yields the Zak phase, indicating the +phase is 0 for W = 400 and 500 nm but  for W = 100, 250, and 700 nm. + +(b)24 + + +Fig. 5. The measured TM-polarized k- and wavelength-resolved total reflectivity mappings of +PmCs with W = (a) 100, (b) 250, (c) 400, (d) 550, and (e) 700 nm taken along the -X direction. +The white dash lines are ±1 and -2 Bloch-like SPPs determined by the phase matching equation. +Two band gaps are formed at the zone center and boundary. The insets are the corresponding +SEM images of the PmCs with the scale bare = 1 µm. + +0..9 +600 +-2SPP +0.8 +6/5 +0.7 +750 ++1SPP +0.6 +825 +0.5 +900 +1 +SPP +(a) +(b) +(c) +(d) +(e +975 +0..4 +80c0L0008060L000800L000s060L0008060L00025 + + +Fig. 6. At the zone boundary, two measured TM-polarized mirror symmetric n = -1 (black +square) and 1 (red circle) (b) reflectivity and (c) TM-TE phase difference spectra for W = 100 +(top) to 700 (bottom). The green and blue solid lines are the best fits determined by CMT. + + + + +26 + + +Fig. 7. (a) The schematic of the heterostructure by joining W = trivial 100 and nontrivial 400 +nm PmCs. The interface is marked by the dash line. (b) The FDTD simulated TM-polarized +reflectivity mapping of the heterostructure taken at the zone boundary along the -X direction, +showing an interface state is found within the gap at  = 640 nm. (c) The wavelength- +dependent near-field intensity mapping simulated at 20 nm above the heterostructure. The +interface is located at x = 0 m, showing strong field localization. The strong fields at 620 and +670 nm arise from the PmC bulk regions. (d) The SEM image of the W = 100 and 400 nm with +the scale bar corresponding to 1 µm. (e) The measured TM-polarized reflectivity mapping of +the heterostructure taken at the zone boundary along the -X direction, showing an interface +state is found within the gap at  = 625 nm. + +W = 400nm +W = 100nm +nontrivial +trivial +0.9 +0.8 +0.7 +0.6 +0.52 +0.9 +0..8 +0..7 +0.6 +575 +575 +(b) +e +600 +600 +625 +625 +650 +650 +675 +675) +700 +(25) +/00 +0.4 +0.65) +0.6 +0.25 +0.30 +0.35 +0.4.0 +0.415 +k (2/) +k (2TN) +700 +(c) +0.9 +680 +0.8 +0..7 +660 +0.6 +0.5 +@) +640 +0.4 +ABl +interface state +0.3 +620 +0.2 +0..1 +W = 400 nm +W = 100 nm +0 +600 +1000 +500 +0 +500 +1000 +x (nrm)27 + + + + + +100 nm 250 nm 400 nm 550 nm 700 nm +FDTD +Zone +center +( +) +Re + (eV) +1.36 +1.37 +1.36 +1.32 +1.30 +( +) +Re − (eV) +1.32 +1.31 +1.33 +1.36 +1.37 +Zone +boundary +( +) +Re + (eV) +2.00 +1.98 +1.84 +1.85 +1.98 +( +) +Re − (eV) +1.82 +1.89 +1.99 +1.96 +1.85 +Experiment +Zone +center +( +) +Re + (eV) +1.36 +1.36 +1.36 +1.33 +1.32 +( +) +Re − (eV) +1.33 +1.32 +1.34 +1.36 +1.36 +Zone +boundary +( +) +Re + (eV) +2.02 +2.01 +1.95 +1.96 +2.02 +( +) +Re − (eV) +1.93 +1.96 +2.02 +2.01 +1.95 + +Table 1. The FDTD and experimental +( +) +Re  at the Brillouin zone center and boundary for +the PmCs with different W. The highlights are the coupled modes located on the +1 SPP band. +If the highlights at the zone center and boundary are both a+ or a− , the Zak phase is 0. If not, +the Zak phase is . + + + +28 + +Supplementary Information +Determination of the band topology of one-dimensional photonic +systems via far-field diffraction + +C. Liu, H.R. Wang, and H.C. Ong +Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s +Republic of China + +A. +Derivation of the connection between the far- and near-fields from one-dimensional +periodic optical system + + +Fig. S1. The schematic of the 2N+1 diffraction orders arising from the coupled mode supported +on 1D periodic leaky system. + +29 + +As shown in Fig. S1, for a one-dimensional optical leaky periodic system that possesses +inversion symmetry in the x-direction, at the Brillouin zone center and boundary, it supports +two Bloch-like coupled modes a above and below the photonic band gap with each dissipates +a total of 2N + 1 mirror symmetric diffraction channels in free space, where N is the highest +diffraction order. For TM- and TE-polarizations, both the near- and far-fields should carry the +same polarization and field symmetry in the x-y plane along the surface. For example, for TM- +polarization, in the far-field at zo above the system, the x-component of the electric field +( , +) +F +x +o +E +x z + is expressed as the superposition of all diffraction orders: +1 +1 +1 +0 +1 +1 +1 +sin +cos +sin +cos +1 +1 +sin +cos +sin +cos +0 +1 +1 +cos +cos +cos +cos +N +N +N o +N +N +N +o +o +N +N +N +o +N +N +N o +i +ik +x +ik +z +i +ik +x +ik +z +N +N +N +N +i +ikz +i +ik +x +ik +z +i +ik +x +ik +z +N +N +N +N +A e +e +e +A +e +e +e +A e e +A +e +e +e +A e +e +e + + + + + + + + + + + + + + + + + +− +− +− +− ++ +− ++ +− ++ +− +− +− +− +− +− ++ +− ++ +− +− +− +− ++ ++ ++ ++ ++ ++ +, (S1) +where An, n, and n are the diffraction amplitude, phase, and angle and the subscript n is the +diffraction order. At the same time, for the near-field, the TM-polarized a+ is a standing wave +with +( ) +( +) +( +) +( +) +zk z +k +z +x +x +x +ˆ +ˆ +E +e +u +x +k sin k x x +k cos k x z +− + ++ +, where kx and kz are the propagation +constants in the x- and z-directions and +( ) +ku +x is the periodic function. Assume +( ) +ku +x is an +even function for simplicity, we see +( ) +x +E +x is an odd function with +( ) +( +) +x +x +E +x +E +x += − +− + +dependence. Therefore, Eq. (S1) should also exhibit +( ) +( +) +F +F +x +x +E +x +E +x += − +− + dependence, yielding +n +n +A +A +− = +, +n +n + + + +− += ++ +, and +0 +0 +A = + that indicate two mirror symmetric diffraction orders have +the same magnitude but are always  out of phase and the normal diffraction order is null. As +a result, Eq. (S1) is rewritten as: +1 +1 +1 +1 +1 +1 +sin +cos +sin +cos +1 +1 +sin +cos +sin +cos +1 +1 +cos +cos +cos +cos +N +N +N o +N +N +N +o +N +N +N +o +N +N +N o +i +ik +x +ik +z +i +ik +x +ik +z +N +N +N +N +i +ik +x +ik +z +i +ik +x +ik +z +N +N +N +N +A e +e +e +A +e +e +e +A +e +e +e +A e +e +e + + + + + + + + + + + + + + + + +− +− +− +− +− +− +− +− +− +− +− +− ++ ++ +− +− +. +(S2) +By matching Eq. (S2) with the outgoing power amplitudes of a+ from CMT, which are +,1 +,2 +0,1 +0,2 +,1 +,2 +1 +2 +N +N +N +N +a + + + + + + +− +− ++ ++ + + + + + + + + ++ + + + + + + ++ + + +, we conclude +( +) +1 +2 +1 +2 +n, +n, +n, +n, + + + + +− +− ++ += − ++ + and +0,1 +0,2 +0 + + ++ += +. Likewise, +for another coupled mode a− where +( ) +( +) +( +) +( +) +zk z +k +z +x +x +x +ˆ +ˆ +E +e +u +x +k cos k x x +k sin k x z +− + ++ +, we see + +30 + +( ) +( +) +x +x +E +x +E +x += +− + and have +n +n +A +A +− = +, +n +n + +− += + and +0 +0 +A  +, indicating two mirror symmetric +orders are in phase and the normal diffraction order is present. Therefore, Eq. (S1) for a− is: +1 +1 +1 +0 +1 +1 +1 +sin +cos +sin +cos +1 +1 +sin +cos +sin +cos +0 +1 +1 +cos +cos +cos +cos +N +N +N o +N +N +N +o +o +N +N +N +o +N +N +N o +i +ik +x +ik +z +i +ik +x +ik +z +N +N +N +N +i +ikz +i +ik +x +ik +z +i +ik +x +ik +z +N +N +N +N +A e +e +e +A +e +e +e +A e e +A +e +e +e +A e +e +e + + + + + + + + + + + + + + + + + +− +− +− +− +− +− +− +− +− +− +− +− +− ++ ++ ++ ++ ++ ++ +. (S3) +We then have +1 +2 +1 +2 +n, +n, +n, +n, + + + + +− +− +− += +− + for the outgoing power amplitudes of a− given as +,1 +,2 +0,1 +0,2 +,1 +,2 +1 +2 +N +N +N +N +a + + + + + + +− +− +− +− + + + + + + + + +− + + + + + + +− + + +. + +Finally, +two +conditions +( +) +1 +2 +1 +2 +n, +n, +n, +n, + + + + +− +− ++ += − ++ + and +1 +2 +1 +2 +n, +n, +n, +n, + + + + +− +− +− += +− + result in +,1 +,2 +n +n + +− += − + and +,1 +,2 +n +n + + +− += − +. +On the other hand, for TE-polarized Bloch-like coupled modes a , at zo in the free space above +the system, the y-component of the far-field electric field +( , +) +F +y +o +E +x z + can be written as: +1 +1 +1 +0 +1 +1 +1 +sin +cos +sin +cos +1 +sin +cos +sin +cos +0 +1 +N +N +N o +N +N +N +o +o +N +N +N +o +N +N +N o +i +ik +x +ik +z +i +ik +x +ik +z +N +N +i +ikz +i +ik +x +ik +z +i +ik +x +ik +z +N +N +A e +e +e +A +e +e +e +A e e +A +e +e +e +A e +e +e + + + + + + + + + + + + + +− +− +− +− ++ +− ++ +− ++ +− +− +− +− +− ++ +− +− +− ++ ++ ++ ++ ++ ++ +. + +(S4) +The near-field of a+ where +( ) +( +) +zk z +k +x +ˆ +E +e +u +x sin k x y +− + +, we have +( ) +( +) +y +y +E +x +E +x += − +− + such that +n +n +A +A +− = +, +n +n + + + +− += ++ +, and +0 +0 +A = +, leading to +( +) +1 +2 +1 +2 +n, +n, +n, +n, + + + + +− +− ++ += − ++ + and +0,1 +0,2 +0 + + ++ += +. +Likewise, for a− where +( ) +( +) +y +y +E +x +E +x += +− +, we have +n +n +A +A +− = +, +n +n + +− += + and +0 +0 +A  +, giving rise +to +1 +2 +1 +2 +n, +n, +n, +n, + + + + +− +− +− += +− +. Therefore, two conditions give the same conclusion that +,1 +,2 +n +n + +− += − + and +,1 +,2 +n +n + + +− += − + regardless of the polarization. As a result, at the zone center for +TM- and TE-polarizations, the outgoing profile is: + +  +( +) +0 +0 +1 +1 +0 +2 +2 +2 +N +N +N , +N +N +, +N +N +N , +N +N +s +s +C s +a +a +s + + + + + + + + + +− +− +− +− +− ++ ++ +− +− +− +− +− ++ + + + + + + + + + + + + + + + + + + + + + + + + += ++ ++ + + + + + + + + + + + + + + + + + + +− +− ++ + + + + + + +, + (S5) + +31 + +where the 1,2 subscripts are now dropped. We see quasi-BIC arises from a+ and it will occur +when +0 +n +n + + +− − += +. However, for the lowest band gap where only the normal diffraction order +is present, quasi-BIC always occur, making it symmetry protected. On the other hand, at the +zone boundary where +0, +s − is always 0, we have for TM- and TE-polarizations: +  +( +) +0 +1 +1 +0 +0 +2 +2 +N +N +N , +N +N +, +N +N +N , +N +N +s +s +C s +a +a +s + + + + + + + + +− +− +− +− +− ++ ++ +− +− +− +− +− ++ + + + + + + + + + + + + + + + + + + + + + + + + += ++ ++ + + + + + + + + + + + + + + + + + + +− +− ++ + + + + + + +. +(S6) +Quasi-BIC occurs depending on the interplay between +n +− and +n + . a+ (a− ) is quasi-BIC if +0 +n +n + + +− − += + ( +0 +n +n + + +− + += +) but dark and bright modes are present if +0 +n +n + + +−  + +. +B. +Simulated near-field patterns of the +1 surface plasmon polariton (SPP) band of 1D +PmCs across the first Brillouin zone +By using the dipole source excitation method, the complex near-field patterns along the +1 SPP +band of 1D Au PmCs with period = 900 nm, groove height = 50 nm and different groove widths +have been simulated. The real and imaginary parts of the surface normal components, Re(Ez) +and Im(Ez), taken at 20 nm above the surface across the Brillouin zone from k = -/P to /P +m-1 are shown in Fig. S2 for groove width W = 100, 250, 400, 550 and 700 nm PmCs. They +will then be used for determining the Zak phase by the Wilson loop method. + +32 + +Fig. S2. The real and imaginary parts of the z-component of the near-field patterns of the PmCs +plotted as a function of k along the +1 SPP band in the first Brillouin zone for different W = +(a) & (b) 100, (c) & (d) 250, (e) & (f) 400, (g) & (h) 550, and (i) & (j) 700 nm. + + + + +(a +(b)33 + +C. +FDTD results of 1D SiO2/Au photonic crystals (PhCs) +Fig. S3(a) shows the unit cell of the PhCs, which has 400 nm thick SiO2 coated on Au surface +with the period P and the groove height H being set at 900 nm and 200 nm whereas the groove +width W varied from 100 and 725 nm with a step size of 125 nm. The corresponding TE- +polarized k-resolved total reflectivity mappings are shown in Fig S3(b) – (f), showing the +dispersive ±1 and -2 photonic bands, which follow the phase matching equation given as +( +) +( +) +2 +2 +sin +D +D +PhC +n +n +m +P + +  += ++ +, where nD is the refractive index of SiO2 and mPhC is the +photonic band. The calculations are superimposed in Fig 3(b). We see mPhC = ±1 photonic +bands cross at k = 0 m-1 and mPhC = +1 and -2 bands cross at k = /P m-1, yielding two energy +band gaps at  = 930 – 1030 nm and 700 – 770 nm at the zone center and boundary. At the +zone center, one symmetry protected quasi-BIC is always found, and it is located on the -1 +band for W = 100 – 475 nm but flips to the +1 band when W increases further. At the same +time, accidental quasi-BICs are also found along the +1 band at different k for all PhCs. + +34 + + +Fig. S3. (a) The FDTD unit cell of the PhC. The simulated TE-polarized k- and wavelength- +resolved total reflectivity mappings of PhCs with W = (b) 100, (c) 225, (d) 350, (e) 475, (f) +600, and (g) 725 nm taken along the -X direction. The white dash lines are calculated by using +the phase-matching equation, indicating ±1 and -2 photonic band are present. At the zone +center and boundary where k = 0 and 0.5, two energy band gaps are formed, featuring two dark +and bright modes are located above or below the gap. Particularly, at k = 0, a symmetry +protected quasi-BIC is observed at either above or below the gap. On the other hand, an +accidentally BIC is observed along the +1 band. + +Air +p +SiO2 +H +W +Au +(b) +C +-2 band ++1 band +-1 band +(d) +(e) +(f) +935 + +We will focus on the modes located on the +1 band at the zone center and boundary and +determine their field symmetries as well as . The reflectivity spectra of the PhCs taken under +normal incidence, i.e., at the zone center, are illustrated in Fig. S4(a), clearly showing only one +single reflectivity dip is present as the bright mode, verifying another coupled mode is quasi- +BIC that does not produce any dip. As quasi-BIC arises solely from a+ for the lowest band +gap, we deduce the coupled mode on the +1 band is symmetric a− for W = 100 – 475 nm PhCs +but becomes asymmetric a+ for W = 600 and 725 nm PhCs. On the other hand, the reflectivity +spectra taken at the zone boundary for all PhCs are shown in Fig. S4(b), showing two bright +and dark modes are present. + +Fig. S4. The TE-polarized total reflectivity spectra of PhCs taken at the zone (a) center and (b) +boundary for different W. At the zone center, only one single reflectivity dip is present as the +bright mode. On the other hand, at the zone boundary, two bright and dark modes are present. + +36 + + +To determine the near-field symmetries of the PhCs at the zone boundary, the two mirror +symmetric diffraction and phase spectra are shown in Fig. S5 and they are fitted with +2 +1, +1, +s +s + − ++ + and +( +) +1, +1, +arg s +s + − ++ from CMT. The best fits are displayed as the solid lines and the +fitted results +( +) +Re  are tabulated in Table S1. in which the highlights are the coupled modes +sitting on the +1 photonic band at the zone center (high energy mode) and boundary (low +energy mode). If the highlights at two regions are either a+ or a− , the Zak phase is 0, but  +when they are different. As a result, we conclude the Zak phase of +1 band for W = 100, 225 +and 600 nm is  but becomes 0 for W = 350, 475 and 725 nm. + + + + + +100 nm +225 nm +350 nm +475 nm +600 nm +725 nm +Zone +center +( +) +Re + (eV) +1.18 +1.19 +1.22 +1.27 +1.33 +1.37 +( +) +Re − (eV) +1.21 +1.26 +1.28 +1.29 +1.29 +1.31 +Zone +boundary +( +) +Re + (eV) +1.62 +1.65 +1.72 +1.78 +1.79 +1.79 +( +) +Re − (eV) +1.67 +1.69 +1.69 +1.71 +1.77 +1.84 + +Table S1. The FDTD +( +) +Re  at the Brillouin zone center and boundary for the PhCs with +different W. The highlights are the coupled modes located on the +1 photonic band. If the +highlights at the zone center and boundary are both a+ or a− , the Zak phase is 0. If not, the +Zak phase is . + + + +37 + + +Fig. S5. At the zone boundary, two TE-polarized mirror symmetric n = -1 (black square) and +1 (red circle) (a) reflectivity and (b) phase spectra of the PhCs for W = 100 (top) to 725 (bottom). +The green and blue solid lines are the best fits determined by CMT. + + + +4 +XC38 + +To verify the Zak phases, we have simulated the real and imaginary parts of the surface normal +components, Re(Ez) and Im(Ez), taken at 20 nm above the surface across the Brillouin zone +from k = -/P to /P m-1 in Fig. S6 for all PhCs. They will then be used for determining the +Zak phase by the Wilson loop method given as +( ) +P +P +X +k dk + + +− +, where +( ) +X +k is +( ) ( ) +( ) ( ) +* +,k +k +unit cell +* +k +,k +unit cell +u +( x ) +i +u +x +x +dx +k +u +x +x u +( x )dx + + + + + + + . The evolutions of the individal phase difference, which is +( ) +X +k +k + , of the +1 band as a fucntion of k with k = 0.04π/P of all PhCs are plotted in Fig. +S7. The integrated areas yield the Zak phases are  for W = 100, 225, and 600 nm and 0 for +W = 350, 475 and 725 nm, and they agree very well with earlier CMT results. + +39 + + +Fig. S6. The real and imaginary parts of the z-component of the near-field patterns of the PhCs +plotted as a function of k along the +1 photonic band in the first Brillouin zone for different W += (a) & (b) 100, (c) & (d) 225, (e) & (f) 350, (g) & (h) 475, (i) & (j) 600, and (k) & (l) 725 nm. + +(C) +(e +(g) +(h) +(i) +(k)40 + + +Fig. S7. The individual phase profiles of the PhCs with different W. The integration yields the +Zak phase, indicating the phase  for W = 100, 225, and 600 nm and 0 for W = 350, 475 and +725 nm. + +D. +Schematic of the Fourier space optical microscope for angle- and wavelength +resolved diffraction mapping and common path interferometry +Fig. S8 shows the schematic of the Fourier space optical microscope. Briefly, a broadband +supercontinuum laser from a nonlinear photonic crystal fiber is collimated and then passed +through a set of linear polarizers, wave plates, and lenses before being focused onto the back +focal plane (BFP) of a 100X objective lens (OB) with numerical aperture = 0.9. The light +exiting from the objective lens is then a collimated beam with well-defined linear polarization. +In addition, by displacing the focused spot across the BFP of the objective lens using a +motorized translation stage, the incident polar angle  of the collimated beam onto the sample +can be varied following sin = d/f, where d is the distance between the focused spot and the +optical axis of the BFP and f is the focal length of the objective lens. In addition, the azimuth +angle  can be varied by a motorized rotation sample stage to align the incident plane to the - +X direction of the PmC. The diffractions from the PmC are then collected by the same objective +lens and are routed through a set of Fourier lens system so that the diffraction orders are +projected onto the momentum space. By placing an aperture at the momentum space to filter + +41 + +out the desired diffraction order, its intensity and phase spectra can be measured by a +spectrometer-based CCD detector and a common path interferometer [1]. +To perform common path interferometry, the 45o linearly polarized collimated beam with the +Jones vector given as +1 +1 +1 +2 +  +  +  + is incident on the PmC. The diffraction order from the PmC +after the aperture can be formulated as: +0 +0 +TM +TE +i +TM +PmC +i +TE +r +e +J +r +e + + + + +=  + + + +, where rTM,TE and TM,TE +are the magnitudes and phases for TM- and TE-polarizations. The diffraction passes through +a quarter wave plate with the fast axis being placed at 45o with respect to the incident plane +and +a +motorized +rotatable +analyzer +with +angle +, +which +are +given +as +2 +( ) +2 +cos +sin +cos +sin +cos +sin +analyzer +J + + + + + + + + + +=  + + + + and +(45 ) +1 +1 +1 +1 +1 +2 +QWP +i +i +J +i +i + +− ++ + + += + + ++ +− + + +. The output vector is +( ) +(45 ) +1 +1 +1 +2 +analyzer +QWP +PmC +J +J +J + + +  +  +  +. After some formulations, the intensities for different  = 0o, +±45o, +and +90o +can +be +written +as: +( +) +2 +2 +2 +0 +1 +1 +( ) +2 +sin +2 +4 +0 +i +TM +TE +TM +TE +TM +TE +r +e +i r +R +r +r +r +r + + + + + ++ += += ++ ++ + + + + +, +( +) +2 +2 +45 +1 +( ) +2 +cos +4 +TM +TE +TM +TE +R +r +r +r +r + + ++ += ++ ++ +, +( +) +2 +2 +45 +1 +( ) +2 +cos +4 +TM +TE +TM +TE +R +r +r +r +r + + +− += ++ +− +, and +( +) +2 +2 +90 +1 +2 +sin +4 +TM +TE +TM +TE +R +r +r +r +r + += ++ +− +, where +TM +TE + + + += +− +. Therefore, the phase difference +between TM- and TE- polarized diffractions can be calculated by: +0 +90 +45 +45 +( ) +( ) +tan ( ) +( ) +( ) +R +R +R +R + + +  + + ++ +− +− += +− +. + +42 + + +Fig. S8. The schematic of the Fourier optical microscope. + +Reference +53. Z.L. Cao, S.L. Wong, S.Y. Wu, H.P. Ho, and H.C. Ong, High performing phase-based +surface plasmon resonance sensing from metallic nanohole arrays, Appl. Phys. Lett. 104, +171116 (2014). + + +P: polarizer +L1,L2,L3,L4: focusing lens +OB: objective lens +BS: beam splitter +BFP: back focal plane \ No newline at end of file diff --git a/DNE2T4oBgHgl3EQfSAfe/content/tmp_files/load_file.txt b/DNE2T4oBgHgl3EQfSAfe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e9ee9bf4e7708adf0817166e7d8abad27ec83a39 --- /dev/null +++ b/DNE2T4oBgHgl3EQfSAfe/content/tmp_files/load_file.txt @@ -0,0 +1,1199 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf,len=1198 +page_content='1 Determination of the Zak phase of one-dimensional photonic systems via far-field diffraction C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Liu*, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wang*, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Onga) Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s Republic of China Bloch waves in 1D periodic systems carry Zak phase, which plays a key role in determining the band topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In general, for systems that possess inversion symmetry, the Zak phase of an isolated band is quantized as 0 or \uf070 and is associated with the spatial field symmetries at the Brillouin zone center and boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The phase is \uf070 if the field symmetries are different but is 0 when they are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Since the radiation losses from leaky systems are strongly associated with the Bloch waves, one may probe the far-field continuum to determine the Zak phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Here, we formulate the diffractions from photonic systems at the zone center and boundary and find their spectral profiles reveal the Bloch wave symmetries and thereby the corresponding Zak phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The field symmetries also generalize the occurrence of bound states in the continuum at high symmetry points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For verification, we have studied the Zak phases of one- dimensional TM plasmonic and TE photonic crystals by electrodynamic simulations and measuring the optical properties of plasmonic crystals using Fourier space diffraction spectroscopy and common path interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In addition, a topological protected interface state is demonstrated when two 0 and \uf070 systems are joined together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The results prove our method provides a simple way for characterizing the band topology of non-Hermitian systems via far-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' These authors equally contributed to this work a) hcong@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='hk 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' INTRODUCTION Topological physics has attracted a widespread of interest not only in condensed matter physics [1-3] but also in other branches such as ultracold atom [4,5], electromagnetism [6-8], mechanics [9], acoustics [10,11], and oceanography [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Much attention in this field is focused on realizing the so-called topologically protected states, which support robust wave propagation against perturbation and disorder [1-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' To produce such states, two systems that are topologically trivial and nontrivial are brought together to facilitate the occurrence of topological phase transition at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' As most of the matters are topologically trivial, the identification and the growth of different classes of topological systems are currently under intensive investigation [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Likewise, developing methods to characterize the topological properties of the systems is equally important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In analogy to the Su-Schrieffer-Heeger (SSH) model, the band topology of one- dimensional (1D) periodic systems is determined by Zak phase, \uf067, which is a geometric phase [15,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For a particular \uf06cth isolated band, \uf067\uf06c emerges when the Bloch wave travels in momentum space adiabatically along the band across the first Brillouin zone from k = -\uf070/P to \uf070/P, where P is the period of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' If systems possess inversion symmetry, \uf067\uf06c is quantized as either 0 or \uf070 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' \uf067 defines the topological invariant of two band systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For systems that support higher order bands, the topology of the band gap of interest is the summation of all \uf067 below that gap, giving rise to a \uf067 summation that is either even or odd multiple of \uf070 for indicating whether the system is topologically trivial or nontrivial [17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' A zero-dimensional interface state is then formed between two odd and even \uf070 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' One notable feature that comes with \uf067\uf06c is the distinctive spatial wave symmetries at the Brillouin zone center and boundary of the band [16-18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The field symmetries, typically even 3 and odd with respect to the unit cell center, are the same for \uf067\uf06c = 0 system but different when \uf067\uf06c = \uf070 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The association between \uf067n and the field symmetry can be understood from the standpoint of Wannier function, which sums the Bloch waves carrying all k along a band [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Consider the Bloch waves at the zone center and boundary that have the same field symmetry, the Wannier function has either the ( ) ( ) W x W x − = or ( ) ( ) W x W x − = − spatial dependence, leading to \uf067\uf06c = ( ) 2 2 x W x dx P \uf070 \uf0a5 −\uf0a5\uf0f2 = 0 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' On the other hand, for the waves that exhibit different spatial symmetries at two high symmetry points, the Wannier function now shows ( ) ( ) W x P W x − + = or ( ) ( ) W x P W x − + = − dependence, which gives \uf067\uf06c = \uf070 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Therefore, instead of tracing the Bloch waves one by one along the band to determine \uf067n, one can simply examine the field symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' However, how to measure the Bloch wave symmetry remains challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' To date, there have been only a few studies focusing on measuring the geometric phase, either Zak or Berry phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Demler and Bloch are among the first to combine Bloch oscillation and interferometry in a dimerized cold atom system to mobilize the Bloch wave across the Brillouin zone and subsequently measure \uf067\uf06c [20,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' They prove \uf067\uf06c = \uf070 evolves when the intercell interaction is stronger than that of intracell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Cardano et al have demonstrated the use of mean displacement method to determine \uf067\uf06c in a chiral Floquet system [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Such method is then extended to other more generalized SSH systems where the next nearest neighbor interaction is strong enough to break the chiral symmetry [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' While most of them trace the Bloch waves, Gorlach et al adopt an alternative approach by probing the spectral positions of the dipolar (bright) and quadrupolar (dark) characteristics of far-field radiations, which scale with the topological invariant of the system as deduced by using temporal coupled mode theory 4 (CMT) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' When the bright and dark radiation bands are at longer and shorter wavelengths, the system is trivial, but becomes nontrivial upon switching places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' However, their method is limited to the lowest band gap at the zone center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Recently, Chan and his coworkers have formulated that the sign of the reflection phase for wavelengths within the \uf06cth band gap can resolves the trivial and nontrivial \uf067\uf06c [17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The determination of \uf067\uf06c via measuring the reflection phase of the band gap is then demonstrated in several photonic and acoustic systems [25-28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Here, we further extend the CMT to formulate the diffractions arising from 1D leaky optical systems and show the mirror symmetric diffraction orders taken at the zone center and boundary directly reveal the near-field symmetries and thereby the corresponding \uf067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' It is found the odd and even near-field symmetries dictate the far-field interferences, shaping the overall radiation profiles including the bound states in the continuum (BICs) [29-35] and Fano resonances [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We find destructive interference always occurs between the diffraction orders of the first band gap at the zone center, resulting in a symmetry-protected quasi-BIC [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' To verify the CMT, we first conduct finite-difference time-domain (FDTD) simulations on 1D Au plasmonic and SiO2/Au photonic crystals which respectively support TM- and TE-polarized surface waves and the results agree very well with the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We then fabricate plasmonic crystals (PmCs) with different geometries and measure their polarization- and angle-resolved diffraction and phase profiles by Fourier space spectroscopy and common path interferometry to study \uf067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Changing the groove width of PmCs leads to band inversion and thus effectively varies the band topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Finally, a topological protected interface state is demonstrated by joining two topological trivial and nontrivial PmCs together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' TEMPORAL COUPLED MODE THEORY 5 At high symmetry points in 1D Brillouin zone, two degenerate but counter propagating Bloch modes interact with each other to yield two coupled modes separated by an energy gap [37,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Such interaction can be described within the framework of CMT [37-40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 1(a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' for an optically thick system that possesses inversion symmetry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' the dynamics of two mode amplitudes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' a1 and a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' taken under TM or TE polarization can be written as: \uf05b \uf05d 1 1 2 2 o c T c o a a d i K s a a dt \uf077 \uf077 \uf077 \uf077 + \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0e9 \uf0f9 = + \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb \uf0eb \uf0fb \uf0eb \uf0fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (1) where o \uf077 and c \uf077 are the complex frequency and coupling constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' which are expressed as ( ) 2 o o a r i \uf077 \uf077 = + \uf047 +\uf047 and c i \uf077 \uf061 \uf062 = + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' where \uf077o is the resonant angular frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' \uf047a and \uf047r are the absorption and radiative decay rates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' and \uf061 and \uf062 are the real and imaginary parts of the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For a given polarization, the discrete incoming power amplitude vector is \uf05b \uf05d 0 T N , , N , s s s s + − + + + = \uf0e9 \uf0f9 \uf0eb \uf0fb , where the subscript N is an integer \uf0b3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 0, s + is denoted as the surface normal power and N , s\uf0b1 + are two mirror symmetric powers defined obliquely with respect to the surface normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 1 0 1 1 2 0 2 2 N , , N , T N , , N , K \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b − − \uf0e9 \uf0f9 = \uf0ea \uf0fa \uf0eb \uf0fb , where 1 N, \uf06b and 2 N, \uf06b are the complex in-coupling constants for inputting energy from the continuum to a1 and a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' N depends on the number of available ports, which is governed by the diffraction equation as ( ) m m P sin sin \uf06c \uf071 \uf066 = − , where m is the diffraction order, \uf071 is the incident polar angle, and \uf066m is the diffraction angle [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For example, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 1(b), for the lowest band gap at the zone center, \uf071 = 0o, such that only one m = 0th propagating order exists in free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For the second band gap at the zone boundary where 2P sin \uf06c \uf071 = , two m = 0th and 1st orders are present at m \uf066 \uf071 = \uf0b1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In general, zone center supports an odd number of ports including 0 \uf06b whereas an even number of ports is found at zone boundary where 0 \uf06b is always zero [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 6 To see how the field symmetry is revealed, we solve the eigenvalues and eigenvectors of the homogeneous part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (1) by diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The complex frequencies of the coupled modes as: ( ) ( ) ( ) 2 o a r i \uf077 \uf077 \uf061 \uf062 \uf0b1 = \uf0b1 + \uf047 +\uf047 \uf0b1 , indicating their spectral positions and decay rates depend on \uf061 and \uf062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For the real part, we see the spectral positions of the coupled modes are determined by the magnitude and sign of \uf061 and they are separated by an energy gap = 2\uf061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' On the other hand, for the imaginary part, one mode has larger decay rate whereas another one has lower, featuring the bright (dipolar) and dark (quadrupolar) modes [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In particular, if 2 0 r \uf062 \uf047 − = , one coupled mode exhibits zero radiation damping, resulting in a quasi-BIC [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The unit eigenvectors are 1 2 1 2 1 2 a a a a a a + − + \uf0e9 \uf0f9 \uf0e9 \uf0f9 = \uf0ea \uf0fa \uf0ea \uf0fa − \uf0eb \uf0fb \uf0eb \uf0fb , which are orthogonal and carry odd and even symmetries with respect to the unit cell center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' As a result, for an isolated energy band, \uf067 = 0 if both the eigenvectors at the zone center and boundary are either a+ or a− but = \uf070 if they are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We study the spatial field symmetries of a\uf0b1 for TM and TE polarized waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Leaky evanescent waves are considered here as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For TM modes such as Bloch-like surface plasmon polaritons (SPPs) propagating in the x-direction, the magnetic fields of a+ are ( )( ) 1 2 x x z ik x ik x k z k ˆ H H Ae u x e e y − − + = − , where A is a constant, kx and kz are the propagation constants in the x- and z-directions, and ( ) ku x is the periodic function [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' ( ) ku x is assumed to be an even function for simplicity as its symmetry does not affect the Zak phase results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The corresponding electric fields are ( ) ( ) ( ) ( ) ( ) 1 2 2 zk z k z x x x H H A ˆ ˆ E e u x k sin k x x k cos k x z i − \uf0d1\uf0b4 + − = = + − \uf077\uf065 \uf077\uf065 , revealing the in-plane x- and out-of-plane z-components are odd and even in the x-direction, or ( ) ( ) x x E x E x = − − and 7 ( ) ( ) z z E x E x = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Likewise, for a− , we have even ( ) ( ) x x E x E x = − and odd ( ) ( ) z z E x E x = − − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Conversely, for TE modes such as waveguide modes, the in-plane electric fields of a+ and a− are ( ) ( ) 2 zk z k x ˆ iAe u x sin k x y − and ( ) ( ) 2 zk z x ˆ Ae u x cos k x y − , giving rise to odd ( ) ( ) y y E x E x = − − and even ( ) ( ) y y E x E x = − , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Therefore, for the in-plane components, the TM and TE polarized a+ and a− are odd and even in the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Once the field symmetries of a\uf0b1 are known, their spectral positions will then be deduced via far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' By using conservation of energy and time reversal symmetry, the outgoing ports are expressed as \uf05b \uf05d \uf05b \uf05d 1 2 a s C s K a − + \uf0e9 \uf0f9 = + \uf0ea \uf0fa \uf0eb \uf0fb , where \uf05b \uf05d 0 T N , , N , s s s s − − − − − = \uf0e9 \uf0f9 \uf0eb \uf0fb and C is the nonresonant scattering matrix [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We find the transformation matrix to be 1 1 1 1 1 2 T T \uf0e9 \uf0f9 = \uf0ea \uf0fa − \uf0eb \uf0fb so that the outgoing fields can now be rewritten as: \uf05b \uf05d \uf05b \uf05d \uf05b \uf05d 1 2 1 2 0 1 0 2 0 1 0 2 1 2 1 2 1 1 2 2 N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' a s C s T K C s a a a \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b − − − − + − + + + − − + − \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0e9 \uf0f9 \uf0ea \uf0fa \uf0ea \uf0fa + − = + = + + \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa + − \uf0eb \uf0fb \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (2) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (2) can be further simplified by using the relationships between n,i \uf06b − and n,i \uf06b , where i = 1 or 2 and n \uf0a3 N is the diffraction order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' As provided in the Supplementary Information [44],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' given the fact that both far- and near-fields should follow the same spatial symmetry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' the radiation patterns of TM a\uf0b1 arising from the interferences between the decay ports should preserve the same ( ) ( ) F F x x E x E x = − − and ( ) ( ) F F x x E x E x = − dependences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' where the superscript F denotes the far-fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' leading to ( ) 1 2 1 2 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' \uf06b \uf06b \uf06b \uf06b − − + = − + and 1 2 1 2 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' \uf06b \uf06b \uf06b \uf06b − − − = − for a+ and a− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Likewise, for TE a\uf0b1 , ( ) ( ) F F y y E x E x = − − and 8 ( ) ( ) F F y y E x E x = − also give ( ) 1 2 1 2 n, n, n, n, \uf06b \uf06b \uf06b \uf06b − − + = − + and 1 2 1 2 n, n, n, n, \uf06b \uf06b \uf06b \uf06b − − − = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' More importantly, both polarizations indicate ,1 ,2 n n \uf06b \uf06b− = − and ,1 ,2 n n \uf06b \uf06b − = − , which agree with the fact that the system should fulfill the inversion symmetry requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' However, 1 n, \uf06b − ( 2 n, \uf06b − ) is not necessarily equal to 1 n, \uf06b ( 2 n, \uf06b ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In addition, for a+ , ( ) 0 1 0 2 0 1 0 2 , , , , \uf06b \uf06b \uf06b \uf06b + = − + implies the normal diffraction order is always missing, resulting in an even number of decay ports at both the zone center and boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Therefore, at the zone center for TM and TE polarizations, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (2) can be reduced as: \uf05b \uf05d ( ) 0 0 1 1 0 2 2 2 N N N , N N , N N N , N N s s C s a a s \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b − − − − − + + − − − − − + \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa = + + \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa − − + \uf0eb \uf0fb \uf0eb \uf0fb \uf0eb \uf0fb , (3) where the 1,2 subscripts are now dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' On the other hand, at the zone boundary, the outgoing fields carry the same analytical form as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (3) except 0, s − = 0 since 0 0 \uf06b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (3) reveals additional information about the occurrence of quasi-BIC at high symmetry points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In general, quasi-BIC occurs when all the decay ports are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Therefore, at the zone center, unless 0 \uf06b = 0, quasi-BIC can only be observed from a+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Particularly, for the lowest zone center band gap where only the N = 0 port is present, an a+ quasi-BIC is always present, making it symmetry protected [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' However, for higher order band gaps, while the normal N = 0 port is still zero, other N > 0 ports are not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Quasi-BIC can still be found if n n \uf06b \uf06b − = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In other words, if all the mirror symmetric decay ports of the uncoupled mode are identical and in-phase, destructive interferences occur everywhere across all diffraction orders, resulting in quasi-BIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Such special condition can only be met for certain tailored system geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' If n n \uf06b \uf06b − \uf0b9 , a+ appears as bright or dark mode depending on the 9 sign of \uf062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' On the other hand, at the zone boundary where 0, s −is always zero, a+ or a− can be quasi-BIC if n n \uf06b \uf06b− = or n n \uf06b \uf06b− = − is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We then explicitly formulate the diffraction orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' By considering only one single incidence port q such that \uf05b \uf05d 0 0 T q, s s + + \uf0e9 \uf0f9 = \uf0eb \uf0fb , the coupled mode amplitudes are ( ) ( ) , 1 2 q q qs a i − + + + − = − \uf06b \uf06b \uf077 \uf077 and ( ) ( ) , 1 2 q q qs a i \uf06b \uf06b \uf077 \uf077 − + − − + = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Two mirror symmetric n \uf0a3 N diffraction orders thus are: ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 2 2 1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 2 2 n n q n n n n q q n n q q n n q q n n q q q s c s s s i i i c i − − − + + − + − − − − + − − − − − + − − − + + + − − − − = − + − = + + + − \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf077 \uf077 \uf077 \uf077 \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf077 \uf077 \uf077 \uf077 (4) where n c\uf0b1 are the complex nonresonant scattering coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We see from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (4) that the radiations from a+ and a− have odd and even symmetries [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' While a− gives two in phase diffraction orders, those from a+ are \uf070 out of phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Therefore, by fitting the magnitude and phase, 2 , , n q s s \uf0b1 − + and ( ) , , arg n q s s \uf0b1 − + , spectra of any pair of oblique mirror diffraction orders at the zone center and boundary with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (4) to determine their relative phase, the spectral positions ( ) Re \uf0b1 \uf077 can be deduced to find out whether a+ or a− is associated with the energy band of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' FINITE-DIFFERENCE TIME DOMAIN SIMULATION We verify the CMT model by FDTD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Two types of optical systems are considered, and they are 1D Au plasmonic and SiO2/Au photonic crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' While the plasmonic crystals (PmCs) support TM-polarized Bloch-like SPPs [45], the photonic crystals (PhCs) excite TE waveguide modes [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We will present the results of PmCs here and those of the 10 PhCs are provided in the Supplementary Information [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For the PmCs, the unit cell is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 2(a), with the period P and groove height H are set at 900 nm and 50 nm, respectively, and the groove width W is varied from 100 and 700 nm with a step size of 150 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The corresponding TM-polarized k- and wavelength-resolved total reflectivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' which sums all the diffraction orders,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' mappings are calculated along the \uf047-X direction in Fig 2(b) – (f),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' showing the presence of the dispersive ±1 and -2 Bloch-like SPP bands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' which follow the phase matching equation given as 2 2 1 1 2 Au SP Au n k P \uf065 \uf065 \uf06c \uf070 \uf0e6 \uf0f6 \uf0e6 \uf0f6 = + \uf0e7 \uf0f7 \uf0e7 \uf0f7 + \uf0e8 \uf0f8 \uf0e8 \uf0f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' where Au \uf065 is the dielectric constant of Au and nSP is the SPP band,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' as illustrated by the dash lines in Fig 2(b) [37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' More importantly, one sees ±1 SPPs cross at k = 0 \uf06dm-1 and +1 and -2 SPPs cross at k = \uf070/P \uf06dm-1, yielding two band gaps at \uf06c = 925 and 650 nm for the zone center and boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In agreement with the CMT model, the coupled modes exhibit dark and bright radiation characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We attempt to determine the Zak phase of the +1 SPP band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' At the zone center for all PmCs, the dark mode is quasi-BIC and located at the +1 band for W = 100 – 400 nm but flips to the -1 band when W increases further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The corresponding reflectivity spectra are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 3(a) for illustration, clearly showing only one single reflectivity dip as the bright mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' As a result, we conclude a+ locates at the +1 band for W = 100 – 400 nm but flips to the -1 band for wider W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' On the other hand, at the zone boundary, we can no longer differentiate the spectral positions of a\uf0b1 simply by examining the total reflectivity spectra because two dark and bright modes are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Since only a pair of mirror symmetric m = 0th and 1st, or n = \uf0b11, diffraction orders is available, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 3(b) & (c) show the simulated 2 1, 1, s s \uf0b1 − + and ( ) 1, 1, arg s s \uf0b1 − + spectra and we fit them by by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (4) to determine the relative phases between the diffraction pairs of two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The best fits are displayed as the solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The corresponding ( ) Re \uf077\uf0b1 of all PmCs are summarized in Table 1, in which the highlights are the 11 coupled modes sitting on the +1 band at the zone center (high energy mode) and boundary (low energy mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' If the highlights at two regions are either a+ or a− , the Zak phase is 0, but \uf070 when they are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' As a result, by comparing the modes at the zone center and boundary of the +1 band, \uf067 = \uf070 for W = 100, 250, 550 nm but \uf067 = 0 for 400 and 700 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' To confirm our findings, we have simulated the near-field intensity profiles at the zone center and boundary of the +1 band by FDTD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 4(a) & (b) for different W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' At the zone center, we see the profiles are even with respect to the groove center for W = 100 – 400 nm but change to odd afterwards [18,47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' On the other hand, the profiles at the zone boundary are odd for W = 100, 250, and 700 nm but are even for 400 and 550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' As a result, the field symmetries indicate \uf067 = \uf070 for W = 100, 250 and 550 nm but 0 for 400 and 700 nm, in consistent with the far-field simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In addition, we have calculated the near-field patterns across the first Brillouin zone for all PmCs in the Supplementary Information [44] and then employ the Wilson loop method to directly determine \uf067\uf06c given as ( ) P P X k dk \uf070 \uf070 −\uf0f2 , where ( ) X k is the Berry connection given as ( ) ( ) ( ) ( ) ,k k unit cell k ,k unit cell u ( x ) i u x x dx k u x x u ( x )dx \uf065 \uf065 \uf0b6 \uf0b6 \uf0f2 \uf0f2 [47,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The evolutions of the individal phase difference, which is ( ) X k k \uf044 , of the +1 band as a fucntion of k with \uf044k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='04π/P are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The integrated areas yield the \uf067\uf06c phases that once again support our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' EXPERIMENTAL VERIFICATION A series of 1D periodic Au rectangular groove PmCs has been fabricated by focused ion beam (FIB) and their scanning electron microscopy (SEM) images are shown in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 5(a) – (e), showing they have P = 900 nm, H = 50 nm, and W varying from 100 to 700 nm [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' After the sample preparation, the PmCs are then transferred to a homebuilt Fourier space 12 optical microscope described in the Supplementary Information for angle- and wavelength- resolved diffraction measurements [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Briefly, a supercontinuum generation laser is illuminated on the sample at a well-defined incident angle \uf071 via the microscope objective lens and the signals from the sample are collected by the same objective lens in which the diffraction orders are projected onto the momentum space [49,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' By using an aperture to filter out the desired diffraction order, a spectrometer-based CCD detector and a common path interferometer are used for measuring the magnitude and phase spectra [51,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' By varying \uf071 sequentially and at the same time measuring the total reflection spectra, we contour plot the TM-polarized reflectivity mappings in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 5(a) – (e) for different W along the \uf047-X direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' They show ±1 and -2 SPP bands are present, and the bands are consistent with the phase-matching equation as illustrated by the dash lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' From the mappings, we see at normal incidence, or the zone center, BIC-like mode is always observed near the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The +1 band has a+ for W = 100 – 400 nm but a− for wider W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' On the other hand, at the zone boundary where +1 and -2 SPPs cross at \uf071 ~ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='5o, we see the dark and bright modes are found and their positions depend on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' To estimate the spectral positions of a\uf0b1 , we measure the corresponding m = 0th and 1st, or n = ±1, reflectivity and TM-TE phase difference spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 6(a) & (b) and fit them by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (4) to determine ( ) Re \uf077\uf0b1 in Table 1, which shows the +1 band is a− for W = 100, 250 and 700 nm is a+ for 400 and 550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Therefore, \uf067 = \uf070 for W = 100, 250 and 550 nm but = 0 for 400 and 700 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Finally, we demonstrate a topologically protected state is formed at the interface between two topological trivial and nontrivial PmCs [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We construct a heterostructure by joining two W = 100 and 400 nm PmCs together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In prior to joining, we have examined by FDTD the field symmetries at the zone center and boundary of two PmCs and determine the \uf067 of the 0, - 1, and +1 SPP bands to be \uf070 , \uf070 and \uf070 for W = 100 nm and \uf070 , \uf070 and 0 for W = 400 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 13 Therefore, the sums of \uf067\uf06c give \uf070 and 0 for W = 100 and 400 nm PmCs, indicating the -2/+1 energy gaps at the zone boundary are topological trivial and nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We then simulate the heterostructure supercell as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 7(a) that consists of 14 unit cells of W = 100 and 400 nm PmCs on the right- and left-handed sides [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 7(b) shows the TM-polarized k- and wavelength-resolved reflectivity mapping at the zone boundary along the \uf047-X direction, clearly demonstrating a localized mode is located at k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='5\uf070/P or θ = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='5o, and \uf06c ~ 640 nm in the mid of the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We also have simulated the wavelength-dependent near-field mapping of the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For different wavelengths, the near-field intensities at 20 nm above the surface is simulated across the heterostructure and then contour plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 7(c), showing the interface is located at x = 0 \uf06dm and the trivial and nontrivial regions are at x > 0 \uf06dm and < 0 \uf06dm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' One sees two strong fields are visible at ~ 620 and 670 nm in the PmC bulk regions away from the interface due to the excitations of the upper and lower coupled modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' However, the strongest field strength is observed at the interface, x = 0 µm, at 640 nm, and it decays rapidly into the bulk regions, signifying the presence of a topologically protected interface state [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We have prepared the heterostructure by FIB and its SEM image is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 7(d) with W = 100 and 400 nm PmCs on the right- and left-hand sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The TM- polarized k- and wavelength-resolved reflectivity mapping of the sample is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 7(e), clearly showing an interface state is found at \uf071 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='5o and \uf06c ~ 625 nm in the +1/-2 band gap at the zone boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' CONCLUSION In summary, we have formulated an analytical model based on temporal CMT to determine the Zak phase of an isolated band in leaky photonic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' At the Brillouin zone center and boundary, as the far- and near-fields of the systems share the same spatial symmetry, the mirror symmetric diffractions are either in or \uf070 out of phase depending on the Bloch wave 14 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Therefore, the near-field symmetries can be probed by studying the diffraction profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In addition, our model generalizes the occurrence of quasi-BIC at the high symmetry points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The interplay between the in-coupling constants of different ports plays a decisive role in manifesting quasi-BICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For verification, we have studied 1D PmCs and PhCs that support TM- and TE-polarized SPP and waveguide modes by FDTD and the results agree very well with the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We also have prepared 1D PmCs by FIB and examined their diffractions by using Fourier space diffraction spectroscopy and common path interferometry for determining the Zak phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In the end, a topological protected interface state is demonstrated by joining two topological trivial and nontrivial PmCs together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' ACKNOWLEDGMENT This research was supported by the Chinese University of Hong Kong through Area of Excellence (AoE/P-02/12) and Innovative Technology Fund Guangdong-Hong Kong Technology Cooperation Funding Scheme (GHP/077/20GD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 15 Reference 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Qi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhang, Topological insulators and superconductors, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 83, 1057 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Hasan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Kane, Topological insulators, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 82, 3045 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Moore, The birth of topological insulators, Nature 464, 194 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Cooper, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Dalibard, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Spielman, Topological bands for ultracold atoms, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 91, 015005 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Yan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhu, Topological quantum matter with cold atoms, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 67, 253 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ozawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Price, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Amo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Goldman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Hafezi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rechtsman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Schuster, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Simon, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zilberberg, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Carusotto, Topological photonics, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 91, 015006 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Joannopoulos and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Soljačić, Topological photonics, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 8, 821 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Khanikaev and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Shvets, Two-dimensional topological photonics, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 11, 763 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Huber, Topological mechanics, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 12, 621 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Gao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Shi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='D Chong, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhang, Topological Acoustics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 114, 114301 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Xiao and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Chan, Topological phases in acoustic and mechanical systems, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 1, 281 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Delplace, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Marston, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Venaille, Topological origin of equatorial waves, Science 358, 1075 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Tang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Po, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Vishwanath, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wan, Comprehensive search for topological materials using symmetry indicators, Nature 566, 486 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 16 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ma, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' He, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Jia, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Xue, Molecular beam epitaxial growth of topological insulators, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 23, 1162 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Su, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Schrieffer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Heeger, Solitons in Polyacetylene, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 42, 1698 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zak, Berry’s Phase for Energy Bands in Solids, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 62, 2747 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Xiao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Chan, Surface impedance and bulk band geometric phases in one-dimensional systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' X 4, 021017 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Xiao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Yang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Sheng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Chan, Geometric phase and band inversion in periodic acoustic systems, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 11, 240 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Asbóth, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Oroszlány, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Pályi, A Short Course on Topological Insulators (Springer, New York, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Abanin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Kitagawa, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Bloch, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Demler, Interferometric approach to measuring band topology in 2D optical lattices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 110, 165304 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Atala, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Aidelsburger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Barreiro, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Abanin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Kitagawa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Demler and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Bloch, Direct measurement of the Zak phase in topological Bloch bands, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 9, 795 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Cardano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' D’Errico1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Dauphin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Maffei, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Piccirillo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' de Lisio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' De Filippis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Cataudella, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Santamato, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Marrucci, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lewenstein, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Massignan, Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 8, 15516 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Jiao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Longhi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Gao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Fu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ren, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Qiao, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Jin, Experimentally detecting quantized Zak phases without chiral symmetry in photonic lattices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 127, 147401 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Gorlach, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ni, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Smirnova, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Korobkin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhirihin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Slobozhanyuk, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Belov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Alù, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Khanikaev, Far-field probing of leaky topological states in all dielectric metasurfaces, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 9, 909 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 17 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Gao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Xiao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Chan, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Tam, Determination of Zak phase by reflection phase in 1D photonic crystals, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 40, 5259 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Xiao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhu, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Chan, Measurement of the Zak phase of photonic bands through the interface states of a metasurface/photonic crystal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' B 93, 041415(R) (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Fan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ding, Zak phases and band properties in acoustic metamaterials with negative modulus or negative density, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' B 94, 174307 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ding, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ren, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Jiang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Chen, Zak phase and band inversion in dimerized one-dimensional locally resonant metamaterials, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' B 97, 195307 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Koshelev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Bogdanov, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Kivshar, Engineering with bound states in the continuum, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' News 31, 38 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Hsu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Stone, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Soljacic, Topological nature of optical bound states in the continuum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 113, 257401 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Hsu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Guan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Shi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zi, Observation of polarization vortices in momentum space, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 120, 186103 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ye, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Gao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Liu, Singular points of polarizations in the momentum space of photonic crystal slabs, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 124, 153904 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Yoda and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Notomi, Generation and annihilation of topologically protected bound states in the continuum and circularly polarized states by symmetry breaking, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 125, 053902 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Hsu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zhen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Stone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Joannopoulos, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Soljacic, Bound states in the continuum, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 1, 16048 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 18 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Doeleman, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Monticone, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' den Hollander, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Alu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Koenderink, Experimental observation of a polarization vortex at an optical bound state in the continuum, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 12 397 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Limonov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rybin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Poddubny, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Kivshar, Fano resonances in photonics, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 11, 543 (2017) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Cao and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ong, Determination of the absorption and radiative decay rates of dark and bright plasmonic modes, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 22, 16112 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Verslegers, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Yu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ruan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Catrysse, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Fan, From electromagnetically induced transparency to superscattering with a single structure: A coupled-mode theory for doubly resonant structures, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 108, 083902 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Haus, Waves and fields in optoelectronics (Prentice-Hall, New Jersey, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Fan, in Optical Fiber Telecommunications V, edited by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Kaminow, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Li, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Willner (Academic Press, Burlington, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Loewen and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Popov, Diffraction gratings and applications (CRC Press, New York, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ropers, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Park, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Stibenz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Steinmeyer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Kim, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lienau, Femtosecond light transmission and subradiant damping in plasmonic crystals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 94 113901 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Billaudeau, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Collin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Sauvan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Bardou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Pardo, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Pelouard, Angle- resolved transmission measurements through anisotropic two-dimensional plasmonic crystals, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 33, 165 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' See Supplementary Materials for the connection between the far- and near-fields from one- dimensional periodic optical system, simulated near-field patterns of the +1 surface plasmon polariton (SPP) band of 1D PmCs across the first Brillouin zone, FDTD simulation results of 1D SiO2/Au photonic crystals (PhCs), and the Fourier space optical 19 microscope for angle- and wavelength resolved diffraction mapping and common path interferometry 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Liu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ong, Generalization of the circular dichroism from metallic arrays that support Bloch-like surface plasmon polaritons, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 15, 024048 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Christ, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Tikhodeev, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Gippius, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Kuhl, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Giessen, Waveguide-plasmon polaritons: Strong coupling of photonic and electronic resonances in a metallic photonic crystal slab, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 91, 183901 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Liu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ong, Realization of topological superlattices and the associated interface states in one-dimensional plasmonic crystals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' B 106, 045401 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Guo, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Jiang, Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology, New.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 21, 093029 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Huang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Yu, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Zare, Surface plasmon resonance imaging using a high numerical aperture microscope objective, Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 79, 2979 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Bleckmann, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Cherpakova, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Linden, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Alberti, Spectral imaging of topological edge states in plasmonic waveguide arrays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' B 96, 045417 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Cao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ho, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ong, High performing phase-based surface plasmon resonance sensing from metallic nanohole arrays, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 104, 171116 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wong and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ong, Phase difference mapping of two-dimensional metallic nanohole arrays, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 100, 233102 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 20 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (a) The schematic shows at the Brillouin zone center and boundary in 1D leaky optical system, two Bloch-like modes a1,2 counter propagate in opposite directions with each supports discrete in-coupling channels 1 2 0 1 2 1 2 N, , , , N, , \uf06b \uf06b \uf06b− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' They interact with each other to form two coupled a\uf0b1 at higher and lower energies separated by an energy band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (b) 0 1 2 0 , , \uf06b \uf0b9 at the zone center but 0 1 2 0 , , \uf06b = at the zone boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Second zone boundary band gap Lowest zone center band gap Lowest zone boundary band gap21 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (a) The unit cell of 1D PmC for FDTD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The simulated TM-polarized k- and wavelength-resolved total reflectivity mappings of PmCs with W = (b) 100, (c) 250, (d) 400, (e) 550, and (f) 700 nm taken along the \uf047-X direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The white dash lines are calculated by using the phase-matching equation, indicating ±1 and -2 Bloch-like SPPs are excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' At the zone center and boundary where k = 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='5, two energy band gaps are formed, featuring two dark and bright modes are located above or below the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Particularly, at k = 0, a quasi- BIC is observed at either above or below the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (a) Air (b) 2 SPP p ↑H +1: SPP W Au 、-1SPP C) (d) (f) e22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The TM-polarized total reflectivity spectra of PmCs taken at the zone center for different W, exhibiting only one single reflectivity dip as the bright mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The red dash line is the band gap center, indicating the quasi-BIC occurs at shorter wavelength for W = 100, 250 and 400 nm but longer wavelength for W = 550 and 700 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' At the zone boundary, two TM- polarized mirror symmetric n = -1 (black square) and 1 (red circle) (b) reflectivity and (c) phase spectra for W = 100 (top) to 700 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The green and blue solid lines are the best fits determined by CMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' XX X23 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The FDTD simulated near-field patterns of the PmCs for different W taken at the Brillouin zone (a) center and (b) boundary, showing their field symmetries are the same for W = 400 and 500 nm but different for W = 100, 250, and 700 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (c) The individual phase profiles determined by the Wilson loop method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The integration yields the Zak phase, indicating the phase is 0 for W = 400 and 500 nm but \uf070 for W = 100, 250, and 700 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (b)24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The measured TM-polarized k- and wavelength-resolved total reflectivity mappings of PmCs with W = (a) 100, (b) 250, (c) 400, (d) 550, and (e) 700 nm taken along the \uf047-X direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The white dash lines are ±1 and -2 Bloch-like SPPs determined by the phase matching equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Two band gaps are formed at the zone center and boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The insets are the corresponding SEM images of the PmCs with the scale bare = 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='.9 600 2SPP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='8 6/5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='7 750 +1SPP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='6 825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='5 900 1 SPP (a) (b) (c) (d) (e 975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='.4 80c0L0008060L000800L000s060L0008060L00025 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' At the zone boundary, two measured TM-polarized mirror symmetric n = -1 (black square) and 1 (red circle) (b) reflectivity and (c) TM-TE phase difference spectra for W = 100 (top) to 700 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The green and blue solid lines are the best fits determined by CMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 26 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (a) The schematic of the heterostructure by joining W = trivial 100 and nontrivial 400 nm PmCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The interface is marked by the dash line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (b) The FDTD simulated TM-polarized reflectivity mapping of the heterostructure taken at the zone boundary along the \uf047-X direction, showing an interface state is found within the gap at \uf06c = 640 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (c) The wavelength- dependent near-field intensity mapping simulated at 20 nm above the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The interface is located at x = 0 \uf06dm, showing strong field localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The strong fields at 620 and 670 nm arise from the PmC bulk regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (d) The SEM image of the W = 100 and 400 nm with the scale bar corresponding to 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (e) The measured TM-polarized reflectivity mapping of the heterostructure taken at the zone boundary along the \uf047-X direction, showing an interface state is found within the gap at \uf06c = 625 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' W = 400nm W = 100nm nontrivial trivial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='7 0.' metadata={'source': 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+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='65) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='415 k (2/) k (2TN) 700 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='9 680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='.7 660 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='5 @) 640 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='4 ABl interface state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='3 620 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='.1 W = 400 nm W = 100 nm 0 600 1000 500 0 500 1000 x (nrm)27 100 nm 250 nm 400 nm 550 nm 700 nm FDTD Zone center ( ) Re \uf077+ (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='30 ( ) Re \uf077− (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='37 Zone boundary ( ) Re \uf077+ (eV) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='98 ( ) Re \uf077− (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='85 Experiment Zone center ( ) Re \uf077+ (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='32 ( ) Re \uf077− (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='36 Zone boundary ( ) Re \uf077+ (eV) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='02 ( ) Re \uf077− (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='93 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='95 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The FDTD and experimental ( ) Re \uf077\uf0b1 at the Brillouin zone center and boundary for the PmCs with different W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The highlights are the coupled modes located on the +1 SPP band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' If the highlights at the zone center and boundary are both a+ or a− , the Zak phase is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' If not, the Zak phase is \uf070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 28 Supplementary Information Determination of the band topology of one-dimensional photonic systems via far-field diffraction C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ong Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s Republic of China A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Derivation of the connection between the far- and near-fields from one-dimensional periodic optical system Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The schematic of the 2N+1 diffraction orders arising from the coupled mode supported on 1D periodic leaky system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 29 As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S1, for a one-dimensional optical leaky periodic system that possesses inversion symmetry in the x-direction, at the Brillouin zone center and boundary, it supports two Bloch-like coupled modes a\uf0b1 above and below the photonic band gap with each dissipates a total of 2N + 1 mirror symmetric diffraction channels in free space, where N is the highest diffraction order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For TM- and TE-polarizations, both the near- and far-fields should carry the same polarization and field symmetry in the x-y plane along the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' for TM- polarization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' in the far-field at zo above the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' the x-component of the electric field ( ,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (S1) where An,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' \uf06an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' and \uf071n are the diffraction amplitude,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' and angle and the subscript n is the diffraction order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' At the same time, for the near-field, the TM-polarized a+ is a standing wave with ( ) ( ) ( ) ( ) zk z k z x x x ˆ ˆ E e u x k sin k x x k cos k x z − \uf0b5 + , where kx and kz are the propagation constants in the x- and z-directions and ( ) ku x is the periodic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Assume ( ) ku x is an even function for simplicity, we see ( ) x E x is an odd function with ( ) ( ) x x E x E x = − − dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (S1) should also exhibit ( ) ( ) F F x x E x E x = − − dependence, yielding n n A A − = , n n \uf06a \uf06a \uf070 − = + , and 0 0 A = that indicate two mirror symmetric diffraction orders have the same magnitude but are always \uf070 out of phase and the normal diffraction order is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' As a result, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (S1) is rewritten as: 1 1 1 1 1 1 sin cos sin cos 1 1 sin cos sin cos 1 1 cos cos cos cos N N N o N N N o N N N o N N N o i ik x ik z i ik x ik z N N N N i ik x ik z i ik x ik z N N N N A e e e A e e e A e e e A e e e \uf06a \uf071 \uf071 \uf06a \uf071 \uf071 \uf06a \uf071 \uf071 \uf06a \uf071 \uf071 \uf071 \uf071 \uf071 \uf071 − − − − − − − − − − − − + + − − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (S2) By matching Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (S2) with the outgoing power amplitudes of a+ from CMT, which are ,1 ,2 0,1 0,2 ,1 ,2 1 2 N N N N a \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b − − + + \uf0e9 \uf0f9 \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa + \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa + \uf0eb \uf0fb , we conclude ( ) 1 2 1 2 n, n, n, n, \uf06b \uf06b \uf06b \uf06b − − + = − + and 0,1 0,2 0 \uf06b \uf06b + = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Likewise, for another coupled mode a− where ( ) ( ) ( ) ( ) zk z k z x x x ˆ ˆ E e u x k cos k x x k sin k x z − \uf0b5 + , we see 30 ( ) ( ) x x E x E x = − and have n n A A − = , n n \uf06a \uf06a− = and 0 0 A \uf0b9 , indicating two mirror symmetric orders are in phase and the normal diffraction order is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (S1) for a− is: 1 1 1 0 1 1 1 sin cos sin cos 1 1 sin cos sin cos 0 1 1 cos cos cos cos N N N o N N N o o N N N o N N N o i ik x ik z i ik x ik z N N N N i ikz i ik x ik z i ik x ik z N N N N A e e e A e e e A e e A e e e A e e e \uf06a \uf071 \uf071 \uf06a \uf071 \uf071 \uf06a \uf06a \uf071 \uf071 \uf06a \uf071 \uf071 \uf071 \uf071 \uf071 \uf071 − − − − − − − − − − − − − + + + + + + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (S3) We then have 1 2 1 2 n, n, n, n, \uf06b \uf06b \uf06b \uf06b − − − = − for the outgoing power amplitudes of a− given as ,1 ,2 0,1 0,2 ,1 ,2 1 2 N N N N a \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b − − − − \uf0e9 \uf0f9 \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa − \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa − \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Finally, two conditions ( ) 1 2 1 2 n, n, n, n, \uf06b \uf06b \uf06b \uf06b − − + = − + and 1 2 1 2 n, n, n, n, \uf06b \uf06b \uf06b \uf06b − − − = − result in ,1 ,2 n n \uf06b \uf06b− = − and ,1 ,2 n n \uf06b \uf06b − = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' for TE-polarized Bloch-like coupled modes a\uf0b1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' at zo in the free space above the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' the y-component of the far-field electric field ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' ) F y o E x z can be written as: 1 1 1 0 1 1 1 sin cos sin cos 1 sin cos sin cos 0 1 N N N o N N N o o N N N o N N N o i ik x ik z i ik x ik z N N i ikz i ik x ik z i ik x ik z N N A e e e A e e e A e e A e e e A e e e \uf06a \uf071 \uf071 \uf06a \uf071 \uf071 \uf06a \uf06a \uf071 \uf071 \uf06a \uf071 \uf071 − − − − + − + − + − − − − − + − − − + + + + + + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (S4) The near-field of a+ where ( ) ( ) zk z k x ˆ E e u x sin k x y − \uf0b5 , we have ( ) ( ) y y E x E x = − − such that n n A A − = , n n \uf06a \uf06a \uf070 − = + , and 0 0 A = , leading to ( ) 1 2 1 2 n, n, n, n, \uf06b \uf06b \uf06b \uf06b − − + = − + and 0,1 0,2 0 \uf06b \uf06b + = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Likewise, for a− where ( ) ( ) y y E x E x = − , we have n n A A − = , n n \uf06a \uf06a− = and 0 0 A \uf0b9 , giving rise to 1 2 1 2 n, n, n, n, \uf06b \uf06b \uf06b \uf06b − − − = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Therefore, two conditions give the same conclusion that ,1 ,2 n n \uf06b \uf06b− = − and ,1 ,2 n n \uf06b \uf06b − = − regardless of the polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' As a result, at the zone center for TM- and TE-polarizations, the outgoing profile is: \uf05b \uf05d ( ) 0 0 1 1 0 2 2 2 N N N , N N , N N N , N N s s C s a a s \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b − − − − − + + − − − − − + \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa = + + \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa − − + \uf0eb \uf0fb \uf0eb \uf0fb \uf0eb \uf0fb , (S5) 31 where the 1,2 subscripts are now dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We see quasi-BIC arises from a+ and it will occur when 0 n n \uf06b \uf06b − − = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' However, for the lowest band gap where only the normal diffraction order is present, quasi-BIC always occur, making it symmetry protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' On the other hand, at the zone boundary where 0, s − is always 0, we have for TM- and TE-polarizations: \uf05b \uf05d ( ) 0 1 1 0 0 2 2 N N N , N N , N N N , N N s s C s a a s \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b \uf06b − − − − − + + − − − − − + \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa = + + \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa \uf0ea \uf0fa − − + \uf0eb \uf0fb \uf0eb \uf0fb \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (S6) Quasi-BIC occurs depending on the interplay between n \uf06b− and n \uf06b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' a+ (a− ) is quasi-BIC if 0 n n \uf06b \uf06b − − = ( 0 n n \uf06b \uf06b − + = ) but dark and bright modes are present if 0 n n \uf06b \uf06b − \uf0b1 \uf0b9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Simulated near-field patterns of the +1 surface plasmon polariton (SPP) band of 1D PmCs across the first Brillouin zone By using the dipole source excitation method, the complex near-field patterns along the +1 SPP band of 1D Au PmCs with period = 900 nm, groove height = 50 nm and different groove widths have been simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The real and imaginary parts of the surface normal components, Re(Ez) and Im(Ez), taken at 20 nm above the surface across the Brillouin zone from k = -\uf070/P to \uf070/P \uf06dm-1 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S2 for groove width W = 100, 250, 400, 550 and 700 nm PmCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' They will then be used for determining the Zak phase by the Wilson loop method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 32 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The real and imaginary parts of the z-component of the near-field patterns of the PmCs plotted as a function of k along the +1 SPP band in the first Brillouin zone for different W = (a) & (b) 100, (c) & (d) 250, (e) & (f) 400, (g) & (h) 550, and (i) & (j) 700 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (a (b)33 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' FDTD results of 1D SiO2/Au photonic crystals (PhCs) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S3(a) shows the unit cell of the PhCs, which has 400 nm thick SiO2 coated on Au surface with the period P and the groove height H being set at 900 nm and 200 nm whereas the groove width W varied from 100 and 725 nm with a step size of 125 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The corresponding TE- polarized k-resolved total reflectivity mappings are shown in Fig S3(b) – (f), showing the dispersive ±1 and -2 photonic bands, which follow the phase matching equation given as ( ) ( ) 2 2 sin D D PhC n n m P \uf06c \uf071 \uf06c = + , where nD is the refractive index of SiO2 and mPhC is the photonic band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The calculations are superimposed in Fig 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' We see mPhC = ±1 photonic bands cross at k = 0 \uf06dm-1 and mPhC = +1 and -2 bands cross at k = \uf070/P \uf06dm-1, yielding two energy band gaps at \uf06c = 930 – 1030 nm and 700 – 770 nm at the zone center and boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' At the zone center, one symmetry protected quasi-BIC is always found, and it is located on the -1 band for W = 100 – 475 nm but flips to the +1 band when W increases further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' At the same time, accidental quasi-BICs are also found along the +1 band at different k for all PhCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 34 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (a) The FDTD unit cell of the PhC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The simulated TE-polarized k- and wavelength- resolved total reflectivity mappings of PhCs with W = (b) 100, (c) 225, (d) 350, (e) 475, (f) 600, and (g) 725 nm taken along the \uf047-X direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The white dash lines are calculated by using the phase-matching equation, indicating ±1 and -2 photonic band are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' At the zone center and boundary where k = 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='5, two energy band gaps are formed, featuring two dark and bright modes are located above or below the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Particularly, at k = 0, a symmetry protected quasi-BIC is observed at either above or below the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' On the other hand, an accidentally BIC is observed along the +1 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Air p SiO2 H W Au (b) C 2 band +1 band 1 band (d) (e) (f) 935 We will focus on the modes located on the +1 band at the zone center and boundary and determine their field symmetries as well as \uf067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The reflectivity spectra of the PhCs taken under normal incidence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=', at the zone center, are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S4(a), clearly showing only one single reflectivity dip is present as the bright mode, verifying another coupled mode is quasi- BIC that does not produce any dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' As quasi-BIC arises solely from a+ for the lowest band gap, we deduce the coupled mode on the +1 band is symmetric a− for W = 100 – 475 nm PhCs but becomes asymmetric a+ for W = 600 and 725 nm PhCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' On the other hand, the reflectivity spectra taken at the zone boundary for all PhCs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S4(b), showing two bright and dark modes are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The TE-polarized total reflectivity spectra of PhCs taken at the zone (a) center and (b) boundary for different W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' At the zone center, only one single reflectivity dip is present as the bright mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' On the other hand, at the zone boundary, two bright and dark modes are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 36 To determine the near-field symmetries of the PhCs at the zone boundary, the two mirror symmetric diffraction and phase spectra are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S5 and they are fitted with 2 1, 1, s s \uf0b1 − + and ( ) 1, 1, arg s s \uf0b1 − + from CMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The best fits are displayed as the solid lines and the fitted results ( ) Re \uf077\uf0b1 are tabulated in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' in which the highlights are the coupled modes sitting on the +1 photonic band at the zone center (high energy mode) and boundary (low energy mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' If the highlights at two regions are either a+ or a− , the Zak phase is 0, but \uf070 when they are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' As a result, we conclude the Zak phase of +1 band for W = 100, 225 and 600 nm is \uf070 but becomes 0 for W = 350, 475 and 725 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 100 nm 225 nm 350 nm 475 nm 600 nm 725 nm Zone center ( ) Re \uf077+ (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='37 ( ) Re \uf077− (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='31 Zone boundary ( ) Re \uf077+ (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='79 ( ) Re \uf077− (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='84 Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The FDTD ( ) Re \uf077\uf0b1 at the Brillouin zone center and boundary for the PhCs with different W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The highlights are the coupled modes located on the +1 photonic band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' If the highlights at the zone center and boundary are both a+ or a− , the Zak phase is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' If not, the Zak phase is \uf070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 37 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' At the zone boundary, two TE-polarized mirror symmetric n = -1 (black square) and 1 (red circle) (a) reflectivity and (b) phase spectra of the PhCs for W = 100 (top) to 725 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The green and blue solid lines are the best fits determined by CMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 4 XC38 To verify the Zak phases, we have simulated the real and imaginary parts of the surface normal components, Re(Ez) and Im(Ez), taken at 20 nm above the surface across the Brillouin zone from k = -\uf070/P to \uf070/P \uf06dm-1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S6 for all PhCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' They will then be used for determining the Zak phase by the Wilson loop method given as ( ) P P X k dk \uf070 \uf070 −\uf0f2 , where ( ) X k is ( ) ( ) ( ) ( ) ,k k unit cell k ,k unit cell u ( x ) i u x x dx k u x x u ( x )dx \uf065 \uf065 \uf0b6 \uf0b6 \uf0f2 \uf0f2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The evolutions of the individal phase difference, which is ( ) X k k \uf044 , of the +1 band as a fucntion of k with \uf044k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='04π/P of all PhCs are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The integrated areas yield the Zak phases are \uf070 for W = 100, 225, and 600 nm and 0 for W = 350, 475 and 725 nm, and they agree very well with earlier CMT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 39 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The real and imaginary parts of the z-component of the near-field patterns of the PhCs plotted as a function of k along the +1 photonic band in the first Brillouin zone for different W = (a) & (b) 100, (c) & (d) 225, (e) & (f) 350, (g) & (h) 475, (i) & (j) 600, and (k) & (l) 725 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' (C) (e (g) (h) (i) (k)40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The individual phase profiles of the PhCs with different W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The integration yields the Zak phase, indicating the phase \uf070 for W = 100, 225, and 600 nm and 0 for W = 350, 475 and 725 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Schematic of the Fourier space optical microscope for angle- and wavelength resolved diffraction mapping and common path interferometry Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S8 shows the schematic of the Fourier space optical microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Briefly, a broadband supercontinuum laser from a nonlinear photonic crystal fiber is collimated and then passed through a set of linear polarizers, wave plates, and lenses before being focused onto the back focal plane (BFP) of a 100X objective lens (OB) with numerical aperture = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The light exiting from the objective lens is then a collimated beam with well-defined linear polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In addition, by displacing the focused spot across the BFP of the objective lens using a motorized translation stage, the incident polar angle \uf071 of the collimated beam onto the sample can be varied following sin\uf071 = d/f, where d is the distance between the focused spot and the optical axis of the BFP and f is the focal length of the objective lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' In addition, the azimuth angle \uf066 can be varied by a motorized rotation sample stage to align the incident plane to the \uf047- X direction of the PmC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The diffractions from the PmC are then collected by the same objective lens and are routed through a set of Fourier lens system so that the diffraction orders are projected onto the momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' By placing an aperture at the momentum space to filter 41 out the desired diffraction order, its intensity and phase spectra can be measured by a spectrometer-based CCD detector and a common path interferometer [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' To perform common path interferometry, the 45o linearly polarized collimated beam with the Jones vector given as 1 1 1 2 \uf0e9 \uf0f9 \uf0ea \uf0fa \uf0eb \uf0fb is incident on the PmC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The diffraction order from the PmC after the aperture can be formulated as: 0 0 TM TE i TM PmC i TE r e J r e \uf071 \uf071 \uf0e9 \uf0f9 = \uf0ea \uf0fa \uf0eb \uf0fb , where rTM,TE and \uf071TM,TE are the magnitudes and phases for TM- and TE-polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The diffraction passes through a quarter wave plate with the fast axis being placed at 45o with respect to the incident plane and a motorized rotatable analyzer with angle \uf078, which are given as 2 ( ) 2 cos sin cos sin cos sin analyzer J \uf078 \uf078 \uf078 \uf078 \uf078 \uf078 \uf078 \uf0e9 \uf0f9 = \uf0ea \uf0fa \uf0eb \uf0fb and (45 ) 1 1 1 1 1 2 QWP i i J i i \uf0b0 − + \uf0e9 \uf0f9 = \uf0ea \uf0fa + − \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The output vector is ( ) (45 ) 1 1 1 2 analyzer QWP PmC J J J \uf078 \uf0b0 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' After some formulations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' the intensities for different \uf078 = 0o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' ±45o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' and 90o can be written as: ( ) 2 2 2 0 1 1 ( ) 2 sin 2 4 0 i TM TE TM TE TM TE r e i r R r r r r \uf06a \uf06c \uf06a \uf0e9 \uf0f9 + = = + + \uf0ea \uf0fa \uf0eb \uf0fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' ( ) 2 2 45 1 ( ) 2 cos 4 TM TE TM TE R r r r r \uf06c \uf06a + = + + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' ( ) 2 2 45 1 ( ) 2 cos 4 TM TE TM TE R r r r r \uf06c \uf06a − = + − ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' and ( ) 2 2 90 1 2 sin 4 TM TE TM TE R r r r r \uf06a = + − ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' where TM TE \uf06a \uf071 \uf071 = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Therefore, the phase difference between TM- and TE- polarized diffractions can be calculated by: 0 90 45 45 ( ) ( ) tan ( ) ( ) ( ) R R R R \uf06c \uf06c \uf06a \uf06c \uf06c \uf06c + − − = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 42 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' The schematic of the Fourier optical microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Reference 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Cao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ho, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Ong, High performing phase-based surface plasmon resonance sensing from metallic nanohole arrays, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' 104, 171116 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} +page_content=' P: polarizer L1,L2,L3,L4: focusing lens OB: objective lens BS: beam splitter BFP: back focal plane' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfSAfe/content/2301.03789v1.pdf'} diff --git a/DdFQT4oBgHgl3EQfPjZG/content/tmp_files/2301.13279v1.pdf.txt b/DdFQT4oBgHgl3EQfPjZG/content/tmp_files/2301.13279v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8c12b5093107e9ffaf05cbc3a350acf85003868 --- /dev/null +++ b/DdFQT4oBgHgl3EQfPjZG/content/tmp_files/2301.13279v1.pdf.txt @@ -0,0 +1,1122 @@ +Learning Coordination Policies over Heterogeneous Graphs for +Human-Robot Teams via Recurrent Neural Schedule Propagation +Batuhan Altundas1, Zheyuan Wang1, Joshua Bishop1 and Matthew Gombolay1 +Abstract— As human-robot collaboration increases in the +workforce, it becomes essential for human-robot teams to +coordinate efficiently and intuitively. Traditional approaches +for human-robot scheduling either utilize exact methods that +are intractable for large-scale problems and struggle to ac- +count for stochastic, time varying human task performance, +or application-specific heuristics that require expert domain +knowledge to develop. We propose a deep learning-based +framework, called HybridNet, combining a heterogeneous +graph-based encoder with a recurrent schedule propagator for +scheduling stochastic human-robot teams under upper- and +lower-bound temporal constraints. The HybridNet’s encoder +leverages Heterogeneous Graph Attention Networks to model +the initial environment and team dynamics while accounting +for the constraints. By formulating task scheduling as a se- +quential decision-making process, the HybridNet’s recurrent +neural schedule propagator leverages Long Short-Term Mem- +ory (LSTM) models to propagate forward consequences of +actions to carry out fast schedule generation, removing the +need to interact with the environment between every task- +agent pair selection. The resulting scheduling policy network +provides a computationally lightweight yet highly expressive +model that is end-to-end trainable via Reinforcement Learning +algorithms. We develop a virtual task scheduling environment +for mixed human-robot teams in a multi-round setting, capable +of modeling the stochastic learning behaviors of human work- +ers. Experimental results showed that HybridNet outperformed +other human-robot scheduling solutions across problem sizes +for both deterministic and stochastic human performance, with +faster runtime compared to pure-GNN-based schedulers. +I. INTRODUCTION +With collaborative robots (cobots) becoming more avail- +able in the industrial and manufacturing environments, robots +and humans increasingly share the same work space to +collaborate on tasks [1]. By removing the cage around tradi- +tional robot platforms and integrating robots into dynamic, +final assembly operations, manufacturers can see improve- +ments in reducing a factory’s footprint and environmental +costs as well as increased productivity [2]. In this paper, +we focus on the problem of multi-agent task allocation and +scheduling [3] with mixed human-robot teams over multiple +iterations of the same task allocation problem. Our work +accounts for and leverages stochastic, time-varying human +task performance to quickly solve task allocation problems +among team members to achieve a high-quality schedule +with respect to the application-specific objective function +*This work was supported in part by the Office of Naval Research +under grant N00014-19-1-2076 and Naval Research Laboratory under grant +N00173-21-1-G009. +1Batuhan Altundas, Zheyuan Wang, Joshua Bishop and Matthew Gom- +bolay are with the Institute for Robotics and Intelligent Machines, Georgia +Institute of Technology, Atlanta, GA 30332, USA {baltundas3, +pjohnwang, jbishop45, mgombolay3}@gatech.edu +while satisfying the temporal constraints (i.e., upper and +lower bound deadline, wait, and task duration constraints). +Compared to task scheduling within multi-robot systems, +the inclusion of human workers makes scheduling even more +challenging because, while robots can be programmed to +carry out certain tasks at a fixed rate, human workers typ- +ically have latent, dynamic, and task-specific proficiencies. +Effective collaboration in human-robot teams requires utiliz- +ing the distinct abilities of each team member to achieve safe, +effective, and fluent execution. For these problems, we must +consider the ability of humans to learn and improve in task +performance over time. To exploit this property, a scheduling +algorithm must reason about a human’s latent performance +characteristics in order to decide whether to assign the best +worker to a task now versus giving more task experience +to a person who is slower but has a greater potential for +fluency at that particular task. However, it is non-trivial to +infer human strengths and weaknesses while ensuring that +the team satisfies requisite scheduling constraints, due to +the uncertainty introduced by variability in task execution +behavior across different individuals, as well as uncertainty +on future task performance affected by human’s learning +effects with practice [4]. Moreover, a lack of consideration +for human preferences and perceived equality may, in the +long run, put efficient behavior and fluent coordination at a +contradiction [5]. +Recent advances in scheduling methods for human-robot +teams have shown a significant improvement in the ability to +dynamically coordinate large-scale teams in final assembly +manufacturing [6], [7]. Prior approaches typically rely on +an assumption of deterministic or static worker-task profi- +ciencies to formulate the scheduling problem as a mixed- +integer linear program (MILP), which is generally NP-hard +[8]. Exact methods are hard to scale and often fail to consider +the time-varying stochastic task proficiencies of human work- +ers over multi-round schedule execution that could result +in significant productivity gains. The heuristic approaches +may be able to determine task assignments; however, such +approaches required domain specific knowledge that takes +years to gain. We desire a scalable algorithmic approach that +can automatically learn to factor in the human behavior for +fast and fluent human-robot teaming. +Advancements in artificial intelligence have fostered the +idea of leveraging deep neural networks (DNNs) to solve +a plethora of problems in operations research [9]. DNNs +can be trained to automatically explore the problem struc- +ture and discover useful representations in high-dimensional +data towards constructing high-quality solutions, without +arXiv:2301.13279v1 [cs.AI] 30 Jan 2023 + +Fig. 1. +Overview of Multi-Round Environment with HybridNet Scheduler. Left: The Multi-Round Scheduling Environment is developed to simulate a +human-robot scheduling problem over multiple iterative rounds of execution, accounting for changes in human task performance. Right: HybridNet consists +of a heterogeneous graph-based encoder to extract high-level embeddings of the problem and a recurrent schedule propagator for fast schedule generation. +hand-crafted feature engineering [10]. Particularly, promis- +ing progress has been made in learning scalable solvers +with graph neural networks via imitation learning (IL) or +reinforcement learning (RL), outperforming state-of-the-art, +approximate methods [11], [12], [13]. +To overcome the limitations of prior work, we propose +a deep learning-based framework, called HybridNet, for +scheduling stochastic human-robot teams under temporal +constraints. Figure 1 shows the overall framework of our +proposed method operating in a multi-round environment. +HybridNet utilizes a heterogeneous graph-based encoder and +a recurrent schedule propagator. The encoder extracts high +level embeddings of the scheduling problem using a hetero- +geneous graph representation of the problem extended from +the simple temporal network (STN) [14]. By formulating +task scheduling as a sequential decision-making process, the +recurrent propagator uses Long Short Term Memory (LSTM) +cells to carry out fast schedule generation. The resulted +policy network provides a computationally lightweight yet +highly expressive model that is end-to-end trainable via +reinforcement learning algorithms. +The primary contributions of our work are: +• We propose a deep learning-based framework, Hybrid- +Net, for human-robot coordination under temporal con- +straints. HybridNet consist of a Heterogeneous Graph- +based encoder and a Recurrent Schedule Propagator. +The encoder extracts relevant information about the +initial environment, while the Propagator generates the +consequential models of each task-agent assignments +based on the initial model. Inspired by the sensory +encoding and recurrent processing of the brain, this +approach allows for fast schedule generation, removing +the need to interact with the environment between every +task-agent pair selection. +• We develop a virtual task scheduling environment for +mixed human-robot teams in a multi-round setting, +capable of modeling the stochastic learning behavior +of human workers. We make our environment OpenAI +gym-compatible and expect it to serve as a testbed to +facilitate the development of human-robot scheduling +algorithms. The implementation is publicly available.1 +• We present a novel policy model that jointly learns how +to pick agents and tasks without interacting with the +environment between intermediate scheduling decisions +and only needs a single reward at the end of schedule. +By factoring in the action space into an agent selec- +tor and a task selector, we enable conditional policy +learning with HybridNet. We account for the state and +agent models when selecting the agents, and combine +the information regarding the tasks, selected agent and +the state for task assignment. As a result, HybridNet is +end-to-end trainable via Policy Gradients algorithms. +• We conducted extensive experiments to validate Hy- +bridNet across a set of problem sizes. Results showed +HybridNet consistently outperformed prior human- +robot scheduling solutions under both deterministic and +stochastic settings. +II. RELATED WORK +A. Multi-Agent Scheduling Problem +Task assignment and scheduling of multi-agent systems is +an optimization problem that has been studied for real world +applications, both for Multi-Robot Task Allocation(MRTA) +problem using traditional techniques [15] and deep learning +based techniques [16] as well as for human-robot collab- +oration [7]. Task Allocation can be formalised by Mixed +Integer Linear Programming (MILP) to capture it’s con- +straints. The exponential complexity of solving the MILP +can be accelerated through constraint programming methods +[7], [17], [18] or heuristic schedulers to leverage better +scalability [19], [20]. Zhang et al. encoded task schedules +as chromosomes for a genetic algorithm that optimized +schedules for heterogeneous human-robot collaboration by +repeatedly crossing over and mutating the solutions to find +the optimal schedule. [21] +1https://github.com/altundasbatu/HybridNet IROS2022 + +Multi-Round Env +HybridNet +Schedule Propagator +[wl|/1] +Encoder +Problem Instance +Input to +Agent +Learning Curve Models +L'STM +LSTM +Sample +Agent +Agent Selector +Embedding +an +Temporal Constraints +Human-Robot Teams +Agent +Layer +Layel +HetGAT Layer +HetGAT +etGAT I +State +LSTM +Agent Index +Learning Curve +State +Repetition Tracker +Estimator + Embeddings +Task +Sample +(Task, Agent) + Round number +Task Selector +Embeddings +a Task +Picked +Single assignment +Evaluate +Step +Reward +/Makespan +Whole Schedule +TrainingGombolay et al. present an algorithm to capture domain +knowledge through scheduling policy requiring domain- +expert demonstrations [22]. Wang et al. propose Schedu- +leNet, a Heterogenous Graph Neural Networks-based model +for task allocation under temporospatial constraints, trained +through Imitation Learning using optimal schedule [23]. +ScheduleNet relies on interactive scheduling scheme, with +constant update of an environment before reaching a com- +plete schedule. These approaches require optimal schedules +generated by other expert systemsto train and have high +computational complexity that make their implementation +costly. +B. Modeling Human-Robot Teams +As advancements in robot capability progress, they be- +come safer and effective to use in conjunction with humans +to complete specialized works. Liu et al. presents a model +of human task completions, showing an increase in the task +efficiency as a result of learning. This paper shows that +prediction of human performance enhances the ability of +the scheduling systems to explicitly reason about the agents’ +capabilities [4]. Prior work on behavioral teaming and the +natural computational intractability of large-scale schedule +optimization suggests that robots can offer a valuable service +by designing and adapting schedules for human teammates. +In our system, we leverage the findings of Liu et al. to +account for humans learning over time, both in problem +generation as part of the environment and a learning curve +predictor as part of the scheduling policy. The human learn- +ing curve follows an exponential function of generic form +over the course of multiple iterations as shown in Equation +1 [4]: +y = c + ke−βi +(1) +where i is the number of iteration the human has previously +executed a task and c, k, β parameters. We further account +for the stochastic-nature of human learning in our environ- +ment. +C. Graph Neural Networks +Graph Neural Networks (GNNs) are a class of deep neural +networks that learn from unstructured data by representing +objects as nodes and relations as edges and aggregating +information from nearby nodes [24]. GNNs have been widely +applied in graph-based problems such as node classification, +link prediction and clustering, and they have shown to +have an impressive performance [25]. The Heterogeneous +Graph Attention Network presented in Wang et al. utilizes +Deep Learning Algorithms to address the Scheduling Prob- +lem, showing improved performance compared to non-Deep +Learning Schedulers such as Earliest-Deadline First (EDF) +[26] and Tercio [7] at the cost of increased computational +complexity [23]. +D. LSTM Based Sequence Prediction +The impact of the LSTM network has been notable +in language modeling [27], speech-to-text transcription[28], +machine translation [29], and other applications that involve +predictive modeling [30], [31], [32]. The advantage of this +lengthier path generated through the recurrent nature of the +neural network is that it affords an opportunity to build a +certain degree of intuition that can prove beneficial during +all phases of the process [30], [33]. +III. HUMAN-ROBOT SCHEDULING PROBLEM +A. Problem Overview +In this paper, we focus on the problem of human-robot task +allocation and scheduling with temporal constraints [15]. We +describe the problem components using a 4-tuple ⟨a, τ, d, w⟩ +form. a represents all agents that belong to the human-robot +team, τ represents all the tasks to be performed. Each task, +τi, and agent, aj, have a task completion duration dur(τi, aj) +and agents are capable of completing a sequence of tasks in +order. d contains the set of deadline constraints, where di ∈ d +specifies the tasks depending on τi [23]. w is the set of wait +constraints where wij ∈ w denotes the wait time between +tasks τi and τj. A Schedule, S, is a sequence of task-agent +pairs ⟨τi, aj⟩ such that S contains all tasks in τ. +B. Multi-Round Scheduling Environment +The Multi-Round Scheduling Environment is developed +to simulate a human-robot scheduling problem over multiple +iterative rounds of execution, accounting for changes in +the task performance of human workers based on previous +round. Each round is a step in the OpenAI Gym-compatible +environment, taking as input the complete set of task-agent +pairs for the scheduling problem, simulating the sequential +assignment of tasks to agents. +Each round’s execution is considered finished when all +the tasks are assigned to one of the agents or if the provided +schedule is determined to be infeasible under the problem +constraints. The environment checks the feasibility of the +provided schedule given the constraints of the problem, +and computes the total duration of task completion of the +schedule if the schedule is feasible. If the schedule does not +satisfy the constraints, it is determined to be infeasible and +the list of tasks that could not been scheduled are returned. +We formulate the Multi-Round Scheduling Environment as +a Partially Observable Markov Decision Process (POMDP) +using a six-tuple ⟨S, A, T, R, Ω, O, γ⟩ below: +• States: The problem state S is a state of the Multi- +Round Environment consistent of the state of the +Agents. +• Actions: Actions at round t within the Multi-Round +Environment refers to a complete set of Task Alloca- +tions made up of a list of task-agent pairs, denoted as +At = [⟨τi1, aj1⟩, ⟨τi2, aj2⟩, ...] to be executed in order. +• Transitions: T corresponds to executing the action in +Multi-Round Scheduling Environment and proceed to +next time step. +• Rewards: Rt is based on the scheduling objective a user +wants to optimize. In Section III-E we show how to +compute Rt when optimizing makespan. +• Observations: Ω is the estimated performance of all the +task-agent pairs, plus the observable constraints. + +• Observation Function: O is handled by the Learning +Curve Estimator explained in the Section III-D. +• Discount factor, γ +C. Agent Models +The Multi-Round Environment stores the Agent informa- +tion, allowing the environment to keep track of each agent +and which tasks it has previously completed. The update of +the Environment happens at the end of each round, allowing +agents to modify themselves based on their internal models. +to update the model based on the selected (task-agent) pairs +for each round. +1) Determinitic Robot Model: We generate the robot task +completion times randomly through uniform distribution. +2) Stochastic Human Model: We generate the human +task completion times randomly based on Equation 1, such +that the Environment can be setup to provide Deterministic +and Stochastic performance for human learning. The task +duration parameters of the human learning model, c, k, β, in +Equation 1 are built from the randomly selected initial task +completion time for round 0. For Stochastic performance, +the standard deviations are used to sample from a Normal +Distribution as presented in Liu et al. [4]. +D. Learning Curve Estimator +The scheduler is given an estimate of the performance of +the human agents for each task based on the information +about the task duration of the previous executions of the +task-agent pair through the Learning Curve Estimator as +part of our OpenAI Gym-like Environment In our paper, +we have implemented a black box model based on the +insights presented in Liu et al.[4] to simulate a Stochastic +Human Learning Estimator. As an Agent completes a task +in multiple rounds, the Agent Model records the task comple- +tion duration, allowing Learning Curve Estimator to predict +the next task-agent duration more accurately. To represent +the increase in accuracy from increase in information, we +implemented a Learning Curve Estimator that generates an +estimate of the human agent performance using the actual +task performance as the mean of a Gaussian Distribution +with noise that exponentially decreases with the number of +repetitions of the same task for that agent in previous rounds. +E. Reward Design +The total reward, Rt, for the schedule generated by the +multi-round scheduling environment is calculated based on +feasible, A′, and infeasible, ˜A′, subsets of task allocations, +such that At = A′ +t∪ ˜A′ +t. Specifically, the reward, Rt, is based +on the expected reward for the feasible subset of task-agent +assignments, Rt(A′ +t), and the reward from the assignment +of the infeasible subset of task-agent assignments, Rt ˜A′ +t... , +based on the point estimate of the reward from assigning +the incomplete task to the agent that will complete it in +the longest possible duration, multiplied by an infeasible +coefficient Ci as shown in equation 2: +Rt = +� +i∈A′ +t +R (τi, ai) + Cimaxaj +� +�� +i∈ ˜ +A′ +R (τi, aj) +� +� +(2) +The Total Schedule Reward, RS, favors schedules with +more feasible task allocations and enables learning from +infeasible explorations during training. +IV. HYBRIDNET SCHEDULING POLICY +As shown in Figure 1, our HybridNet framework consists +of a heterogeneous graph-based encoder to learn high level +embeddings of the scheduling problem, and a recurrent +schedule propagator to generate the team schedule sequen- +tially. This hybrid network architecture enables directly +learning useful features from the problem structure, owing to +the expressiveness of heterogeneous graph neural networks, +and at the same time efficiently constructing the schedule +with our LSTM-based propagator. As a result, HybridNet +does not require interacting with the environment between +every task-agent pair selection, which is necessary but com- +putationally expensive in prior work [16], [23]. +We denote the policy learned by HybridNet as πθ(A|S), +with θ representing the parameters of the neural network. At +round t, an action takes the form of an ordered sequence +of scheduling decisions, At = {d1, d2, ..., dn}, di = ⟨τi, aj⟩, +where a latter decision, di, is conditioned on its former ones, +d1:i−1. Then, the policy can be factorized as +pθ(At|St) = +n +� +i=1 +pθ(di|St, d1:i−1) +(3) +Using +the +Recurrent +Schedule +Propagator, +HybridNet +recursively +computes +the +conditional +probability, +pθ(di|St, d1:i−1), +for +sampling +a +task-agent +pair. +At +the end, the network collects all the decisions and sends to +the environment for execution. +A. Heterogeneous Graph Encoder +We build our Encoder using the heterogeneous graph at- +tention (HetGAT) layer proposed in [23] that has been shown +effective in representation learning of multi-agent scheduling +problems. At the start of each round for a given human-robot +scheduling problem, the heterogeneous graph representation +is built by extending from the simple temporal network +(STN) that encodes the temporal constraints to include agent +nodes and a state summary node. The metagraph of the +resulted graph is shown in Figure 2, which summarizes +all the node types and edge types. Then, a HetGAT layer +computes the output node features by performing per-edge- +type message passing followed by per-node-type feature +reduction, while utilizing a feature-dependent and structure- +free attention mechanism. We refer interested readers to [23] +for full details of implementing a HetGAT layer. +By stacking several HetGAT layers sequentially, we con- +struct the Encoder that utilizes multi-layer structure to extract +high-level embeddings of each node that will be send to +the propagator for schedule generation. We follow the same + +Fig. 2. +Metagraph of the heterogeneous graph built from the STN by +adding agent and state summary nodes. +hyper-parameters for HetGAT layers as provided in Wang et +al. [23] +B. Recurrent Schedule Propagator +The HetGAT layers are computationally complex and +require interactive scheduling to generate the initial model. +By utilizing an LSTM based Recurrent Predictor, we prop- +agate forward consequences of each task-agent assignment, +recreating the encoded information about the environment +without relying on the initial HetGAT Layer, significantly +reducing the computational complexity of our scheduler. +The Recurrent Schedule Propagator takes as input the +Task, State and Agent embeddings generated by the Het- +erogeneous Graph Encoder and sequentially generates task- +agent pairs based on the encoded information. To predict the +consecutive encoding of state and agents, we use an LSTM +Model to recursively generate the Agent and State after +each assignment of a task to an agent, without interacting +with the Environment, outputting the sequential task-agent +assignment for the complete set of tasks. The pseudo-code +for scheduling generation with HybridNet is presented in +Algorithm 1. +As di = ⟨τi, aj⟩, we further factor pθ(di|St, d1:i−1) into +an agent selector and a task selector. That is, πfactor(d|·) = +πagent(aj|·) · πtask(τi|aj, ·). This factorization allows the +policy to capture the underlying composite and conditional +nature of the scheduling decisions, where the task to schedule +is strongly dependent on the picked agent. +The Agent Selector selects the new agent for the next deci- +sion d based on the state and agent information. Specifically, +the concatenated state-agent embeddings are processed by a +feed-forward neural network, fa, to compute the likelihood +of selecting each agent for the next task-agent pair, using +Equation 4. A softmax operation is performed to convert +the raw predictions into a probability distribution. After +the selection of the agent, the agent embedding of the +chosen agent is updated based on the selected task and state +embeddings, as state change only happens for the assigned +agent. This approach allows the agent selector to consider +how busy each agent is, based on the inherent information +Algorithm 1 Psuedocode for Schedule Generation +Input: graph g, features f, unscheduled-Tasks u +Output: schedule +1: schedule = [ ], i = 1 +2: (ha1, ca1, ht1, ct1, hs1, cs1) ← Encoder(g, f) +3: while |u| ̸= 0 do +4: +pai ← AgentSelector(hsi, hai) +5: +ai ← Sampling(pai) +6: +pti ← TaskSelector(hti, hsi, ai) +7: +ti ← Sampling(pti−1) +8: +schedule.append(⟨ti, ai⟩) +9: +unscheduledTasks.remove(ti) +10: +if |unscheduledTasks| == 0 then +11: +return schedule +12: +end if +13: +i ← i + 1 +14: +hsi, csi ← LSTMs((hti−1[ti], hai−1[ai]), +hai−1, cai−1) +15: +hai, cai ← LSTMa((hti−1[ti], hai−1[ai]), +hai−1, cai−1) +16: end while +presented in the embeddings. +πagent(aj|s) = softmaxi(fa([haj||hs])) +(4) +Next, the Schedule Propagator uses the Task Selector to +assign the task for the selected agent based on the state, agent +and unscheduled task embeddings. As shown in Equation 5, +the Task Selector concatenates the state, selected agent and +the unscheduled task embeddings and passes the combined +information to a feedforward neural network, fτ, to calculate +the likelihood of the task being assigned to the selected +agent. After assigning to an agent for execution, the tasks +are removed from the list of unscheduled tasks. Since the +calculation of likelihood of each task is independent of each +other up to the last softmax operation, the model is scalable +and can be used for differentproblem sizes. +πtask(τi|aj, s) = softmaxi(fτ([hτi||haj||hs])) +(5) +The key component of the Schedule Propagator is the use +of LSTM. As shown in line 12 of Algorithm 1, after each +task-agent pair selection, the state and agent embeddings are +updated using the state LSTM and agent LSTM, respectively. +The LSTM Cell stores the hidden and cell data from the +previous step of the task allocation and predicts the next +step based on the input using the Equation 6 [33]. +ft = σ(Wf[ht−1, xt] + bf) +it = σ(Wi[ht, xt] + bi) +˜ct = tanh(Wc[ht−1, xt] + bc) +ct = ftct−1 + it˜ct +ot = σ(Wo[ht−1, xt] + bo +ht = ottanh(ct) +(6) +Where the Encoder produces initial hidden state, h1 and +initial cell state c1 as an output in the form of [h1, c1]. +During testing, we utilize a batched sampling strategy for +further performance gains. Specifically, we generate multiple +schedules for the same task allocation problem every round. + +communicate +Agent +assignedTo +State +takeTime +UseTime +Task +in +temporalWe select the best performing schedule by computing the +estimated makespan utilizing the Learning Curve Estimator +and provide it to the Multi-Round Environment. More sam- +pling improves solution quality at increased computation. +C. Stochastic Policy Learning +We train HybridNet in multi-round scheduling environ- +ments using Policy Gradient methods that seek to directly +optimize the model parameters based on rewards received +from the environment [34]. Specifically, we compute the +gradient of the model using the sum of the log likelihood +of Agent and Task Selectors, as shown in Equation 7: +∇θJ(θ) = Eπ( +T +� +t +Aπθ +t (st, ⟨τi, ai⟩) +∇θ(logπθ(τi|ai, st) + logπθ(ai|st)) +(7) +In Equation 7, the advantage term, At is estimated by sub- +tracting a “baseline” from the total future reward calculated +in Equation 2. We calculate the “baseline” using the reward +generated for the same task-allocation problem from multiple +batches executed in multiple sequential rounds in the Multi- +Round Environment. Each element of the batch solves the +same scheduling problem and the environment is updated +to account for the task-allocation of the previous round, +updating the agent models. The gradients were calculated +from Equation 7 to updated the model weights. +Due to the combinatorial nature of the task scheduling +problem, plus the stochasticity in human proficiency, learning +a helpful value function as a baseline for computing the +advantage term is non-trivial. Instead, we investigate two +more accessible and efficient alternatives: +• Step-based Baseline: During gradient estimation, the +baseline value subtracted is set as the average return +value across training episodes in the current batch. +• Greedy Rollout Baseline: Greedy Rollout Baseline uses, +πgreedy(A|S), a deterministic greedy version of the Hy- +bridNet scheduler, to collect rewards in the environment. +Its weights, θgreedy, are updated periodically by copying +the weights from the current learner, πθ(A|S). +V. EXPERIMENTAL RESULTS +A. Data Generation +We generate scheduling problems with deadline and wait +constraints under different scales. For all scales, the deadline +constraints are randomly generated for approximately 25% of +the tasks from a range of [1, 5N] where N is the number of +tasks. Approximately 25% of the tasks have wait constraints, +and the duration of non-zero wait constraints is sampled from +U([1, 10]). Task durations are clamped to 10 to 100. +1) Small Scale: The small data set has 9 to 11 tasks with 2 +robots and 2 humans in a team. We generated 2000 Training +Problems and 200 Test Problems. +2) Medium Scale: The medium data set has 18 to 22 +tasks with 2 robots and 2 humans in a team. We generated +2000 Training Problems and 200 Test Problems to inspect +the scalability of our trained model. +3) Large Scale: The large data set is defined as problems +with 36 to 44 tasks chosen at random with 2 robots and 2 +humans in a team. We have generated 200 Test Problems +to evaluated the HybridNet performance with zero training +problems (i.e., zero-shot transfer to from the smaller scale +datasets to the Large Scale dataset). +To simulate the stochastic learning of human agents, for +each Data Set noise is introduced to the Human Agent +models by simulating the natural distribution of the c, k, β +parameters of Equation 1. This allows for each Data Set to +simulate Deterministic and Stochastic Human Performance. +The stochastic model is clipped to fall within the specified +range of task durations. +B. Benchmarking +We benchmark HybridNet against the following methods: +• EDF: A ubiquitous heuristic algorithm, earliest deadline +first (EDF), that selects from a list of available tasks the +one with the earliest deadline, assigning it to the first +available agent. +• Genetic Algorithm: An Evolutionary Optimization Al- +gorithm that uses Post-Processing on the Schedule +Generated by EDF [21]. Genetic algorithm creates new +schedules based on the initial schedule through iterative +randomized mutations by swapping task allocations +and task orders [4]. Each generation selects the top +performing schedules, sorted on feasibility and total +schedule completion time, and used as the baseline for +creating new mutations. The Genetic Algorithm was run +for 10 generation with 90 baseline schedules, 10 task +allocation and 10 task order swapping mutations. +Furthermore, we evaluate the functionality of the Re- +current Schedule Propagator by comparing it against the +following HybridNet variant: +• HetGAT: We implement a HetGAT Scheduler based on +the Encoder of HybridNet. After each task-agent pair +assignment, instead of using the LSTM Cells to update +the task, agent and state embeddings, it directly interacts +with the environment to model the consequences of +action with a new heterogeneous graph and re-computes +those information from it. +We evaluate HybridNet on three metrics: 1) Proportion +of problems solved; 2) Adjusted makespan: determined by +the average of the makespan of feasible schedules and the +maximum possible makespan of the infeasible schedules; +and 3) Runtime statistics. Runtime statistics for training and +execution is compared for HybridNet and HetGAT Scheduler +to model their computational complexity. Because HetGAT +Scheduler relies on interactive scheduling through the envi- +ronment after every task-agent pair allocation, we only train +and evaluate it for Deterministic Human Performance. +C. Model Details +We implement HybridNet and HetGAT using PyTorch [35] +and Deep Graph Library [36]. The HybridNet Encoder used +in training/testing is constructed by stacking three multi-head + +TABLE I +EVALUATION RESULTS: ADJUSTED MAKESPAN AND FEASIBILITY WITH DETERMINISTIC HUMAN TASK PROFICIENCY COMPARING BENCHMARKS +WITH HYBRIDNET TRAINED ON SMALL AND MEDIUM SCALES, WITH SCHEDULES SAMPLED FROM SIZES 8 AND 16 +Training +Methods +Small +Medium +Large +Makespan +Feasibility (%) +Makespan +Feasibility (%) +Makespan +Feasibility (%) +- +EDF +239.31 +73.00 +1109.85 +15.00 +2535.89 +1.00 +- +Genetic Algorithm +302.42 ± 0.77 +74.10 ± 0.30 +1180.07 ±2.54 +16.60 ± 0.70 +2542.79 ± 0.06 +1.00 ± 0.00 +Step-based +HetGAT 8 +257.20 ± 0.18 +86.29 ± 0.08 +751.27 ± 1.29 +50.17 ± 0.14 +2123.96 ± 5.66 +17.12 ± 0.27 +HetGAT 16 +249.69 ± 0.30 +86.51 ± 0.09 +723.57 ± 0.94 +50.29 ± 0.11 +2081.65 ± 5.45 +17.15 ± 0.16 +Greedy +HetGAT 8 +261.15 ± 0.09 +85.59 ± 0.10 +784.32 ± 0.52 +53.28 ± 0.17 +2017.25 ± 2.16 +23.98 ± 0.14 +HetGAT 16 +255.70 ± 0.23 +86.05 ± 0.15 +765.79 ± 0.96 +53.41 ± 0.08 +1983.73 ± 1.59 +23.84 ± 0.01 +Step-based +HybridNet Small 8 +260.22 ± 0.15 +86.93 ± 0.10 +770.48 ± 1.07 +59.11 ± 0.35 +2005.80 ± 2.33 +30.65 ± 0.39 +HybridNet Small 16 +252.57 ± 0.49 +87.08 ± 0.10 +746.35 ± 0.52 +60.89 ± 0.36 +1953.65 ± 3.76 +33.24 ± 0.61 +Greedy +HybridNet Small 8 +266.74 ± 0.31 +84.65 ± 0.32 +758.96 ± 2.27 +61.09 ± 0.43 +2049.32 ± 3.73 +28.74 ± 0.45 +HybridNet Small 16 +258.17 ± 0.45 +85.13 ± 0.20 +723.35 ± 1.70 +63.68 ± 0.49 +1973.15 ± 2.91 +32.46 ± 0.40 +Step-based +HybridNet Medium 8 +- +- +722.85 ± 0.61 +64.69 ± 0.29 +2010.86 ± 1.97 +30.86 ± 0.45 +HybridNet Medium 16 +- +- +697.40 ± 2.04 +66.25 ± 0.51 +1944.72 ± 4.10 +33.88 ± 0.49 +Greedy +HybridNet Medium 8 +- +- +692.01 ± 3.69 +68.33 ± 0.66 +2011.78 ± 5.08 +30.58 ± 0.87 +HybridNet Medium 16 +- +- +659.01 ± 0.89 +71.00 ± 0.45 +1936.97 ± 4.68 +34.66 ± 0.74 +TABLE II +EVALUATION RESULTS: ADJUSTED MAKESPAN AND FEASIBILITY WITH STOCHASTIC HUMAN TASK PROFICIENCY +Methods +Small +Medium +Large +Makespan +Feasibility (%) +Makespan +Feasibility (%) +Makespan +Feasibility (%) +EDF +227.81± 6.17 +75.65 ± 1.21 +1071.02± 20.65 +17.30 ± 1.12 +2524.92± 8.95 +1.15 ± 0.23 +Genetic Algorithm +283.79 ± 10.39 +77.45 ± 2.05 +1149.42 ± 12.14 +19.55 ± 1.31 +2541.20 ± 3.54 +1.05 ± 0.15 +HybridNet Small +298.81 ± 0.96 +79.54 ± 0.52 +881.16 ± 2.89 +48.89 ± 1.09 +2141.80 ± 5.12 +23.51 ± 0.96 +HybridNet Medium +- +- +859.99 ± 4.82 +51.94 ± 1.32 +2174.57 ± 8.53 +22.31 ± 0.94 +TABLE III +EVALUATION RESULTS: RUNTIME PERFORMANCE ON SINGLE PROBLEM +Methods +HetGAT8 +HybridNet8 +HybridNet16 +Training Time (s) +Small +184.52 ± 18.00 +19.97 ± 0.91 +- +Medium +354.77 ± 38.31 +22.40 ± 6.52 +- +Evaluation Time (s) +Small +22.91 ± 5.85 +10.94 ± 0.99 +18.95 ± 3.53 +Medium +70.12 ± 8.67 +14.77 ± 1.42 +22.30 ± 7.55 +Large +123.76 ± 32.32 +18.84 ± 7.38 +27.78 ± 16.52 +HetGAT layers (the first two use concatenation, and the last +one uses averaging). The feature dimension of hidden layers += 64, and the number of heads = 8. The Recurrent Propagator +utilizes a LSTMCell of size 32 followed by a fully-connected +layer and a softmax layer. We set γ = 0.99, batch size = +8 and used Adam optimizer [37] with a learning rate of +2 × 10−3, and a weight decay of 5 × 10−4. We employed a +learning rate decay of 0.5 every 4000 epochs. We evaluate the +models using a batch size of 8 and 16. For the Multi-Round +Environment, the infeasible reward coefficient Ci = 2.0 and +total round number = 4. Both training and evaluation were +conducted on a Quadro RTX 8000 GPU. +D. Evaluation Results +Table I shows the evaluation performance with Deter- +ministic Human Proficiency in different scales. The Deter- +ministic Human Proficiency means that during training and +evaluation, human learning curve is known and execution +is deterministic for every agent. In Table I, “Small” and +“Medium” term after model name denotes the data scale the +model was trained on and the number following it denotes +the batch size for schedule sampling. The results show that +HybridNet outperforms both EDF and Genetic Algorithm in +adjusted makespan and percentage of feasibility. HybridNet +trained on Small scale problems generalizes for both Medium +and Large scale problems with similar or slightly worse +performance than HybridNet trained on Medium. HybridNet +and HetGAT performs similarly on all scales. This shows that +HybridNet is capable of learning high performance policies +by leveraging the Recurrent Schedule Propagator and without +requiring interaction with the Environment. +We provide the runtimes of training and evaluation for +HetGAT and HybridNET in Table III. HybridNet is approx- +imately 10 times faster in training compared to HetGAT +Model and at least 2 times faster during evaluation for +same batch size. EDF and Genetic Algorithm were evalu- +ated through the CPU without GPU acceleration, making it +infeasible to accurately compare the performance of the Deep +Learning Models to the Traditional Models. +We show that for HybridNet, step-based training has better +performance over the greedy baseline, while for HetGAT +model, greedy baseline training is better. We also observed +that greedy baseline training reached convergence faster than +step-based training (4500 epochs vs. 19000 epochs). Further +investigation is worthwhile. +Table II shows the evaluation performance with Stochas- +tic Human Proficiency in different scales. The Stochastic +Human Proficiency is presented as randomness in both the +actual human execution within Multi-Round Environment +and uncertainty within the Learning Curve Estimator used +for schedule generation. The results show that HybridNet +outperforms the EDF and Genetic Algorithm across different +data scales. The largest performance gap was observed on +large dataset (23.51% vs. 1.15%). Here, HetGAT model is +not included as it requires interaction with the environment +after every task-agent assignment to observe the outcome, +which is not available until the whole schedule is generated +and sent to the Stochastic Environment for execution to +emulate real-world scenarios. +VI. CONCLUSIONS +We present a deep learning-based framework, called +HybridNet, combining a heterogeneous graph-based en- +coder with a recurrent schedule propagator, for scheduling + +stochastic human-robot teams under temporal constraints. +The resulting policy network provides a computationally +lightweight yet highly expressive model that is end-to-end +trainable via reinforcement learning algorithms. We devel- +oped a multi-round task scheduling environment for stochas- +tic human-robot teams and conducted extensive experiments, +showing that HybridNet outperforms other human-robot +scheduling solutions across problem sizes. Future research +includes integrating the learning-based human estimator into +HybridNet, transfer learning across optimizing different ob- +jective functions, and deploying the trained network in a real- +world scenario. +REFERENCES +[1] Z. Yan, N. Jouandeau, and A. A. Cherif, “A survey and analysis +of multi-robot coordination,” International Journal of Advanced +Robotic Systems, vol. 10, no. 12, p. 399, 2013. [Online]. Available: +https://doi.org/10.5772/57313 +[2] C. Heyer, “Human-robot interaction and future industrial robotics ap- +plications,” in 2010 IEEE/RSJ International Conference on Intelligent +Robots and Systems. +IEEE, 2010, pp. 4749–4754. +[3] E. Nunes, M. Manner, H. Mitiche, and M. Gini, “A taxonomy for task +allocation problems with temporal and ordering constraints,” Robotics +and Autonomous Systems, vol. 90, pp. 55–70, 2017. +[4] R. Liu, M. Natarajan, and M. Gombolay, “Human-robot team coordi- +nation with dynamic and latent human task proficiencies: Scheduling +with learning curves,” 2020. +[5] M. C. Gombolay, R. A. Gutierrez, S. G. Clarke, G. F. Sturla, +and J. A. Shah, “Decision-making authority, team efficiency and +human worker satisfaction in mixed human–robot teams,” Autonomous +Robots, vol. 39, no. 3, pp. 293–312, 2015. +[6] E. Nunes and M. Gini, “Multi-robot auctions for allocation of tasks +with temporal constraints,” in Twenty-Ninth AAAI Conference on +Artificial Intelligence, 2015, pp. 2110–2116. +[7] M. C. Gombolay, R. J. Wilcox, and J. A. Shah, “Fast scheduling of +robot teams performing tasks with temporospatial constraints,” IEEE +Transactions on Robotics, vol. 34, no. 1, pp. 220–239, 2018. +[8] M. M. Solomon, “On the worst-case performance of some heuristics +for the vehicle routing and scheduling problem with time window +constraints,” Networks, vol. 16, no. 2, pp. 161–174, 1986. +[9] Y. Bengio, A. Lodi, and A. Prouvost, “Machine learning for com- +binatorial optimization: a methodological tour d’horizon,” European +Journal of Operational Research, vol. 290, no. 2, pp. 405–421, 2021. +[10] Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” nature, vol. +521, no. 7553, pp. 436–444, 2015. +[11] E. Khalil, H. Dai, Y. Zhang, B. Dilkina, and L. Song, “Learning +combinatorial optimization algorithms over graphs,” in Advances in +Neural Information Processing Systems, 2017, pp. 6348–6358. +[12] W. Kool, H. van Hoof, and M. Welling, “Attention, learn to solve +routing problems!” in International Conference on Learning Repre- +sentations, 2019. +[13] T. Ma, P. Ferber, S. Huo, J. Chen, and M. Katz, “Online planner +selection with graph neural networks and adaptive scheduling,” +Proceedings of the AAAI Conference on Artificial Intelligence, +vol. 34, no. 04, pp. 5077–5084, Apr. 2020. [Online]. Available: +https://ojs.aaai.org/index.php/AAAI/article/view/5949 +[14] R. Dechter, I. Meiri, and J. Pearl, “Temporal constraint networks,” +Artificial intelligence, vol. 49, no. 1-3, pp. 61–95, 1991. +[15] E. Nunes, M. Manner, H. Mitiche, and M. Gini, “A taxonomy for task +allocation problems with temporal and ordering constraints,” Robotics +and Autonomous Systems, vol. 90, pp. 55–70, Apr. 2017, publisher +Copyright: © 2016 Elsevier B.V. +[16] Z. Wang and M. Gombolay, “Learning scheduling policies for multi- +robot coordination with graph attention networks,” IEEE Robotics and +Automation Letters, vol. 5, no. 3, pp. 4509–4516, 2020. +[17] J. F. Benders, “Partitioning procedures for solving mixed-variables +programming problems,” Numerische Mathematik, vol. 4, no. 1, +pp. 238–252, Dec. 1962. [Online]. Available: https://doi.org/10.1007/ +bf01386316 +[18] H. Ren and L. Tang, “An improved hybrid milp/cp algorithm frame- +work for the job-shop scheduling,” 2009 IEEE International Confer- +ence on Automation and Logistics, pp. 890–894, 2009. +[19] E. Castro and S. Petrovic, “Combined mathematical programming +and heuristics for a radiotherapy pre-treatment scheduling problem,” +Journal of Scheduling, vol. 15, no. 3, pp. 333–346, May 2011. +[Online]. Available: https://doi.org/10.1007/s10951-011-0239-8 +[20] J. Chen and R. Askin, “Project selection, scheduling and resource allo- +cation with time dependent returns,” European Journal of Operational +Research, vol. 193, pp. 23–34, 02 2009. +[21] S. Zhang, Y. Chen, J. Zhang, and Y. Jia, “Real-time adaptive assembly +scheduling in human-multi-robot collaboration according to human +capability*,” 2020 IEEE International Conference on Robotics and +Automation (ICRA), pp. 3860–3866, 2020. +[22] M. C. Gombolay, “Apprenticeship scheduling for human-robot teams,” +in Proceedings of the Thirtieth AAAI Conference on Artificial Intelli- +gence, ser. AAAI’16. +AAAI Press, 2016, p. 2497–2498. +[23] Z. Wang, C. Liu, and M. Gombolay, “Heterogeneous graph attention +networks for scalable multi-robot scheduling with temporospatial +constraints,” Autonomous Robots, vol. 46, no. 1, pp. 249–268, 2022. +[24] F. +Scarselli, +M. +Gori, +A. +C. +Tsoi, +M. +Hagenbuchner, +and +G. Monfardini, “The graph neural network model,” Trans. Neur. +Netw., vol. 20, no. 1, p. 61–80, jan 2009. [Online]. Available: +https://doi.org/10.1109/TNN.2008.2005605 +[25] K. Xu, W. Hu, J. Leskovec, and S. Jegelka, “How powerful are +graph neural networks?” in International Conference on Learning +Representations, 2019. +[26] J. Singh, “An algorithm to reduce the time complexity of earliest +deadline first scheduling algorithm in real-time system,” 2010. +[27] M. Sundermeyer, R. Schl¨uter, and H. Ney, “Lstm neural networks for +language modeling,” in Thirteenth annual conference of the interna- +tional speech communication association, 2012. +[28] A. Graves, S. Fern´andez, and J. Schmidhuber, “Bidirectional lstm +networks for improved phoneme classification and recognition,” in +International conference on artificial neural networks. Springer, 2005, +pp. 799–804. +[29] J. Zhou, G. Cui, S. Hu, Z. Zhang, C. Yang, Z. Liu, L. Wang, C. Li, +and M. Sun, “Graph neural networks: A review of methods and +applications,” 2021. +[30] A. Sherstinsky, “Fundamentals of recurrent neural network (RNN) +and +long +short-term +memory +(LSTM) +network,” +CoRR, +vol. +abs/1808.03314, 2018. [Online]. Available: http://arxiv.org/abs/1808. +03314 +[31] P. Malhotra, A. Ramakrishnan, G. Anand, L. Vig, P. Agarwal, and +G. Shroff, “Lstm-based encoder-decoder for multi-sensor anomaly +detection,” arXiv preprint arXiv:1607.00148, 2016. +[32] A. Ycart, E. Benetos, et al., “A study on lstm networks for polyphonic +music sequence modelling.” +ISMIR, 2017. +[33] Y. Yu, X. Si, C. Hu, and J. Zhang, “A review of recurrent neural +networks: Lstm cells and network architectures,” Neural Computation, +vol. 31, no. 7, pp. 1235–1270, 2019. +[34] R. S. Sutton, S. Singh, and D. McAllester, “Comparing policy-gradient +algorithms,” IEEE Transactions on Systems, Man, and Cybernetics, +2000. +[35] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, +T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, +E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, +L. Fang, J. Bai, and S. Chintala, “Pytorch: An imperative style, high- +performance deep learning library,” in Advances in Neural Information +Processing Systems, H. Wallach, H. Larochelle, A. Beygelzimer, +F. d'Alch´e-Buc, E. Fox, and R. Garnett, Eds., vol. 32. +Curran +Associates, Inc., 2019. +[36] M. Wang, D. Zheng, Z. Ye, Q. Gan, M. Li, X. Song, J. Zhou, C. Ma, +L. Yu, Y. Gai, T. Xiao, T. He, G. Karypis, J. Li, and Z. Zhang, “Deep +graph library: A graph-centric, highly-performant package for graph +neural networks,” 2020. +[37] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimiza- +tion,” 2017. + diff --git a/DdFQT4oBgHgl3EQfPjZG/content/tmp_files/load_file.txt b/DdFQT4oBgHgl3EQfPjZG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3721bf5d7b1206b290dc32157c1420c2d87edf7e --- /dev/null +++ b/DdFQT4oBgHgl3EQfPjZG/content/tmp_files/load_file.txt @@ -0,0 +1,780 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf,len=779 +page_content='Learning Coordination Policies over Heterogeneous Graphs for Human-Robot Teams via Recurrent Neural Schedule Propagation Batuhan Altundas1, Zheyuan Wang1, Joshua Bishop1 and Matthew Gombolay1 Abstract— As human-robot collaboration increases in the workforce, it becomes essential for human-robot teams to coordinate efficiently and intuitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Traditional approaches for human-robot scheduling either utilize exact methods that are intractable for large-scale problems and struggle to ac- count for stochastic, time varying human task performance, or application-specific heuristics that require expert domain knowledge to develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We propose a deep learning-based framework, called HybridNet, combining a heterogeneous graph-based encoder with a recurrent schedule propagator for scheduling stochastic human-robot teams under upper- and lower-bound temporal constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The HybridNet’s encoder leverages Heterogeneous Graph Attention Networks to model the initial environment and team dynamics while accounting for the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' By formulating task scheduling as a se- quential decision-making process, the HybridNet’s recurrent neural schedule propagator leverages Long Short-Term Mem- ory (LSTM) models to propagate forward consequences of actions to carry out fast schedule generation, removing the need to interact with the environment between every task- agent pair selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The resulting scheduling policy network provides a computationally lightweight yet highly expressive model that is end-to-end trainable via Reinforcement Learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We develop a virtual task scheduling environment for mixed human-robot teams in a multi-round setting, capable of modeling the stochastic learning behaviors of human work- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Experimental results showed that HybridNet outperformed other human-robot scheduling solutions across problem sizes for both deterministic and stochastic human performance, with faster runtime compared to pure-GNN-based schedulers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' INTRODUCTION With collaborative robots (cobots) becoming more avail- able in the industrial and manufacturing environments, robots and humans increasingly share the same work space to collaborate on tasks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' By removing the cage around tradi- tional robot platforms and integrating robots into dynamic, final assembly operations, manufacturers can see improve- ments in reducing a factory’s footprint and environmental costs as well as increased productivity [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' In this paper, we focus on the problem of multi-agent task allocation and scheduling [3] with mixed human-robot teams over multiple iterations of the same task allocation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Our work accounts for and leverages stochastic, time-varying human task performance to quickly solve task allocation problems among team members to achieve a high-quality schedule with respect to the application-specific objective function This work was supported in part by the Office of Naval Research under grant N00014-19-1-2076 and Naval Research Laboratory under grant N00173-21-1-G009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 1Batuhan Altundas, Zheyuan Wang, Joshua Bishop and Matthew Gom- bolay are with the Institute for Robotics and Intelligent Machines, Georgia Institute of Technology, Atlanta, GA 30332, USA {baltundas3, pjohnwang, jbishop45, mgombolay3}@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='edu while satisfying the temporal constraints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=', upper and lower bound deadline, wait, and task duration constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Compared to task scheduling within multi-robot systems, the inclusion of human workers makes scheduling even more challenging because, while robots can be programmed to carry out certain tasks at a fixed rate, human workers typ- ically have latent, dynamic, and task-specific proficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Effective collaboration in human-robot teams requires utiliz- ing the distinct abilities of each team member to achieve safe, effective, and fluent execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' For these problems, we must consider the ability of humans to learn and improve in task performance over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' To exploit this property, a scheduling algorithm must reason about a human’s latent performance characteristics in order to decide whether to assign the best worker to a task now versus giving more task experience to a person who is slower but has a greater potential for fluency at that particular task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' However, it is non-trivial to infer human strengths and weaknesses while ensuring that the team satisfies requisite scheduling constraints, due to the uncertainty introduced by variability in task execution behavior across different individuals, as well as uncertainty on future task performance affected by human’s learning effects with practice [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Moreover, a lack of consideration for human preferences and perceived equality may, in the long run, put efficient behavior and fluent coordination at a contradiction [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Recent advances in scheduling methods for human-robot teams have shown a significant improvement in the ability to dynamically coordinate large-scale teams in final assembly manufacturing [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Prior approaches typically rely on an assumption of deterministic or static worker-task profi- ciencies to formulate the scheduling problem as a mixed- integer linear program (MILP), which is generally NP-hard [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Exact methods are hard to scale and often fail to consider the time-varying stochastic task proficiencies of human work- ers over multi-round schedule execution that could result in significant productivity gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The heuristic approaches may be able to determine task assignments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' however, such approaches required domain specific knowledge that takes years to gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We desire a scalable algorithmic approach that can automatically learn to factor in the human behavior for fast and fluent human-robot teaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Advancements in artificial intelligence have fostered the idea of leveraging deep neural networks (DNNs) to solve a plethora of problems in operations research [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' DNNs can be trained to automatically explore the problem struc- ture and discover useful representations in high-dimensional data towards constructing high-quality solutions, without arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='13279v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='AI] 30 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Overview of Multi-Round Environment with HybridNet Scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Left: The Multi-Round Scheduling Environment is developed to simulate a human-robot scheduling problem over multiple iterative rounds of execution, accounting for changes in human task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Right: HybridNet consists of a heterogeneous graph-based encoder to extract high-level embeddings of the problem and a recurrent schedule propagator for fast schedule generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' hand-crafted feature engineering [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Particularly, promis- ing progress has been made in learning scalable solvers with graph neural networks via imitation learning (IL) or reinforcement learning (RL), outperforming state-of-the-art, approximate methods [11], [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' To overcome the limitations of prior work, we propose a deep learning-based framework, called HybridNet, for scheduling stochastic human-robot teams under temporal constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Figure 1 shows the overall framework of our proposed method operating in a multi-round environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' HybridNet utilizes a heterogeneous graph-based encoder and a recurrent schedule propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The encoder extracts high level embeddings of the scheduling problem using a hetero- geneous graph representation of the problem extended from the simple temporal network (STN) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' By formulating task scheduling as a sequential decision-making process, the recurrent propagator uses Long Short Term Memory (LSTM) cells to carry out fast schedule generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The resulted policy network provides a computationally lightweight yet highly expressive model that is end-to-end trainable via reinforcement learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The primary contributions of our work are: We propose a deep learning-based framework, Hybrid- Net, for human-robot coordination under temporal con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' HybridNet consist of a Heterogeneous Graph- based encoder and a Recurrent Schedule Propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The encoder extracts relevant information about the initial environment, while the Propagator generates the consequential models of each task-agent assignments based on the initial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Inspired by the sensory encoding and recurrent processing of the brain, this approach allows for fast schedule generation, removing the need to interact with the environment between every task-agent pair selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We develop a virtual task scheduling environment for mixed human-robot teams in a multi-round setting, capable of modeling the stochastic learning behavior of human workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We make our environment OpenAI gym-compatible and expect it to serve as a testbed to facilitate the development of human-robot scheduling algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The implementation is publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='1 We present a novel policy model that jointly learns how to pick agents and tasks without interacting with the environment between intermediate scheduling decisions and only needs a single reward at the end of schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' By factoring in the action space into an agent selec- tor and a task selector, we enable conditional policy learning with HybridNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We account for the state and agent models when selecting the agents, and combine the information regarding the tasks, selected agent and the state for task assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' As a result, HybridNet is end-to-end trainable via Policy Gradients algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We conducted extensive experiments to validate Hy- bridNet across a set of problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Results showed HybridNet consistently outperformed prior human- robot scheduling solutions under both deterministic and stochastic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Multi-Agent Scheduling Problem Task assignment and scheduling of multi-agent systems is an optimization problem that has been studied for real world applications, both for Multi-Robot Task Allocation(MRTA) problem using traditional techniques [15] and deep learning based techniques [16] as well as for human-robot collab- oration [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Task Allocation can be formalised by Mixed Integer Linear Programming (MILP) to capture it’s con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The exponential complexity of solving the MILP can be accelerated through constraint programming methods [7], [17], [18] or heuristic schedulers to leverage better scalability [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' encoded task schedules as chromosomes for a genetic algorithm that optimized schedules for heterogeneous human-robot collaboration by repeatedly crossing over and mutating the solutions to find the optimal schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [21] 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content="com/altundasbatu/HybridNet IROS2022 Multi-Round Env HybridNet Schedule Propagator [wl|/1] Encoder Problem Instance Input to Agent Learning Curve Models L'STM LSTM Sample Agent Agent Selector Embedding an Temporal Constraints Human-Robot Teams Agent Layer Layel HetGAT Layer HetGAT etGAT I State LSTM Agent Index Learning Curve State Repetition Tracker Estimator Embeddings Task Sample (Task," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Agent) Round number Task Selector Embeddings a Task Picked Single assignment Evaluate Step Reward /Makespan Whole Schedule TrainingGombolay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' present an algorithm to capture domain knowledge through scheduling policy requiring domain- expert demonstrations [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' propose Schedu- leNet, a Heterogenous Graph Neural Networks-based model for task allocation under temporospatial constraints, trained through Imitation Learning using optimal schedule [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' ScheduleNet relies on interactive scheduling scheme, with constant update of an environment before reaching a com- plete schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' These approaches require optimal schedules generated by other expert systemsto train and have high computational complexity that make their implementation costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Modeling Human-Robot Teams As advancements in robot capability progress, they be- come safer and effective to use in conjunction with humans to complete specialized works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' presents a model of human task completions, showing an increase in the task efficiency as a result of learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' This paper shows that prediction of human performance enhances the ability of the scheduling systems to explicitly reason about the agents’ capabilities [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Prior work on behavioral teaming and the natural computational intractability of large-scale schedule optimization suggests that robots can offer a valuable service by designing and adapting schedules for human teammates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' In our system, we leverage the findings of Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' to account for humans learning over time, both in problem generation as part of the environment and a learning curve predictor as part of the scheduling policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The human learn- ing curve follows an exponential function of generic form over the course of multiple iterations as shown in Equation 1 [4]: y = c + ke−βi (1) where i is the number of iteration the human has previously executed a task and c, k, β parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We further account for the stochastic-nature of human learning in our environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Graph Neural Networks Graph Neural Networks (GNNs) are a class of deep neural networks that learn from unstructured data by representing objects as nodes and relations as edges and aggregating information from nearby nodes [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' GNNs have been widely applied in graph-based problems such as node classification, link prediction and clustering, and they have shown to have an impressive performance [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The Heterogeneous Graph Attention Network presented in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' utilizes Deep Learning Algorithms to address the Scheduling Prob- lem, showing improved performance compared to non-Deep Learning Schedulers such as Earliest-Deadline First (EDF) [26] and Tercio [7] at the cost of increased computational complexity [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' LSTM Based Sequence Prediction The impact of the LSTM network has been notable in language modeling [27], speech-to-text transcription[28], machine translation [29], and other applications that involve predictive modeling [30], [31], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The advantage of this lengthier path generated through the recurrent nature of the neural network is that it affords an opportunity to build a certain degree of intuition that can prove beneficial during all phases of the process [30], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' HUMAN-ROBOT SCHEDULING PROBLEM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Problem Overview In this paper, we focus on the problem of human-robot task allocation and scheduling with temporal constraints [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We describe the problem components using a 4-tuple ⟨a, τ, d, w⟩ form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' a represents all agents that belong to the human-robot team, τ represents all the tasks to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Each task, τi, and agent, aj, have a task completion duration dur(τi, aj) and agents are capable of completing a sequence of tasks in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' d contains the set of deadline constraints, where di ∈ d specifies the tasks depending on τi [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' w is the set of wait constraints where wij ∈ w denotes the wait time between tasks τi and τj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' A Schedule, S, is a sequence of task-agent pairs ⟨τi, aj⟩ such that S contains all tasks in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Multi-Round Scheduling Environment The Multi-Round Scheduling Environment is developed to simulate a human-robot scheduling problem over multiple iterative rounds of execution, accounting for changes in the task performance of human workers based on previous round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Each round is a step in the OpenAI Gym-compatible environment, taking as input the complete set of task-agent pairs for the scheduling problem, simulating the sequential assignment of tasks to agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Each round’s execution is considered finished when all the tasks are assigned to one of the agents or if the provided schedule is determined to be infeasible under the problem constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The environment checks the feasibility of the provided schedule given the constraints of the problem, and computes the total duration of task completion of the schedule if the schedule is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' If the schedule does not satisfy the constraints, it is determined to be infeasible and the list of tasks that could not been scheduled are returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We formulate the Multi-Round Scheduling Environment as a Partially Observable Markov Decision Process (POMDP) using a six-tuple ⟨S, A, T, R, Ω, O, γ⟩ below: States: The problem state S is a state of the Multi- Round Environment consistent of the state of the Agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Actions: Actions at round t within the Multi-Round Environment refers to a complete set of Task Alloca- tions made up of a list of task-agent pairs, denoted as At = [⟨τi1, aj1⟩, ⟨τi2, aj2⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='] to be executed in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Transitions: T corresponds to executing the action in Multi-Round Scheduling Environment and proceed to next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Rewards: Rt is based on the scheduling objective a user wants to optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' In Section III-E we show how to compute Rt when optimizing makespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Observations: Ω is the estimated performance of all the task-agent pairs, plus the observable constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Observation Function: O is handled by the Learning Curve Estimator explained in the Section III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Discount factor, γ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Agent Models The Multi-Round Environment stores the Agent informa- tion, allowing the environment to keep track of each agent and which tasks it has previously completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The update of the Environment happens at the end of each round, allowing agents to modify themselves based on their internal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' to update the model based on the selected (task-agent) pairs for each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 1) Determinitic Robot Model: We generate the robot task completion times randomly through uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 2) Stochastic Human Model: We generate the human task completion times randomly based on Equation 1, such that the Environment can be setup to provide Deterministic and Stochastic performance for human learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The task duration parameters of the human learning model, c, k, β, in Equation 1 are built from the randomly selected initial task completion time for round 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' For Stochastic performance, the standard deviations are used to sample from a Normal Distribution as presented in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Learning Curve Estimator The scheduler is given an estimate of the performance of the human agents for each task based on the information about the task duration of the previous executions of the task-agent pair through the Learning Curve Estimator as part of our OpenAI Gym-like Environment In our paper, we have implemented a black box model based on the insights presented in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [4] to simulate a Stochastic Human Learning Estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' As an Agent completes a task in multiple rounds, the Agent Model records the task comple- tion duration, allowing Learning Curve Estimator to predict the next task-agent duration more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' To represent the increase in accuracy from increase in information, we implemented a Learning Curve Estimator that generates an estimate of the human agent performance using the actual task performance as the mean of a Gaussian Distribution with noise that exponentially decreases with the number of repetitions of the same task for that agent in previous rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Reward Design The total reward, Rt, for the schedule generated by the multi-round scheduling environment is calculated based on feasible, A′, and infeasible, ˜A′, subsets of task allocations, such that At = A′ t∪ ˜A′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Specifically, the reward, Rt, is based on the expected reward for the feasible subset of task-agent assignments, Rt(A′ t), and the reward from the assignment of the infeasible subset of task-agent assignments, Rt ˜A′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' , based on the point estimate of the reward from assigning the incomplete task to the agent that will complete it in the longest possible duration, multiplied by an infeasible coefficient Ci as shown in equation 2: Rt = � i∈A′ t R (τi, ai) + Cimaxaj � �� i∈ ˜ A′ R (τi, aj) � � (2) The Total Schedule Reward, RS, favors schedules with more feasible task allocations and enables learning from infeasible explorations during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' HYBRIDNET SCHEDULING POLICY As shown in Figure 1, our HybridNet framework consists of a heterogeneous graph-based encoder to learn high level embeddings of the scheduling problem, and a recurrent schedule propagator to generate the team schedule sequen- tially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' This hybrid network architecture enables directly learning useful features from the problem structure, owing to the expressiveness of heterogeneous graph neural networks, and at the same time efficiently constructing the schedule with our LSTM-based propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' As a result, HybridNet does not require interacting with the environment between every task-agent pair selection, which is necessary but com- putationally expensive in prior work [16], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We denote the policy learned by HybridNet as πθ(A|S), with θ representing the parameters of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' At round t, an action takes the form of an ordered sequence of scheduling decisions, At = {d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=', dn}, di = ⟨τi, aj⟩, where a latter decision, di, is conditioned on its former ones, d1:i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Then, the policy can be factorized as pθ(At|St) = n � i=1 pθ(di|St, d1:i−1) (3) Using the Recurrent Schedule Propagator, HybridNet recursively computes the conditional probability, pθ(di|St, d1:i−1), for sampling a task-agent pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' At the end, the network collects all the decisions and sends to the environment for execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Heterogeneous Graph Encoder We build our Encoder using the heterogeneous graph at- tention (HetGAT) layer proposed in [23] that has been shown effective in representation learning of multi-agent scheduling problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' At the start of each round for a given human-robot scheduling problem, the heterogeneous graph representation is built by extending from the simple temporal network (STN) that encodes the temporal constraints to include agent nodes and a state summary node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The metagraph of the resulted graph is shown in Figure 2, which summarizes all the node types and edge types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Then, a HetGAT layer computes the output node features by performing per-edge- type message passing followed by per-node-type feature reduction, while utilizing a feature-dependent and structure- free attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We refer interested readers to [23] for full details of implementing a HetGAT layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' By stacking several HetGAT layers sequentially, we con- struct the Encoder that utilizes multi-layer structure to extract high-level embeddings of each node that will be send to the propagator for schedule generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We follow the same Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Metagraph of the heterogeneous graph built from the STN by adding agent and state summary nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' hyper-parameters for HetGAT layers as provided in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [23] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Recurrent Schedule Propagator The HetGAT layers are computationally complex and require interactive scheduling to generate the initial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' By utilizing an LSTM based Recurrent Predictor, we prop- agate forward consequences of each task-agent assignment, recreating the encoded information about the environment without relying on the initial HetGAT Layer, significantly reducing the computational complexity of our scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The Recurrent Schedule Propagator takes as input the Task, State and Agent embeddings generated by the Het- erogeneous Graph Encoder and sequentially generates task- agent pairs based on the encoded information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' To predict the consecutive encoding of state and agents, we use an LSTM Model to recursively generate the Agent and State after each assignment of a task to an agent, without interacting with the Environment, outputting the sequential task-agent assignment for the complete set of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The pseudo-code for scheduling generation with HybridNet is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' As di = ⟨τi, aj⟩, we further factor pθ(di|St, d1:i−1) into an agent selector and a task selector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' That is, πfactor(d|·) = πagent(aj|·) · πtask(τi|aj, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' This factorization allows the policy to capture the underlying composite and conditional nature of the scheduling decisions, where the task to schedule is strongly dependent on the picked agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The Agent Selector selects the new agent for the next deci- sion d based on the state and agent information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Specifically, the concatenated state-agent embeddings are processed by a feed-forward neural network, fa, to compute the likelihood of selecting each agent for the next task-agent pair, using Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' A softmax operation is performed to convert the raw predictions into a probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' After the selection of the agent, the agent embedding of the chosen agent is updated based on the selected task and state embeddings, as state change only happens for the assigned agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' This approach allows the agent selector to consider how busy each agent is, based on the inherent information Algorithm 1 Psuedocode for Schedule Generation Input: graph g, features f, unscheduled-Tasks u Output: schedule 1: schedule = [ ], i = 1 2: (ha1, ca1, ht1, ct1, hs1, cs1) ← Encoder(g, f) 3: while |u| ̸= 0 do 4: pai ← AgentSelector(hsi, hai) 5: ai ← Sampling(pai) 6: pti ← TaskSelector(hti, hsi, ai) 7: ti ← Sampling(pti−1) 8: schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='append(⟨ti, ai⟩) 9: unscheduledTasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='remove(ti) 10: if |unscheduledTasks| == 0 then 11: return schedule 12: end if 13: i ← i + 1 14: hsi, csi ← LSTMs((hti−1[ti], hai−1[ai]), hai−1, cai−1) 15: hai, cai ← LSTMa((hti−1[ti], hai−1[ai]), hai−1, cai−1) 16: end while presented in the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' πagent(aj|s) = softmaxi(fa([haj||hs])) (4) Next, the Schedule Propagator uses the Task Selector to assign the task for the selected agent based on the state, agent and unscheduled task embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' As shown in Equation 5, the Task Selector concatenates the state, selected agent and the unscheduled task embeddings and passes the combined information to a feedforward neural network, fτ, to calculate the likelihood of the task being assigned to the selected agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' After assigning to an agent for execution, the tasks are removed from the list of unscheduled tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Since the calculation of likelihood of each task is independent of each other up to the last softmax operation, the model is scalable and can be used for differentproblem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' πtask(τi|aj, s) = softmaxi(fτ([hτi||haj||hs])) (5) The key component of the Schedule Propagator is the use of LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' As shown in line 12 of Algorithm 1, after each task-agent pair selection, the state and agent embeddings are updated using the state LSTM and agent LSTM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The LSTM Cell stores the hidden and cell data from the previous step of the task allocation and predicts the next step based on the input using the Equation 6 [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' ft = σ(Wf[ht−1, xt] + bf) it = σ(Wi[ht, xt] + bi) ˜ct = tanh(Wc[ht−1, xt] + bc) ct = ftct−1 + it˜ct ot = σ(Wo[ht−1, xt] + bo ht = ottanh(ct) (6) Where the Encoder produces initial hidden state, h1 and initial cell state c1 as an output in the form of [h1, c1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' During testing, we utilize a batched sampling strategy for further performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Specifically, we generate multiple schedules for the same task allocation problem every round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' communicate Agent assignedTo State takeTime UseTime Task in temporalWe select the best performing schedule by computing the estimated makespan utilizing the Learning Curve Estimator and provide it to the Multi-Round Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' More sam- pling improves solution quality at increased computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Stochastic Policy Learning We train HybridNet in multi-round scheduling environ- ments using Policy Gradient methods that seek to directly optimize the model parameters based on rewards received from the environment [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Specifically, we compute the gradient of the model using the sum of the log likelihood of Agent and Task Selectors, as shown in Equation 7: ∇θJ(θ) = Eπ( T � t Aπθ t (st, ⟨τi, ai⟩) ∇θ(logπθ(τi|ai, st) + logπθ(ai|st)) (7) In Equation 7, the advantage term, At is estimated by sub- tracting a “baseline” from the total future reward calculated in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We calculate the “baseline” using the reward generated for the same task-allocation problem from multiple batches executed in multiple sequential rounds in the Multi- Round Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Each element of the batch solves the same scheduling problem and the environment is updated to account for the task-allocation of the previous round, updating the agent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The gradients were calculated from Equation 7 to updated the model weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Due to the combinatorial nature of the task scheduling problem, plus the stochasticity in human proficiency, learning a helpful value function as a baseline for computing the advantage term is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Instead, we investigate two more accessible and efficient alternatives: Step-based Baseline: During gradient estimation, the baseline value subtracted is set as the average return value across training episodes in the current batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Greedy Rollout Baseline: Greedy Rollout Baseline uses, πgreedy(A|S), a deterministic greedy version of the Hy- bridNet scheduler, to collect rewards in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Its weights, θgreedy, are updated periodically by copying the weights from the current learner, πθ(A|S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' EXPERIMENTAL RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Data Generation We generate scheduling problems with deadline and wait constraints under different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' For all scales, the deadline constraints are randomly generated for approximately 25% of the tasks from a range of [1, 5N] where N is the number of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Approximately 25% of the tasks have wait constraints, and the duration of non-zero wait constraints is sampled from U([1, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Task durations are clamped to 10 to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 1) Small Scale: The small data set has 9 to 11 tasks with 2 robots and 2 humans in a team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We generated 2000 Training Problems and 200 Test Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 2) Medium Scale: The medium data set has 18 to 22 tasks with 2 robots and 2 humans in a team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We generated 2000 Training Problems and 200 Test Problems to inspect the scalability of our trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 3) Large Scale: The large data set is defined as problems with 36 to 44 tasks chosen at random with 2 robots and 2 humans in a team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We have generated 200 Test Problems to evaluated the HybridNet performance with zero training problems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=', zero-shot transfer to from the smaller scale datasets to the Large Scale dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' To simulate the stochastic learning of human agents, for each Data Set noise is introduced to the Human Agent models by simulating the natural distribution of the c, k, β parameters of Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' This allows for each Data Set to simulate Deterministic and Stochastic Human Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The stochastic model is clipped to fall within the specified range of task durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Benchmarking We benchmark HybridNet against the following methods: EDF: A ubiquitous heuristic algorithm, earliest deadline first (EDF), that selects from a list of available tasks the one with the earliest deadline, assigning it to the first available agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Genetic Algorithm: An Evolutionary Optimization Al- gorithm that uses Post-Processing on the Schedule Generated by EDF [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Genetic algorithm creates new schedules based on the initial schedule through iterative randomized mutations by swapping task allocations and task orders [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Each generation selects the top performing schedules, sorted on feasibility and total schedule completion time, and used as the baseline for creating new mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The Genetic Algorithm was run for 10 generation with 90 baseline schedules, 10 task allocation and 10 task order swapping mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Furthermore, we evaluate the functionality of the Re- current Schedule Propagator by comparing it against the following HybridNet variant: HetGAT: We implement a HetGAT Scheduler based on the Encoder of HybridNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' After each task-agent pair assignment, instead of using the LSTM Cells to update the task, agent and state embeddings, it directly interacts with the environment to model the consequences of action with a new heterogeneous graph and re-computes those information from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We evaluate HybridNet on three metrics: 1) Proportion of problems solved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 2) Adjusted makespan: determined by the average of the makespan of feasible schedules and the maximum possible makespan of the infeasible schedules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' and 3) Runtime statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Runtime statistics for training and execution is compared for HybridNet and HetGAT Scheduler to model their computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Because HetGAT Scheduler relies on interactive scheduling through the envi- ronment after every task-agent pair allocation, we only train and evaluate it for Deterministic Human Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Model Details We implement HybridNet and HetGAT using PyTorch [35] and Deep Graph Library [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The HybridNet Encoder used in training/testing is constructed by stacking three multi-head TABLE I EVALUATION RESULTS: ADJUSTED MAKESPAN AND FEASIBILITY WITH DETERMINISTIC HUMAN TASK PROFICIENCY COMPARING BENCHMARKS WITH HYBRIDNET TRAINED ON SMALL AND MEDIUM SCALES, WITH SCHEDULES SAMPLED FROM SIZES 8 AND 16 Training Methods Small Medium Large Makespan Feasibility (%) Makespan Feasibility (%) Makespan Feasibility (%) EDF 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='31 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='00 1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='85 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='00 2535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='00 Genetic Algorithm 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='77 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='30 1180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='07 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='54 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='70 2542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='00 Step-based HetGAT 8 257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='18 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='08 751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='27 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='29 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='14 2123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='96 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='66 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='27 HetGAT 16 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='30 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='09 723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='94 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='11 2081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='65 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='45 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='16 Greedy HetGAT 8 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='09 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='10 784.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='52 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='17 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='25 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='16 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='14 HetGAT 16 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='23 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='15 765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='96 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='08 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='73 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='59 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='01 Step-based HybridNet Small 8 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='15 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='10 770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='48 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='07 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='35 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='80 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='33 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='39 HybridNet Small 16 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='49 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='10 746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='52 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='36 1953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='65 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='76 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='61 Greedy HybridNet Small 8 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='31 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='32 758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='96 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='27 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='43 2049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='32 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='73 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='45 HybridNet Small 16 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='45 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='20 723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='35 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='70 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='49 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='15 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='91 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='40 Step-based HybridNet Medium 8 722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='61 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='29 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='86 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='97 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='45 HybridNet Medium 16 697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='40 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='04 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='51 1944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='72 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='10 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='49 Greedy HybridNet Medium 8 692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='01 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='69 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='66 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='78 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='08 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='87 HybridNet Medium 16 659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='89 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='45 1936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='97 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='68 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='74 TABLE II EVALUATION RESULTS: ADJUSTED MAKESPAN AND FEASIBILITY WITH STOCHASTIC HUMAN TASK PROFICIENCY Methods Small Medium Large Makespan Feasibility (%) Makespan Feasibility (%) Makespan Feasibility (%) EDF 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='81± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='17 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='65 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='21 1071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='02± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='65 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='30 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='12 2524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='92± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='23 Genetic Algorithm 283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='79 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='39 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='45 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='05 1149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='42 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='14 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='55 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='31 2541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='20 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='15 HybridNet Small 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='96 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='52 881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='16 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='89 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='89 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='09 2141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='80 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='12 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='96 HybridNet Medium 859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='99 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='82 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='94 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='32 2174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='57 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='53 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='94 TABLE III EVALUATION RESULTS: RUNTIME PERFORMANCE ON SINGLE PROBLEM Methods HetGAT8 HybridNet8 HybridNet16 Training Time (s) Small 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='52 ± 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='00 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='91 Medium 354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='77 ± 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='31 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='40 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='52 Evaluation Time (s) Small 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='91 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='85 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='99 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='95 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='53 Medium 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='12 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='67 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='77 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='42 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='30 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='55 Large 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='76 ± 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='32 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='84 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='38 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='78 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='52 HetGAT layers (the first two use concatenation, and the last one uses averaging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The feature dimension of hidden layers = 64, and the number of heads = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The Recurrent Propagator utilizes a LSTMCell of size 32 followed by a fully-connected layer and a softmax layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We set γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='99, batch size = 8 and used Adam optimizer [37] with a learning rate of 2 × 10−3, and a weight decay of 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We employed a learning rate decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='5 every 4000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We evaluate the models using a batch size of 8 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' For the Multi-Round Environment, the infeasible reward coefficient Ci = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='0 and total round number = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Both training and evaluation were conducted on a Quadro RTX 8000 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Evaluation Results Table I shows the evaluation performance with Deter- ministic Human Proficiency in different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The Deter- ministic Human Proficiency means that during training and evaluation, human learning curve is known and execution is deterministic for every agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' In Table I, “Small” and “Medium” term after model name denotes the data scale the model was trained on and the number following it denotes the batch size for schedule sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The results show that HybridNet outperforms both EDF and Genetic Algorithm in adjusted makespan and percentage of feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' HybridNet trained on Small scale problems generalizes for both Medium and Large scale problems with similar or slightly worse performance than HybridNet trained on Medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' HybridNet and HetGAT performs similarly on all scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' This shows that HybridNet is capable of learning high performance policies by leveraging the Recurrent Schedule Propagator and without requiring interaction with the Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We provide the runtimes of training and evaluation for HetGAT and HybridNET in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' HybridNet is approx- imately 10 times faster in training compared to HetGAT Model and at least 2 times faster during evaluation for same batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' EDF and Genetic Algorithm were evalu- ated through the CPU without GPU acceleration, making it infeasible to accurately compare the performance of the Deep Learning Models to the Traditional Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We show that for HybridNet, step-based training has better performance over the greedy baseline, while for HetGAT model, greedy baseline training is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We also observed that greedy baseline training reached convergence faster than step-based training (4500 epochs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 19000 epochs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Further investigation is worthwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Table II shows the evaluation performance with Stochas- tic Human Proficiency in different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The Stochastic Human Proficiency is presented as randomness in both the actual human execution within Multi-Round Environment and uncertainty within the Learning Curve Estimator used for schedule generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The results show that HybridNet outperforms the EDF and Genetic Algorithm across different data scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The largest performance gap was observed on large dataset (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='51% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='15%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Here, HetGAT model is not included as it requires interaction with the environment after every task-agent assignment to observe the outcome, which is not available until the whole schedule is generated and sent to the Stochastic Environment for execution to emulate real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' CONCLUSIONS We present a deep learning-based framework, called HybridNet, combining a heterogeneous graph-based en- coder with a recurrent schedule propagator, for scheduling stochastic human-robot teams under temporal constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' The resulting policy network provides a computationally lightweight yet highly expressive model that is end-to-end trainable via reinforcement learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' We devel- oped a multi-round task scheduling environment for stochas- tic human-robot teams and conducted extensive experiments, showing that HybridNet outperforms other human-robot scheduling solutions across problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Future research includes integrating the learning-based human estimator into HybridNet, transfer learning across optimizing different ob- jective functions, and deploying the trained network in a real- world scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' REFERENCES [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Yan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Jouandeau, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Cherif, “A survey and analysis of multi-robot coordination,” International Journal of Advanced Robotic Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 399, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='5772/57313 [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Heyer, “Human-robot interaction and future industrial robotics ap- plications,” in 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' IEEE, 2010, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 4749–4754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Nunes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Manner, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Mitiche, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gini, “A taxonomy for task allocation problems with temporal and ordering constraints,” Robotics and Autonomous Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 90, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 55–70, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Natarajan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gombolay, “Human-robot team coordi- nation with dynamic and latent human task proficiencies: Scheduling with learning curves,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gombolay, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gutierrez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Clarke, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Sturla, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Shah, “Decision-making authority, team efficiency and human worker satisfaction in mixed human–robot teams,” Autonomous Robots, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 39, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 293–312, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Nunes and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gini, “Multi-robot auctions for allocation of tasks with temporal constraints,” in Twenty-Ninth AAAI Conference on Artificial Intelligence, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 2110–2116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gombolay, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Wilcox, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Shah, “Fast scheduling of robot teams performing tasks with temporospatial constraints,” IEEE Transactions on Robotics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 220–239, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Solomon, “On the worst-case performance of some heuristics for the vehicle routing and scheduling problem with time window constraints,” Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 161–174, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [9] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Bengio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Lodi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Prouvost, “Machine learning for com- binatorial optimization: a methodological tour d’horizon,” European Journal of Operational Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 290, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 405–421, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' LeCun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Bengio, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Hinton, “Deep learning,” nature, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 521, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 7553, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 436–444, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Khalil, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Dai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Dilkina, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Song, “Learning combinatorial optimization algorithms over graphs,” in Advances in Neural Information Processing Systems, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 6348–6358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [12] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Kool, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' van Hoof, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Welling, “Attention, learn to solve routing problems!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' in International Conference on Learning Repre- sentations, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [13] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Ma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Ferber, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Huo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Chen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Katz, “Online planner selection with graph neural networks and adaptive scheduling,” Proceedings of the AAAI Conference on Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 04, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 5077–5084, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Available: https://ojs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='org/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='php/AAAI/article/view/5949 [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Dechter, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Meiri, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Pearl, “Temporal constraint networks,” Artificial intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 49, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 1-3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 61–95, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [15] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Nunes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Manner, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Mitiche, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gini, “A taxonomy for task allocation problems with temporal and ordering constraints,” Robotics and Autonomous Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 90, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 55–70, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 2017, publisher Copyright: © 2016 Elsevier B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [16] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Wang and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gombolay, “Learning scheduling policies for multi- robot coordination with graph attention networks,” IEEE Robotics and Automation Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 4509–4516, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Benders, “Partitioning procedures for solving mixed-variables programming problems,” Numerische Mathematik, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 238–252, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='1007/ bf01386316 [18] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Ren and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Tang, “An improved hybrid milp/cp algorithm frame- work for the job-shop scheduling,” 2009 IEEE International Confer- ence on Automation and Logistics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 890–894, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [19] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Castro and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Petrovic, “Combined mathematical programming and heuristics for a radiotherapy pre-treatment scheduling problem,” Journal of Scheduling, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 333–346, May 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='1007/s10951-011-0239-8 [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Chen and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Askin, “Project selection, scheduling and resource allo- cation with time dependent returns,” European Journal of Operational Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 193, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 23–34, 02 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Zhang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Jia, “Real-time adaptive assembly scheduling in human-multi-robot collaboration according to human capability*,” 2020 IEEE International Conference on Robotics and Automation (ICRA), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 3860–3866, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gombolay, “Apprenticeship scheduling for human-robot teams,” in Proceedings of the Thirtieth AAAI Conference on Artificial Intelli- gence, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' AAAI’16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' AAAI Press, 2016, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 2497–2498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [23] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Liu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gombolay, “Heterogeneous graph attention networks for scalable multi-robot scheduling with temporospatial constraints,” Autonomous Robots, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 46, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 249–268, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [24] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Scarselli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gori, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Tsoi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Hagenbuchner, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Monfardini, “The graph neural network model,” Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Neur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 61–80, jan 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='1109/TNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='2005605 [25] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Hu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Leskovec, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Jegelka, “How powerful are graph neural networks?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' in International Conference on Learning Representations, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Singh, “An algorithm to reduce the time complexity of earliest deadline first scheduling algorithm in real-time system,” 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Sundermeyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Schl¨uter, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Ney, “Lstm neural networks for language modeling,” in Thirteenth annual conference of the interna- tional speech communication association, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Graves, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Fern´andez, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Schmidhuber, “Bidirectional lstm networks for improved phoneme classification and recognition,” in International conference on artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Springer, 2005, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 799–804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Zhou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Cui, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Hu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Li, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Sun, “Graph neural networks: A review of methods and applications,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Sherstinsky, “Fundamentals of recurrent neural network (RNN) and long short-term memory (LSTM) network,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' abs/1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='03314, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Available: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='org/abs/1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 03314 [31] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Malhotra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Ramakrishnan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Anand, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Vig, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Agarwal, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Shroff, “Lstm-based encoder-decoder for multi-sensor anomaly detection,” arXiv preprint arXiv:1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content='00148, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Ycart, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Benetos, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=', “A study on lstm networks for polyphonic music sequence modelling.” ISMIR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [33] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Si, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Hu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Zhang, “A review of recurrent neural networks: Lstm cells and network architectures,” Neural Computation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 1235–1270, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [34] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Sutton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Singh, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' McAllester, “Comparing policy-gradient algorithms,” IEEE Transactions on Systems, Man, and Cybernetics, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Paszke, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gross, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Massa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Lerer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Bradbury, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Chanan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Killeen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Lin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gimelshein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Antiga, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Desmaison, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Kopf, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' DeVito, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Raison, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Tejani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Chilamkurthy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Steiner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Fang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Bai, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Chintala, “Pytorch: An imperative style, high- performance deep learning library,” in Advances in Neural Information Processing Systems, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Larochelle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Beygelzimer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=" d'Alch´e-Buc, E." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Fox, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Garnett, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Zheng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Ye, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Song, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Zhou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Ma, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Gai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Xiao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' He, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Karypis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Li, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' Zhang, “Deep graph library: A graph-centric, highly-performant package for graph neural networks,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFQT4oBgHgl3EQfPjZG/content/2301.13279v1.pdf'} +page_content=' P.' metadata={'source': 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+sw1@wellesley.edu +Siman Shen +Grinnell College +Grinnell, IA, USA +shenlisa@grinnell.edu +Yi-Hao Peng +HCI Institute, Carnegie Mellon +University +Pittsburgh, PA, USA +yihaop@cs.cmu.edu +Jeffrey Nichols +Snooty Bird LLC +USA +jwnichls@gmail.com +Jeffrey P. Bigham +HCI Institute, Carnegie Mellon +University +Pittsburgh, PA, USA +jbigham@cs.cmu.edu +ABSTRACT +Modeling user interfaces (UIs) from visual information allows sys- +tems to make inferences about the functionality and semantics +needed to support use cases in accessibility, app automation, and +testing. Current datasets for training machine learning models are +limited in size due to the costly and time-consuming process of +manually collecting and annotating UIs. We crawled the web to +construct WebUI, a large dataset of 400,000 rendered web pages +associated with automatically extracted metadata. We analyze the +composition of WebUI and show that while automatically extracted +data is noisy, most examples meet basic criteria for visual UI mod- +eling. We applied several strategies for incorporating semantics +found in web pages to increase the performance of visual UI un- +derstanding models in the mobile domain, where less labeled data +is available: (i) element detection, (ii) screen classification and (iii) +screen similarity. +KEYWORDS +Dataset; UI Modeling; Computer Vision; Transfer Learning; Web +Semantics; Computational Interaction +ACM Reference Format: +Jason Wu, Siyan Wang, Siman Shen, Yi-Hao Peng, Jeffrey Nichols, and Jeffrey +P. Bigham. 2023. WebUI: A Dataset for Enhancing Visual UI Understanding +with Web Semantics. In Proceedings of the 2023 CHI Conference on Human Fac- +tors in Computing Systems (CHI ’23), April 23–28, 2023, Hamburg, Germany. +ACM, New York, NY, USA, 14 pages. https://doi.org/10.1145/3544548.3581158 +1 +INTRODUCTION +Computational modeling of user interfaces (UIs) allows us to under- +stand design decisions [15, 28], improve their accessibility [55], and +automate their usage [7, 31, 32]. Often, these systems must interact +with UIs in environments with incomplete or missing metadata (e.g., +Permission to make digital or hard copies of part or all of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for third-party components of this work must be honored. +For all other uses, contact the owner/author(s). +CHI ’23, April 23–28, 2023, Hamburg, Germany +© 2023 Copyright held by the owner/author(s). +ACM ISBN 978-1-4503-9421-5/23/04. +https://doi.org/10.1145/3544548.3581158 +mobile apps authored with inaccessible UI toolkits). This presents +many challenges since it necessitates that they reliably identify and +reason about the functionality of the UI to support downstream +applications. Visual modeling of UIs, which has shown to be a +promising solution, predicts information directly from a screen- +shot using machine learning models and introduces no additional +dependencies. +Building the datasets needed to train accurate visual models +involves collecting a large number of screenshots paired with their +underlying semantic or structural representations. Recent efforts to +collect datasets [15, 55] for data-driven modeling have focused on +mobile apps, which are typically manually crawled and annotated +by crowdworkers since they are often difficult to automate. This +process is both time-consuming and expensive — prior work has +estimated that collecting a dataset of 72,000 app screens from 10,000 +apps took 5 months and cost $20,000 [15]. Because of this, datasets +for visual UI modeling are limited in size and can be prohibitively +expensive to keep updated. +The web presents a possible solution to UI data scarcity since +web pages are a promising source of data to bootstrap and enhance +visual UI understanding. In contrast to mobile UIs, web UIs (i.e., web +pages) are much easier to crawl since they are authored in a unified +parsable language (i.e., HTML) that typically exposes semantics +(e.g., links and listeners) necessary for automated navigation. The +same web page can also be viewed in many different viewports +and display settings, which makes it possible to collect a large +dataset of UIs rendered on a variety of devices (e.g., a smartphone or +tablet). In addition, web browsers offer several facilities to extract +visual, semantic, and stylistic information programmatically. In +particular, web conventions, such as the semantic HTML and the +ARIA initiatives, while not always adopted, constitute a large, if +potentially noisy, source of annotations for UI elements. Finally, +the web offers a virtually unlimited supply of data and has already +been employed as a data source for large-scale machine learning +[23, 52, 53]. We explore the possibility of automatically collecting +and labeling a large dataset of web UIs to support visual UI modeling +in other domains (e.g., mobile). Compared to previous web datasets +[28], our dataset is much larger, more recent, and contains semantic +information needed to support common visual UI understanding +tasks. +arXiv:2301.13280v1 [cs.HC] 30 Jan 2023 + +CHI ’23, April 23–28, 2023, Hamburg, Germany +Wu et al. +In this paper, we show that a large dataset of automatically +collected web pages can improve the performance of visual UI +Understanding models through transfer learning techniques, and +we verify this phenomenon for three tasks. We first describe the +platform that we built to crawl websites automatically and scrape +relevant visual, semantic, and style data. Our crawler visited a +total of approximately 400,000 web pages using different simulated +devices. WebUI, the resulting dataset is an order of magnitude larger +than other publicly available datasets [28]. Next, we analyzed our +dataset’s composition and estimated data quality using several +automated metrics: (i) element size, (ii) element occlusion, and +(iii) layout responsiveness. We found that most websites met basic +criteria for visual UI modeling. Finally, we propose a framework +for incorporating web semantics to enhance the performance of +existing visual UI understanding approaches. We apply it to three +tasks in the literature: (i) element detection, (ii) screen classification +and (iii) video screen similarity and show that incorporating web +data improves performance in other target domains, even when +labels are unavailable. +To summarize, our paper makes the following contributions: +(1) The WebUI dataset, which consists of 400,000 web pages +each accessed with multiple simulated devices. We collected +WebUI using automated web crawling and automatically +associated web pages with visual, semantic, and stylistic +information that can generalize to UIs of other platforms. +(2) An analyis of the composition and quality of examples in +WebUI for visual UI modeling in terms of (i) element size, (ii) +element occlusion, and (iii) website layout responsiveness. +(3) A demonstration of the usefulness of the WebUI dataset +through three applications from the literature: (i) element +detection, (ii) screen classification and (iii) screen similarity. +We show that incorporating web data can lead to perfor- +mance improvements when used in a transfer learning set- +ting, and we verified its improvement for our three tasks. We +envision that similar approaches can be used for other tasks +common in visual UI understanding. Furthermore, we show +that models trained on only web data can often be directly +applied to other domains (e.g., Android app screens). +All code, models, and data will be released to the public to encourage +further research in this area. +2 +RELATED WORK +2.1 +Datasets for UI Modeling +There have been several datasets collected to support UI modeling, +mostly in the mobile domain. Several datasets have been collected to +support training specialized models [26, 40, 44] . The AMP dataset +consists of 77k screens from 4,068 iOS apps and was originally used +to train Screen Recognition, an enhanced screen reader [55], but +has also been extended with additional pairwise annotations to +support automated crawling applications [20]. +The largest publicly available dataset Rico, which consists of +72K app screens from 9.7K Android apps, was collected using a +combination of automated and human crawling [15]. It captures +aspects of user interfaces that are static (e.g., app screenshots) and +dynamic (e.g., animations and user interaction traces). Rico has +served as the primary source of data for much UI understanding +research and it has been extended and re-labeled to support many +downstream applications, such as natural language interaction [7, +32, 49] and UI retrieval for design [6, 15]. +Nevertheless, Rico has several weaknesses [14]. Several works +have identified labeling errors and noise (e.g., nodes in the view +hierarchy do not match up with the screenshot). To this end, efforts +have been made to repair and filter examples. Enrico first randomly +sampled 10,000 examples from Rico then cleaned and provided +additional annotations for 1460 of them [29]. The VINS dataset [6] is +a dataset for UI element detection that was created by collecting and +manually taking screenshots from several sources, including Rico. +The Clay dataset (60K app screens) was generated by denoising +Rico through a pipeline of automated machine learning models and +human annotators to provide element labels [30]. Rico and other +manually annotated datasets are expensive to create and update, and +thus, models trained on them may exhibit degraded performance +on newer design guidelines (e.g., Material Design is an updated +design look for Android). For example, Rico was collected in early +2017 and has yet to see any update. Finally, many of these datasets +focus on one particular platform (e.g., mobile phone) and therefore +may learn visual patterns specific to the screen dimensions. For +example, “hamburger menus” are usually used in mobile apps while +desktop apps may use navigation bars. +In our work, we scrape the web for examples of UIs, which +addresses some drawbacks (high cost, difficult to update, device- +dependent) of current datasets but not others (dataset noise). The +closest to our work is Webzeitgeist [28], which also used automated +crawling to mine the design of web pages. To support design mining +and machine learning applications, Webzeitgeist crawled 103,744 +webpages and associated web elements with extracted properties +such as HTML tag, size, font, and color. This work is primarily used +for data-driven design applications and does not attempt to transfer +semantics to other domains. We also collect multiple views of each +website and query the browser for accessibility metadata, which +can further facilitate UI modeling applications. +2.2 +Applications of UI Datasets +Applications that operate and improve existing UIs must reliably +identify their composition and functionality. Originally, many relied +on pixel-based or heuristic matching [1, 18, 43, 54]. The introduc- +tion of large UI datasets, such as those previously discussed, have +provided the opportunity to learn more robust computational mod- +els, especially those from visual data. The goal of this paper is to +improve the performance of these computational models by lever- +aging a large body of web data and its associated semantics. There +have been many efforts to learn the semantics of UIs [37, 49, 50]. In +this paper, we focus on three modeling tasks at the (i) element (ele- +ment detection), (ii) screen (screen classification), and (iii) app-level +(screen similarity). +Element detection identifies the location and type of UI widgets +from a screenshot and has applications in accessibility metadata +repair [55], design search [6], and software testing [12, 51]. Labeled +datasets for element detection exist [6, 15, 30, 55]; however they +are quite small compared to other datasets for object detection [36] +which contain an order of magnitude more examples (330K). We +found that incorporating our web UI dataset (400K examples) in a + +WebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics +CHI ’23, April 23–28, 2023, Hamburg, Germany +pre-training phase led to performance benefits. Other work involves +modeling UIs at a higher level (e.g., screen-level) to reason about the +design categorization [29] and purpose [49] of a screen. Similarly, +datasets with screen-level annotations of UIs are much smaller than +others used in the CV literature [17] so we used additional web +data to improve accuracy. Finally, we investigated screen similarity, +a task that reasons about multiple UI inputs (e.g., frames of a video +recording), where no publicly available labeled data exists. We +found that models trained on related web semantics (e.g., URL +similarity) were able to successfully generalize to mobile screens. +In summary, our paper shows that applying examples from the +web and relevant machine learning techniques can improve the +performance of computational models that depend on UI data. +2.3 +Related Machine Learning Approaches +We briefly introduce and summarize three machine learning ap- +proaches that we apply in our paper. Broadly, they fall under a body +of research known as “transfer learning” which uses knowledge +from learning one task (e.g., web pages) to improve performance +on another (e.g., mobile app screens). +Inductive transfer learning is a technique that improves model +performance by first “pre-training” a model on a related task, typi- +cally where a lot of data is available [42]. Once the model converges +on the first task, its weights are used as a starting point when train- +ing on the target task. Labeled data is required for both the source +and target domains, although it is possible that there are fewer +target examples. +In some cases, labeled data are missing for either the source or +target domains. If source labels are unavailable, semi-supervised +learning (SSL) can be applied to take advantage of unlabeled data to +improve performance [9]. For example, WebUI doesn’t contain any +labels for screen type (e.g., login screen, register screen), but we’d +like to use it to improve prediction accuracy on a small number of +annotated Android app screens. In our work, we apply a form of +SSL known as “self-learning” [9], where a UI classification model it- +eratively improves its performance by generating pseudo-labels for +an unlabeled dataset, then re-training itself using high-confidence +samples. +Finally, to support use-cases where target labels are unavailable, +we apply unsupervised domain adaptation (UDA) [22]. In many +cases, visual UI models trained on web data can be directly used +on any screenshot (including Android and iOS apps), and UDA +improves the performance and robustness of models to domain +changes. This type of knowledge transfer is particularly interesting +because it enables us to explore the feasibility of new UI under- +standing tasks (without manually annotating a large number of +examples) and bring some benefits of web semantics (e.g., semantic +HTML) to other platforms. +3 +WEBUI DATASET +We introduce the WebUI dataset, which we construct and release +to support UI modeling. The WebUI dataset is composed of 400,000 +web pages automatically crawled from the web. We stored screen- +shots and corresponding metadata from the browser engine, which +serve as annotations of UI element semantics. Because the collec- +tion process is highly automated, our final dataset is an order of +Database +Crawling +Coordinator +Crawler +Web +workers +assign URLs +to worker +send back +crawled URLs +Request and +collect data +Figure 1: Overview of our crawling architecture. A crawl- +ing coordinator contains a queue of URLs to crawl and as- +signs them to workers in a crawler pool. Workers asyn- +chronously process URLs by visiting them in a automated +browser, scraping relevant metadata, then uploading them +to a cloud database. +magnitude larger than other publicly available ones (Figure 4) and +can be more easily updated over time. +In this section, we give an overview of our web crawling architec- +ture, analyze the composition of our dataset, and provide evidence +that it can support visual UI modeling for other platforms. +3.1 +Web UI Crawler +3.1.1 +Crawling Architecture. To collect our dataset, we implemented +a parallelizable cloud-based web crawler. Our crawler consists of +(i) a crawling coordinator server that keeps track of visited and +queued URLs, (ii) a pool of crawler workers that scrapes URLs using +a headless browser, and (iii) a database service that stores uploaded +artifacts from the workers. The crawler worker is implemented +using a headless framework [3] for interfacing with the Chrome +browser. Each crawler worker repeatedly requests a URL from the +coordinator server, which keeps global data structures for visited +and upcoming URLs. The crawler worker includes some simple +heuristics to automatically dismiss certain types of popups (e.g., +GDPR cookie warnings) to help it access page content. +We seeded our coordinator using a list of websites that we hy- +pothesized would lead to diverse examples of web pages (e.g., link +aggregation websites and design blogs) and ones that we expected to +have high-quality accessibility metadata (e.g., government websites). +A full list of our seed websites can be found in the supplementary +materials. +We explored several crawling policies and eventually settled on +one that encourages diverse exploration by inversely weighting the +probability of visiting a URL by its similarity to the visited set. For +example, if the crawler previously visited http://example.com/user/ +alpha, it would be less likely to subsequently visit http://example. +com/user/beta. We set a minimum probability so that it is possible to +re-visit links to support additional types of analysis (e.g., temporal +changes). The coordinator organizes upcoming (i.e., queued) URLs +by their hostname, (i) selects a hostname randomly with uniform +probability, and then (ii) selects a URL using its assigned probability. +Empirically, we found this technique to be effective at avoiding + +CHI ’23, April 23–28, 2023, Hamburg, Germany +Wu et al. +1280x720 +1366x768 +1536x864 +1920x1080 +iPhone +iPad +Figure 2: Screenshots from a web page accessed using 6 dif- +ferent devices: 4 desktop resolutions, a smartphone, and a +tablet. By requesting a responsive web page at different reso- +lutions, we induce several layout variations (e.g., navigation +and hero button). +crawler traps, which are websites that cause automated crawlers to +get stuck in endless loops navigating within the same site. +3.1.2 +Data Collected from a Web Page. We used a pool of crawler +workers to crawl web pages in parallel, and we visited each URL +with multiple simulated devices. We collected several types of se- +mantic information by querying the rendering and accessibility +engine. We set a timeout limit of 6 minutes for each URL, so some +web pages were not visited by all simulated devices. +Simulated Devices. We sampled each web page with 6 sim- +ulated devices: 4 of the most common desktop resolutions [4], a +tablet, and a mobile phone. Devices are simulated by setting the +browser window resolution and user agent to match the goal device, +both of which may affect the page’s content and rendering. +Screenshots. Our crawler worker captured two types of screen- +shots (i.e., visual data) from websites. We captured a viewport +screenshot, with fixed image dimensions, and a full-page screenshot, +with variable height. Images were saved using lossy compression +to save storage. While compression can introduce some artifacts, +previous work [19] suggests that the effect on deep learning model +performance is minimal. +Accessibility Tree. We used a browser automation library to +query Chrome’s developer tools to retrieve an accessibility tree +for each page [2]. The accessibility tree is a tree-based represen- +tation of a web page that is shown to assistive technology, such +as screen readers. The tree contains accessibility objects, which +usually correspond to UI elements and can be queried for properties +(e.g., clickability, headings). +Compared to the DOM tree, the accessibility tree is simplified by +removing redundant nodes (e.g.,
tags that are only used for +styling) and automatically populated with semantic information +via associated ARIA attributes or inferred from the node’s contents. +The browser generates the accessibility tree using a combination of +HTML tags, ARIA attributes, and event listeners (e.g., click handlers) +to create a more consistent semantic representation of the UI. For +instance, there are multiple ways to create a button (e.g., a styled +div) and the accessibility tree is intended to unify all of these to a +single button tag. +Layout and Computed Style. For each element in the accessi- +bility tree, we stored layout information from the rendering engine. +Specifically, we retrieved 4 bounding boxes relevant to the “box +model”: (i) the content bounding box, (ii) the padding bounding +# of elements (in thousands) +0 +25000 +50000 +75000 +100000 +125000 +text +link +list item +image +heading +paragraph +line break +generic +grid cell +button +Frequency of Common Element Types +Figure 3: 10 most common element types in the WebUI +dataset. Element types are based on automatically computed +roles, which are not mutually exclusive. Text is the most +common type, but many types offer semantic information +about what text is used for e.g, a heading, paragraph or link. +# of UIs +0 +100,000 +200,000 +300,000 +400,000 +500,000 +Enrico +VINS +Clay +Rico +Screen +Recognition +Webzeitgeist +WebUI +UI Dataset Size +Figure 4: Comparison of WebUI to existing UI datasets. We- +bUI contains nearly 400,000 web pages and is nearly one or- +der of magnitude larger than existing datasets available for +download (Enrico, VINS, Clay, Rico). Each web page also con- +tains multiple screenshots captured using 6 simulated de- +vices. +box, (iii) the border bounding box, and (iv) the margin bounding +box. Each element was also associated with its computed style in- +formation, which included font size, background color and other +CSS properties. +3.2 +Dataset Composition +The WebUI dataset contains 400K web UIs captured over a period +of 3 months and cost about $500 to crawl. We grouped web pages +together by their domain name, then generated training (70%), +validation (10%), and testing (20%) splits. This ensured that similar +pages from the same website must appear in the same split. We +created four versions of the training dataset. Three of these splits +were generated by randomly sampling a subset of the training split: +Web-7k, Web-70k, Web-350k. We chose 70k as a baseline size, since +it is approximately the size of existing UI datasets [15, 55]. We +also generated an additional split (Web-7k-Resampled) to provide a +small, higher quality split for experimentation. Web-7k-Resampled +was generated using a class-balancing sampling technique, and +we removed screens with possible visual defects (e.g., very small, +occluded, or invisible elements). More information about how this +set was generated can be found in the appendix. The validation and +test split was always kept the same. +3.2.1 +Comparison to Existing Datasets. WebUI is an order of magni- +tude larger than existing datasets used for UI understanding (Figure +4) and provides rich semantic and style information not found in +mobile datasets. WebUI focuses on the static properties of web +pages and does not store page loading times or element animations. + +业Libera.Chat +About +Contribute +ChannelNamespaces +Guides +FAQ +Connect +Providinga communityplatform forFreeand open- +source software and peer directed projects. +Connectby pointingyour IRC clientto +irc.libera.chat:6697 (TLS) +Choosing an IRC client +Libera.Chat +Channel Namespaces +Happy Birthday, Libera Chat! +19thMay2022byLiberastaff +Helloeveryone,todaywe celebratethe anniversary of Libera.Chat going public!业Libera.Chat +About +Contribute +Channel Namespaces +Guides +FAQ +Connect +Providing a community platform for free and open- +source software and peer directed projects. +Connect by pointing your IRC client to +irc.libera.chat:6697 (TLS) +Choosing anIRC client +Libera.Chat +Channel Namespaces +Happy Birthday, Libera Chat! +19thMay 2022 by Libera staff +Hello everyone, today we celebrate the anniversary of Libera.Chat going public! +Wherewearecomingfrom +Exactly one year ago Libera.Chat was unveiled as a real time communication and +collaboration servicefor freeandopen-sourcesoftware,peer-directed projects, +openly licensed content and collaboration.Starting from scratch wemanaged, just +within a fewmonths, tobecome the largest IRC network.业Libera.Chat +About +Contribute +Channel Namespaces +Guides +FAQ +Connect +Providinga communityplatform forFreeandopen- +source softwareandpeer directedprojects. +Connect bypointingyour IRC clientto +irc.libera.chat:6697 (TLS) +Choosing an IRC client +Libera.Chat +Channel Namespaces +Happy Birthday, Libera Chat! +19thMay2022by Libera staff +Hello everyone, today we celebrate the anniversary of Libera.Chat going public! +Wherewearecomingfrom业Libera.Chat +About +Contribute +Channel Namespaces +Guides +FAQ +Connect +Providing a community platform For Free +and open +source software and peer directed projects. +Connect by pointing your IRC client to +irc.libera.chat:6697 (TLS) +Choosing an IRC client +Libera.Chat +Channel Namespaces +Happy Birthday, Libera Chat! +19th May 2022 by Libera staff +Hello everyone, today we celebrate the anniversary of Libera.Chat going public! +Wherewearecomingfrom +Exactly one year ago Libera.Chat was unveiled as a real time communication and +collaboration service for free and open-source software, peer-directed projects, +openly licensed content and collaboration. Starting from scratch we managed, just +within a fewmonths, to become the largestIRC network. +Starting from scratch, we managed to gain around 5o o00 users in just a month and a +half, a number which has been mostly steady since. with regard to channels we had +roughly 15 00o channels formed within half a month, compared to the usercount this +number is still growing, but the curve Flattened itself a bit. You can see detailed +graphs over at https://netsplit.de/networks/statistics.php?net=Libera.Chat +We also saw many communities and projects migrating over to Libera from other +places in the first few days, counting 250 in just one week and 500 after a monthWLibera.Chat +Navigation +Libera.Chat +Providing a community platform For Free +and open-source software and peer +directed projects. +Connect by pointing your IRC client to +irc.libera.chat:6697 (TLS) +ChoosinganIRC client +Channel Namespaces +Happy Birthday, +Libera Chat! +19th May 2022 by Libera staffLibera.Chat +About +Contribute +Channel Namespaces +Guides +FAQ +Connect +Providing a community platform for Free and open- +source software and peer directed projects. +Connect by pointing your IRC client to +irc.libera.chat:6697 (TLS) +Choosing an IRC client +Libera.Chat +Channel Namespaces +Happy Birthday, Libera Chat! +19th May 2022 by Libera staff +Hello everyone, today we celebrate the anniversary of Libera.Chat going public! +Where we are coming from +Exactly one year ago Libera.Chat was unveiled as a real time communication and +collaboration service for Free and open-source software, peer-directed projects, +openly licensed content and collaboration. Starting From scratch we managed, just +withinafewmonths,tobecomethelargestIRCnetwork. +Starting From scratch,we managed to gain around 5oooo users in just amonth and a +half, a number which has been mostly steady since. With regard to channels we had +roughly 15ooo channels formed within half a month, compared to the usercount this +number is still growing,but the curve flattened itself a bit.You can see detailed +graphs over at https://netsplit.de/networks/statistics.php?net=Libera.Chat +We also saw many communities and projects migrating overto Libera from other +places in the first fewdays, counting 25o in just one week and5o0 aftera month. +Today we are hosting roughly 95o projects and communities, and that number is still +growing. We are hoping to reach the 1oooth registration soon! +All these communites are quitediverse.Libera.Chat services arenot onlyused by +major Free/open source operating systems and well known,world wide operating +institutions such as the Wikimedia Foundation; we also have local Linux User Groups, +the hackspace around the corner hacking on whimsical gadgets and liberating your +hardware or someones scratch-your-own-itch image viewer that call Libera.Chat their +home.WebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics +CHI ’23, April 23–28, 2023, Hamburg, Germany +We analyzed the makeup of web UIs and compared them to +mobile UIs. The distribution of UI types (e.g. Login, News, Search) +in WebUI are also likely to be different than mobile data, since +many web pages are primarily hypertext documents. We extracted +elements from the accessibility tree and categorized them using +their computed accessibility role and the role of any singleton +parents. For example, a clickable image is created in HTML by +surrounding an image () element with an anchor element +(). Thus, it is possible for elements to be assigned to multiple +classes. Figure 3 shows the frequency of element types in our dataset. +Similar to prior work [55], we find that text is the most common +element in our dataset. However, we find limited overlap between +the rest of the label set, possibly due to the nature of web data and +the mutually exclusive nature of existing label sets. On average, +there were 60 elements on a web UI, 30 of which were visible in the +viewport. This is more than the number of elements on mobile app +screens, which prior work estimated to be around 25 per screen, +although this may in part be due to differences in segmentation +(e.g., a single Rich Text Field on Android can contain differently +formatted text while on HTML they would broken up into different +tags). On average, there were also more clickable elements per web +page (20 on web pages vs 15 “interactable" elements on Android +apps), likely due to the prevalence of hyperlinks on the web. +3.2.2 +Dataset Quality. Compared to manually labeled examples, +automatically extracted annotations can contain errors that impact +modeling performance. We conducted an analysis on a small, ran- +domly sampled data from our dataset (1000 web pages). While there +are numerous possible defects, we focus on three that we believe are +most relevant to data quality: (i) element size, (ii) element occlusion, +and (iii) website responsiveness. Our analysis is primarily focused +on quantifying possible defects but not reparing them. Previous +work [30, 44] has explored automated methods for correcting mis- +matched labels and occluded elements, and we expect the overall +quality of WebUI could be improved if these were applied.. +Element Size. Element size refers to the dimensions of an anno- +tated object in an image. For example, if a bounding box annotation +surrounds an object that is too small relative to the image resolution, +it may be difficult for a model to identify the object. The average +area of bounding boxes in our data is approximately 14000𝑝𝑥2, but +this may have been influenced by short segments of text. The Web +Content Accessibility Guidelines (WCAG) guideline for target size +also recommends that interactable elements have a minimum size +of 44 by 44 pixels, so that they can be easily selected by users. In our +dataset, one third of interactable elements (e.g., elements tagged as +links or button) were smaller than this threshold. +Element Occlusion. Element occlusion occurs when one object +partially or completely covers another in a screenshot. Occluded el- +ements are detrimental to visual modeling since they may represent +targets that can be impossible to predict correctly. We quantified the +occlusion rate by counting the number of screens with overlapping +leaf elements. We found that 18% of screens in our sampled split +contained overlapping leaf elements. However, of the overlapping +elements, only a third of them were occluded by more than 20% of +their total area. +Responsive Websites. Website responsiveness relates to how +well a web page adapts to different screen viewports. Since we sim- +ulated multiple devices for each web page, responsive websites are +likely to produce more variation in their layouts than unresponsive +ones. To measure responsiveness, we automatically computed met- +rics included in the Chrome Lighthouse tool for estimating layout +responsiveness: (i) responsiveness of content width to window size +and (ii) the use of a viewport meta tag, which is needed for proper +mobile rendering. From our analysis we found that 70% and 80% of +processed web pages met the first, and second criteria, respectively. +In summary, our analysis suggests that most web pages in our +dataset meet some basic requirements for visual UI modeling. Given +the reliance of our data collection on extracted accessibility meta- +data, we expect high quality examples to adhere to good accessibility +practices, such as those outlined by WCAG. However, considering +the inaccessibility of the web and that many criteria are difficult +to verify automatically, we also expect many web pages to vio- +late some of these criteria. There are other desirable properties for +dataset quality that we did not check, e.g., the accurate use of se- +mantic HTML tags, ARIA tags, and tightness of element bounding +boxes. These properties were harder to verify automatically, since +they require knowledge of developer intention and associated tasks. +In our analysis, we only attempt to identify possible defects, and +we did not attempt to remove or repair samples. This could be a +direction for future work to improve dataset quality [8, 30]. +4 +TRANSFERRING SEMANTICS FROM WEB +DATA +We hypothesized that web data is similar and relevant to modeling +other types of UIs from their pixels. In this paper, we are specif- +ically interested in the mobile domain, as mobile apps often lack +metadata and can only be reliably understood from their visual +appearance. In many cases, manually-annotated mobile datasets +are small, and in some cases, labels are completely unavailable. We +used transfer learning to apply our dataset to three existing tasks +in the UI understanding literature: (i) element detection, (ii) screen +classification, and (iii) screen similarity. Table 1 shows downstream +applications where UI understanding tasks can benefit from web +data. Because each task contains different constraints (e.g., presence +of labeled target data) it is difficult to apply a single strategy to +serve all use-cases. For example, inductive transfer learning typi- +cally requires labels in both the pre-training and fine-tuning phase +is impossible to apply to a setting where target labels are unavail- +able (e.g., screen similarity). We expect our three transfer learning +strategies to be applicable to most future use-cases, since they span +all combinations of labeled data availability (Table 1). +4.1 +Element Detection +Element detection requires a machine learning model to identify +the locations and types of UI elements from a screenshot. Often +these models are based on object detection frameworks. +Element detection is an example of a task where labeled data is +available in both the source and target domain (albeit fewer exam- +ples of mobile screens), so it is possible to employ inductive transfer +learning. The WebUI dataset contains the locations of elements that +we scraped from the website accessibility tree. Element types are + +CHI ’23, April 23–28, 2023, Hamburg, Germany +Wu et al. +Table 1: Table of strategies for transferring semantics from web pages to other types of UIs. We explored scenarios where +labeled data is missing in either domain by applying three strategies: (i) finetuning, (ii) semi-supervised learning, and (iii) +domain adaptation. +Approach +Finetuning +Semi-supervised Learning +Domain Adaptation +Application +Element Detection +Screen Classification +Screen Similarity +Web (Source) +Y +N +Y +Mobile (Target) +Y +Y +N +Web +Data +VINS +Element +Detector +Element +Detector +Step 1: +Pre-training +Step 3: +Fine-tuning +Step 2: Weight initialization +Figure 5: We applied inductive transfer learning to improve +the performance of a element detection model. First, we pre- +trained the model on web pages to predict the location of +nodes in the accessibility tree. Then, we used the weights of +the web model to initialize the downstream model. Finally, +we fine-tuned the downstream model on a smaller dataset +consisting of mobile app screens. +inferred from the HTML tags and the ARIA labels [2]. We show that +this training strategy results in improvements to element detection +performance. +4.1.1 +Model Implementation. We primarily followed the details +provided by VINS [6] to implement our element detection model. +The VINS dataset, which we used for training, is composed of +4800 annotated UI screenshots from various sources such as design +wireframes, Android apps, and iOS apps. Since the authors did not +release official data splits, we randomly partitioned the data into +training (70%), validation (15%), and testing (15%) sets. This specific +split ratio was chosen since it has been used in other UI modeling +work [50]. The paper identifies 11 primary UI component classes; +however the released raw dataset includes a total of 22 class labels. +For the extraneous labels, we either tried to merge them with the +11 primary labels (e.g., “Remember Me" merged with “Check Box") +or assigned them to an “Other" class (e.g., “Map") if no good fit was +found. Instead of the SSD object detection model [38] used by VINS, +we opted to start from the more recent FCOS model architecture +[48], since we found it was easier to modify to support multi-label +training. Previous element detection work [6, 12, 55] trained models +to assign one class label (e.g., Button, Text field) to each detected +element in the screenshot. To take advantage of multiple, nested +definitions of web elements in our dataset, we trained the object +detection model to predict multiple labels for each bounding box. +Figure 5 illustrates the overall training process. In the pre-training +phase, the element detection model is trained on a split of the We- +bUI dataset. Due to cost and time constraints, we trained all element +detection models for a maximum of 5 days. We also used early stop- +ping on the validation metric to reduce the chance of overfitting. +Afterwards, a specific part of the model was re-initialized (the ob- +ject classification head) to match the number of classes in the VINS +dataset before it was fine-tuned. We found it difficult to modify the +original SSD architecture to support the multi-label pre-training, +so we only followed the original training from scratch procedure +described in the paper as a baseline. +4.1.2 +Results. Table 2 shows the performance of each model con- +figuration on the VINS test set, and we show that our updated +configurations lead to significant performance improvements. Our +primary performance metric for this task was the mean average pre- +cision (mAP), which is a standard metric used for object detection +models that takes into the accuracy of bounding box location (i.e., +how closely the predicted box overlaps with ground truth) and clas- +sification (prediction of object type). The mAP score is calculated +by computing an individual average precision (AP) score for each +possible element class (e.g., Text, Check Box), which represents the +object detector’s accuracy in detecting each object class. The AP +scores are averaged to produce the mAP score. We calculated the +mAP score over classes that could be mapped to the original label +set in the paper [6] i.e., we excluded the “Other" class where there +was no clear mapping to the original set. We calculated the un- +weighted mean between class APs, which assigns equal importance +to common and rare element types. Our best model configuration +performed 0.14 better than the baseline in terms of mAP score. +While the largest source of improvement over the baseline con- +figuration (SSD) came from the updated FCOS model architecture, +our fine-tuning procedure contributed to gains as well. Specifi- +cally, we note that pre-training with more examples led to better +performance (around 0.04 mAP). Depending on the downstream +application of the element detection model, this improvement could +lead to better user experience but would require further validation. +For example, a screen reader [55] does not require tight bounding +boxes; however, it would benefit from detecting more (small) el- +ements on the screen. Query-based design search [6] could also +retrieve more relevant examples. +Although we followed the original training procedure as closely +as possible, we were unable to reach the mAP score reported in the +original VINS paper. This can be attributed to (i) our use of different +randomized splits and (ii) differences in mappings between class +labels from the raw data to the 11 primary classes, which were not +provided in the previously released code. Nevertheless, since we +used the same splits and class mappings across all of our model +configurations, we expect the relative performance improvements +to be consistent. +We also investigated the zero-shot performance of element de- +tectors trained only on web data (i.e., without fine-tuning). It is +difficult to compute performance quantitatively, since the label sets + +WebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics +CHI ’23, April 23–28, 2023, Hamburg, Germany +Table 2: Element detection performance (11 object classes) +for different model configurations. Pre-training on more +web screens led to better performance on mobile screens af- +ter fine-tuning. +Model Configuration +mAP +SSD (Random Init.) +0.6737 +FCOS (Random Init.) +0.7739 +FCOS (Pre-trained on Web7k) +0.7877 +FCOS (Pre-trained on Web7k-Resampled) +0.7961 +FCOS (Pre-trained on Web70k) +0.7921 +FCOS (Pre-trained on Web350k) +0.8115 +between the web and mobile datasets do not directly overlap. How- +ever, we provide qualitative evidence that zero-shot learning could +be successful. Figure 6 shows the output of a web model when run +on mobile app screens from Rico. We conducted minimal prepro- +cessing, such as cropping out the Android system notification bar +and the navigation soft buttons. In many cases, the web analogs +of mobile text and image elements are detected accurately, which +suggests that some element classes have consistent appearance +across platforms. Interestingly, some web classes such as links and +headings are also detected in the image, which could be used to infer +new semantics such as clickability [47] and navigation landmarks. +4.2 +Screen Classification +Classifying screen type or functionality from a screenshot can +be useful for design analysis and automation. Previously, small +amounts of data have been collected and annotated for this purpose. +Enrico [29] is an example of a dataset (1460 samples, subset of Rico +[15]) where each screenshot is assigned to one of 20 mutually- +exclusive design categories. Because of the dataset’s small size, it is +challenging to train accurate deep learning classification models. +While our web dataset is large, it also does not have the screen- +type annotations, and thus it is not possible to employ the same +pre-training strategy that was used for element detection. +Instead, we applied a semi-supervised learning technique known +as self-training [9]. Self-training is a process that improves model +performance by iteratively labeling and re-training on a large source +of unlabeled data. We investigated the effects of using WebUI as the +unlabeled dataset and show that doing so improves overall screen +classification accuracy. +4.2.1 +Model Implementation. Figure 7 shows our procedure for +incorporating WebUI data into our model training via self-training. +First, we trained screen classifier based on the VGG-16 archi- +tecture with batch normalization and dropout [45], as described +by the Enrico paper [29]. Since official training, validation, and +testing splits were not provided, we randomly generated our own +(70%/15%/15%). This model was trained only on data from the Enrico +training split and served as the teacher classifier. Next, the teacher +model was used to generate “soft" pseudo-labels for screenshots +in the WebUI dataset, where each sample was mapped to a vector +containing probabilities for each class. We followed the procedure +used by Yalniz et al. [53] to keep only the top K most confident +Table 3: Classification accuracy (across 20 classes) for dif- +ferent configurations of our screen classification model. In- +creasing the amount of data used with our semi-supervised +learning method led to increased accuracy. +Model Configuration +Accuracy +VGG-16 +0.4737 +Noisy ResNet-50 +0.4649 +Noisy ResNet-50 (Rico) +0.4956 +Noisy ResNet-50 (Web7k) +0.4864 +Noisy ResNet-50 (Web7k-Resampled) +0.4868 +Noisy ResNet-50 (Web70k) +0.5175 +Noisy ResNet-50 (Web350k) +0.5263 +labels for each class. To select K, we first randomly sampled a small +subset of 1000 web pages from our dataset and performed a param- +eter search to find the optimal value. Based on our experiments, +we found that a value of 10% of the total dataset size led to good +performance (e.g., we set K=700 for the Web-7k split). Finally, we +trained a student classifier on a combination of the original and +automatically generated labels. We employed a specific type of +self-training known as Noisy Student Training [52], which involves +injecting noise into the student model’s training process so that it +becomes more robust. Two types of noise are used in this process: +(i) input noise, which is implemented via random data augmenta- +tion techniques and (ii) model noise, which is implemented with +dropout [46] and stochastic depth [27]. Because stochastic depth +can only be applied to model architectures with residual blocks, we +used an architecture based on ResNet-50 [25] instead of VGG-16. +4.2.2 +Results. Overall, we found that applying self-training to in- +corporate additional unlabeled data led to consistent performance +improvements (Table 3). The best classifier using WebUI data was +5% more accurate than the baseline model, which was only trained +with the Enrico dataset. Our baseline VGG-16 model performed +considerably worse than the originally reported results [29] but +achieved similar accuracy to another reproduction of the work [35]. +The performance difference could be attributed to differences in +randomized splits. Since we used the same splits across all condi- +tions, we expect relative performance differences to be consistent. +To investigate the effects of using a new model architecture, we also +trained a Noisy ResNet-50 (architecture used by the student model) +on the Enrico dataset. The resulting classifier performed relatively +poorly (worse than the baseline model), since the modifications +introduced (dropout and stochastic depth) require more data to +train effectively. +The primary source of improvement stems from the inclusion of +additional unlabeled data during the training process, which led to +a more generalizable student model. We observed that the small size +of the Enrico dataset (1460 samples) quickly led to overfitting during +training and limited overall performance. Semi-supervised learning +techniques, such as self-training, allow training on a much larger +number of examples. We found that model accuracy improved when +we incorporated more unlabeled examples, both from WebUI and +Rico. + +CHI ’23, April 23–28, 2023, Hamburg, Germany +Wu et al. +Figure 6: Output of our element detection models run on two app screens. In many cases, detections from our web-only model +(Blue) coincide with ones from our fine-tuned model (Orange), which suggests some zero-shot transfer capabilities. Predicted +tags from the web-only model also provide additional metadata corresponding to clickability (link) and heading prediction +(heading); however, the predicted bounding boxes are often less tight than the fine-tuned model. +Enrico +Web +Data +Teacher +Classifier +Student +Classifier +Step 1:
 +Training +Step 3: +Noisy Training +Step 2:
 +Pseudo-labels +Figure 7: We applied semi-supervised learning to boost +screen classification performance using unlabeled web data. +First, a teacher classifier is trained using a “gold" dataset of +labeled mobile screens. Then, the teacher classifier is used +to generate a “silver" dataset of pseudo-labels by running it +on a large, unlabeled data source (e.g., web data). Finally, the +“gold" and “silver" datasets are combined when training a +student classifier, which is larger and regularized with noise +to improve generalization. This process can be repeated; +however, we only perform one iteration. +4.3 +Screen Similarity +Web +Data +Similarity +Model +RICO +UI Similarity +Mobile Examples +Unsup. Domain +Adaptation +Figure 8: We used unsupervised domain adaptation (UDA) to +train a screen similarity model that predicts relationships +between pairs of web pages and mobile app screens. The +training uses web data to learn similarity between screen- +shots using their associated URLs. Unlabeled data from +Rico is used to train an domain-adversarial network, which +guides the main model to learn features that transferrable +from web pages to mobile screens. +Identifying variations within the same screen and detecting tran- +sitions to new screens are useful for replaying user interaction +traces, processing bug reports [13], and automated app testing +[33, 34]. To model these properties and understand how multiple +screens from an application relate to each other, previous work +[20, 34] has sought to differentiate between distinct UIs and varia- +tions of the same UI. For example, the same checkout screen may +appear different based on the number and types of products added +to the cart. Common screen interactions such as scrolling and in- +teraction with expandable widgets (e.g., menus, dialogs, keyboards, +and notifications) may also alter the visual appearance of a screen. +Visual prediction reduces system reliance on accessibility metadata, +which may be missing or incomplete, and further extends the ap- +plications of these models, as they can process video recordings of +user interactions (e.g., reproducing bug reports) [5, 13]. +Previous work [20] opted to manually annotate a dataset of +more than one thousand iPhone applications that were manually +“crawled" by crowdworkers; however, the dataset was not released +to the public. As a weak source of annotation, we used web page +URLs to automatically label page relations. Since no labeled data is +available in the mobile domain, we employed domain-adversarial +network training [22], a type of unsupervised domain adaptation +(UDA), to encourage the model to learn transferrable features from +the web domain that might apply to the mobile domain. Note that +while it is possible to apply the semi-supervised learning strategy +(which was used for the screen classification task) in reverse, it may +be less effective, since the unlabeled dataset (mobile UIs) is smaller +than the labeled dataset. +4.3.1 +Model Implementation. We followed previous work [20] and +used a ResNet-18 [25] model trained as a siamese network [24]. The +siamese network uses the same model to encode two inputs, then +compares them in feature space (i.e., their embeddings) to decide if +they are different variations of the same UI screen. Our approach is +different from the method proposed by previous work [13], which + +img +Weather Alert +SevereThunderstorm +inyour.city. +Text,heading +Text,paragraph +sendyouweatheralertsbasedon +ooatio +rowid +img +Text,heading +vedlleAle +Text +AlertiLocatior +Text,button +DoneImage +WeatherAlert +SevereThunderstorm +inyour.city. +Never miss a weather alert +Text +Wewouldalsoliketosendyouweatneralertsbasedona +locationormultiplelocationsyouprovide +Switch +Text +veatner Alers +Text +Text +Icon +ocatior +anFanciscoA +Text Button +Doneimg,link,Text +Text,heading +Sun & Moon +Text,heading +Amino +img +Loving Pokemon sun and moon?Thenjointhiscommunity +Text,link +Sign Up +Text +Log In +Text +JuPyuuT +.+ I:.-I. +vicend +PrIvacvIPolcIIcon +Image +Text +bun +& +Moon +Text +m +Image +Loving Pokemon sun and moon?Thenjointhiscommunity +Text Button +Sign Up +4x +Text +ainnok'dn6imm +ermeof Service andPrivacvPoliciWebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics +CHI ’23, April 23–28, 2023, Hamburg, Germany +Table 4: Classification performance (same-screen vs new- +screen) of our screen similarity models evaluated on pairs +of screens from our web data. Performance increased when +the model was trained on more data and slightly decreased +when trained with the UDA objective. +Model Configuration +F1-Score +ResNet-18 (Web7k) +0.7097 +ResNet-18 UDA (Web7k) +0.7184 +ResNet-18 (Web7k-Resampled) +0.7368 +ResNet-18 UDA (Web7k-Resampled) +0.7191 +ResNet-18 (Web70k) +0.8222 +ResNet-18 UDA (Web70k) +0.8193 +ResNet-18 (Web350k) +0.9630 +ResNet-18 UDA (Web350k) +0.9500 +applies random data augmentations (e.g., blurring, rotation, trans- +lation) to screenshots to create same-screen pairs. Instead, we ran- +domly sampled pairs of screenshots from our web data for training, +with balanced probability for same-screen and new-screen pairs. +Same-screen pairs were generated by finding screenshots with the +same URL but accessed at different times or simulating page scrolls +on a full-page screen capture by sliding a window vertically along +the image. Note that occasionally, simulated page scrolls and access- +ing the same web page at different times still produced identical or +nearly identical screenshots, so in our test set, we filtered these out +using perceptual hashing. Different-screen pairs were generated +both by sampling screenshots from within the same domain but +with different URL path, and by sampling screenshots from other +domains. +The domain-adversarial training process seeks to simultaneously +accomplish two objectives: (i) learn an embedding space where two +screenshots are from the same screen if their distance is less than a +threshold, and (ii) learn an encoding function that applies to both +the web and mobile domains. The first objective is related to the +primary task of distinguishing same-screen pairs from new-screen +pairs and is achieved with a pairwise margin-based loss [20]. The +second objective aims to align the feature distributions of the two +domains by maximizing the error rate of a domain classifier, which +is a network that tries to classify whether a sample is from a web +or mobile UI. For this task, we used only web page screenshots cap- +tured on simulated smartphones, to make the domain classification +objective more challenging. +4.3.2 +Results. Since one of the assumptions of our problem is that +labeled examples of same-screen and new-screen pairs are unavail- +able for mobile apps, we used two alternative methods to evaluate +our screen similarity model: (i) quantitative evaluation on labeled +pairs of web screens and (ii) qualitative evaluation on a set of unla- +beled Android interaction videos. +Table 4 shows the quantitative performance of our models evalu- +ated on pairs of web pages from our dataset. Overall, training with +more data led to significantly better performance, an increase of +over 20%. The inclusion of a domain adaptation objective sometimes +led to a slight drop in classification performance since it introduces +additional constraints in the learning process. We qualitatively eval- +uated our model’s performance characteristics on mobile screens +by using them to segment videos of mobile app interaction. We +used a dataset of screen recordings of bug reproductions [13] for 6 +open-source Android apps and applied our model by sequentially +sampling frames from the video and evaluating whether a new +screen was reached. Note our sampling process differs from other +previous work [7, 15] that segmented crawls at recording time us- +ing accessibility metadata, because we do not have accessibility +metadata corresponding to the previously collected recordings used +in our analysis. Figure 9 shows an example of a usage video pro- +cessed by our model. While the web model was effective detecting +some types of transitions that occurred in mobile apps, it was less +effective at others, such as software keyboards and dialogs, which +do not occur frequently in the WebUI dataset. We include more +model-generated segmentations of the bug reproduction dataset in +supplementary material. +In this work, we applied unsupervised domain adaptation, which +does not require any labels from the target domain. Other domain +adaptation strategies exist, and some are able to incorporate small +amounts of labeled data, which we expect could improve the accu- +racy of our model by contributing transition types unique to mobile +apps. +5 +DISCUSSION +5.1 +Performance Impact of Web Data +Empirically, we showed that automatically crawled and annotated +web pages, like those available in WebUI, can effectively support +common visual modeling tasks for other domains (e.g., mobile apps) +through transfer learning strategies. In cases where a small amount +of labeled mobile data was available, as in element detection and +screen classification, incorporating web data led to better perfor- +mance. Even when labeled data was completely unavailable, as in +screen similarity, models trained only on web data could often be +directly applied to mobile app screens. Our results suggest that the +size of current UI datasets may be a limiting factor, since model +performance increases consistently when trained on larger splits +of data. Our observations and analysis of WebUI’s composition +showed that web pages can differ from mobile app screens in terms +of complexity (i.e., average number of on-screen elements) and +element types. However, the performance improvements from our +machine learning experiments suggest that web and mobile UIs are +similar enough to transfer some types of semantics between them. +We currently only explored three examples, although we believe +that other UI modeling works [11, 47, 50] can also benefit from +similar approaches. We did not evaluate all possible applications of +WebUI in our paper, due to time and cost constraints. However, the +three experiments we conducted cover all possibilities of source and +target domain labels (1), so similar transfer learning techniques are +likely to apply. Future work that builds upon WebUI can conduct +more detailed evaluations of other downstream tasks. +One specific area that we believe is promising for future work +is automated design verification [41], which could benefit from a +large volume of web pages containing paired visual and stylistic +information. Our highly automated data collection process also +allows WebUI to be more easily updated in the future by re-visiting + +CHI ’23, April 23–28, 2023, Hamburg, Germany +Wu et al. +Droid Weight +AntennaPod +Time Tracker +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +Token +GrowTracker +GNUCash +Figure 9: Examples of interaction videos segmented by our best models trained with UDA (Red) and without UDA (Blue). Videos +are sampled at 1 fps. The output of both models contain errors, however, we found that the adapted UDA model generally +produced better segmentations. Common errors include oversegmentation due to app dialogs and soft keyboards, which do +not occur in the WebUI dataset. +the same list of URLs. An updated version of the dataset could also +facilitate longitudinal analysis of the design [14] and accessibil- +ity [21] of web UIs. Nevertheless, WebUI is currently unlikely to +support other types of modeling, such as user interaction mining +[15, 16], that require realistic interaction traces, since our crawling +strategy was largely based on random link traversal. +5.2 +Improved Automated Crawling +Our crawler was unable to access much of the “deep web" (i.e., +large part of the web that cannot be indexed), and thus our dataset +contains few, if any, web pages that are not publicly accessible +or protected by authentication flows. It also did not attempt to +interact with all elements on a web page and conducted a very +limited exploration of any JavaScript-enabled functionality that +might have been present. Trends in web and app development, +such as the creation of Progressive Web Apps (PWAs), suggest that +this type of functionality will become more common, and traditional +link-based traversal may become less effective at exploring UI states. +To improve automated crawling and data collection, our crawler +could benefit from a semantic understanding of web pages. For +example, it could detect page functionality to explore states that +require human input and either execute automated routines (e.g. +detecting login fields) or employ crowdsourcing [15] to allow it to +proceed in more complex scenarios. Our currently trained models +could augment or improve this process by identifying tasks associ- +ated with web pages (e.g., screen classification) or by augmenting +potentially noisy labels provided by the automatically generated +accessibility tree. In turn, the crawler could explore more of the +web, leading to higher quality and more diverse data. If repeated +iteratively, this process would constitute a form of Never-Ending +Learning [39], a machine learning paradigm where models learn +continuously over long periods of time. Instead of learning from a +fixed dataset, models could constantly improve itself by encounter- +ing new content and designs, both of which are important due to +the dynamic nature of UIs. +5.3 +Generalized UI Understanding +Our experiments show that incorporating web data is most effec- +tive for improving visual UI modeling in transfer learning settings +where a limited amount of target labels are available for fine-tuning. +A logical next step is to obtain similar benefits without any addi- +tional labeled data. To this end, we identified several strategies for +improving generalization. First, unlike existing UI datasets that +contain examples from one device type, we intentionally simulated +multiple viewports and devices during data collection. The decom- +position of one-hot labels (where each element type is assigned + +<23> +31 +《23> +23 +A welan +Gime Trac +Add tas +Add ta: +Add tas +Ad ask +① 7123:hange三 +三 +三 +三 +三 +Fn, May 15 +Fr, May 1 5 +15:21 +ay 20 +0 7123WebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics +CHI ’23, April 23–28, 2023, Hamburg, Germany +exactly one type) into combinations of multi-hot tags (each element +can be assigned multiple labels) may also be useful, since it avoids +the problem of platform-specific element types. Figure 6 demon- +strates the zero-shot transfer capabilities of models trained only +on web data by successfully detecting and classifying elements on +Android app screens. While the label sets of web and Android data +do not directly overlap, the web model outputs reasonable analogs +(e.g., Text, link) for Android widgets (e.g., Text Button). Finally, our +screen similarity model shows how unsupervised domain adaptation +can improve the transferrability of learned features across domains +through an explicit machine learning objective. +A long-term goal of our automated data collection and modeling +efforts is achieving a more generalized understanding of UIs — a +single model that could be used to predict semantics for any UI. This +is challenging due to differing design guidelines and paradigms, but +it could ultimately lead to a better understanding of how to solve +UI problems across platforms. +6 +CONCLUSION +In this paper, we introduced WebUI, a dataset of approximately +400,000 web pages paired with visual, semantic, and style informa- +tion to support visual UI modeling. Unlike most existing datasets for +UI research that depend on costly and time-consuming human ex- +ploration and annotation, WebUI was collected with a web crawler +that uses existing metadata, such as the accessibility tree and com- +puted styles, as noisy labels for visual prediction. Our highly auto- +mated process allowed us to collect an order of magnitude more +UIs than other publicly released datasets and often associates more +information (e.g., clickability, responsiveness) with each example. +We demonstrated the utility of our dataset by incorporating it into +three visual UI modeling tasks in the mobile domain: (i) element de- +tection, (ii) screen classification, and (iii) screen similarity. In cases +where a small amount of labeled mobile data exists, incorporating +web data led to increased performance, and in cases without any +labeled mobile data, we found that models trained on web pages +could often generalize to mobile app screens. In summary, our work +shows that the web constitutes a large source of data that can more +sustainably be crawled and mined for supporting visual UI research +and modeling. +ACKNOWLEDGMENTS +This work was funded in part by an NSF Graduate Research Fel- +lowship. +REFERENCES +[1] 2022. AutoIt Function PixelSearch. https://www.autoitscript.com/autoit3/docs/ +functions/PixelSearch.htm. +[2] 2022. Chrome DevTools engineering blog Full Accessibility Tree in Chrome +DevTools. https://developer.chrome.com/blog/full-accessibility-tree/. Accessed: +2022-09-15. +[3] 2022. Puppeteer - Chrome. https://developer.chrome.com/docs/puppeteer/. Ac- +cessed: 2022-09-15. +[4] 2022. +What is the ideal screen size for responsive design? +https://www. +browserstack.com/guide/ideal-screen-sizes-for-responsive-design. Accessed: +2022-09-15. +[5] Carlos Bernal-Cárdenas, Nathan Cooper, Madeleine Havranek, Kevin Moran, +Oscar Chaparro, Denys Poshyvanyk, and Andrian Marcus. 2022. Translating +Video Recordings of Complex Mobile App UI Gestures Into Replayable Scenarios. +IEEE Transactions on Software Engineering (2022). +[6] Sara Bunian, Kai Li, Chaima Jemmali, Casper Harteveld, Yun Fu, and Magy Seif +Seif El-Nasr. 2021. Vins: Visual search for mobile user interface design. In Pro- +ceedings of the 2021 CHI Conference on Human Factors in Computing Systems. +1–14. +[7] Andrea Burns, Deniz Arsan, Sanjna Agrawal, Ranjitha Kumar, Kate Saenko, and +Bryan A Plummer. 2022. Interactive Mobile App Navigation with Uncertain or +Under-specified Natural Language Commands. arXiv preprint arXiv:2202.02312 +(2022). +[8] Tsung-Hsiang Chang, Tom Yeh, and Rob Miller. 2011. Associating the visual +representation of user interfaces with their internal structures and metadata. In +Proceedings of the 24th annual ACM symposium on User interface software and +technology. 245–256. +[9] Olivier Chapelle, Bernhard Scholkopf, and Alexander Zien. 2009. Semi-supervised +learning. IEEE Transactions on Neural Networks 20, 3 (2009), 542–542. +[10] Nitesh V Chawla, Kevin W Bowyer, Lawrence O Hall, and W Philip Kegelmeyer. +2002. SMOTE: synthetic minority over-sampling technique. Journal of artificial +intelligence research 16 (2002), 321–357. +[11] Jieshan Chen, Amanda Swearngin, Jason Wu, Titus Barik, Jeffrey Nichols, and +Xiaoyi Zhang. 2022. Towards Complete Icon Labeling in Mobile Applications. In +CHI Conference on Human Factors in Computing Systems. 1–14. +[12] Jieshan Chen, Mulong Xie, Zhenchang Xing, Chunyang Chen, Xiwei Xu, Liming +Zhu, and Guoqiang Li. 2020. Object detection for graphical user interface: Old +fashioned or deep learning or a combination?. In proceedings of the 28th ACM +joint meeting on European Software Engineering Conference and Symposium on +the Foundations of Software Engineering. 1202–1214. +[13] Nathan Cooper, Carlos Bernal-Cárdenas, Oscar Chaparro, Kevin Moran, and +Denys Poshyvanyk. 2021. It takes two to tango: Combining visual and textual +information for detecting duplicate video-based bug reports. In 2021 IEEE/ACM +43rd International Conference on Software Engineering (ICSE). IEEE, 957–969. +[14] Biplab Deka, Bardia Doosti, Forrest Huang, Chad Franzen, Joshua Hibschman, +Daniel Afergan, Yang Li, Ranjitha Kumar, Tao Dong, and Jeffrey Nichols. 2021. +An Early Rico Retrospective: Three Years of Uses for a Mobile App Dataset. +In Artificial Intelligence for Human Computer Interaction: A Modern Approach. +Springer, 229–256. +[15] Biplab Deka, Zifeng Huang, Chad Franzen, Joshua Hibschman, Daniel Afergan, +Yang Li, Jeffrey Nichols, and Ranjitha Kumar. 2017. Rico: A mobile app dataset +for building data-driven design applications. In Proceedings of the 30th Annual +ACM Symposium on User Interface Software and Technology. 845–854. +[16] Biplab Deka, Zifeng Huang, and Ranjitha Kumar. 2016. ERICA: Interaction +mining mobile apps. In Proceedings of the 29th annual symposium on user interface +software and technology. 767–776. +[17] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. 2009. Imagenet: +A large-scale hierarchical image database. In 2009 IEEE conference on computer +vision and pattern recognition. Ieee, 248–255. +[18] Morgan Dixon and James Fogarty. 2010. Prefab: implementing advanced behav- +iors using pixel-based reverse engineering of interface structure. In Proceedings +of the SIGCHI Conference on Human Factors in Computing Systems. 1525–1534. +[19] Samuel Dodge and Lina Karam. 2016. Understanding how image quality af- +fects deep neural networks. In 2016 eighth international conference on quality of +multimedia experience (QoMEX). IEEE, 1–6. +[20] Shirin Feiz, Jason Wu, Xiaoyi Zhang, Amanda Swearngin, Titus Barik, and Jeffrey +Nichols. 2022. Understanding Screen Relationships from Screenshots of Smart- +phone Applications. In 27th International Conference on Intelligent User Interfaces. +447–458. +[21] Raymond Fok, Mingyuan Zhong, Anne Spencer Ross, James Fogarty, and Ja- +cob O Wobbrock. 2022. A Large-Scale Longitudinal Analysis of Missing Label +Accessibility Failures in Android Apps. In CHI Conference on Human Factors in +Computing Systems. 1–16. +[22] Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo +Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. 2016. +Domain-adversarial training of neural networks. The journal of machine learning +research 17, 1 (2016), 2096–2030. +[23] Leo Gao, Stella Biderman, Sid Black, Laurence Golding, Travis Hoppe, Charles +Foster, Jason Phang, Horace He, Anish Thite, Noa Nabeshima, et al. 2020. The +pile: An 800gb dataset of diverse text for language modeling. arXiv preprint +arXiv:2101.00027 (2020). +[24] Raia Hadsell, Sumit Chopra, and Yann LeCun. 2006. Dimensionality reduction +by learning an invariant mapping. In 2006 IEEE Computer Society Conference on +Computer Vision and Pattern Recognition (CVPR’06), Vol. 2. IEEE, 1735–1742. +[25] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. 2016. Deep residual +learning for image recognition. In Proceedings of the IEEE conference on computer +vision and pattern recognition. 770–778. +[26] Zecheng He, Srinivas Sunkara, Xiaoxue Zang, Ying Xu, Lijuan Liu, Nevan Wich- +ers, Gabriel Schubiner, Ruby Lee, and Jindong Chen. 2021. Actionbert: Leveraging +user actions for semantic understanding of user interfaces. In Proceedings of the +AAAI Conference on Artificial Intelligence, Vol. 35. 5931–5938. +[27] Gao Huang, Yu Sun, Zhuang Liu, Daniel Sedra, and Kilian Q Weinberger. 2016. +Deep networks with stochastic depth. In European conference on computer vision. + +CHI ’23, April 23–28, 2023, Hamburg, Germany +Wu et al. +Springer, 646–661. +[28] Ranjitha Kumar, Arvind Satyanarayan, Cesar Torres, Maxine Lim, Salman Ahmad, +Scott R Klemmer, and Jerry O Talton. 2013. Webzeitgeist: design mining the web. +In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. +ACM, New York, NY, USA, 3083–3092. +[29] Luis A Leiva, Asutosh Hota, and Antti Oulasvirta. 2020. Enrico: A dataset for +topic modeling of mobile UI designs. In 22nd International Conference on Human- +Computer Interaction with Mobile Devices and Services. 1–4. +[30] Gang Li, Gilles Baechler, Manuel Tragut, and Yang Li. 2022. Learning to Denoise +Raw Mobile UI Layouts for Improving Datasets at Scale. In CHI Conference on +Human Factors in Computing Systems. 1–13. +[31] Toby Jia-Jun Li, Amos Azaria, and Brad A Myers. 2017. SUGILITE: creating +multimodal smartphone automation by demonstration. In Proceedings of the 2017 +CHI conference on human factors in computing systems. 6038–6049. +[32] Yang Li, Jiacong He, Xin Zhou, Yuan Zhang, and Jason Baldridge. 2020. Mapping +natural language instructions to mobile UI action sequences. arXiv preprint +arXiv:2005.03776 (2020). +[33] Yuanchun Li, Ziyue Yang, Yao Guo, and Xiangqun Chen. 2017. Droidbot: a +lightweight ui-guided test input generator for android. In 2017 IEEE/ACM 39th +International Conference on Software Engineering Companion (ICSE-C). IEEE, 23– +26. +[34] Yuanchun Li, Ziyue Yang, Yao Guo, and Xiangqun Chen. 2019. Humanoid: A +deep learning-based approach to automated black-box android app testing. In +2019 34th IEEE/ACM International Conference on Automated Software Engineering +(ASE). IEEE, 1070–1073. +[35] Paul Pu Liang, Yiwei Lyu, Xiang Fan, Zetian Wu, Yun Cheng, Jason Wu, Leslie +Chen, Peter Wu, Michelle A Lee, Yuke Zhu, et al. 2021. +Multibench: Mul- +tiscale benchmarks for multimodal representation learning. +arXiv preprint +arXiv:2107.07502 (2021). +[36] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva +Ramanan, Piotr Dollár, and C Lawrence Zitnick. 2014. Microsoft coco: Common +objects in context. In European conference on computer vision. Springer, 740–755. +[37] Thomas F Liu, Mark Craft, Jason Situ, Ersin Yumer, Radomir Mech, and Ranjitha +Kumar. 2018. Learning design semantics for mobile apps. In Proceedings of the +31st Annual ACM Symposium on User Interface Software and Technology. 569–579. +[38] Wei Liu, Dragomir Anguelov, Dumitru Erhan, Christian Szegedy, Scott Reed, +Cheng-Yang Fu, and Alexander C Berg. 2016. Ssd: Single shot multibox detector. +In European conference on computer vision. Springer, 21–37. +[39] Tom Mitchell, William Cohen, Estevam Hruschka, Partha Talukdar, Bishan Yang, +Justin Betteridge, Andrew Carlson, Bhavana Dalvi, Matt Gardner, Bryan Kisiel, +et al. 2018. Never-ending learning. Commun. ACM 61, 5 (2018), 103–115. +[40] Kevin Moran, Carlos Bernal-Cárdenas, Michael Curcio, Richard Bonett, and +Denys Poshyvanyk. 2018. Machine learning-based prototyping of graphical user +interfaces for mobile apps. IEEE Transactions on Software Engineering 46, 2 (2018), +196–221. +[41] Kevin Moran, Boyang Li, Carlos Bernal-Cárdenas, Dan Jelf, and Denys Poshy- +vanyk. 2018. Automated reporting of GUI design violations for mobile apps. In +Proceedings of the 40th International Conference on Software Engineering. 165–175. +[42] Sinno Jialin Pan and Qiang Yang. 2009. A survey on transfer learning. IEEE +Transactions on knowledge and data engineering 22, 10 (2009), 1345–1359. +[43] Richard S. Schwerdtfeger. 1991. Making the GUI Talk. ftp://service.boulder.ibm. +com/sns/sr-os2/sr2doc/guitalk.txt. +[44] Vinoth Pandian Sermuga Pandian, Sarah Suleri, and Matthias Jarke. 2021. Synz: +Enhanced synthetic dataset for training ui element detectors. In 26th International +Conference on Intelligent User Interfaces-Companion. 67–69. +[45] Karen Simonyan and Andrew Zisserman. 2014. Very deep convolutional networks +for large-scale image recognition. arXiv preprint arXiv:1409.1556 (2014). +[46] Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan +Salakhutdinov. 2014. Dropout: a simple way to prevent neural networks from +overfitting. The journal of machine learning research 15, 1 (2014), 1929–1958. +[47] Amanda Swearngin and Yang Li. 2019. Modeling mobile interface tappability +using crowdsourcing and deep learning. In Proceedings of the 2019 CHI Conference +on Human Factors in Computing Systems. 1–11. +[48] Zhi Tian, Chunhua Shen, Hao Chen, and Tong He. 2019. Fcos: Fully convolutional +one-stage object detection. In Proceedings of the IEEE/CVF international conference +on computer vision. 9627–9636. +[49] Bryan Wang, Gang Li, Xin Zhou, Zhourong Chen, Tovi Grossman, and Yang +Li. 2021. Screen2words: Automatic mobile UI summarization with multimodal +learning. In The 34th Annual ACM Symposium on User Interface Software and +Technology. 498–510. +[50] Jason Wu, Xiaoyi Zhang, Jeff Nichols, and Jeffrey P Bigham. 2021. Screen Parsing: +Towards Reverse Engineering of UI Models from Screenshots. In The 34th Annual +ACM Symposium on User Interface Software and Technology. 470–483. +[51] Mulong Xie, Sidong Feng, Zhenchang Xing, Jieshan Chen, and Chunyang Chen. +2020. UIED: a hybrid tool for GUI element detection. In Proceedings of the 28th +ACM Joint Meeting on European Software Engineering Conference and Symposium +on the Foundations of Software Engineering. 1655–1659. +[52] Qizhe Xie, Minh-Thang Luong, Eduard Hovy, and Quoc V Le. 2020. +Self- +training with noisy student improves imagenet classification. In Proceedings of +the IEEE/CVF conference on computer vision and pattern recognition. 10687–10698. +[53] I Zeki Yalniz, Hervé Jégou, Kan Chen, Manohar Paluri, and Dhruv Mahajan. 2019. +Billion-scale semi-supervised learning for image classification. arXiv preprint +arXiv:1905.00546 (2019). +[54] Tom Yeh, Tsung-Hsiang Chang, and Robert C Miller. 2009. Sikuli: using GUI +screenshots for search and automation. In Proceedings of the 22nd annual ACM +symposium on User interface software and technology. 183–192. +[55] Xiaoyi Zhang, Lilian de Greef, Amanda Swearngin, Samuel White, Kyle Murray, +Lisa Yu, Qi Shan, Jeffrey Nichols, Jason Wu, Chris Fleizach, et al. 2021. Screen +recognition: Creating accessibility metadata for mobile applications from pixels. +In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems. +1–15. +A +ADDITIONAL DATASET SAMPLES +We provide additional samples from the WebUI (Figure 10) to sup- +plement the example in the paper (Figure 2). Our example gallery +shows several different types of websites, including login, landing, +product, portfolio, and informational pages. Each website is cap- +tured using different simulated devices, which shows, among other +things, how content responds to screen size. We also computed the +percentile-rank of each web page’s class distribution. +B +CLASS IMBALANCE ANALYSIS +This section describes analysis of class imbalance of WebUI and +its effect on transfer learning applications. Similar to other UI +datasets[55], WebUI exhibits an imbalance of UI element classes, +where some types of elements (e.g., text) appear much more fre- +quently than others (e.g., images). Several aspects of WebUI (e.g., +finer-grain text segmentation, multi-hot labels, and prevalence of +documents on the web) also contributed to class imbalance. +We used a frequency-based resampling method to generate the +Web7k-Resampled, which resulted in more examples of infrequent +element types. Our technique assigned weights to samples to in- +crease the representation of UIs containing rare or infrequent ele- +ment types, and we resampled based on the 10 element types shown +in Figure 3. Algorithm 1 provides an overview of our resampling +technique. Note that unlike some class-balancing algorithms (e.g., +SMOTE [10]), our technique does not generate additional synthetic +samples and does not include the same screen more than once. +Web7k-Resampled contains proportionally more examples of +many infrequent classes (Figure 3). Figure 11 shows the proportional +increase in screens containing each element type. Figure 12 shows +the proportional increase in the total number of elements for each +type. +The results from our performance evaluations in the main paper +suggest that this resampled split leads to improvements for each +of our three tasks when compared to a randomly sampled subset +of the same size. Notably, the element detector model resampled +7k split outperformed the one trained on 70k random split, which +suggests that element balancing was particularly useful for tasks +where elements types are directly predicted. Tests with other two +tasks (screen classification and screen similarity) also led to im- +provements for the resampled models; however, the gains were +more modest. The improvements could be because the element +distribution in the resampled split is closer to that of the target data. +In addition, we provide a deeper analysis of the Element Detection + +WebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics +CHI ’23, April 23–28, 2023, Hamburg, Germany +Figure 10: Samples from WebUI accessed with different simulated devices. For each screen, we compute its element type dis- +tribution (normalized to 1). Then, we computed the percentile-rank of the top 10 classes with respect to the entire dataset. For +example, the bottom row’s button class has a percentile-rank of 90, meaning the web page’s relative frequency of is greater +than 90% of others in the dataset. + +TheOpenTracingpeject isarchivedLearmore +MioratetrQpenTelemietry.today +U OPENTRACING +DOSSGUIDS +PDOUEETGETMIVAIVEOCITHEEBLOG REGST +SAYHIOWGUTER +bash +VendarnautralAesand +instrumiemtationfofcistnbutedtracing +Librariesavailableinglanguages +So ZivScrintJaa.Pyahon RuhyPliObiectiwcrCHiC +The latestfromourblog +Openi Tracing hasbeen Archived +toove +DADPWEARD +MOTANSASA +All advertisingisfirstcome,first +choiceof placement, and mustbe +paid in advance via PayPal. jpg or +aifTormatprefered.Adaltional +single-pageads arehalf-price. +Websitesizesandpricesbelow +Starneadvertisingratesand +sizesare,perquarterlyissue +$75/fullpage(5"wx8*h) +$40/half-page (5"w x3.8"h): +$15/business-card size (2*x 3.5 +eitherway! +Banneradvertisingis +availableatthebottomofevery +pageonthesite,justabovethe +copyrightandadvertising links +CustomsizescanDeaanqed +Leff sidebaradvertising is +availableonanypagewitha +menu sidebar.Leftsidebarads +are square orrectangular,120 +- +pixels wide. +Riaht sidebaradvertisingis +availableonlyonselectedpaqes +see +ndetpageasanexample)text +link +listitem +image +heading +paragraph +linebreak +generic +gridcell +button +0 +20 +40 +60 +80 +100NAYA +To.search.:typeandhitentef. +D +FAMILY +CENTER +ABOUT +SERVICES +SUPPORTUS +POLICY-ADVOCACY +EVENTSBLOG +SUPPORTUS +WAYSTOGIVE +DONATENOW +Your qenerous donation limpacts the lives of 10.o00peopleeach yearwith wraparound services thathelpour +WAYSTOGIVE +community thrive. +LEADERSHIPGIVING +Todonateonline,clickhereNAYA +FAMILY +Tosearch.typeandhitenter. +CENTER +ABOUT +SERVICES +SUPPORTUS +POLICY-ADVOCACY +EVENTS +BLOG +SUPPORTUS +WAYSTOGIVE +JONATENOW +Yourgenerousdonationi impactsthe livesof10.0o0peopleeachyearwithwraporoundservicesthathelpour +WAYSTOGIVE +comumunitythrve +LEADERSHIPGIVING +Todonateonline,clickhete, +GIRCLE +MAKEADONATIONBY CHECK +OLR PARTNERS AND +MakeyourCheckpayableto:NAYAFamilyCenterNAYA +o sealcLnype anct hitereetC +FAMILY +CENTER +ABOUTSERVICES +SUPPORTUS +POLICY-ADVOCACYEVENTSBLOG +SUPPORTUS +WAYSTOGIVE +DONATENGW +WAYSTOGIVE +community tithe +LEADERSHIPGIVING +To donateonline,cick here. +CIRCLE +MAKEASONATIONBYCHECK +DURPARTNERS AND +MakeyourcheckpayabletoNAYAFamilyCenter +SUPPUHTERS +Sendato: +REIURNTO. +NAYA Famlly Certer +NEERGHDKIKOO +5135-NE Columbia Blvd +Portand,OR97218NAYA +FAMILY +CENTER +ABOUTSERVICESSUPPORTUSPOLICY-ADVOCACYEVENTSBLOG +SUPPORTUS +WAYSTOGIVE +DONATENOA +committythit +wourgsemusdonatoninpcts +WAYSTOGME +LEADERSEIPOIVING +Be danateonint,chice bere +MAKEABONATIONBYCHECK +CUREPARINERSANU +SUPFOHTEHS +Senditto: +RETEFNTO +Sn Nt Cohrta aid +WoAPanilyCeter +NIRCHIKIKOD) +Partirid,O81 97218 +Atm Develagnsert +MONTHLYGIYING +catamuficbankwithdrawalatbycharpingyoureredtcare +werecesessunmaryufyeuramislgiisglustanieaprarinjpunuaigmjtasugendarstspyurrecierg +sfng.NyFamlyCenbertheoppertunityto.tolanceogrmourosthrooghoutheyewr.Fertaxerproetyou +mattuia +natDeutlomentDataSaeclwlidNAYA +CENTER +SAMILY +TSEEEEE +SUPPORTOS +WAYSTOGIVE +NSUOM +MANAYA +FAMILY +CENTER +ABOUT SERVICESSUPPORTUSPOLICY-ADVOCACY EVENTSBLOG +SUPPORTUS +WAYSTOGIVE +TDNATEIIOW +hurgenerousdoatignimgectsthelivesgrLe,ooopeeoleeodyearwitharapartunfservcesfathelpaur +WAYSTOGIVE +TADERSHIP GIVING +To.donat +RAL +MAKEADONATIONEYCHECK +OURPARINERSAND +Makeyoorcheckgeyableto:NAVAFamiy Certe +SUFPOBTERS +Send ito +RETURNTO +wYATantyCentm +NERCHOKIKOO +5135MEColusbiaBive +Poitlid,ORST238 +AllrtDevelepment +MONTHEYGIMNG +obetyou +gitstortinewthephoetallerenailtoLisaHarmn.DeveloonerttetaSpecislisttaitonavoatusroo +5032as.a177x295.To.males1ecaminggit.cicthm +BIETSONHONORORMEMORY +Rementeralnvedonewthiagitmadenthesrhonerer.men +noryGftsareackoowedjed througtscantsest +mtherecipientotyourchoice.lomalsaDibutegit,tftrhat +STOCINS +Aconistionofshoresheldformorettanayesriseigibletertesaneincame.taxdeductionasgttsotcas, +shosld yoeitemiae,Alto,ithedoratod.sharesincreasedinvolue.Feuavoidmecaptaqainstaxmatweulidbe +uwdstunpausstieeshaess +Ce +Torecelveareceiptandtoensyrethstyourtinsterisprocessedcotretty.pleaeletinxneefyeurstoctgint +infonatisn +Datoenitext +link +listitem +image +heading +paragraph +linebreak +generic +gridcell +button +0 +20 +40 +60 +80 +100The OpeiTracingprojectis-archivedLearn more +Migrale to OpenTelenetry todayl +COPENTRACING +DOGUT电NVCVRY +SAYHIGNITTER +bash +enwormewtraApisano +instrurmentatiomtordistnoutecftragimg +Librariesavallableinganguages +SoaveScritt.lavePahon/Buby PHeDbimttiw.fteCn +The:Iotestfrom.ourtatog +Cieniracing has bren Archived +DHDWUARD +HOTAN +SHPROSETheOpenTracngpmjeclisrarchived.Lcanman +oletoDpeniFemetiytodoy! 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Because the more money and tech +available to [create]a work,the less imagination there wil be in it. +http/www.mty.com/shured/movies/interview/m/moore_alun.CGo3ttext +link +listitem +image +heading +paragraph +linebreak +generic +gridcell +button +0 +20 +40 +60 +80 +100text +link +listitem +image +heading +paragraph +linebreak +generic +gridcell +button +0 +20 +40 +60 +80 +100text +link +listitem +image +heading +paragraph +linebreak +generic +gridcell +button +0 +20 +40 +60 +80 +100text +link +listitem +image +heading +paragraph +linebreak +generic +button +0 +20 +40 +60 +80 +100text +link +listitem +image +heading +paragraph +linebreak +generic +gridcell +button +0 +20 +40 +60 +80 +100text +link +listitem +image +heading +paragraph +linebreak +generic +gridcell +button +0 +20 +40 +60 +80 +100$bocoup +SERVIES +XEONDUDE +BLOB +ABOUTUS.+ +Search +a +WESTANDIN +INCLUSIVETECHNOLOGY +CONSULTING +WEPARTNERWITHTECHCOMPANIES +We are ateam of product and standards developers.woridng +with pirtners on engineering,design,and organizational +ANDNONPROFITSTOINCREASE +stateyejectsthataligthoumisstonndaluesWea +ACCESSIBILITY.INCLUSION.AND +commitedtomalingtecspacessafefrmaginalizedpeople +JUSTICEONANDTHROUGHTHEWEB +inourprojects-anidon our tean.Read theCodeot Contha +which governs out worktext +link +listitem +image +heading +paragraph +linebreak +generic +gridcell +button +0 +20 +40 +60 +80 +100ComicPress +Home +Blog +Contact Us +WiteFerUs +ComiePress By Counsellr +Lounch.Updates +A newer, Betfer. Comic Theme Is Coming Soon! +Gecriotihedontaunctht +ErtallAdidress* +Suorat +Cainnot Wait?ComicPress +Hame +Blog +ContotrUs +Wiae Fer Us +ComicPressByCounsellr +LiinchUpdotes +Anewer,BetterComic ThemelsComing Soonl +GntNothelfonfund +EiallAddrtiss" +Cannot Wait? +ToGetThe Clossic Version On WordPress.org Click HmeComicPress +Heme +CostactUs +Wite For Us +ComicPress By Counsell +LanechUpdates +Anewur,Buttur Comic Theme h Coming Soon! +GaNotfedanlaunn +EmalAddres +Sohmit +Cannot Wait? +ToGetThe Clasuc Version OnWordPresorg ClidkHery +yCoumsellWerdiirestSermemComicPress +Han +Cootist U, +Wesfetlh +ComicPress By Counsellr +Liaanknilute +Aoe;BethrConeThenehCiigSoosl +EmEAdT +TeGetThrCaneVer +Latest From The WordPress Blogbocoup +ONTACTUS +Search +QComicPress +三Menu +ComiePressBy Counsell +Anewer,Better ComicTheme ls Coming +Soonl +Get Notified or aunch! +Email Addlress* +SubmitSbocoup +WESTANDINSOLIDARITYWITHTHEMOVEMENTFORBLACKLIVESComicPress +Have +ig +Cofaet Ui +Wit Fer Us +ComicPress ByCounsellr +LounchUpdotai +Arenir.BeterComeThemeComngSoe +EmatATm +TGThCletioOWardgtext +link +listitem +image +heading +paragraph +linebreak +generic +gridcell +button +0 +20 +40 +60 +80 +100Scienceficton.&fantasyPoetryAssociation +Badvortisinaisfrtcometirstchoo +rquarte +Banner.advertisingis availableathebottom ofevery page on thesite. just above the +warfSu +46ex.60-bonner:s150ryear.perpepe:$4n/year.site-wide +00250rectangl +Fcod/rearste-wide +are square or rectangular.120 pixeis widSciencefiction.&fantasuPoetryAssociatior +gfformatpn +size(2"x3.s"eitherway +nsizescanbearange +etn-e0bumer-stscyesrperpepe:s0/yesrste-aide +aresquareor.rectanguiar,120pixelswide +12080 +$75/yearserpape +120±90onxtontosL +debat.Left sidebat ad +120.a60 +2120×240CHI ’23, April 23–28, 2023, Hamburg, Germany +Wu et al. +Table 5: Average Precision (AP) of each element class (excluding the “Other" class) for the Element Detection task. +Element Type +SSD (Random) +FCOS (Random) +FCOS (Web7k) +FCOS (Web7k-Re.) +FCOS (Web70k) +FCOS (Web350k) +Background Image +0.85 +0.88 +0.86 +0.91 +0.85 +0.93 +Checked View +0.06 +0.28 +0.31 +0.34 +0.32 +0.38 +Icon +0.72 +0.73 +0.75 +0.75 +0.75 +0.77 +Input Field +0.22 +0.59 +0.7 +0.60 +0.72 +0.69 +Image +0.73 +0.8 +0.77 +0.82 +0.78 +0.82 +Text +0.66 +0.83 +0.89 +0.84 +0.9 +0.85 +Text Button +0.57 +0.9 +0.94 +0.94 +0.95 +0.94 +Page Indicator +0.83 +0.76 +0.83 +0.76 +0.79 +0.8 +Pop-Up Window +0.85 +0.83 +0.8 +0.85 +0.78 +0.83 +Sliding Menu +0.95 +0.98 +0.96 +0.98 +0.96 +0.97 +Switch +0.97 +0.93 +0.86 +0.97 +0.91 +0.94 +mAP +0.67 +0.77 +0.79 +0.80 +0.79 +0.81 +Algorithm 1: Pseudo-code for the frequency-based resam- +pling algorithm used to generate the Web7k-Resampled +split. +1 function SampleSplit (𝑁,𝐶,𝑆); +Input +:Number of samples to choose 𝑁, list of element +classes 𝐶, and list of samples 𝑆 +Output:Resampled subset of 𝑆 +/* Vector containing total frequencies for 𝑐 ∈ 𝐶 +*/ +2 𝑓𝐶 ← total # of elements in 𝑆 for each class +/* Matrix where rows are 𝑠 ∈ 𝑆 and columns are +normalized frequency of 𝑐 ∈ 𝐶 for 𝑠 +*/ +3 𝑓𝑆 ← frequency of classes 𝑐 ∈ 𝐶 (columns) for 𝑠 ∈ 𝑆 (rows) +/* Assign sampling weights to 𝑐 ∈ 𝐶 inversely +proportional to frequency +*/ +4 𝑤𝐶 ← [ +1 +𝑓𝐶 [𝑐] | 𝑐 ∈ 𝐶] +5 samples ← [] +/* Repeat until desired split size is reached +*/ +6 while len(samples) < 𝑁 do +7 +𝑐𝑠 ← Sample(𝐶,𝑤𝐶) +8 +𝑤𝑠 ← [𝑓𝑆 [𝑠,𝑐𝑠] | 𝑠 ∈ 𝑆] +9 +sample ← SampleWithoutReplace(𝑆,𝑤𝑠) +10 +add sample to samples +11 end +12 return samples +Relative Freq. Change +0 +1 +2 +3 +text +link +list item +image +heading +paragraph +line break +generic +grid cell +button +Change in Screen Frequency after Resampling +Figure 11: We calculated the change in frequency (expressed +as a ratio) of screens containing at least one of each element +type after resampling. For example, the number of screens +containing at least one image element is 2.7x more than in +the randomly sampled set. +Relative Freq. Change +0.0 +0.5 +1.0 +1.5 +2.0 +text +link +list item +image +heading +paragraph +line break +generic +grid cell +button +Change in Element Frequency after Resampling +Figure 12: We calculated the change in frequency (expressed +as a ratio) of total number of elements after resampling. +For example, the average screen in the resampled split con- +tains 1.3x more images. Note that is possible for most el- +ement classes to increase in frequency (while not having +other classes experience a proportional decrease) because el- +ement classes are not mutually exclusive, and the resampled +split contains more elements that are assigned multiple tags. +class, which is most likely to be affected by element type imbal- +ance. Table 5 shows that the Web7k-resampled split has higher AP +for classes like "Text Button" and "Image", which had increased +representation after resampling. + diff --git a/DtFQT4oBgHgl3EQfPzZf/content/tmp_files/load_file.txt b/DtFQT4oBgHgl3EQfPzZf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b5163dbce32628bdf9b553859d0b7ae3f68cd410 --- /dev/null +++ b/DtFQT4oBgHgl3EQfPzZf/content/tmp_files/load_file.txt @@ -0,0 +1,1444 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf,len=1443 +page_content='WebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics Jason Wu HCI Institute, Carnegie Mellon University Pittsburgh, PA, USA jsonwu@cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='edu Siyan Wang Wellesley College Wellesley, MA, USA sw1@wellesley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='edu Siman Shen Grinnell College Grinnell, IA, USA shenlisa@grinnell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='edu Yi-Hao Peng HCI Institute, Carnegie Mellon University Pittsburgh, PA, USA yihaop@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='edu Jeffrey Nichols Snooty Bird LLC USA jwnichls@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='com Jeffrey P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Bigham HCI Institute, Carnegie Mellon University Pittsburgh, PA, USA jbigham@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='edu ABSTRACT Modeling user interfaces (UIs) from visual information allows sys- tems to make inferences about the functionality and semantics needed to support use cases in accessibility, app automation, and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Current datasets for training machine learning models are limited in size due to the costly and time-consuming process of manually collecting and annotating UIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We crawled the web to construct WebUI, a large dataset of 400,000 rendered web pages associated with automatically extracted metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We analyze the composition of WebUI and show that while automatically extracted data is noisy, most examples meet basic criteria for visual UI mod- eling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We applied several strategies for incorporating semantics found in web pages to increase the performance of visual UI un- derstanding models in the mobile domain, where less labeled data is available: (i) element detection, (ii) screen classification and (iii) screen similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' KEYWORDS Dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' UI Modeling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Computer Vision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Transfer Learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Web Semantics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Computational Interaction ACM Reference Format: Jason Wu, Siyan Wang, Siman Shen, Yi-Hao Peng, Jeffrey Nichols, and Jeffrey P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Bigham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' WebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the 2023 CHI Conference on Human Fac- tors in Computing Systems (CHI ’23), April 23–28, 2023, Hamburg, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' ACM, New York, NY, USA, 14 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1145/3544548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='3581158 1 INTRODUCTION Computational modeling of user interfaces (UIs) allows us to under- stand design decisions [15, 28], improve their accessibility [55], and automate their usage [7, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Often, these systems must interact with UIs in environments with incomplete or missing metadata (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Copyrights for third-party components of this work must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For all other uses, contact the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' CHI ’23, April 23–28, 2023, Hamburg, Germany © 2023 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' ACM ISBN 978-1-4503-9421-5/23/04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1145/3544548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='3581158 mobile apps authored with inaccessible UI toolkits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' This presents many challenges since it necessitates that they reliably identify and reason about the functionality of the UI to support downstream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Visual modeling of UIs, which has shown to be a promising solution, predicts information directly from a screen- shot using machine learning models and introduces no additional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Building the datasets needed to train accurate visual models involves collecting a large number of screenshots paired with their underlying semantic or structural representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Recent efforts to collect datasets [15, 55] for data-driven modeling have focused on mobile apps, which are typically manually crawled and annotated by crowdworkers since they are often difficult to automate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' This process is both time-consuming and expensive — prior work has estimated that collecting a dataset of 72,000 app screens from 10,000 apps took 5 months and cost $20,000 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Because of this, datasets for visual UI modeling are limited in size and can be prohibitively expensive to keep updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The web presents a possible solution to UI data scarcity since web pages are a promising source of data to bootstrap and enhance visual UI understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In contrast to mobile UIs, web UIs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', web pages) are much easier to crawl since they are authored in a unified parsable language (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', HTML) that typically exposes semantics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', links and listeners) necessary for automated navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The same web page can also be viewed in many different viewports and display settings, which makes it possible to collect a large dataset of UIs rendered on a variety of devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', a smartphone or tablet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In addition, web browsers offer several facilities to extract visual, semantic, and stylistic information programmatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In particular, web conventions, such as the semantic HTML and the ARIA initiatives, while not always adopted, constitute a large, if potentially noisy, source of annotations for UI elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Finally, the web offers a virtually unlimited supply of data and has already been employed as a data source for large-scale machine learning [23, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We explore the possibility of automatically collecting and labeling a large dataset of web UIs to support visual UI modeling in other domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', mobile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Compared to previous web datasets [28], our dataset is much larger, more recent, and contains semantic information needed to support common visual UI understanding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='13280v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='HC] 30 Jan 2023 CHI ’23, April 23–28, 2023, Hamburg, Germany Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In this paper, we show that a large dataset of automatically collected web pages can improve the performance of visual UI Understanding models through transfer learning techniques, and we verify this phenomenon for three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We first describe the platform that we built to crawl websites automatically and scrape relevant visual, semantic, and style data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our crawler visited a total of approximately 400,000 web pages using different simulated devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' WebUI, the resulting dataset is an order of magnitude larger than other publicly available datasets [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Next, we analyzed our dataset’s composition and estimated data quality using several automated metrics: (i) element size, (ii) element occlusion, and (iii) layout responsiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We found that most websites met basic criteria for visual UI modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Finally, we propose a framework for incorporating web semantics to enhance the performance of existing visual UI understanding approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We apply it to three tasks in the literature: (i) element detection, (ii) screen classification and (iii) video screen similarity and show that incorporating web data improves performance in other target domains, even when labels are unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' To summarize, our paper makes the following contributions: (1) The WebUI dataset, which consists of 400,000 web pages each accessed with multiple simulated devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We collected WebUI using automated web crawling and automatically associated web pages with visual, semantic, and stylistic information that can generalize to UIs of other platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' (2) An analyis of the composition and quality of examples in WebUI for visual UI modeling in terms of (i) element size, (ii) element occlusion, and (iii) website layout responsiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' (3) A demonstration of the usefulness of the WebUI dataset through three applications from the literature: (i) element detection, (ii) screen classification and (iii) screen similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We show that incorporating web data can lead to perfor- mance improvements when used in a transfer learning set- ting, and we verified its improvement for our three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We envision that similar approaches can be used for other tasks common in visual UI understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Furthermore, we show that models trained on only web data can often be directly applied to other domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', Android app screens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' All code, models, and data will be released to the public to encourage further research in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1 Datasets for UI Modeling There have been several datasets collected to support UI modeling, mostly in the mobile domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Several datasets have been collected to support training specialized models [26, 40, 44] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The AMP dataset consists of 77k screens from 4,068 iOS apps and was originally used to train Screen Recognition, an enhanced screen reader [55], but has also been extended with additional pairwise annotations to support automated crawling applications [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The largest publicly available dataset Rico, which consists of 72K app screens from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='7K Android apps, was collected using a combination of automated and human crawling [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' It captures aspects of user interfaces that are static (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', app screenshots) and dynamic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', animations and user interaction traces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Rico has served as the primary source of data for much UI understanding research and it has been extended and re-labeled to support many downstream applications, such as natural language interaction [7, 32, 49] and UI retrieval for design [6, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Nevertheless, Rico has several weaknesses [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Several works have identified labeling errors and noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', nodes in the view hierarchy do not match up with the screenshot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' To this end, efforts have been made to repair and filter examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Enrico first randomly sampled 10,000 examples from Rico then cleaned and provided additional annotations for 1460 of them [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The VINS dataset [6] is a dataset for UI element detection that was created by collecting and manually taking screenshots from several sources, including Rico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The Clay dataset (60K app screens) was generated by denoising Rico through a pipeline of automated machine learning models and human annotators to provide element labels [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Rico and other manually annotated datasets are expensive to create and update, and thus, models trained on them may exhibit degraded performance on newer design guidelines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', Material Design is an updated design look for Android).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, Rico was collected in early 2017 and has yet to see any update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Finally, many of these datasets focus on one particular platform (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', mobile phone) and therefore may learn visual patterns specific to the screen dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, “hamburger menus” are usually used in mobile apps while desktop apps may use navigation bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In our work, we scrape the web for examples of UIs, which addresses some drawbacks (high cost, difficult to update, device- dependent) of current datasets but not others (dataset noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The closest to our work is Webzeitgeist [28], which also used automated crawling to mine the design of web pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' To support design mining and machine learning applications, Webzeitgeist crawled 103,744 webpages and associated web elements with extracted properties such as HTML tag, size, font, and color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' This work is primarily used for data-driven design applications and does not attempt to transfer semantics to other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We also collect multiple views of each website and query the browser for accessibility metadata, which can further facilitate UI modeling applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2 Applications of UI Datasets Applications that operate and improve existing UIs must reliably identify their composition and functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Originally, many relied on pixel-based or heuristic matching [1, 18, 43, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The introduc- tion of large UI datasets, such as those previously discussed, have provided the opportunity to learn more robust computational mod- els, especially those from visual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The goal of this paper is to improve the performance of these computational models by lever- aging a large body of web data and its associated semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' There have been many efforts to learn the semantics of UIs [37, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In this paper, we focus on three modeling tasks at the (i) element (ele- ment detection), (ii) screen (screen classification), and (iii) app-level (screen similarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Element detection identifies the location and type of UI widgets from a screenshot and has applications in accessibility metadata repair [55], design search [6], and software testing [12, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Labeled datasets for element detection exist [6, 15, 30, 55];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' however they are quite small compared to other datasets for object detection [36] which contain an order of magnitude more examples (330K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We found that incorporating our web UI dataset (400K examples) in a WebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics CHI ’23, April 23–28, 2023, Hamburg, Germany pre-training phase led to performance benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Other work involves modeling UIs at a higher level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', screen-level) to reason about the design categorization [29] and purpose [49] of a screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Similarly, datasets with screen-level annotations of UIs are much smaller than others used in the CV literature [17] so we used additional web data to improve accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Finally, we investigated screen similarity, a task that reasons about multiple UI inputs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', frames of a video recording), where no publicly available labeled data exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We found that models trained on related web semantics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', URL similarity) were able to successfully generalize to mobile screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In summary, our paper shows that applying examples from the web and relevant machine learning techniques can improve the performance of computational models that depend on UI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='3 Related Machine Learning Approaches We briefly introduce and summarize three machine learning ap- proaches that we apply in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Broadly, they fall under a body of research known as “transfer learning” which uses knowledge from learning one task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', web pages) to improve performance on another (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', mobile app screens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Inductive transfer learning is a technique that improves model performance by first “pre-training” a model on a related task, typi- cally where a lot of data is available [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Once the model converges on the first task, its weights are used as a starting point when train- ing on the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Labeled data is required for both the source and target domains, although it is possible that there are fewer target examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In some cases, labeled data are missing for either the source or target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' If source labels are unavailable, semi-supervised learning (SSL) can be applied to take advantage of unlabeled data to improve performance [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, WebUI doesn’t contain any labels for screen type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', login screen, register screen), but we’d like to use it to improve prediction accuracy on a small number of annotated Android app screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In our work, we apply a form of SSL known as “self-learning” [9], where a UI classification model it- eratively improves its performance by generating pseudo-labels for an unlabeled dataset, then re-training itself using high-confidence samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Finally, to support use-cases where target labels are unavailable, we apply unsupervised domain adaptation (UDA) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In many cases, visual UI models trained on web data can be directly used on any screenshot (including Android and iOS apps), and UDA improves the performance and robustness of models to domain changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' This type of knowledge transfer is particularly interesting because it enables us to explore the feasibility of new UI under- standing tasks (without manually annotating a large number of examples) and bring some benefits of web semantics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', semantic HTML) to other platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 3 WEBUI DATASET We introduce the WebUI dataset, which we construct and release to support UI modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The WebUI dataset is composed of 400,000 web pages automatically crawled from the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We stored screen- shots and corresponding metadata from the browser engine, which serve as annotations of UI element semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Because the collec- tion process is highly automated, our final dataset is an order of Database Crawling Coordinator Crawler Web workers assign URLs to worker send back crawled URLs Request and collect data Figure 1: Overview of our crawling architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' A crawl- ing coordinator contains a queue of URLs to crawl and as- signs them to workers in a crawler pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Workers asyn- chronously process URLs by visiting them in a automated browser, scraping relevant metadata, then uploading them to a cloud database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' magnitude larger than other publicly available ones (Figure 4) and can be more easily updated over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In this section, we give an overview of our web crawling architec- ture, analyze the composition of our dataset, and provide evidence that it can support visual UI modeling for other platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1 Web UI Crawler 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1 Crawling Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' To collect our dataset, we implemented a parallelizable cloud-based web crawler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our crawler consists of (i) a crawling coordinator server that keeps track of visited and queued URLs, (ii) a pool of crawler workers that scrapes URLs using a headless browser, and (iii) a database service that stores uploaded artifacts from the workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The crawler worker is implemented using a headless framework [3] for interfacing with the Chrome browser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Each crawler worker repeatedly requests a URL from the coordinator server, which keeps global data structures for visited and upcoming URLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The crawler worker includes some simple heuristics to automatically dismiss certain types of popups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', GDPR cookie warnings) to help it access page content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We seeded our coordinator using a list of websites that we hy- pothesized would lead to diverse examples of web pages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', link aggregation websites and design blogs) and ones that we expected to have high-quality accessibility metadata (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', government websites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' A full list of our seed websites can be found in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We explored several crawling policies and eventually settled on one that encourages diverse exploration by inversely weighting the probability of visiting a URL by its similarity to the visited set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, if the crawler previously visited http://example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='com/user/ alpha, it would be less likely to subsequently visit http://example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' com/user/beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We set a minimum probability so that it is possible to re-visit links to support additional types of analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', temporal changes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The coordinator organizes upcoming (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', queued) URLs by their hostname, (i) selects a hostname randomly with uniform probability, and then (ii) selects a URL using its assigned probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Empirically, we found this technique to be effective at avoiding CHI ’23, April 23–28, 2023, Hamburg, Germany Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1280x720 1366x768 1536x864 1920x1080 iPhone iPad Figure 2: Screenshots from a web page accessed using 6 dif- ferent devices: 4 desktop resolutions, a smartphone, and a tablet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' By requesting a responsive web page at different reso- lutions, we induce several layout variations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', navigation and hero button).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' crawler traps, which are websites that cause automated crawlers to get stuck in endless loops navigating within the same site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2 Data Collected from a Web Page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We used a pool of crawler workers to crawl web pages in parallel, and we visited each URL with multiple simulated devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We collected several types of se- mantic information by querying the rendering and accessibility engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We set a timeout limit of 6 minutes for each URL, so some web pages were not visited by all simulated devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Simulated Devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We sampled each web page with 6 sim- ulated devices: 4 of the most common desktop resolutions [4], a tablet, and a mobile phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Devices are simulated by setting the browser window resolution and user agent to match the goal device, both of which may affect the page’s content and rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Screenshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our crawler worker captured two types of screen- shots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', visual data) from websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We captured a viewport screenshot, with fixed image dimensions, and a full-page screenshot, with variable height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Images were saved using lossy compression to save storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' While compression can introduce some artifacts, previous work [19] suggests that the effect on deep learning model performance is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Accessibility Tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We used a browser automation library to query Chrome’s developer tools to retrieve an accessibility tree for each page [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The accessibility tree is a tree-based represen- tation of a web page that is shown to assistive technology, such as screen readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The tree contains accessibility objects, which usually correspond to UI elements and can be queried for properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', clickability, headings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Compared to the DOM tree, the accessibility tree is simplified by removing redundant nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=',
tags that are only used for styling) and automatically populated with semantic information via associated ARIA attributes or inferred from the node’s contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The browser generates the accessibility tree using a combination of HTML tags, ARIA attributes, and event listeners (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', click handlers) to create a more consistent semantic representation of the UI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For instance, there are multiple ways to create a button (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', a styled div) and the accessibility tree is intended to unify all of these to a single button tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Layout and Computed Style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For each element in the accessi- bility tree, we stored layout information from the rendering engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Specifically, we retrieved 4 bounding boxes relevant to the “box model”: (i) the content bounding box, (ii) the padding bounding # of elements (in thousands) 0 25000 50000 75000 100000 125000 text link list item image heading paragraph line break generic grid cell button Frequency of Common Element Types Figure 3: 10 most common element types in the WebUI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Element types are based on automatically computed roles, which are not mutually exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Text is the most common type, but many types offer semantic information about what text is used for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g, a heading, paragraph or link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' # of UIs 0 100,000 200,000 300,000 400,000 500,000 Enrico VINS Clay Rico Screen Recognition Webzeitgeist WebUI UI Dataset Size Figure 4: Comparison of WebUI to existing UI datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We- bUI contains nearly 400,000 web pages and is nearly one or- der of magnitude larger than existing datasets available for download (Enrico, VINS, Clay, Rico).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Each web page also con- tains multiple screenshots captured using 6 simulated de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' box, (iii) the border bounding box, and (iv) the margin bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Each element was also associated with its computed style in- formation, which included font size, background color and other CSS properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2 Dataset Composition The WebUI dataset contains 400K web UIs captured over a period of 3 months and cost about $500 to crawl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We grouped web pages together by their domain name, then generated training (70%), validation (10%), and testing (20%) splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' This ensured that similar pages from the same website must appear in the same split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We created four versions of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Three of these splits were generated by randomly sampling a subset of the training split: Web-7k, Web-70k, Web-350k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We chose 70k as a baseline size, since it is approximately the size of existing UI datasets [15, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We also generated an additional split (Web-7k-Resampled) to provide a small, higher quality split for experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Web-7k-Resampled was generated using a class-balancing sampling technique, and we removed screens with possible visual defects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', very small, occluded, or invisible elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' More information about how this set was generated can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The validation and test split was always kept the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1 Comparison to Existing Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' WebUI is an order of magni- tude larger than existing datasets used for UI understanding (Figure 4) and provides rich semantic and style information not found in mobile datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' WebUI focuses on the static properties of web pages and does not store page loading times or element animations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 业Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat About Contribute ChannelNamespaces Guides FAQ Connect Providinga communityplatform forFreeand open- source software and peer directed projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Connectby pointingyour IRC clientto irc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='chat:6697 (TLS) Choosing an IRC client Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat Channel Namespaces Happy Birthday, Libera Chat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 19thMay2022byLiberastaff Helloeveryone,todaywe celebratethe anniversary of Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat going public!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='业Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat About Contribute Channel Namespaces Guides FAQ Connect Providing a community platform for free and open- source software and peer directed projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Connect by pointing your IRC client to irc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='chat:6697 (TLS) Choosing anIRC client Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat Channel Namespaces Happy Birthday, Libera Chat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 19thMay 2022 by Libera staff Hello everyone, today we celebrate the anniversary of Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat going public!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Wherewearecomingfrom Exactly one year ago Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat was unveiled as a real time communication and collaboration servicefor freeandopen-sourcesoftware,peer-directed projects, openly licensed content and collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Starting from scratch wemanaged, just within a fewmonths, tobecome the largest IRC network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='业Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat About Contribute Channel Namespaces Guides FAQ Connect Providinga communityplatform forFreeandopen- source softwareandpeer directedprojects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Connect bypointingyour IRC clientto irc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='chat:6697 (TLS) Choosing an IRC client Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat Channel Namespaces Happy Birthday, Libera Chat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 19thMay2022by Libera staff Hello everyone, today we celebrate the anniversary of Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat going public!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Wherewearecomingfrom业Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat About Contribute Channel Namespaces Guides FAQ Connect Providing a community platform For Free and open source software and peer directed projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Connect by pointing your IRC client to irc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='chat:6697 (TLS) Choosing an IRC client Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat Channel Namespaces Happy Birthday, Libera Chat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 19th May 2022 by Libera staff Hello everyone, today we celebrate the anniversary of Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat going public!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Wherewearecomingfrom Exactly one year ago Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat was unveiled as a real time communication and collaboration service for free and open-source software, peer-directed projects, openly licensed content and collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Starting from scratch we managed, just within a fewmonths, to become the largestIRC network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Starting from scratch, we managed to gain around 5o o00 users in just a month and a half, a number which has been mostly steady since.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' with regard to channels we had roughly 15 00o channels formed within half a month, compared to the usercount this number is still growing, but the curve Flattened itself a bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' You can see detailed graphs over at https://netsplit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='de/networks/statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='net=Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat We also saw many communities and projects migrating over to Libera from other places in the first few days, counting 250 in just one week and 500 after a monthWLibera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat Navigation Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat Providing a community platform For Free and open-source software and peer directed projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Connect by pointing your IRC client to irc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='chat:6697 (TLS) ChoosinganIRC client Channel Namespaces Happy Birthday, Libera Chat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 19th May 2022 by Libera staffLibera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat About Contribute Channel Namespaces Guides FAQ Connect Providing a community platform for Free and open- source software and peer directed projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Connect by pointing your IRC client to irc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='chat:6697 (TLS) Choosing an IRC client Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat Channel Namespaces Happy Birthday, Libera Chat!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 19th May 2022 by Libera staff Hello everyone, today we celebrate the anniversary of Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat going public!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Where we are coming from Exactly one year ago Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat was unveiled as a real time communication and collaboration service for Free and open-source software, peer-directed projects, openly licensed content and collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Starting From scratch we managed, just withinafewmonths,tobecomethelargestIRCnetwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Starting From scratch,we managed to gain around 5oooo users in just amonth and a half, a number which has been mostly steady since.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' With regard to channels we had roughly 15ooo channels formed within half a month, compared to the usercount this number is still growing,but the curve flattened itself a bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='You can see detailed graphs over at https://netsplit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='de/networks/statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='net=Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat We also saw many communities and projects migrating overto Libera from other places in the first fewdays, counting 25o in just one week and5o0 aftera month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Today we are hosting roughly 95o projects and communities, and that number is still growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We are hoping to reach the 1oooth registration soon!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' All these communites are quitediverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat services arenot onlyused by major Free/open source operating systems and well known,world wide operating institutions such as the Wikimedia Foundation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' we also have local Linux User Groups, the hackspace around the corner hacking on whimsical gadgets and liberating your hardware or someones scratch-your-own-itch image viewer that call Libera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Chat their home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='WebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics CHI ’23, April 23–28, 2023, Hamburg, Germany We analyzed the makeup of web UIs and compared them to mobile UIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The distribution of UI types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Login, News, Search) in WebUI are also likely to be different than mobile data, since many web pages are primarily hypertext documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We extracted elements from the accessibility tree and categorized them using their computed accessibility role and the role of any singleton parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, a clickable image is created in HTML by surrounding an image () element with an anchor element ().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Thus, it is possible for elements to be assigned to multiple classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Figure 3 shows the frequency of element types in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Similar to prior work [55], we find that text is the most common element in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' However, we find limited overlap between the rest of the label set, possibly due to the nature of web data and the mutually exclusive nature of existing label sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' On average, there were 60 elements on a web UI, 30 of which were visible in the viewport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' This is more than the number of elements on mobile app screens, which prior work estimated to be around 25 per screen, although this may in part be due to differences in segmentation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', a single Rich Text Field on Android can contain differently formatted text while on HTML they would broken up into different tags).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' On average, there were also more clickable elements per web page (20 on web pages vs 15 “interactable" elements on Android apps), likely due to the prevalence of hyperlinks on the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2 Dataset Quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Compared to manually labeled examples, automatically extracted annotations can contain errors that impact modeling performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We conducted an analysis on a small, ran- domly sampled data from our dataset (1000 web pages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' While there are numerous possible defects, we focus on three that we believe are most relevant to data quality: (i) element size, (ii) element occlusion, and (iii) website responsiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our analysis is primarily focused on quantifying possible defects but not reparing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Previous work [30, 44] has explored automated methods for correcting mis- matched labels and occluded elements, and we expect the overall quality of WebUI could be improved if these were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='. Element Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Element size refers to the dimensions of an anno- tated object in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, if a bounding box annotation surrounds an object that is too small relative to the image resolution, it may be difficult for a model to identify the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The average area of bounding boxes in our data is approximately 14000𝑝𝑥2, but this may have been influenced by short segments of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The Web Content Accessibility Guidelines (WCAG) guideline for target size also recommends that interactable elements have a minimum size of 44 by 44 pixels, so that they can be easily selected by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In our dataset, one third of interactable elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', elements tagged as links or button) were smaller than this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Element Occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Element occlusion occurs when one object partially or completely covers another in a screenshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Occluded el- ements are detrimental to visual modeling since they may represent targets that can be impossible to predict correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We quantified the occlusion rate by counting the number of screens with overlapping leaf elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We found that 18% of screens in our sampled split contained overlapping leaf elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' However, of the overlapping elements, only a third of them were occluded by more than 20% of their total area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Responsive Websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Website responsiveness relates to how well a web page adapts to different screen viewports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Since we sim- ulated multiple devices for each web page, responsive websites are likely to produce more variation in their layouts than unresponsive ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' To measure responsiveness, we automatically computed met- rics included in the Chrome Lighthouse tool for estimating layout responsiveness: (i) responsiveness of content width to window size and (ii) the use of a viewport meta tag, which is needed for proper mobile rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' From our analysis we found that 70% and 80% of processed web pages met the first, and second criteria, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In summary, our analysis suggests that most web pages in our dataset meet some basic requirements for visual UI modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Given the reliance of our data collection on extracted accessibility meta- data, we expect high quality examples to adhere to good accessibility practices, such as those outlined by WCAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' However, considering the inaccessibility of the web and that many criteria are difficult to verify automatically, we also expect many web pages to vio- late some of these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' There are other desirable properties for dataset quality that we did not check, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', the accurate use of se- mantic HTML tags, ARIA tags, and tightness of element bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' These properties were harder to verify automatically, since they require knowledge of developer intention and associated tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In our analysis, we only attempt to identify possible defects, and we did not attempt to remove or repair samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' This could be a direction for future work to improve dataset quality [8, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 4 TRANSFERRING SEMANTICS FROM WEB DATA We hypothesized that web data is similar and relevant to modeling other types of UIs from their pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In this paper, we are specif- ically interested in the mobile domain, as mobile apps often lack metadata and can only be reliably understood from their visual appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In many cases, manually-annotated mobile datasets are small, and in some cases, labels are completely unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We used transfer learning to apply our dataset to three existing tasks in the UI understanding literature: (i) element detection, (ii) screen classification, and (iii) screen similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Table 1 shows downstream applications where UI understanding tasks can benefit from web data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Because each task contains different constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', presence of labeled target data) it is difficult to apply a single strategy to serve all use-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, inductive transfer learning typi- cally requires labels in both the pre-training and fine-tuning phase is impossible to apply to a setting where target labels are unavail- able (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', screen similarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We expect our three transfer learning strategies to be applicable to most future use-cases, since they span all combinations of labeled data availability (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1 Element Detection Element detection requires a machine learning model to identify the locations and types of UI elements from a screenshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Often these models are based on object detection frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Element detection is an example of a task where labeled data is available in both the source and target domain (albeit fewer exam- ples of mobile screens), so it is possible to employ inductive transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The WebUI dataset contains the locations of elements that we scraped from the website accessibility tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Element types are CHI ’23, April 23–28, 2023, Hamburg, Germany Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Table 1: Table of strategies for transferring semantics from web pages to other types of UIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We explored scenarios where labeled data is missing in either domain by applying three strategies: (i) finetuning, (ii) semi-supervised learning, and (iii) domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Approach Finetuning Semi-supervised Learning Domain Adaptation Application Element Detection Screen Classification Screen Similarity Web (Source) Y N Y Mobile (Target) Y Y N Web Data VINS Element Detector Element Detector Step 1: Pre-training Step 3: Fine-tuning Step 2: Weight initialization Figure 5: We applied inductive transfer learning to improve the performance of a element detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' First, we pre- trained the model on web pages to predict the location of nodes in the accessibility tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Then, we used the weights of the web model to initialize the downstream model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Finally, we fine-tuned the downstream model on a smaller dataset consisting of mobile app screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' inferred from the HTML tags and the ARIA labels [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We show that this training strategy results in improvements to element detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1 Model Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We primarily followed the details provided by VINS [6] to implement our element detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The VINS dataset, which we used for training, is composed of 4800 annotated UI screenshots from various sources such as design wireframes, Android apps, and iOS apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Since the authors did not release official data splits, we randomly partitioned the data into training (70%), validation (15%), and testing (15%) sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' This specific split ratio was chosen since it has been used in other UI modeling work [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The paper identifies 11 primary UI component classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' however the released raw dataset includes a total of 22 class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For the extraneous labels, we either tried to merge them with the 11 primary labels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', “Remember Me" merged with “Check Box") or assigned them to an “Other" class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', “Map") if no good fit was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Instead of the SSD object detection model [38] used by VINS, we opted to start from the more recent FCOS model architecture [48], since we found it was easier to modify to support multi-label training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Previous element detection work [6, 12, 55] trained models to assign one class label (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', Button, Text field) to each detected element in the screenshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' To take advantage of multiple, nested definitions of web elements in our dataset, we trained the object detection model to predict multiple labels for each bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Figure 5 illustrates the overall training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In the pre-training phase, the element detection model is trained on a split of the We- bUI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Due to cost and time constraints, we trained all element detection models for a maximum of 5 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We also used early stop- ping on the validation metric to reduce the chance of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Afterwards, a specific part of the model was re-initialized (the ob- ject classification head) to match the number of classes in the VINS dataset before it was fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We found it difficult to modify the original SSD architecture to support the multi-label pre-training, so we only followed the original training from scratch procedure described in the paper as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2 Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Table 2 shows the performance of each model con- figuration on the VINS test set, and we show that our updated configurations lead to significant performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our primary performance metric for this task was the mean average pre- cision (mAP), which is a standard metric used for object detection models that takes into the accuracy of bounding box location (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', how closely the predicted box overlaps with ground truth) and clas- sification (prediction of object type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The mAP score is calculated by computing an individual average precision (AP) score for each possible element class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', Text, Check Box), which represents the object detector’s accuracy in detecting each object class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The AP scores are averaged to produce the mAP score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We calculated the mAP score over classes that could be mapped to the original label set in the paper [6] i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', we excluded the “Other" class where there was no clear mapping to the original set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We calculated the un- weighted mean between class APs, which assigns equal importance to common and rare element types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our best model configuration performed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='14 better than the baseline in terms of mAP score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' While the largest source of improvement over the baseline con- figuration (SSD) came from the updated FCOS model architecture, our fine-tuning procedure contributed to gains as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Specifi- cally, we note that pre-training with more examples led to better performance (around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='04 mAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Depending on the downstream application of the element detection model, this improvement could lead to better user experience but would require further validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, a screen reader [55] does not require tight bounding boxes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' however, it would benefit from detecting more (small) el- ements on the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Query-based design search [6] could also retrieve more relevant examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Although we followed the original training procedure as closely as possible, we were unable to reach the mAP score reported in the original VINS paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' This can be attributed to (i) our use of different randomized splits and (ii) differences in mappings between class labels from the raw data to the 11 primary classes, which were not provided in the previously released code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Nevertheless, since we used the same splits and class mappings across all of our model configurations, we expect the relative performance improvements to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We also investigated the zero-shot performance of element de- tectors trained only on web data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', without fine-tuning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' It is difficult to compute performance quantitatively, since the label sets WebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics CHI ’23, April 23–28, 2023, Hamburg, Germany Table 2: Element detection performance (11 object classes) for different model configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Pre-training on more web screens led to better performance on mobile screens af- ter fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Model Configuration mAP SSD (Random Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='6737 FCOS (Random Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='7739 FCOS (Pre-trained on Web7k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='7877 FCOS (Pre-trained on Web7k-Resampled) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='7961 FCOS (Pre-trained on Web70k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='7921 FCOS (Pre-trained on Web350k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='8115 between the web and mobile datasets do not directly overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' How- ever, we provide qualitative evidence that zero-shot learning could be successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Figure 6 shows the output of a web model when run on mobile app screens from Rico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We conducted minimal prepro- cessing, such as cropping out the Android system notification bar and the navigation soft buttons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In many cases, the web analogs of mobile text and image elements are detected accurately, which suggests that some element classes have consistent appearance across platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Interestingly, some web classes such as links and headings are also detected in the image, which could be used to infer new semantics such as clickability [47] and navigation landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2 Screen Classification Classifying screen type or functionality from a screenshot can be useful for design analysis and automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Previously, small amounts of data have been collected and annotated for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Enrico [29] is an example of a dataset (1460 samples, subset of Rico [15]) where each screenshot is assigned to one of 20 mutually- exclusive design categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Because of the dataset’s small size, it is challenging to train accurate deep learning classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' While our web dataset is large, it also does not have the screen- type annotations, and thus it is not possible to employ the same pre-training strategy that was used for element detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Instead, we applied a semi-supervised learning technique known as self-training [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Self-training is a process that improves model performance by iteratively labeling and re-training on a large source of unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We investigated the effects of using WebUI as the unlabeled dataset and show that doing so improves overall screen classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1 Model Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Figure 7 shows our procedure for incorporating WebUI data into our model training via self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' First, we trained screen classifier based on the VGG-16 archi- tecture with batch normalization and dropout [45], as described by the Enrico paper [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Since official training, validation, and testing splits were not provided, we randomly generated our own (70%/15%/15%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' This model was trained only on data from the Enrico training split and served as the teacher classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Next, the teacher model was used to generate “soft" pseudo-labels for screenshots in the WebUI dataset, where each sample was mapped to a vector containing probabilities for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We followed the procedure used by Yalniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [53] to keep only the top K most confident Table 3: Classification accuracy (across 20 classes) for dif- ferent configurations of our screen classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In- creasing the amount of data used with our semi-supervised learning method led to increased accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Model Configuration Accuracy VGG-16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='4737 Noisy ResNet-50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='4649 Noisy ResNet-50 (Rico) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='4956 Noisy ResNet-50 (Web7k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='4864 Noisy ResNet-50 (Web7k-Resampled) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='4868 Noisy ResNet-50 (Web70k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='5175 Noisy ResNet-50 (Web350k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='5263 labels for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' To select K, we first randomly sampled a small subset of 1000 web pages from our dataset and performed a param- eter search to find the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Based on our experiments, we found that a value of 10% of the total dataset size led to good performance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', we set K=700 for the Web-7k split).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Finally, we trained a student classifier on a combination of the original and automatically generated labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We employed a specific type of self-training known as Noisy Student Training [52], which involves injecting noise into the student model’s training process so that it becomes more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Two types of noise are used in this process: (i) input noise, which is implemented via random data augmenta- tion techniques and (ii) model noise, which is implemented with dropout [46] and stochastic depth [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Because stochastic depth can only be applied to model architectures with residual blocks, we used an architecture based on ResNet-50 [25] instead of VGG-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2 Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Overall, we found that applying self-training to in- corporate additional unlabeled data led to consistent performance improvements (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The best classifier using WebUI data was 5% more accurate than the baseline model, which was only trained with the Enrico dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our baseline VGG-16 model performed considerably worse than the originally reported results [29] but achieved similar accuracy to another reproduction of the work [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The performance difference could be attributed to differences in randomized splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Since we used the same splits across all condi- tions, we expect relative performance differences to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' To investigate the effects of using a new model architecture, we also trained a Noisy ResNet-50 (architecture used by the student model) on the Enrico dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The resulting classifier performed relatively poorly (worse than the baseline model), since the modifications introduced (dropout and stochastic depth) require more data to train effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The primary source of improvement stems from the inclusion of additional unlabeled data during the training process, which led to a more generalizable student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We observed that the small size of the Enrico dataset (1460 samples) quickly led to overfitting during training and limited overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Semi-supervised learning techniques, such as self-training, allow training on a much larger number of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We found that model accuracy improved when we incorporated more unlabeled examples, both from WebUI and Rico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' CHI ’23, April 23–28, 2023, Hamburg, Germany Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Figure 6: Output of our element detection models run on two app screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In many cases, detections from our web-only model (Blue) coincide with ones from our fine-tuned model (Orange), which suggests some zero-shot transfer capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Predicted tags from the web-only model also provide additional metadata corresponding to clickability (link) and heading prediction (heading);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' however, the predicted bounding boxes are often less tight than the fine-tuned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Enrico Web Data Teacher Classifier Student Classifier Step 1: Training Step 3: Noisy Training Step 2: Pseudo-labels Figure 7: We applied semi-supervised learning to boost screen classification performance using unlabeled web data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' First, a teacher classifier is trained using a “gold" dataset of labeled mobile screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Then, the teacher classifier is used to generate a “silver" dataset of pseudo-labels by running it on a large, unlabeled data source (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', web data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Finally, the “gold" and “silver" datasets are combined when training a student classifier, which is larger and regularized with noise to improve generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' This process can be repeated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' however, we only perform one iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='3 Screen Similarity Web Data Similarity Model RICO UI Similarity Mobile Examples Unsup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Domain Adaptation Figure 8: We used unsupervised domain adaptation (UDA) to train a screen similarity model that predicts relationships between pairs of web pages and mobile app screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The training uses web data to learn similarity between screen- shots using their associated URLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Unlabeled data from Rico is used to train an domain-adversarial network, which guides the main model to learn features that transferrable from web pages to mobile screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Identifying variations within the same screen and detecting tran- sitions to new screens are useful for replaying user interaction traces, processing bug reports [13], and automated app testing [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' To model these properties and understand how multiple screens from an application relate to each other, previous work [20, 34] has sought to differentiate between distinct UIs and varia- tions of the same UI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, the same checkout screen may appear different based on the number and types of products added to the cart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Common screen interactions such as scrolling and in- teraction with expandable widgets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', menus, dialogs, keyboards, and notifications) may also alter the visual appearance of a screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Visual prediction reduces system reliance on accessibility metadata, which may be missing or incomplete, and further extends the ap- plications of these models, as they can process video recordings of user interactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', reproducing bug reports) [5, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Previous work [20] opted to manually annotate a dataset of more than one thousand iPhone applications that were manually “crawled" by crowdworkers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' however, the dataset was not released to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' As a weak source of annotation, we used web page URLs to automatically label page relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Since no labeled data is available in the mobile domain, we employed domain-adversarial network training [22], a type of unsupervised domain adaptation (UDA), to encourage the model to learn transferrable features from the web domain that might apply to the mobile domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Note that while it is possible to apply the semi-supervised learning strategy (which was used for the screen classification task) in reverse, it may be less effective, since the unlabeled dataset (mobile UIs) is smaller than the labeled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1 Model Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We followed previous work [20] and used a ResNet-18 [25] model trained as a siamese network [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The siamese network uses the same model to encode two inputs, then compares them in feature space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', their embeddings) to decide if they are different variations of the same UI screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our approach is different from the method proposed by previous work [13], which img Weather Alert SevereThunderstorm inyour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Text,heading Text,paragraph sendyouweatheralertsbasedon ooatio rowid img Text,heading vedlleAle Text AlertiLocatior Text,button DoneImage WeatherAlert SevereThunderstorm inyour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Never miss a weather alert Text Wewouldalsoliketosendyouweatneralertsbasedona locationormultiplelocationsyouprovide Switch Text veatner Alers Text Text Icon ocatior anFanciscoA Text Button Doneimg,link,Text Text,heading Sun & Moon Text,heading Amino img Loving Pokemon sun and moon?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Thenjointhiscommunity Text,link Sign Up Text Log In Text JuPyuuT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='+ I:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' vicend PrIvacvIPolcIIcon Image Text bun & Moon Text m Image Loving Pokemon sun and moon?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content="Thenjointhiscommunity Text Button Sign Up 4x Text ainnok'dn6imm ermeof Service andPrivacvPoliciWebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics CHI ’23, April 23–28, 2023, Hamburg, Germany Table 4: Classification performance (same-screen vs new- screen) of our screen similarity models evaluated on pairs of screens from our web data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Performance increased when the model was trained on more data and slightly decreased when trained with the UDA objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Model Configuration F1-Score ResNet-18 (Web7k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='7097 ResNet-18 UDA (Web7k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='7184 ResNet-18 (Web7k-Resampled) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='7368 ResNet-18 UDA (Web7k-Resampled) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='7191 ResNet-18 (Web70k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='8222 ResNet-18 UDA (Web70k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='8193 ResNet-18 (Web350k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='9630 ResNet-18 UDA (Web350k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='9500 applies random data augmentations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', blurring, rotation, trans- lation) to screenshots to create same-screen pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Instead, we ran- domly sampled pairs of screenshots from our web data for training, with balanced probability for same-screen and new-screen pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Same-screen pairs were generated by finding screenshots with the same URL but accessed at different times or simulating page scrolls on a full-page screen capture by sliding a window vertically along the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Note that occasionally, simulated page scrolls and access- ing the same web page at different times still produced identical or nearly identical screenshots, so in our test set, we filtered these out using perceptual hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Different-screen pairs were generated both by sampling screenshots from within the same domain but with different URL path, and by sampling screenshots from other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The domain-adversarial training process seeks to simultaneously accomplish two objectives: (i) learn an embedding space where two screenshots are from the same screen if their distance is less than a threshold, and (ii) learn an encoding function that applies to both the web and mobile domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The first objective is related to the primary task of distinguishing same-screen pairs from new-screen pairs and is achieved with a pairwise margin-based loss [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The second objective aims to align the feature distributions of the two domains by maximizing the error rate of a domain classifier, which is a network that tries to classify whether a sample is from a web or mobile UI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For this task, we used only web page screenshots cap- tured on simulated smartphones, to make the domain classification objective more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2 Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Since one of the assumptions of our problem is that labeled examples of same-screen and new-screen pairs are unavail- able for mobile apps, we used two alternative methods to evaluate our screen similarity model: (i) quantitative evaluation on labeled pairs of web screens and (ii) qualitative evaluation on a set of unla- beled Android interaction videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Table 4 shows the quantitative performance of our models evalu- ated on pairs of web pages from our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Overall, training with more data led to significantly better performance, an increase of over 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The inclusion of a domain adaptation objective sometimes led to a slight drop in classification performance since it introduces additional constraints in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We qualitatively eval- uated our model’s performance characteristics on mobile screens by using them to segment videos of mobile app interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We used a dataset of screen recordings of bug reproductions [13] for 6 open-source Android apps and applied our model by sequentially sampling frames from the video and evaluating whether a new screen was reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Note our sampling process differs from other previous work [7, 15] that segmented crawls at recording time us- ing accessibility metadata, because we do not have accessibility metadata corresponding to the previously collected recordings used in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Figure 9 shows an example of a usage video pro- cessed by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' While the web model was effective detecting some types of transitions that occurred in mobile apps, it was less effective at others, such as software keyboards and dialogs, which do not occur frequently in the WebUI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We include more model-generated segmentations of the bug reproduction dataset in supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In this work, we applied unsupervised domain adaptation, which does not require any labels from the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Other domain adaptation strategies exist, and some are able to incorporate small amounts of labeled data, which we expect could improve the accu- racy of our model by contributing transition types unique to mobile apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 5 DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1 Performance Impact of Web Data Empirically, we showed that automatically crawled and annotated web pages, like those available in WebUI, can effectively support common visual modeling tasks for other domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', mobile apps) through transfer learning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In cases where a small amount of labeled mobile data was available, as in element detection and screen classification, incorporating web data led to better perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Even when labeled data was completely unavailable, as in screen similarity, models trained only on web data could often be directly applied to mobile app screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our results suggest that the size of current UI datasets may be a limiting factor, since model performance increases consistently when trained on larger splits of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our observations and analysis of WebUI’s composition showed that web pages can differ from mobile app screens in terms of complexity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', average number of on-screen elements) and element types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' However, the performance improvements from our machine learning experiments suggest that web and mobile UIs are similar enough to transfer some types of semantics between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We currently only explored three examples, although we believe that other UI modeling works [11, 47, 50] can also benefit from similar approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We did not evaluate all possible applications of WebUI in our paper, due to time and cost constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' However, the three experiments we conducted cover all possibilities of source and target domain labels (1), so similar transfer learning techniques are likely to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Future work that builds upon WebUI can conduct more detailed evaluations of other downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' One specific area that we believe is promising for future work is automated design verification [41], which could benefit from a large volume of web pages containing paired visual and stylistic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our highly automated data collection process also allows WebUI to be more easily updated in the future by re-visiting CHI ’23, April 23–28, 2023, Hamburg, Germany Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Droid Weight AntennaPod Time Tracker 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Token GrowTracker GNUCash Figure 9: Examples of interaction videos segmented by our best models trained with UDA (Red) and without UDA (Blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Videos are sampled at 1 fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The output of both models contain errors, however, we found that the adapted UDA model generally produced better segmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Common errors include oversegmentation due to app dialogs and soft keyboards, which do not occur in the WebUI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' the same list of URLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' An updated version of the dataset could also facilitate longitudinal analysis of the design [14] and accessibil- ity [21] of web UIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Nevertheless, WebUI is currently unlikely to support other types of modeling, such as user interaction mining [15, 16], that require realistic interaction traces, since our crawling strategy was largely based on random link traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2 Improved Automated Crawling Our crawler was unable to access much of the “deep web" (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', large part of the web that cannot be indexed), and thus our dataset contains few, if any, web pages that are not publicly accessible or protected by authentication flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' It also did not attempt to interact with all elements on a web page and conducted a very limited exploration of any JavaScript-enabled functionality that might have been present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Trends in web and app development, such as the creation of Progressive Web Apps (PWAs), suggest that this type of functionality will become more common, and traditional link-based traversal may become less effective at exploring UI states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' To improve automated crawling and data collection, our crawler could benefit from a semantic understanding of web pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, it could detect page functionality to explore states that require human input and either execute automated routines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' detecting login fields) or employ crowdsourcing [15] to allow it to proceed in more complex scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our currently trained models could augment or improve this process by identifying tasks associ- ated with web pages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', screen classification) or by augmenting potentially noisy labels provided by the automatically generated accessibility tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In turn, the crawler could explore more of the web, leading to higher quality and more diverse data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' If repeated iteratively, this process would constitute a form of Never-Ending Learning [39], a machine learning paradigm where models learn continuously over long periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Instead of learning from a fixed dataset, models could constantly improve itself by encounter- ing new content and designs, both of which are important due to the dynamic nature of UIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='3 Generalized UI Understanding Our experiments show that incorporating web data is most effec- tive for improving visual UI modeling in transfer learning settings where a limited amount of target labels are available for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' A logical next step is to obtain similar benefits without any addi- tional labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' To this end, we identified several strategies for improving generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' First, unlike existing UI datasets that contain examples from one device type, we intentionally simulated multiple viewports and devices during data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The decom- position of one-hot labels (where each element type is assigned <23> 31 《23> 23 A welan Gime Trac Add tas Add ta: Add tas Ad ask ① 7123:hange三 三 三 三 三 Fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' May 15 Fr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' May 1 5 15:21 ay 20 0 7123WebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics CHI ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' April 23–28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Germany exactly one type) into combinations of multi-hot tags (each element can be assigned multiple labels) may also be useful,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' since it avoids the problem of platform-specific element types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Figure 6 demon- strates the zero-shot transfer capabilities of models trained only on web data by successfully detecting and classifying elements on Android app screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' While the label sets of web and Android data do not directly overlap, the web model outputs reasonable analogs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', Text, link) for Android widgets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', Text Button).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Finally, our screen similarity model shows how unsupervised domain adaptation can improve the transferrability of learned features across domains through an explicit machine learning objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' A long-term goal of our automated data collection and modeling efforts is achieving a more generalized understanding of UIs — a single model that could be used to predict semantics for any UI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' This is challenging due to differing design guidelines and paradigms, but it could ultimately lead to a better understanding of how to solve UI problems across platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 6 CONCLUSION In this paper, we introduced WebUI, a dataset of approximately 400,000 web pages paired with visual, semantic, and style informa- tion to support visual UI modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Unlike most existing datasets for UI research that depend on costly and time-consuming human ex- ploration and annotation, WebUI was collected with a web crawler that uses existing metadata, such as the accessibility tree and com- puted styles, as noisy labels for visual prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our highly auto- mated process allowed us to collect an order of magnitude more UIs than other publicly released datasets and often associates more information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', clickability, responsiveness) with each example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We demonstrated the utility of our dataset by incorporating it into three visual UI modeling tasks in the mobile domain: (i) element de- tection, (ii) screen classification, and (iii) screen similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In cases where a small amount of labeled mobile data exists, incorporating web data led to increased performance, and in cases without any labeled mobile data, we found that models trained on web pages could often generalize to mobile app screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In summary, our work shows that the web constitutes a large source of data that can more sustainably be crawled and mined for supporting visual UI research and modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was funded in part by an NSF Graduate Research Fel- lowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' REFERENCES [1] 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' AutoIt Function PixelSearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='autoitscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='com/autoit3/docs/ functions/PixelSearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='htm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [2] 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Chrome DevTools engineering blog Full Accessibility Tree in Chrome DevTools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='chrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='com/blog/full-accessibility-tree/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Accessed: 2022-09-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [3] 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Puppeteer - Chrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='chrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='com/docs/puppeteer/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Ac- cessed: 2022-09-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [4] 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' What is the ideal screen size for responsive design?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' browserstack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='com/guide/ideal-screen-sizes-for-responsive-design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Accessed: 2022-09-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [5] Carlos Bernal-Cárdenas, Nathan Cooper, Madeleine Havranek, Kevin Moran, Oscar Chaparro, Denys Poshyvanyk, and Andrian Marcus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Translating Video Recordings of Complex Mobile App UI Gestures Into Replayable Scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' IEEE Transactions on Software Engineering (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [6] Sara Bunian, Kai Li, Chaima Jemmali, Casper Harteveld, Yun Fu, and Magy Seif Seif El-Nasr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Vins: Visual search for mobile user interface design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Pro- ceedings of the 2021 CHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [7] Andrea Burns, Deniz Arsan, Sanjna Agrawal, Ranjitha Kumar, Kate Saenko, and Bryan A Plummer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Interactive Mobile App Navigation with Uncertain or Under-specified Natural Language Commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='02312 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [8] Tsung-Hsiang Chang, Tom Yeh, and Rob Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Associating the visual representation of user interfaces with their internal structures and metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the 24th annual ACM symposium on User interface software and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 245–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [9] Olivier Chapelle, Bernhard Scholkopf, and Alexander Zien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' IEEE Transactions on Neural Networks 20, 3 (2009), 542–542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [10] Nitesh V Chawla, Kevin W Bowyer, Lawrence O Hall, and W Philip Kegelmeyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' SMOTE: synthetic minority over-sampling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Journal of artificial intelligence research 16 (2002), 321–357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [11] Jieshan Chen, Amanda Swearngin, Jason Wu, Titus Barik, Jeffrey Nichols, and Xiaoyi Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Towards Complete Icon Labeling in Mobile Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In CHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [12] Jieshan Chen, Mulong Xie, Zhenchang Xing, Chunyang Chen, Xiwei Xu, Liming Zhu, and Guoqiang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Object detection for graphical user interface: Old fashioned or deep learning or a combination?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='. In proceedings of the 28th ACM joint meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1202–1214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [13] Nathan Cooper, Carlos Bernal-Cárdenas, Oscar Chaparro, Kevin Moran, and Denys Poshyvanyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' It takes two to tango: Combining visual and textual information for detecting duplicate video-based bug reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In 2021 IEEE/ACM 43rd International Conference on Software Engineering (ICSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' IEEE, 957–969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [14] Biplab Deka, Bardia Doosti, Forrest Huang, Chad Franzen, Joshua Hibschman, Daniel Afergan, Yang Li, Ranjitha Kumar, Tao Dong, and Jeffrey Nichols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' An Early Rico Retrospective: Three Years of Uses for a Mobile App Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Artificial Intelligence for Human Computer Interaction: A Modern Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Springer, 229–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [15] Biplab Deka, Zifeng Huang, Chad Franzen, Joshua Hibschman, Daniel Afergan, Yang Li, Jeffrey Nichols, and Ranjitha Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Rico: A mobile app dataset for building data-driven design applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the 30th Annual ACM Symposium on User Interface Software and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 845–854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [16] Biplab Deka, Zifeng Huang, and Ranjitha Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' ERICA: Interaction mining mobile apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the 29th annual symposium on user interface software and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 767–776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [17] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Imagenet: A large-scale hierarchical image database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In 2009 IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Ieee, 248–255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [18] Morgan Dixon and James Fogarty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Prefab: implementing advanced behav- iors using pixel-based reverse engineering of interface structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1525–1534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [19] Samuel Dodge and Lina Karam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Understanding how image quality af- fects deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In 2016 eighth international conference on quality of multimedia experience (QoMEX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' IEEE, 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [20] Shirin Feiz, Jason Wu, Xiaoyi Zhang, Amanda Swearngin, Titus Barik, and Jeffrey Nichols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Understanding Screen Relationships from Screenshots of Smart- phone Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In 27th International Conference on Intelligent User Interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 447–458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [21] Raymond Fok, Mingyuan Zhong, Anne Spencer Ross, James Fogarty, and Ja- cob O Wobbrock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' A Large-Scale Longitudinal Analysis of Missing Label Accessibility Failures in Android Apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In CHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [22] Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Domain-adversarial training of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The journal of machine learning research 17, 1 (2016), 2096–2030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [23] Leo Gao, Stella Biderman, Sid Black, Laurence Golding, Travis Hoppe, Charles Foster, Jason Phang, Horace He, Anish Thite, Noa Nabeshima, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The pile: An 800gb dataset of diverse text for language modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' arXiv preprint arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='00027 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [24] Raia Hadsell, Sumit Chopra, and Yann LeCun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Dimensionality reduction by learning an invariant mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' IEEE, 1735–1742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [25] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [26] Zecheng He, Srinivas Sunkara, Xiaoxue Zang, Ying Xu, Lijuan Liu, Nevan Wich- ers, Gabriel Schubiner, Ruby Lee, and Jindong Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Actionbert: Leveraging user actions for semantic understanding of user interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 5931–5938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [27] Gao Huang, Yu Sun, Zhuang Liu, Daniel Sedra, and Kilian Q Weinberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Deep networks with stochastic depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In European conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' CHI ’23, April 23–28, 2023, Hamburg, Germany Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Springer, 646–661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [28] Ranjitha Kumar, Arvind Satyanarayan, Cesar Torres, Maxine Lim, Salman Ahmad, Scott R Klemmer, and Jerry O Talton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Webzeitgeist: design mining the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' ACM, New York, NY, USA, 3083–3092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [29] Luis A Leiva, Asutosh Hota, and Antti Oulasvirta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Enrico: A dataset for topic modeling of mobile UI designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In 22nd International Conference on Human- Computer Interaction with Mobile Devices and Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [30] Gang Li, Gilles Baechler, Manuel Tragut, and Yang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Learning to Denoise Raw Mobile UI Layouts for Improving Datasets at Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In CHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [31] Toby Jia-Jun Li, Amos Azaria, and Brad A Myers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' SUGILITE: creating multimodal smartphone automation by demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the 2017 CHI conference on human factors in computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 6038–6049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [32] Yang Li, Jiacong He, Xin Zhou, Yuan Zhang, and Jason Baldridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Mapping natural language instructions to mobile UI action sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='03776 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [33] Yuanchun Li, Ziyue Yang, Yao Guo, and Xiangqun Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Droidbot: a lightweight ui-guided test input generator for android.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In 2017 IEEE/ACM 39th International Conference on Software Engineering Companion (ICSE-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' IEEE, 23– 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [34] Yuanchun Li, Ziyue Yang, Yao Guo, and Xiangqun Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Humanoid: A deep learning-based approach to automated black-box android app testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In 2019 34th IEEE/ACM International Conference on Automated Software Engineering (ASE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' IEEE, 1070–1073.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [35] Paul Pu Liang, Yiwei Lyu, Xiang Fan, Zetian Wu, Yun Cheng, Jason Wu, Leslie Chen, Peter Wu, Michelle A Lee, Yuke Zhu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Multibench: Mul- tiscale benchmarks for multimodal representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='07502 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [36] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Microsoft coco: Common objects in context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In European conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Springer, 740–755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [37] Thomas F Liu, Mark Craft, Jason Situ, Ersin Yumer, Radomir Mech, and Ranjitha Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Learning design semantics for mobile apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the 31st Annual ACM Symposium on User Interface Software and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 569–579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [38] Wei Liu, Dragomir Anguelov, Dumitru Erhan, Christian Szegedy, Scott Reed, Cheng-Yang Fu, and Alexander C Berg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Ssd: Single shot multibox detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In European conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Springer, 21–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [39] Tom Mitchell, William Cohen, Estevam Hruschka, Partha Talukdar, Bishan Yang, Justin Betteridge, Andrew Carlson, Bhavana Dalvi, Matt Gardner, Bryan Kisiel, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Never-ending learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' ACM 61, 5 (2018), 103–115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [40] Kevin Moran, Carlos Bernal-Cárdenas, Michael Curcio, Richard Bonett, and Denys Poshyvanyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Machine learning-based prototyping of graphical user interfaces for mobile apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' IEEE Transactions on Software Engineering 46, 2 (2018), 196–221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [41] Kevin Moran, Boyang Li, Carlos Bernal-Cárdenas, Dan Jelf, and Denys Poshy- vanyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Automated reporting of GUI design violations for mobile apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the 40th International Conference on Software Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 165–175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [42] Sinno Jialin Pan and Qiang Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' A survey on transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' IEEE Transactions on knowledge and data engineering 22, 10 (2009), 1345–1359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [43] Richard S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Schwerdtfeger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Making the GUI Talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' ftp://service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='boulder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' com/sns/sr-os2/sr2doc/guitalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='txt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [44] Vinoth Pandian Sermuga Pandian, Sarah Suleri, and Matthias Jarke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Synz: Enhanced synthetic dataset for training ui element detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In 26th International Conference on Intelligent User Interfaces-Companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 67–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [45] Karen Simonyan and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Very deep convolutional networks for large-scale image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' arXiv preprint arXiv:1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1556 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [46] Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Dropout: a simple way to prevent neural networks from overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The journal of machine learning research 15, 1 (2014), 1929–1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [47] Amanda Swearngin and Yang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Modeling mobile interface tappability using crowdsourcing and deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [48] Zhi Tian, Chunhua Shen, Hao Chen, and Tong He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Fcos: Fully convolutional one-stage object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF international conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 9627–9636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [49] Bryan Wang, Gang Li, Xin Zhou, Zhourong Chen, Tovi Grossman, and Yang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Screen2words: Automatic mobile UI summarization with multimodal learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In The 34th Annual ACM Symposium on User Interface Software and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 498–510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [50] Jason Wu, Xiaoyi Zhang, Jeff Nichols, and Jeffrey P Bigham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Screen Parsing: Towards Reverse Engineering of UI Models from Screenshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In The 34th Annual ACM Symposium on User Interface Software and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 470–483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [51] Mulong Xie, Sidong Feng, Zhenchang Xing, Jieshan Chen, and Chunyang Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' UIED: a hybrid tool for GUI element detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the 28th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1655–1659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [52] Qizhe Xie, Minh-Thang Luong, Eduard Hovy, and Quoc V Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Self- training with noisy student improves imagenet classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 10687–10698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [53] I Zeki Yalniz, Hervé Jégou, Kan Chen, Manohar Paluri, and Dhruv Mahajan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Billion-scale semi-supervised learning for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' arXiv preprint arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='00546 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [54] Tom Yeh, Tsung-Hsiang Chang, and Robert C Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Sikuli: using GUI screenshots for search and automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the 22nd annual ACM symposium on User interface software and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 183–192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' [55] Xiaoyi Zhang, Lilian de Greef, Amanda Swearngin, Samuel White, Kyle Murray, Lisa Yu, Qi Shan, Jeffrey Nichols, Jason Wu, Chris Fleizach, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Screen recognition: Creating accessibility metadata for mobile applications from pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' A ADDITIONAL DATASET SAMPLES We provide additional samples from the WebUI (Figure 10) to sup- plement the example in the paper (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our example gallery shows several different types of websites, including login, landing, product, portfolio, and informational pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Each website is cap- tured using different simulated devices, which shows, among other things, how content responds to screen size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We also computed the percentile-rank of each web page’s class distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' B CLASS IMBALANCE ANALYSIS This section describes analysis of class imbalance of WebUI and its effect on transfer learning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Similar to other UI datasets[55], WebUI exhibits an imbalance of UI element classes, where some types of elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', text) appear much more fre- quently than others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Several aspects of WebUI (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', finer-grain text segmentation, multi-hot labels, and prevalence of documents on the web) also contributed to class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' We used a frequency-based resampling method to generate the Web7k-Resampled, which resulted in more examples of infrequent element types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Our technique assigned weights to samples to in- crease the representation of UIs containing rare or infrequent ele- ment types, and we resampled based on the 10 element types shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Algorithm 1 provides an overview of our resampling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Note that unlike some class-balancing algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=', SMOTE [10]), our technique does not generate additional synthetic samples and does not include the same screen more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Web7k-Resampled contains proportionally more examples of many infrequent classes (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Figure 11 shows the proportional increase in screens containing each element type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Figure 12 shows the proportional increase in the total number of elements for each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The results from our performance evaluations in the main paper suggest that this resampled split leads to improvements for each of our three tasks when compared to a randomly sampled subset of the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Notably, the element detector model resampled 7k split outperformed the one trained on 70k random split, which suggests that element balancing was particularly useful for tasks where elements types are directly predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Tests with other two tasks (screen classification and screen similarity) also led to im- provements for the resampled models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' however, the gains were more modest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' The improvements could be because the element distribution in the resampled split is closer to that of the target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' In addition, we provide a deeper analysis of the Element Detection WebUI: A Dataset for Enhancing Visual UI Understanding with Web Semantics CHI ’23, April 23–28, 2023, Hamburg, Germany Figure 10: Samples from WebUI accessed with different simulated devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For each screen, we compute its element type dis- tribution (normalized to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Then, we computed the percentile-rank of the top 10 classes with respect to the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, the bottom row’s button class has a percentile-rank of 90, meaning the web page’s relative frequency of is greater than 90% of others in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' TheOpenTracingpeject 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professionellenKlang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Mit unserem Playlist- Manager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='trefft ihrden GeschmackalleuererGaste und eurePartywird ein Erfolg Musikanlagemietenbocoup SERVICES RECENTWORK BLOG ABOUTUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content="- Search Q CONTACT US WESTANDINSOLIDARITYWITHTHEMOVEMENTFORBLACKLIVES INCLUSIVETECHNOLOGY CONSULTINGMOMENT STARTSETTE MUSIKANLAGEMIETEN PLAYLIST-MANAGER ZUBEHOR SOFUNKTIONIERY'S eue Hier gibt's wasauf die Ohren!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Wir haben den.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='let me:down so far, that there is sn iw ieis between irnagination and money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Because the more money and tech available to [create]a work,the less imagination there wil be in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' http/www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='mty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='com/shured/movies/interview/m/moore_alun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='81 Algorithm 1: Pseudo-code for the frequency-based resam- pling algorithm used to generate the Web7k-Resampled split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' 1 function SampleSplit (𝑁,𝐶,𝑆);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Input :Number of samples to choose 𝑁,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' list of element classes 𝐶,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' and list of samples 𝑆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='Output:Resampled subset of 𝑆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='/* Vector containing total frequencies for 𝑐 ∈ 𝐶 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='2 𝑓𝐶 ← total # of elements in 𝑆 for each class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='/* Matrix where rows are 𝑠 ∈ 𝑆 and columns are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='normalized frequency of 𝑐 ∈ 𝐶 for 𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='3 𝑓𝑆 ← frequency of classes 𝑐 ∈ 𝐶 (columns) for 𝑠 ∈ 𝑆 (rows) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='/* Assign sampling weights to 𝑐 ∈ 𝐶 inversely ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='proportional to frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='4 𝑤𝐶 ← [ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='𝑓𝐶 [𝑐] | 𝑐 ∈ 𝐶] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='5 samples ← [] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='/* Repeat until desired split size is reached ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='6 while len(samples) < 𝑁 do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='𝑐𝑠 ← Sample(𝐶,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='𝑤𝐶) 8 𝑤𝑠 ← [𝑓𝑆 [𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='𝑐𝑠] | 𝑠 ∈ 𝑆] 9 sample ← SampleWithoutReplace(𝑆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='𝑤𝑠) 10 add sample to samples 11 end 12 return samples Relative Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Change 0 1 2 3 text link list item image heading paragraph line break generic grid cell button Change in Screen Frequency after Resampling Figure 11: We calculated the change in frequency (expressed as a ratio) of screens containing at least one of each element type after resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, the number of screens containing at least one image element is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='7x more than in the randomly sampled set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Relative Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Change 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='0 text link list item image heading paragraph line break generic grid cell button Change in Element Frequency after Resampling Figure 12: We calculated the change in frequency (expressed as a ratio) of total number of elements after resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' For example, the average screen in the resampled split con- tains 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content='3x more images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Note that is possible for most el- ement classes to increase in frequency (while not having other classes experience a proportional decrease) because el- ement classes are not mutually exclusive, and the resampled split contains more elements that are assigned multiple tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' class, which is most likely to be affected by element type imbal- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} +page_content=' Table 5 shows that the Web7k-resampled split has higher AP for classes like "Text Button" and "Image", which had increased representation after resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFQT4oBgHgl3EQfPzZf/content/2301.13280v1.pdf'} diff --git a/E9E3T4oBgHgl3EQfVgpZ/content/tmp_files/2301.04460v1.pdf.txt b/E9E3T4oBgHgl3EQfVgpZ/content/tmp_files/2301.04460v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..47d2d785b735cff54eb818d3662a0f4dbdf1a319 --- /dev/null +++ b/E9E3T4oBgHgl3EQfVgpZ/content/tmp_files/2301.04460v1.pdf.txt @@ -0,0 +1,1663 @@ +Fast spline detection in high density microscopy data +Albert Alonso +Julius B. Kirkegaard +January 12, 2023 +Abstract +Computer-aided analysis of biological microscopy data has seen a massive improvement with the utilization of +general-purpose deep learning techniques. Yet, in microscopy studies of multi-organism systems, the problem of +collision and overlap remains challenging. This is particularly true for systems composed of slender bodies such as +crawling nematodes, swimming spermatozoa, or the beating of eukaryotic or prokaryotic flagella. Here, we develop +a novel end-to-end deep learning approach to extract precise shape trajectories of generally motile and overlapping +splines. +Our method works in low resolution settings where feature keypoints are hard to define and detect. +Detection is fast and we demonstrate the ability to track thousands of overlapping organisms simultaneously. +While our approach is agnostic to area of application, we present it in the setting of and exemplify its usability on +dense experiments of crawling Caenorhabditis elegans. The model training is achieved purely on synthetic data, +utilizing a physics-based model for nematode motility, and we demonstrate the model’s ability to generalize from +simulations to experimental videos. +1 +Introduction +Large-scale, high-throughput quantification of microscopy data have increasingly become possible with the aid of +computer vision [1–6]. +In particular within the last decade, deep learning techniques [7–9] have improved and +enabled accurate image analysis of microscopy data in a broad range of areas including cell counting [10, 11], cell +segmentation [12–14], nucleus detection [6, 15], sub-cellular segmentation [16], drug discovery [17], cancer detection +[18–20], and the identification of infectious diseases [21, 22]. Detection models serve as the fundamental operation +in tracking procedures, and combined with suitable tracking algorithms, these can achieve morphologically resolved +organism tracks that can accurately quantify organism motility [23], the application of which ranges from fundamental +neuroscience [24–26] and the circuitry of simple organisms [27–30] to drug discovery [31–35]. +Multi-organism detection can be achieved at increasing levels of fidelity: at the crudest, only center-of-mass +locations or bounding boxes are predicted [36] which does enable tracking of organisms but provide little morpho- +logical information. In contrast, pixel-wise segmentation models [12] and pose estimation using keypoints [37] reveal +accurate shape dynamics when employed on high-resolution data. However, these methods rely on high definition +objects, as segmentation and prediction is highly sensible to noise. In particular for organisms that are long and +slender, pixel-wise segmentation fails at low resolution as correct predictions require sub-pixel accuracy. Moreover, +at high densities, these methods may fail due to their inability to properly handle overlap between organisms. +Here, we consider the problem of studying slender organisms at low resolution and high density with the goal to +enable both accurate identity tracking and quantification of shape dynamics. This problem has traditionally been +approached by employing pixel-wise segmentation and subsequent skeletonization procedures [38–43], an approach +that requires ad-hoc procedures to solve the problem of correctly identifying overlapping organisms [44], the com- +binatorial complexity of which blows up at high densities. To this end we abandon pixel-wise output and instead +construct a neural network architecture that predicts, potentially overlapping, splines directly [45–47]. Our method +enables both accurate shape prediction and tracking in dense experiments of slender objects. This is applicable to a +1 +arXiv:2301.04460v1 [cs.CV] 11 Jan 2023 + +broad class of systems [Fig. 1], including tracking of nematode worms [48–50], spiral or elongated bacteria [51–54], +spermatozoa [55, 56], the flagella of both eukaryotes [42, 43] and prokaryotes [57], and freely swimming flagella such +those of microgametes [58]. +a +b +c +d +Figure 1: Microscopy images of different microorganisms whose slender structure and frequent overlaps makes them +hard to detect using classical approaches. a. C. elegans motility experiment from the dataset of this paper. b. +Motile, flexuous, thin, spiral-shaped B. pilosicoli bacteria. Still from Ref. [53]. c. Beating flagella of the green alga +C. reinhardtii, provided by Kirsty Wan, University of Exeter. d. Swimming human spermatozoa. From dataset in +Ref. [56]. +Our method relies on recent advances in deep learning [59–63] and extends these by few simple ideas: We found +that humans are better at correctly resolving overlap between moving bodies when given access to videos rather +than still micrographs. Thus, to allow the neural network to encode the identity of individual bodies as a function +of their motion, the input to our neural network is taken to be short video clips rather than single frames. Our +network outputs multiple independent predictions, and for each produces (1) the spline representing the centre-line +of an object, (2) an estimated confidence score for the prediction, and (3) a latent vector, the space of which we +induce a metric on that measures whether two predictions are trying to predict the same body. To train the network, +each output quantity is associated with a specific loss term, where, importantly, the spline loss term is permutation- +invariant in the labels. To resolve overlap, we do non-max suppression [36], but rather than measuring distances +between spline predictions, we use the latent space output, which allows two predictions to be kept even though +they are close in physical space. This enables correct predictions for data in which objects overlap very closely. Our +method is further tailored to support the subsequent tracking process, which must link uniquely predictions from +frame to frame. To that end, we not only predict the object location at a single timepoint, but also predict consecutive +past and future splines. Using these time-resolved predictions in the linking process enables high-precision tracking +even through dense regions. +Our method is principally applicable to all microscopy datasets that involve slender bodies. In this paper, we +focus on its applications for tracking dense experiments of crawling C. elegans worms, a popular model system +2 + +业in neuroscience [64], human diseases [65], drug discovery [32], motor control [66], memory [67], and ageing [68]. +Studies of C. elegans often rely on phenotypic assays that measure the motility of the nematode worms as function +of some environmental condition or treatment [35, 69–81], the throughput of which can be massively increased if +overlap between organisms can be tolerated. Likewise, resolving identities of organisms during overlap is crucial +for studies of interactions between organisms [82]. Previous work on tracking C. elegans have generally employed +classical computer vision approaches to accurately track single or a few high-definition worms [39, 83–86], or many +low-resolution worms at non-overlapping densities [40, 87, 88], in some cases by utilizing a computational model of +the worm motion for hypothesis tracking [39, 83]. Recently, deep learning techniques have been utilized to track +C. elegans worms using e.g. bounding box predictions [89–91] and fully resolved centre-line splines in the case of +isolated worms [92], allowing for detection also during periods of self-overlap. +With this paper, we publish a dataset of videos of motile C. elegans worms imaged at a wide range of densities. +The dataset includes ∼ 1,500 labelled splines that we use to evaluate, but not train, our detection model. +We +demonstrate that our model can be trained exclusively using synthetically generated data and yet generalizes well +to real videos. Our method leverages the parallel capabilities of convolutional neural networks and is thus able to +handle thousands of detections in a single pass, resulting in real-time detection at ∼ 90 Hz at 512 × 512 resolution +on a single GPU. The code is open source and available at https://github.com/kirkegaardlab/deeptangle. +2 +Results +2.1 +Architecture +Figure 2 illustrates the overall structure of our approach. Our model is based on single-stage detection models [36, +59] that output many candidate predictions per target in a single forward pass and rely on a score system to prune +until a single candidate is left for each target object. The performance of such single-stage models have been shown +to enable accurate real-time bounding box detection [62]. The backbone of our neural network [Fig. 2a] consists +of convolutional residual networks [60] with the small modification that we employ average pooling rather than +max-pooling to avoid translational invariance in the spline predictions, which need to be accurate to a sub-pixel +degree. +We take the input to our model to be a stack consecutive frames in order to provide the model with a temporal +context [Fig. 2c]. This has previously been shown to improve the detection of e.g. partially hidden objects [93]. In +particular, in present case of motile slender objects where dynamic crossings and overlap between objects are very +common, a temporal context can provide the necessary information to resolve the problem of correct identification. +Furthermore, the temporal context allows the output of our model to include information on the motion of the +splines, which we will further exploit for tracking purposes. +The backbone of our neural network performs a 162-fold reduction in resolution when mapping the input images +to feature space, from which the network outputs multiple anchored predictions. We choose the resulting number +of candidates to be considerably larger than the number of objects in the frame, thus ensuring that all objects have +suggestions. The anchored approach further means that the only restriction on input size is that its dimensions +be divisible by 16, and, in particular, it allows training at a certain resolution H × W and subsequent inference at +another H′ × W ′ without loss of accuracy. +The output of our model is composed of spline predictions, confidence scores and latent vectors: +Spline predictions +We choose to represent the centre-line of the slender bodies of interest by arrays consisting of k +equidistant points [Fig. 2d]. These coordinate arrays, which we refer to as splines, become high-precision descriptors +even for complex shapes when k is chosen large. To reduce the complexity of predicting k points, we embed the +spline representation with a principal component (PCA) transform A, the dimension κ of which can be much smaller +than k [94]. The PCA components λ represent shape, and addition hereto, the network also predicts the offset x0 of +3 + +Neural Network +I +[ +H +, +W +, +T +] +fθ +(λ, x0) +[ +M +, +W +T +, +m +] +z +[ +M +, +W +T +, +K +, +2 +] +p +[ +M +, +D +] +s +[ +M +] +qϕ +x = x0+Aλ +a +Latent Space +Emergence +of clusters +1. Score Prunning +rl +Latent Space +Best spline +remaining +2. Suppression +Latent Space +3. Repeat +b +Splines coordinates +Synthetic +I− +I+ +I +Real +I− +I+ +I +L(lx, ls, lp) +ˆz +Neural +Network +Backpropagation + + +z +p +s + + + + +x− +x +x+ + + +Splines z +Filtering +x = + + +(x0,0, y0,0) +. . . +(x0,k, y0,k) +... +... +... +(xn,0, yn,0) +. . . +(xn,k, yn,k) + + +Predictions +Spline points +Visualization +Visualization +c +• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••x +• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••x +. +. ψi +• +(xi, yi) +• +• +ds +d +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +•• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +direct distance +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +•• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +flip distance +e +Figure 2: +(a) Structure of the detection method. Trainable neural networks are colored in gray, and represent +the convolutional neural network f(I; θ) and the latent space encoder q(λ, x0; φ). (b) Procedure to prune unfiltered +predictions to final detections with the use of the encoded latent space vectors. (c) Method overview from the input +clip I (we use a stack of 11 frames in this work) to the final matrix of splines x. The target frames [I−, I, I+] (center +frames from the clip, orange) are explicitly shown for both the synthetic and real videos. Additionally, the training +setup is represented using lighter color arrows; from synthetic data to loss backpropagation. After detection, direct +visualization of the predicted splines x is possible. (d) Diagram with a spline descriptor composed of k equidistant +points along the skeleton of the nematode. (e) Visual representation of the two distances used in Eq. (1), the +minimum of which corresponds to correct head-tail alignment and is the one that will be used in the model. +4 + +CCthe spline, the internal calculation of which is done in a local coordinate system defined by the anchor points. Thus, +instead of predicting 2k floating point values per spline, the network needs only output κ + 2. +The temporal context of the input image stack permits output spline prediction also for the non-central images. +In our approach, we predict a set of three splines z = [x−, x, x+] corresponding to the three central frames [I−, I, I+] +of the input stack [Fig. 2c]. We consider the central spline x the main output, whereas the past x− and future x+ +splines are considered auxiliary predictions whose main purpose lie on their use during the latent space encoding as +well as the tracking procedure. +We define the similarity measure between two splines by the standard Euclidean distance. In the case of splines +that look symmetric from either end, we exploit this symmetry and employ the flip-invariant distance defined by +d2(x, x′) = min +� +k +� +i=1 +(xi − x′ +i)2, +k +� +i=1 +(xi − x′ +k−i+1)2� +, +(1) +as illustrated in Figure 2e. +Likewise, we define a distance between two collections of consecutive splines z, z′ by their weighted average +d2 +s = � +t ωt d2(zt, z′ +t), where the weights can be adjusted to give focus to central predictions, and for the present case +we choose ω = 2ω− = 2ω+. +The neural network is trained to minimize the distance d2 +s between predictions and labels. To do so, we let the +independent predictors specialize for different shapes. This is achieved by using a permutation-invariant loss such +that the total loss is computed as a sum over the labels only, each using the predictor that best match the labels. +Thus many spline prediction will not contribute to the spline loss. +Confidence scores +Each independent prediction of the network includes a confidence score s, which is used to +filter out bad candidates. In bounding box or mask detection, intersection over union (IoU) is commonly used to +evaluate the accuracy of a prediction, however, this metric does not generalize well to spline predictions when there +is overlap. Instead, we introduce a custom metric to define the goodness of a spline set z by comparing it to its label +ˆz, +ˆs = exp +� +−d2 +s(z,ˆz)/σ2 +s +� +. +(2) +Here, σs is a parameter that sets the scale over which the score varies. The metric is sensitive to perturbations on +accurate predictions, i.e. predictions close to labels where ds → 0, but loses sensitivity the worse the predictions is. +This is a useful feature as correct scoring for good predictions is crucial for choosing the best one, whereas low-scoring +predictions are discarded in any case and their relative scoring therefore unimportant. +The score prediction is trained using L2 loss. To avoid conflicting backwards error propagation between this task +and that of spline prediction (as scoring bad predictions is easier), we stop the gradient flow in the computational +graph on the last layer of the score-predicting part of fθ [Fig. 2a] such that it does not interfere with the accuracy +of the predicted splines. +Latent space for candidates suppression +Finally, we need to ensure that there is only one prediction per +object. Bounding box detectors let the user decide the fraction of overlap between prediction boxes of the same class +that should be considered to be targeting the same object. As our method must work at high densities, this task is +complicated by the fact that two predictions might be very close, even completely overlapping in the central frame, +and yet represent different objects. The task of choosing a suitable cutoff distance is therefore difficult, and we make +this a trainable task. We do so by embedding each prediction in a low-dimensional latent space in which comparison +between predictions is cheap, thus allowing efficient and fast candidate suppression also at high densities. +5 + +Our method computes the latent vectors p for predictions using an auxiliary neural network, qφ which acts +directly on the eigenvalues λ and offsets x0 rather than the more redundant spline coordinate points. We induce a +Euclidean metric on the latent space with the interpretation that two predictions i, j are predicting the same object +with probability +P(i ↔ j) = +� +exp +� +−||pi − pj||2� +if ||x0i − x0j|| ≤ σl, +0 +otherwise. +(3) +Here, σl is a real-space visibility cutoff that prevents far predictions to interact in the encoded space, thus avoiding +the need to scale the dimensionality of the latent space with the number of candidates or the input size. We note +that when using the flip-invariant metric ds on splines, we explicitly construct the latent space encoder to likewise +be flip-invariant (see Methods). +To train the latent space, we make the assumption that during training predictors are ‘trying’ to predict the +label closest to the prediction spline. Combined with the probability interpretation, this allows us to use binary +cross entropy as a loss function for the probability defined in Eq. (3). To avoid wrong clustering between undefined +close-by predictions, the loss contribution of each prediction is scaled by the product of their real scores ˆsiˆsj, this +ensuring that the network focus its attention of good predictions that will not be filtered out. Finally, since the +encoder should not to alter the performance of the spline suggestions, the loss on the latent space representations +only updates the weights qφ of the encoder, but is trained concurrently with the main model. +We employ non-max suppression to choose the best prediction of each object, but with distances measured in +latent space, as illustrated in Fig. 2b. Concretely: Once all the predictions whose score is lower than a threshold +τs have been discarded, multiple candidates are likely to still remain for each target object. The lack of low score +predictions expose clusters in the latent space that correspond to single objects. We sort the remaining predictions +by their score, automatically accepting the highest scored one. Once a prediction i is accepted, all predictions j +that have a high probability P(i ↔ j) > τo of being the same object are removed. This is equivalent to setting an +exclusion radius rl in the latent space as shown in Fig. 2b. We keep iterating on the remaining predictions, pruning +the latent space until all candidates have been iterated. The final number of accepted predictions should equal the +number of objects in the frame. +Detection on dense C. elegans experiments +To evaluate our approach, we study microscopy videos of crawling C. elegans worms. We are particularly interested +in videos captured at much higher densities than those typically used in motility experiments. Thus we evaluate our +model on wide-field videos captured under approximately uniform illumination [40], exemplified in Fig. 3a. In our +dataset, the number of nematode worms vary ranging from from ∼ 400 with a small probability of overlap occurring +( ≈ 0.05 average overlaps per worm) to extremely densely packed plates with up to ∼ 6,000 nematodes, where there +is, on average, one overlap per worm. This means that in the dense plates, detection methods that stop tracking +after contact between worms happens are rendered completely ineffective. +Defining worm density ρ as the number of worms in a region per square millimeter, we find, as expected, a +linear relation between the average amount of overlap per worm and the density [Fig. +4a]. +Due to the spatial +heterogeneity of the worm distribution inside the plate, higher densities can be observed when considering small +regions. On 100 mm2 scales, the highest density in the dataset is ρ ∼ 2.5 mm−1, but this jumps to an extreme +ρ ∼ 3.5 mm−1 when considering 10 mm2 regions, where humans begin to struggle to correctly identify worms. For +quantitative evaluation of our model, ∼ 200 random regions of the videos were sampled and hand-labelled resulting +in ∼ 1,500 labelled worm splines. A sample of frames are shown in Fig. 3b to provide a sense of the different densities +encountered in the evaluation dataset, with the predictions of the model overlaid. +6 + +2736 x 2192 +a +b +Figure 3: Showcase of the capabilities of the method. (a) Detected splines predicted on an entire densely populated +well plate with a single forward pass through the neural network. Inset shows a zoom-in section to demonstrate +the accuracy of detection across the entire plate (except near borders, where the plate interferes). The total plate +contains around 6,000 splines. (b) Close up evaluation of different experimental clips with different densities of +worms. +Simulation-based training. +To train our network, we implement a physics-based synthetic dataset generator to +exploit perfectly defined labels. This approach removes the need for a supervised dataset, and also allows labelled +videos in situations where manual labeling may not be reliable, or where the subjectivity of the human labellers can +result in inconsistent labels. Physics-based synthetic datasets have successfully been used to train systems on similar +conditions, for instance where manual labelling may introduce unnecessary noise or bias to the model [16]. Our +in-silico data generator has two main components: a physics-based model for the organism and a synthetic frame +generator. +In-silico worms are generated on demand every training step which removes the possibility of overfiting to the +generated frames. +In order to train the model to work effectively with a range of worm densities, we generate +batches with different numbers of worms in a uniform manner, without bias towards low or high worm counts. This +teaches the model to handle a variety of densities without overfitting to any specific case. And to make the model +more robust, training also happens on densities whose manual annotation would be extremely challenging. The +simulation and video synthesis are implemented in a GPU framework which enables fast end-to-end training without +the performance penalization of data transferring between the accelerator and the host machine. +We base the worm simulation on resistive force theory, as it has previously been shown to correctly predict the +position of the skeleton for short spans of times [95]. Since the network only perceives the frames surrounding the +target frames, we found the total duration of the clip to be short enough that a linear crawling model approximation +fits our needs. The physics-based model should encapsulate all types of organism behavior. This can be achieved by +7 + +1 cmp= 0.19mm-2p= 0.79mm-2p= 0.91mm-2p= 1.57mm-2p= 1.97mm-2p = 3.28mm-2 +1 mm• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +b +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Worm density in clip (mm−2) +0.0 +0.5 +1.0 +1.5 +Average overlaps on a worm +a +c +ρ = 2.3mm−2 δadtw = 0.6 px +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Worm density in clip (mm−2) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Double prediction +artefacts +Double prediction +artefacts +Error distance δadtw dependance with density +d +0 +1 +2 +3 +4 +Worm density in clip (mm−2) +0.996 +0.997 +0.998 +0.999 +1.000 +True Positive rate +e +0 +1 +2 +3 +4 +Worm density in clip (mm−2) +0.00 +0.02 +0.04 +0.06 +0.08 +False Negative rate +f +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Score threshold τs +0.46 +0.48 +0.50 +0.52 +0.54 +0.56 +Average error distance δadtw +τo = 0.1 +τo = 0.3 +τo = 0.5 +τo = 0.7 +g +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Score threshold τs +0.95 +0.96 +0.97 +0.98 +0.99 +1.00 +Average TP rate +h +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Score threshold τs +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Average FN rate +i +Figure 4: +(a) Average number of overlaps counted on frames of pixel size 512 × 512 with different densities of +worms (N = 90). (b) Illustration of the asymmetric dynamic time warping distance error corresponding to the +average value of the orange euclidean distances between the prediction (green) and the labelled points (white). (c) +Example frame with manually labelled points (white) and models predictions (colored). The metric is only evaluated +in the lighter area of size 100 × 100. (d) Quantified accuracy of the detections by showing the distance to the +manually labelled splines. Distributions for different densities are shown. The violin plots represent the 99 percentile +of the data whereas outliers are plotted individually. +(e–f) Rates for True Positive and False Negative on the +manually annotated dataset. (g–i) Performance of the model with different combinations of score (τs) and overlap +(τo) thresholds. N = 1,420. +8 + +oversampling the behavior, i.e. by making the simulations more diverse in the behavior than reality and thus hope +to include all types of real behavior as well. Details on the worm simulation and video synthesis can be found in the +methods section. +Despite the potential for physics-based simulations to be used for synthetic training data, discrepancies with real +data may lead to inaccuracies when applied to real microscopy images. This reality gap can be the result of an +overly simplified motility model or physics model, or the result of imprecise video synthesis. The gap may be further +increased by the fact that the model relies on the PCA transformation matrix A obtained on synthetic data, where the +number of PCA components used have been chosen to accurately reproduce all synthetic patterns, but not necessarily +to generalize to out-of-sample videos. Thus we find that our model is limited to accurate skeleton predictions only on +shapes that resemble those produced by our simulations, and the goal of the simulations is therefore to reproduce a +broad spectrum of possible motility patterns. Likewise, we find that our model is susceptible to the brightness of the +videos, and accordingly we adjust the real videos to increase their resemblance to the training data (see Methods). +Metrics +Despite being trained exclusively on synthetic data, the model’s inference performance is very good on real +clips. From visual inspection, no immediate discrepancies are observed between detections in low density clips and +at high density [Fig. 3b]. Likewise, per design, the network accuracy is independent on the input clip dimensions, +and the parallel structure of convolutions permits the use of large videos covering thousands of nematodes to be +processed simultaneously in a single forward pass [Fig. 3a]. We note, however, that even though no quality impact +on detections is observed when using large fields-of-view clips, there can be a dependency if non-uniform illumination +is used as different sections of the frame may have different requirements for preprocessing. +For a quantitative assessment of the method accuracy, we compare to the manually labelled dataset, an example +of which alongside the model predictions can be seen in Fig. 4c. As the predictions are densely defined splines +(here, ∼ 50 points), we introduce a custom metric to suitably evaluate the accuracy of the predictions using labels +with lower fidelity. The metric used must be shift-invariant, as having points anywhere along the spline should yield +zero error regardless of whether the label points precisely coincide with the prediction points or not. Likewise, label +points should be monotonically assigned along the spline in order to avoid artificially reducing the error for strongly +bent or self-coiling worms. Finally, it must be robust against the subjectivity of the labellers, as manual annotations +might miss or avoid spots where visibility is low such as the end-points of the worms. +To satisfy all these requirements, we introduce a metric based on the dynamical time warping (DTW) distance +used to measure similarity between temporal curves. In our modified version, asymmetric DTW, summation only +runs over label points. Thus, the metric δadtw is defined as follows: Let d(i, j) be the Euclidean distance between +label point i and prediction line segment j, then +δadtw = min +α +1 +N +N +� +i=1 +d(i, α(i)), +(4) +where α : [1, N] → [1, M] is a monotonic (non-decreasing or non-increasing) assignments of the N label points to the +M prediction line segments. A visual representation of the metric is shown in Figure 4b, and the O(NM) algorithm +for its calculation is detailed in the Methods section. +The results of evaluating the trained model on the labelled dataset are shown in Fig. 4. For reliable comparisons, +we first solve the assignment algorithm for the label-prediction pairs. This means that in the case of two completely +overlapped worms, two predictions need to be present to not count as a miss, and likewise, two predictions cannot be +considered to target the same label. We find an average error of δadtw ≈ 0.54 px with no strong dependency between +accuracy and density of worms [Fig. 4d], with the exception of a slight increase in error for extremely dense clips +(∼ 3.5 mm−2). The average error corresponds to less than the width of a worm (≈ 2 px ≈ 50 µm), and part of this +can be attributed to the fact that human accuracy is also near the half-pixel level [Fig. 4c]. Some outliers can be seen +however, which can mostly be attributed to an artefact of the model, where the network mistakes a single long worm +9 + +for two overlapping shorter predictions. This effect seems particularly sensitive to incorrect intensity normalization +of the videos. +Let σϵ be a cutoff distance above which we no longer consider the predictions to be targeting the closest label. For +all the figures in Figure 4, this cutoff is assumed to be σϵ = 3.0 px, and we observe no significant changes by tuning +it within the range of sensible values. We define the True Positive (TP) rate as the fraction of predictions that both +gets assigned a label and this label is within the the distance σϵ of the prediction. Figure 4e shows that the model +rarely predicts a spline where there is nothing with a TP rate of 0.999. Nevertheless, there are some predictions that +do not get assigned a label which can be attributed to the double-prediction artefacts just mentioned. The likelihood +of this happening decreases with density, but the rate is so low that it is almost negligible. Similarly, we define the +False Negative (FN) rate as the fraction of labels that are not assigned a prediction closer than σϵ. Fig. 4f shows +that the model in general manages a low FN rate at around ∼ 0.015, but that this increases to a rate of ∼ 0.06 +at extreme densities such as ρ ≥ 3.0 mm−2, where clusters tend to be densely packed and manual labeling likewise +becomes challenging. +The filtering part of the model depends on the previously introduced thresholds τs and τo. The score threshold, +0 < τs < 1, is used to prune low score predictions [Fig. 2b(1)], while the overlap threshold, 0 < τo < 1, is used to +decide the probability of two independent predictions to be targeting the same object [Fig. 2b(2)]. Throughout this +paper, we have set these to τs = τo = 0.5. However, due to their relevance in modifying the filtering process, we +evaluate how different combinations of thresholds may alter the performance results. Figures 4g–i show the average +performance obtained across all densities when filtering the predictions with variable thresholds. In spite of some +dependency between worm density and TP/FN rates, we consider the average metric to be a good indicative of the +performance on each case. +Fig. 4g shows the effect of the thresholds on accuracy. No significant dependency on the thresholds is observed. +This can be explained by the fact that accuracy is determined by the best predictors only, which are not discarded +until a high τs is used, and ones those are removed, τo becomes irrelevant. Further, the fact that there is no notable +difference between different values of τo indicates that the clusters are highly compact. +In contrast, Fig. 4h shows that the TP rate has a stronger dependency on τo at low τs because low score predictions +do not form compact clusters, and therefore a larger exclusion radius is required to discard them. Finally, Fig. 4i +shows that misses only begin to occur once the best predictions are discarded, and a strong dependence on the τs is +not observed before that point. +2.2 +Tracking from consecutive detections +Motility assays require not only accurate detections but also the ability to link these across frames to form time- +resolved tracks of individual organisms. This is challenging at high densities where we have the breakdown of the +assumption that the closest detected object to the previous frame corresponds to the same identity. In general, +greedy approaches to particle tracking such as assigning directly the closest particle in consecutive frames frequently +leads to failed tracks. Instead, the process of tracking can be efficiently formulated as a set of linear assignment +problems [96]. Naturally, here we can expand upon particle tracking by using a metric that measures distances not +between center-of-mass of the worms, but between the full splines as defined in Eq. (1). This works well for most +predictions, but can fail for fast-moving worms or in dense clusters. +A separate approach to tracking is Kalman filtering. This would require separate detection of entry and exit +events of worms, as well as a probabilistic model for worm motility, which would most likely have to be highly non- +linear. Kalman filtering is viable for the tracking of few organisms, but for present large-scale systems we require a +more efficient approach. As previously mentioned, splines from adjacent frames are also predicted in order to embed +temporal information into the latent vector. We propose a directed metric that leverages both past x− and future +x+ spline predictions [Fig. 5a]. Thus to find a mapping σ from one frame to the next, we solve +10 + +xi +xj +x+ +i +x+ +j +xk +xl +x− +k +x− +l +d(x+ +i (t1), xk(t2)) + d(x+ +j (t1), xl(t2)) + d(xi(t1), x− +k (t2)) + d(xj(t1), x− +l (t2)) +Minimal assignment distance +Forward distance df +Backward distance db +a +No constrains +Midpoint cutoff +Final assignment +t0 +t1 +t2 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +t0 +t1 +t2 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +t0 +t1 +t2 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +b +d +1 +25 +49 +Position (k) +0 +3 +6 +9 +12 +15 +18 +21 +24 +27 +Time (s) +1 +25 +49 +Position (k) +-π +0 +π +e +f +0.0 +0.5 +1.0 +1.5 +2.0 +Worm density in clip (mm−2) +0.90 +0.95 +1.00 +Average track integrity +Directed +Normal +c +Only 135 tracks +All tracks (∼ 6000) +All tracks +y +x +t +h +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Worm density in clip (mm−2) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +mm/s +σM ∼ 1/ +√ +N +SE (σM) on CoM speed (vmm) +g +Figure 5: +(a) Illustration of the directed distance used to assign consecutive detections the same identity. The +simplified drawing shows two independent predictions at adjacent frames and showcases how the assignment scheme +computes the identity by comparing future-present and past-present distances and choosing the assignment that +minimizes their sum. (b) Diagram showcasing how using a location cutoff simplifies the assignment problem. Nodes +represent independent detections at each frame whereas edge values are given by the directed distance measure. The +assignment happens by minimizing the sum of edges at each timestep. (c) Comparison of using the straightforward +spline distance and the proposed directed approach. The accuracy is evaluated by measuring the integrity of the +tracks. +In contrast to other metrics in this paper, this plot has been obtained using synthetic worms as long- +term, accurate tracks are required to evaluate the tracking integrity (See Methods for details on Tracking integrity). +(d) Qualitative example of 30 s trajectories of the center of mass of the nematodes in a dense experiment. The +still background image represent the last frame of the video. To improve the visualization, a small subset of the +trajectories are shown. In contrast, a corner of the frame is used to display all the trajectories to showcase the +density of simultaneous tracks. (e) Two samples of the spline angle ψ of two randomly sampled nematodes from (d). +(f) Undulations corresponding to 30 s of the detections relative to the center of mass coordinate of nine randomly +sampled nematodes from (d). (g) Standard error value of the measurements of the center of mass speed as a function +of density. (h) Showcase of the possible throughput of the method, by simultaneously tracking more than 6,000 tracks +from a full dense plate. A small window on the tracks is shown to showcase their continuity. +11 + +CCσ = arg min +σ +�� +i +d(xi(t), x− +σi(t′)) + d(x+ +i (t), xσi(t′) +� +. +(5) +Identity assignment can be seen as a network flow global optimization where nodes represent detections and edges +carry cost of assignment. To avoid having to perform all possible combinations of assignments, we include a physical +distance threshold on the midpoint of the central spline. This threshold significantly simplifies the assignment scheme +and improves the runtime of the filtering process [Fig. 5b]. +To quantify the performance of these methods, we define the tracking integrity ι as a scalar that indicates how +consistent the assignment of a label to a prediction is along the tracked video. Perfect tracks have ι = 1, whereas +labels that gets assigned two different identities for half of the duration of the video have ι = 1 +2, and so on (see +Methods for a detailed definition). We evaluate this on synthetically generated videos of 10 seconds (200 frames) +that have perfectly labelled tracks, the results of which are shown in Fig. 5c. On videos with densities up to 2.0 +mm−2, we achieve an average integrity of ι ≈ 0.97. This a ∼ 30 % improvement of the error over using direct spline +assignment defined in Eq. (1). We observe that the integrity is almost perfect at low densities, but drops to ι ≈ 0.93 +at the highest densities. +When applied to high density videos of C. elegans, the tracking method is able to keep track of individual worms +as they pass through clusters of other worms [Fig. 5d]. In contrast to pixel-level classification of worms, our approach +outputs splines directly, and thus subsequent analysis is straightforward. For instance, one may directly study the +worm undulations [Fig. 5f] or extract the worm spline angle ψ = arctan(y(s, t) − y0(t), x(s, t) − x0(t)) to provide +insight into the movement patterns and kinematics of the worm [Fig. 5e]. +One of the key advantages of our methods is its ability to collect a larger number of samples compared to +traditional techniques, while still obtaining reliable results. As the standard error decreases with the number of +samples, using our methods allows for metrics to be gathered with less uncertainty while still requiring the same +experimental setup. For instance, Figure 5g) shows how the error of estimating the average speed of the center +of mass of the nematodes decreases with density. This advantage can be extended to tracking large numbers of +nematodes in crowded environments, such as extremely dense petri dishes where more than 6,000 concurrent tracks +can be simultaneously computed [Fig. 5h]. Thus, with our method, we are able to collect a larger number of samples +and obtain more precise and reliable results, even in challenging conditions. +3 +Discussion +We have introduced a novel deep learning approach for detecting and tracking slender bodies, such as crawling +nematodes, in microscopy data. The presented convolutional neural network architecture is capable of accurately +detecting a large number of overlapping organisms, a task that can be particularly challenging for standard methods +such as bounding boxes and pixel-level classifiers due to the issue of occlusion and overlap. To address this, we have +implemented a latent space encoding which allow us to filter by non-maximum suppression and effectively handle +overlapping objects. Not only is our method capable of accurately detecting and tracking slender bodies, but it also +demonstrates strong scalability, performing well across a range of input frame sizes and densities of bodies. This +makes it an ideal tool for a variety of experimental settings where splines are useful descriptors, including studies of +crawling nematodes, swimming spermatozoa and beating eukaryotic or prokaryotic flagella. +Besides a suitable detector model, labeled training data is also needed. We have demonstrated that relying on a +physics-based model to generate synthetic data is adequate to train our network to perform well on real data. This is +a key achievement as it means that applications of our system for different experimental studies do not require large +datasets to be procured, but rather the implementation of a suitable simulation. Our approach for synthetic data +generation relies on over-sampling the behavior of the worms. This is naturally a trade-off as too extreme behavior +can lead to datasets that are too hard for the neural network to replicate. For our model, we found that we slightly +12 + +undersampled certain worms shapes such as strong coiling, which the model therefore could struggle with identifying. +Though we did not look into this here, an interesting avenue for future research would be to bootstrap synthetic +motility models on small datasets of real organisms. In a similar fashion, the frame-generator procedure should +oversample the textures, pixel intensities and noise of real videos. Here, it could be interesting to study whether style +transfer [15] or diffusion models [97] could be used to further reduce the gap between training and inference data. +For tracking, we introduced a directed metric that employs past and future spline predictions to link them across +time. At very high densities this may still fail, in particular because the directed metric yields little advantage if +predictions are missing in some frames. A potential way to improve on this could come from utilizing the latent space +encoding as well. This would require temporal continuity in the latent space representation, which is achievable by +modifying the associated loss function. This should enhance the integrity of tracking, as it could potentially be used +to resolve issues such as switches by leveraging the separation of closely physical predictions with different temporal +behaviour that characterises the latent encoding. We believe that these suggestions might be fruitful avenues for +further research for improving deep learning models for dense detection of splines. +In this paper, we have proposed a new approach for fast and precise detection and tracking of slender bodies in +microscopy data. Its speed and accurate performance across a range of densities and sizes, combined with the ability +to handle overlapping objects, makes it a valuable tool for a variety of experimental settings where precise tracking +is essential for obtaining quantitative metrics. +4 +Methods +Convolutional neural network +Most of the weights of the network are at the feature detection convolutional +network whose backbone is made of four ResNet groups consisting of 2, 4, 4, 2 blocks with strides 1, 2, 1, 2, respectively. +We modify the original ResNet architecture by replacing the initial max-pooling layer for an average-pool layer to +avoid translational invariance. The final shape of the feature space is [H/16, W/16, C], with C being the number of +candidates each cell proposes. We have set C = 8 for this project in order to fulfil the condition of M ≫ N even +at high densities. All in all, there will always be C candidates per each cell regardless of input size, which leads to +a large number of candidates to be sorted in the filtering process. The head of the convolutional neural network is +composed of two fully connected layers of 512 and C · (3(m + 2) + 1) cells, respectively, with batch normalization +in-between. Due to the orientation invariance of the loss function on the spline predictions, it is possible that the +splines in the predicted set x−, x, x+ are not aligned. To remedy this, we aligned them by comparing with the +eigenvalues of the flipped spline. In order to get the flipped eigenvalues λf, we use +λf = A−1JAλ +(6) +where A is the PCA transformation matrix and J is the exchange matrix. +Latent space encoder +The encoder qφ is composed of two fully connected layers with batch normalization in- +between. The input of the encoder is the vector of size 3(m + 2) characterizing the splines predictions and the +output is D floating point values, corresponding to the coordinates of p in the D-dimensional latent space. We have +found D = 8 to be a well-performing dimension in our experiments. Due to the orientation invariance of the splines +predictions, we need to construct the encoder to cluster those splines regardless of orientations as well. To do so, the +input values are expanded to include those of the flipped splines λ → (λ, λf) and both are fed to the same layer. To +ensure symmetry, the output is then summed before passing through the last layer. In doing so, the encoder becomes +independent of spline orientation. +13 + +Input clips pre-processing +The images used to train the model have dark (small pixel intensity) background, +as we employ zero-padded convolutional layers. This is relevant for real recordings, where a negative flip may be +necessary to match the network requirements. During training, generated clips are normalized using a 1–99 percentile +normalization. For real clips, we have found that accuracy is improved if we apply CLAHE (adaptive histogram +equalization) before prediction. Likewise, a simple intensity correction factor µ may need to be applied to the videos +in order to match the pixel profile of the simulated data. For our dataset, we use correction factors of µ ≈ 1.2, to +get the best results. Note that we match real data to the synthetic as this avoids the need to retrain the network for +different experimental setups. +Loss functions +Spline descriptors are trained as a regression problem. Thus, the loss contribution is given by +the custom distance defined in Eq. (1). To enforce specialization on the predictors, and due to the number of +predictions M being considerable larger than the number of bodies N, only the best predictors are accounted for in +the loss. Nevertheless, there may be labels ˆx completely or partially outside the frame at tc, despite being inside at +t0. To make sure not to punish bad predictions at the boundaries for not matching invisible splines, instead of using +the number of simulated bodies N, the subset of bodies completely inside the frame Nv is used and the final loss +expression is given by: +lx = 1 +Nv +Nv +� +i +min +m d2 +s(zm, ˆzi) +(7) +The score L2 loss is computed as the difference of the values predicted and the score the spline proposals should +have. Thus, using Eq. (2), we train the predicted score of all predictions using: +ls = 1 +M +M +� +i +� +exp +� +− min +n +d2 +s(zi, ˆzn) +σs +� +− s +�2 +(8) +Finally, the loss function for the latent space encoder is a modified cross entropy loss scaled by the product of scores. +Denote Pi,j = P(i ↔ j) as defined in Eq. +(3), then the encoder loss is defined as an average over all pairs of +predictions ⟨i, j⟩ that are physically within the cutoff σl, +lp = 1 +S ⟨ˆsiˆsj(tij log (Pi,j) + (1 − ti,j) log (1 − Pi,j))⟩⟨i,j⟩ , +(9) +where S = � ˆsiˆsj, and ti,j indicates whether i and j are targeting the same label k, and is set by +tij = +� +1 +if ki = kj +0 +otherwise +(10) +with ki, kj being the closest labels to the predictions zi, zj respectively. +Training details +Training has been done from scratch, i.e. without the use of a pretrained backbone. During +training, the frame size for the input clips used was 256×256, but due to the anchored approach this does not +constrain inference to happen at the same resolution. Synthetic input is generated on demand and on device rather +than using a fixed pre-generated dataset. Thus, the network never sees the same frame twice and there is no host- +to-device data transfer. As mentioned in the main text, all networks are trained simultaneously, despite the weights +of each one depending on different cost functions. The code has been written in Jax using Haiku and training has +been carried on a cluster of 8 × NVIDIA A5000’s. +14 + +Inference +Inference happens at any resolution whose dimensions are multiple of 16. The input frames need to be +slightly pre-process as described in the previous sections. Candidate predictions are chosen using a score threshold, +and non-maximum suppression in latent space is used for filtering. Due to the sequential nature of the filtering +process, the implementation is written to use the CPU using numba. +Worm simulation +Worm trajectories are computed by employing a resisitve force theory crawling model used to +predict rigid body motions of C. elegans from the undulations [95]. Thus, we ensure that from a given set of generated +undulations, the produced motions will match those of real worms. From empirical observations, we propose a simple +equation (Eq. (11)) to generate the undulation of the worms. We define the motions by the spline angle ψk(s) with +s ∈ [0, 1] [Fig. 2d], and decompose this into a linear combination: +ψ(s) = ψu(s, t) + ψs(s, t). +(11) +This logically separates the worm undulations into two types of motion: one corresponding to a sinusoidal motion +ψs and one in which the whole body bends ψu. These we define by +ψu(s, t) = A cos +�2π +T t + ρ1 +� +cos (kusk + ρ2) +(12) +ψs(s, t) = ˜A cos +�2π +T t + kssk + ρ3 +� +(13) +where ˜A = 1 +2 (1 + | sin (2πt) |) A and the rest of parameters are sampled from random distributions. Although many +improvements for the above equations can be suggested, we prefer to keep the model simple. +Once the values of the parameters for ψ are generated all for the timesteps of the simulation, the positional +coordinates are obtained using +⃗x(s, t) = L +� s +0 +�cos (ψ(s′, t) + γ) +sin (ψ(s′, t) + γ) +� +ds′ +(14) +where γ is a random orientation and L is the length of the worm (also sampled). Once the skeleton is defined, the +rigid body motions are predicted by solving [95] +⃗F = +� L +0 +⃗f ds = 0, +(15) +⃗τ = +� L +0 +(⃗x − ⃗xCoM) × ⃗f ds = 0, +(16) +where the force ⃗f can be calculated from the spline velocity ⃗U = ∂t⃗x + V + Ω × (⃗x − ⃗xCoM) by +⃗f = αt (ˆt · ⃗U) ˆt + αn (ˆn · ⃗U) ˆn. +(17) +Here, V and Ω are the center-of-mass velocity and rotational velocity (that we are solving for), and αt and αn = α αt +is the tangential and normal drag coefficients, which is also sampled for (α > 1). We did not find a need for using a +non-linear force theory. The simulation is run with Python 3.9 using the Jax library. +15 + +Video synthesis +Given the labels for the splines positions, synthetic videos are generated to be used as input +during training. In order to add width to each worm, we vary the local body radius r by a function of the form +r(s) = ˜R |sin(arccos(as + b))| +(18) +The pixel values of those circles are calculated with anti-aliasing. +Once the worms have been rendered, noise +artefacts such as uneven background, blurring, Gaussian noise, etc. are added to replicate the observed conditions +of real experiments. During training, standard augmentation techniques are applied as well. In the same manner as +the motions simulation and the neural network training, frame generation is also written in Python using the Jax +library in order to leverage GPU capabilities. +Experimental dataset +Videos of crawling C. elegans were filmed using the protocol described in Ref. [40]. +Manually annotated dataset +The evaluation dataset is annotated using a custom tool that can be found at +https://github.com/kirkegaardlab/deeptanglelabel. Around ∼ 1,500 splines have been annotated and this +dataset (videos and labels) is included in the SI. +Asymmetric dynamic time-warped error distance +In order to evaluate the manually labelled dataset, we +introduce an error distance that compares the similarity between two curves by calculating a distance between each +point on one curve and the nearest segment on the other. The error distance used is a variation of the dynamic +time warping distance, which is widely used for comparing time series data. We note that, just as is the case for the +dynamic time warping distance, this is not a true distance in the mathematical sense. +Algorithm 1: Algorithm for asymmetric dynamic time warping +Data: Label curve defined by N points {pi}, and prediction curve defined by M line segments {sj}. +Result: The asymmetric dynamically time-warped distance from label to prediction. +Initialize matrices C, D with size [N, M]. +for i = 1 to N do +for j = 1 to M do +Di,j ← distance from point to segment(pi, sj) +C1,1 ← D1,1 +for i = 2 to N do +Ci,1 = Ci−1,1 + Di,1 +for j = 2 to M do +C1,j = min (C1,j−1, D1,j−1) +for i = 2 to N do +for j = 2 to M do +Ci,j = min (Ci,j−1, Ci−1,j + Di,j) +return CN,M/N +Tracking implementation +Tracking is done by sequentially predicting individual frames. For better performance, +batching of frames allows for parallel detections and can drastically reduce execution time. Nevertheless, due to the +requirement of including surrounding frames for each detection, a considerable increase in memory usage is observed. +Once a collection of spline detections is obtained, each prediction is adapted in order to make it work with the +TrackPy Python library. Due to the peculiarity of our distance metric, we implement a custom neighbor strategy +(see Code Availability) that avoids the assumption of a symmetric distance function. +16 + +It may happen that some detection artefacts appear during the sequential detection performed on tracking. We +have implemented a quick check on the resulting tracks to make sure not to have stubs, and fix obvious branching +of tracks due to these artefact. Slight increase in integrity is observed on dense clips. +Tracking integrity +Given a true label of a track of length N, we associate to this track at each time point i a +prediction identity Ii. We may then define the integrity of the track as ι = 1/N 2 �N +i=1 +�N +j=1[Ii = Ij]. For instance, +if a label is given identities I = [1, 1, 1, 5, 5, 5, 3, 3, 3] during the track, i.e. there have been two identity swaps, we find +ι = 1 +3, which has the interpretation that the track was correct for a third of the time. This measure will in general +scale like ι ∼ N −1, as longer tracks will have higher likelihood of identity swaps. +Acknowledgments +Video of C. elegans were provided by Celia Raimondi, Sunehera Sarwat and Michele Perni. +This work was supported by the Novo Nordisk Foundation, Grant Agreement NNF20OC0062047. +References +[1] +Nikhil R Pal and Sankar K Pal. “A review on image segmentation techniques”. In: Pattern recognition 26.9 +(1993). Publisher: Elsevier, pp. 1277–1294. +[2] +Dinesh D Patil and Sonal G Deore. “Medical image segmentation: a review”. In: International Journal of +Computer Science and Mobile Computing 2.1 (2013), pp. 22–27. +[3] +Dzung L Pham, Chenyang Xu, and Jerry L Prince. “A survey of current methods in medical image segmenta- +tion”. In: Annual review of biomedical engineering 2.3 (2000), pp. 315–337. +[4] +Anne E Carpenter et al. “CellProfiler: image analysis software for identifying and quantifying cell phenotypes”. +In: Genome biology 7.10 (2006). Publisher: Springer, pp. 1–11. +[5] +Rainer Pepperkok and Jan Ellenberg. “High-throughput fluorescence microscopy for systems biology”. In: +Nature reviews Molecular cell biology 7.9 (2006). Publisher: Nature Publishing Group, pp. 690–696. +[6] +Juan C Caicedo et al. “Data-analysis strategies for image-based cell profiling”. In: Nature methods 14.9 (2017). +Publisher: Nature Publishing Group, pp. 849–863. +[7] +Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. “Deep learning”. In: Nature 521.7553 (2015). Publisher: +Nature Publishing Group, pp. 436–444. +[8] +Fuyong Xing et al. “Deep learning in microscopy image analysis: A survey”. In: IEEE transactions on neural +networks and learning systems 29.10 (2017). Publisher: IEEE, pp. 4550–4568. +[9] +Erick Moen et al. “Deep learning for cellular image analysis”. In: Nature methods 16.12 (2019). Publisher: +Nature Publishing Group, pp. 1233–1246. +[10] +David A Van Valen et al. “Deep learning automates the quantitative analysis of individual cells in live-cell +imaging experiments”. In: PLoS computational biology 12.11 (2016). Publisher: Public Library of Science San +Francisco, CA USA, e1005177. +[11] +Thorsten Falk et al. “U-Net: deep learning for cell counting, detection, and morphometry”. In: Nature methods +16.1 (2019). Publisher: Nature Publishing Group, pp. 67–70. +[12] +Olaf Ronneberger, Philipp Fischer, and Thomas Brox. “U-net: Convolutional networks for biomedical image +segmentation”. In: International Conference on Medical image computing and computer-assisted intervention. +Springer, 2015, pp. 234–241. +[13] +Carsen Stringer et al. “Cellpose: a generalist algorithm for cellular segmentation”. In: Nature methods 18.1 +(2021). Publisher: Nature Publishing Group, pp. 100–106. +17 + +[14] +Noah F Greenwald et al. “Whole-cell segmentation of tissue images with human-level performance using large- +scale data annotation and deep learning”. In: Nature biotechnology 40.4 (2022). Publisher: Nature Publishing +Group, pp. 555–565. +[15] +Reka Hollandi et al. “nucleAIzer: a parameter-free deep learning framework for nucleus segmentation using +image style transfer”. In: Cell Systems 10.5 (2020). Publisher: Elsevier, pp. 453–458. +[16] +Arif Ahmed Sekh et al. “Physics-based machine learning for subcellular segmentation in living cells”. In: Nature +Machine Intelligence 3.12 (Dec. 2021). Number: 12 Publisher: Nature Publishing Group, pp. 1071–1080. issn: +2522-5839. doi: 10.1038/s42256-021-00420-0. url: https://www.nature.com/articles/s42256-021- +00420-0 (visited on 10/28/2022). +[17] +Paul Lang et al. “Cellular imaging in drug discovery”. In: Nature Reviews Drug Discovery 5.4 (2006). Publisher: +Nature Publishing Group, pp. 343–356. +[18] +Mitko Veta, Paul J van Diest, and Josien PW Pluim. “Cutting out the middleman: measuring nuclear area +in histopathology slides without segmentation”. In: International conference on medical image computing and +computer-assisted intervention. Springer, 2016, pp. 632–639. +[19] +Andre Esteva et al. “Dermatologist-level classification of skin cancer with deep neural networks”. In: nature +542.7639 (2017). Publisher: Nature Publishing Group, pp. 115–118. +[20] +Nicolas Coudray et al. “Classification and mutation prediction from non–small cell lung cancer histopathology +images using deep learning”. In: Nature medicine 24.10 (2018). Publisher: Nature Publishing Group, pp. 1559– +1567. +[21] +Mahdieh Poostchi et al. “Image analysis and machine learning for detecting malaria”. In: Translational Research +194 (2018). Publisher: Elsevier, pp. 36–55. +[22] +Vibor Laketa. “Microscopy in infectious disease research—Imaging across scales”. In: Journal of molecular +biology 430.17 (2018). Publisher: Elsevier, pp. 2612–2625. +[23] +Gordon J Berman. “Measuring behavior across scales”. In: BMC biology 16.1 (2018). Publisher: Springer, +pp. 1–11. +[24] +John W Krakauer et al. “Neuroscience needs behavior: correcting a reductionist bias”. In: Neuron 93.3 (2017). +Publisher: Elsevier, pp. 480–490. +[25] +Gopal P Sarma et al. “OpenWorm: overview and recent advances in integrative biological simulation of +Caenorhabditis elegans”. In: Philosophical Transactions of the Royal Society B 373.1758 (2018). Publisher: +The Royal Society, p. 20170382. +[26] +Kelsey M Hallinen et al. “Decoding locomotion from population neural activity in moving C. elegans”. In: Elife +10 (2021). Publisher: eLife Sciences Publications Limited, e66135. +[27] +Linda Turner et al. “Visualizing flagella while tracking bacteria”. In: Biophysical journal 111.3 (2016). Pub- +lisher: Elsevier, pp. 630–639. +[28] +Marco Polin et al. “Chlamydomonas swims with two “gears” in a eukaryotic version of run-and-tumble locomo- +tion”. In: Science 325.5939 (2009). Publisher: American Association for the Advancement of Science, pp. 487– +490. +[29] +Liang Li, Edward C Cox, and Henrik Flyvbjerg. “‘Dicty dynamics’: Dictyostelium motility as persistent random +motion”. In: Physical biology 8.4 (2011). Publisher: IOP Publishing, p. 046006. +[30] +Andrew Adamatzky. “Neuroscience without neurons”. In: AIP Conference Proceedings 2425.1 (2022). eprint: +https://aip.scitation.org/doi/pdf/10.1063/5.0082008, p. 390001. doi: 10.1063/5.0082008. url: https:// +aip.scitation.org/doi/abs/10.1063/5.0082008. +18 + +[31] +David Kokel and Randall T Peterson. “Using the zebrafish photomotor response for psychotropic drug screen- +ing”. In: Methods in cell biology. Vol. 105. Elsevier, 2011, pp. 517–524. +[32] +Linda P O’Reilly et al. “C. elegans in high-throughput drug discovery”. In: Advanced drug delivery reviews 69 +(2014). Publisher: Elsevier, pp. 247–253. +[33] +Demetrio Raldua and Benjamin Pina. “In vivo zebrafish assays for analyzing drug toxicity”. In: Expert opinion +on drug metabolism & toxicology 10.5 (2014). Publisher: Taylor & Francis, pp. 685–697. +[34] +Adam Michael Stewart, Robert Gerlai, and Allan V Kalueff. “Developing highER-throughput zebrafish screens +for in-vivo CNS drug discovery”. In: Frontiers in behavioral neuroscience 9 (2015). Publisher: Frontiers Media +SA, p. 14. +[35] +Michele Perni et al. “A natural product inhibits the initiation of α-synuclein aggregation and suppresses its +toxicity”. In: Proceedings of the National Academy of Sciences 114.6 (2017). Publisher: National Acad Sciences, +E1009–E1017. +[36] +Joseph Redmon et al. “You only look once: Unified, real-time object detection”. In: Proceedings of the IEEE +conference on computer vision and pattern recognition. 2016, pp. 779–788. +[37] +Talmo D. Pereira et al. “Fast animal pose estimation using deep neural networks”. In: Nature Methods 16.1 (Jan. +2019). Number: 1 Publisher: Nature Publishing Group, pp. 117–125. issn: 1548-7105. doi: 10.1038/s41592- +018-0234-5. url: https://www.nature.com/articles/s41592-018-0234-5 (visited on 10/10/2022). +[38] +Wei Geng et al. “Automatic tracking, feature extraction and classification of C. elegans phenotypes”. In: IEEE +transactions on biomedical engineering 51.10 (2004). Publisher: IEEE, pp. 1811–1820. +[39] +Nicolas Roussel et al. “A computational model for C. elegans locomotory behavior: application to multiworm +tracking”. In: IEEE transactions on biomedical engineering 54.10 (2007). Publisher: IEEE, pp. 1786–1797. +[40] +Michele Perni et al. “Massively parallel C. elegans tracking provides multi-dimensional fingerprints for pheno- +typic discovery”. In: Journal of neuroscience methods 306 (2018). Publisher: Elsevier, pp. 57–67. +[41] +Jirapat Likitlersuang et al. “C. elegans tracking and behavioral measurement”. In: JoVE (Journal of Visualized +Experiments) 69 (2012), e4094. +[42] +Veikko F Geyer et al. “Cell-body rocking is a dominant mechanism for flagellar synchronization in a swimming +alga”. In: Proceedings of the National Academy of Sciences 110.45 (2013). Publisher: National Acad Sciences, +pp. 18058–18063. +[43] +Kirsty Y Wan, Kyriacos C Leptos, and Raymond E Goldstein. “Lag, lock, sync, slip: the many ‘phases’ of cou- +pled flagella”. In: Journal of the Royal Society Interface 11.94 (2014). Publisher: The Royal Society, p. 20131160. +[44] +Nikzad Babaii Rizvandi et al. “Skeleton analysis of population images for detection of isolated and overlapped +nematode C. elegans”. In: 2008 16th European signal processing conference. IEEE, 2008, pp. 1–5. +[45] +Pascal Laube, Matthias O Franz, and Georg Umlauf. “Deep learning parametrization for B-spline curve ap- +proximation”. In: 2018 International Conference on 3D Vision (3DV). IEEE, 2018, pp. 691–699. +[46] +Jun Gao et al. “Deepspline: Data-driven reconstruction of parametric curves and surfaces”. In: arXiv preprint +arXiv:1901.03781 (2019). +[47] +Soham Mandal and Virginie Uhlmann. “Splinedist: Automated cell segmentation with spline curves”. In: 2021 +IEEE 18th International Symposium on Biomedical Imaging (ISBI). IEEE, 2021, pp. 1082–1086. +[48] +Greg J Stephens et al. “Dimensionality and dynamics in the behavior of C. elegans”. In: PLoS computational +biology 4.4 (2008). Publisher: Public Library of Science San Francisco, USA, e1000028. +[49] +Andr´e EX Brown et al. “A dictionary of behavioral motifs reveals clusters of genes affecting Caenorhabditis +elegans locomotion”. In: Proceedings of the National Academy of Sciences 110.2 (2013). Publisher: National +Acad Sciences, pp. 791–796. +19 + +[50] +Tosif Ahamed, Antonio C Costa, and Greg J Stephens. “Capturing the continuous complexity of behaviour in +Caenorhabditis elegans”. In: Nature Physics 17.2 (2021). Publisher: Nature Publishing Group, pp. 275–283. +[51] +Noel R Krieg, Joseph P Tomelty, and J Scott Wells Jr. “Inhibitio of Flagellar Coordination in Spirillum +volutans”. In: Journal of Bacteriology 94.5 (1967). Publisher: Am Soc Microbiol, pp. 1431–1436. +[52] +Dhruv K Vig and Charles W Wolgemuth. “Swimming dynamics of the Lyme disease spirochete”. In: Physical +review letters 109.21 (2012). Publisher: APS, p. 218104. +[53] +David J. Hampson. “The Spirochete Brachyspira pilosicoli, Enteric Pathogen of Animals and Humans”. In: +Clinical Microbiology Reviews 31.1 (Nov. 29, 2017), e00087–17. doi: 10.1128/CMR.00087-17. url: https: +//journals.asm.org/doi/10.1128/CMR.00087-17 (visited on 01/09/2023). +[54] +Navish Wadhwa and Howard C Berg. “Bacterial motility: machinery and mechanisms”. In: Nature Reviews +Microbiology 20.3 (2022). Publisher: Nature Publishing Group, pp. 161–173. +[55] +David M Woolley et al. “A study of synchronisation between the flagella of bull spermatozoa, with related +observations”. In: Journal of Experimental Biology 212.14 (2009). Publisher: Company of Biologists, pp. 2215– +2223. +[56] +Trine B. Haugen et al. “VISEM: a multimodal video dataset of human spermatozoa”. In: Proceedings of the +10th ACM Multimedia Systems Conference. MMSys ’19. New York, NY, USA: Association for Computing +Machinery, June 18, 2019, pp. 261–266. isbn: 9781450362979. doi: 10.1145/3304109.3325814. url: https: +//doi.org/10.1145/3304109.3325814 (visited on 01/09/2023). +[57] +Linda Turner, William S Ryu, and Howard C Berg. “Real-time imaging of fluorescent flagellar filaments”. In: +Journal of bacteriology 182.10 (2000). Publisher: Am Soc Microbiol, pp. 2793–2801. +[58] +Laurence G Wilson, Lucy M Carter, and Sarah E Reece. “High-speed holographic microscopy of malaria +parasites reveals ambidextrous flagellar waveforms”. In: Proceedings of the National Academy of Sciences 110.47 +(2013). Publisher: National Acad Sciences, pp. 18769–18774. +[59] +Joseph Redmon and Ali Farhadi. “Yolov3: An incremental improvement”. In: arXiv preprint arXiv:1804.02767 +(2018). +[60] +Kaiming He et al. “Deep residual learning for image recognition”. In: Proceedings of the IEEE conference on +computer vision and pattern recognition. 2016, pp. 770–778. +[61] +Kaiming He et al. “Mask r-cnn”. In: Proceedings of the IEEE international conference on computer vision. +2017, pp. 2961–2969. +[62] +Tsung-Yi Lin et al. “Focal loss for dense object detection”. In: Proceedings of the IEEE international conference +on computer vision. 2017, pp. 2980–2988. +[63] +Jiuxiang Gu et al. “Recent advances in convolutional neural networks”. In: Pattern recognition 77 (2018). +Publisher: Elsevier, pp. 354–377. +[64] +Piali Sengupta and Aravinthan DT Samuel. “Caenorhabditis elegans: a model system for systems neuroscience”. +In: Current opinion in neurobiology 19.6 (2009). Publisher: Elsevier, pp. 637–643. +[65] +Maria Markaki and Nektarios Tavernarakis. “Caenorhabditis elegans as a model system for human diseases”. +In: Current opinion in biotechnology 63 (2020). Publisher: Elsevier, pp. 118–125. +[66] +Jesse M Gray, Joseph J Hill, and Cornelia I Bargmann. “A circuit for navigation in Caenorhabditis elegans”. +In: Proceedings of the National Academy of Sciences 102.9 (2005). Publisher: National Acad Sciences, pp. 3184– +3191. +[67] +Catherine H Rankin, Christine DO Beck, and Catherine M Chiba. “Caenorhabditis elegans: a new model +system for the study of learning and memory”. In: Behavioural brain research 37.1 (1990). Publisher: Elsevier, +pp. 89–92. +20 + +[68] +Michael R Klass. “A method for the isolation of longevity mutants in the nematode Caenorhabditis elegans +and initial results”. In: Mechanisms of ageing and development 22.3 (1983). Publisher: Elsevier, pp. 279–286. +[69] +CH Opperman and S Chang. “Effects of Aldicarb and Fenamiphos on Acetycholinesterase and Motility of +Caenorhabditis elegans”. In: Journal of Nematology 23.1 (1991). Publisher: Society of Nematologists, p. 20. +[70] +Sheng Fong et al. “Energy crisis precedes global metabolic failure in a novel Caenorhabditis elegans Alzheimer +Disease model”. In: Scientific reports 6.1 (2016). Publisher: Nature Publishing Group, pp. 1–9. +[71] +Amy L Lee et al. “A new Caenorhabditis elegans model of human huntingtin 513 aggregation and toxicity in +body wall muscles”. In: PloS one 12.3 (2017). Publisher: Public Library of Science San Francisco, CA USA, +e0173644. +[72] +Michele Perni et al. “Multistep inhibition of α-synuclein aggregation and toxicity in vitro and in vivo by +trodusquemine”. In: ACS chemical biology 13.8 (2018). Publisher: ACS Publications, pp. 2308–2319. +[73] +D Dipon Ghosh et al. “C. elegans discriminates colors to guide foraging”. In: Science 371.6533 (2021). Publisher: +American Association for the Advancement of Science, pp. 1059–1063. +[74] +James F Morley et al. “The threshold for polyglutamine-expansion protein aggregation and cellular toxicity +is dynamic and influenced by aging in Caenorhabditis elegans”. In: Proceedings of the National Academy of +Sciences 99.16 (2002). Publisher: National Acad Sciences, pp. 10417–10422. +[75] +Jesse M Gray et al. “Oxygen sensation and social feeding mediated by a C. elegans guanylate cyclase homo- +logue”. In: Nature 430.6997 (2004). Publisher: Nature Publishing Group, pp. 317–322. +[76] +Linjiao Luo et al. “Olfactory behavior of swimming C. elegans analyzed by measuring motile responses to +temporal variations of odorants”. In: Journal of neurophysiology 99.5 (2008). Publisher: American Physiological +Society, pp. 2617–2625. +[77] +Terence I. Moy et al. “High Throughput Screen for Novel Antimicrobials using a Whole Animal Infection +Model”. In: ACS chemical biology 4.7 (July 17, 2009), pp. 527–533. issn: 1554-8929. doi: 10.1021/cb900084v. +url: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2745594/ (visited on 12/09/2022). +[78] +Annelie Persson et al. “Natural variation in a neural globin tunes oxygen sensing in wild Caenorhabditis +elegans”. In: Nature 458.7241 (2009). Publisher: Nature Publishing Group, pp. 1030–1033. +[79] +Raphael Sznitman et al. “Multi-environment model estimation for motility analysis of Caenorhabditis elegans”. +In: PLoS One 5.7 (2010). Publisher: Public Library of Science San Francisco, USA, e11631. +[80] +Steven D Buckingham, Frederick A Partridge, and David B Sattelle. “Automated, high-throughput, motility +analysis in Caenorhabditis elegans and parasitic nematodes: Applications in the search for new anthelmintics”. +In: International Journal for Parasitology: Drugs and Drug Resistance 4.3 (2014). Publisher: Elsevier, pp. 226– +232. +[81] +Jeong-Hoon Hahm et al. “C. elegans maximum velocity correlates with healthspan and is maintained in worms +with an insulin receptor mutation”. In: Nature communications 6.1 (2015). Publisher: Nature Publishing Group, +pp. 1–7. +[82] +He-Peng Zhang et al. “Collective motion and density fluctuations in bacterial colonies”. In: Proceedings of the +National Academy of Sciences 107.31 (2010). Publisher: National Acad Sciences, pp. 13626–13630. +[83] +Nicolas Roussel et al. “Robust tracking and quantification of C. elegans body shape and locomotion through +coiling, entanglement, and omega bends”. In: Worm. Vol. 3. Issue: 4. Taylor & Francis, 2014, e982437. +[84] +Sijie Jason Wang and Zhao-Wen Wang. “Track-a-worm, an open-source system for quantitative assessment of +C. elegans locomotory and bending behavior”. In: PloS one 8.7 (2013). Publisher: Public Library of Science +San Francisco, USA, e69653. +21 + +[85] +Zhaoyang Feng et al. “An imaging system for standardized quantitative analysis of C. elegans behavior”. In: +BMC bioinformatics 5.1 (2004). Publisher: Springer, pp. 1–6. +[86] +Ebraheem Fontaine, Alan H Barr, and Joel W Burdick. “Tracking of multiple worms and fish for biological +studies”. In: ICCV Workshop on Dynamical Vision. Citeseer, 2007. +[87] +Daniel Ramot et al. “The Parallel Worm Tracker: a platform for measuring average speed and drug-induced +paralysis in nematodes”. In: PloS one 3.5 (2008). Publisher: Public Library of Science San Francisco, USA, +e2208. +[88] +Nicholas A Swierczek et al. “High-throughput behavioral analysis in C. elegans”. In: Nature methods 8.7 (2011). +Publisher: Nature Publishing Group, pp. 592–598. +[89] +Kathleen Bates, Kim N. Le, and Hang Lu. “Deep learning for robust and flexible tracking in behavioral studies +for C. elegans”. In: PLOS Computational Biology 18.4 (Apr. 8, 2022). Publisher: Public Library of Science, +e1009942. issn: 1553-7358. doi: 10.1371/journal.pcbi.1009942. url: https://journals.plos.org/ +ploscompbiol/article?id=10.1371/journal.pcbi.1009942 (visited on 10/20/2022). +[90] +Shoubhik Chandan Banerjee, Khursheed Ahmad Khan, and Rati Sharma. Deep-Worm-Tracker: Deep Learning +Methods for Accurate Detection and Tracking for Behavioral Studies in C. elegans. Pages: 2022.08.18.504475 +Section: New Results. Aug. 19, 2022. doi: 10.1101/2022.08.18.504475. url: https://www.biorxiv.org/ +content/10.1101/2022.08.18.504475v1 (visited on 10/20/2022). +[91] +Anthony D Fouad et al. “High-throughput imaging of Caenorhabditis elegans aging using collective activity +monitoring”. In: bioRxiv (2021). Publisher: Cold Spring Harbor Laboratory. +[92] +Laetitia Hebert et al. “WormPose: Image synthesis and convolutional networks for pose estimation in C. +elegans”. In: PLoS computational biology 17.4 (2021). Publisher: Public Library of Science San Francisco, CA +USA, e1008914. +[93] +Sara Beery et al. “Context R-CNN: Long Term Temporal Context for Per-Camera Object Detection”. In: +Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2020, pp. 13075– +13085. url: https : / / openaccess . thecvf . com / content _ CVPR _ 2020 / html / Beery _ Context _ R - CNN _ +Long_Term_Temporal_Context_for_Per-Camera_Object_Detection_CVPR_2020_paper.html (visited on +12/05/2022). +[94] +B. Moghaddam and A. Pentland. “Probabilistic visual learning for object detection”. In: Proceedings of IEEE +International Conference on Computer Vision. Proceedings of IEEE International Conference on Computer +Vision. June 1995, pp. 786–793. doi: 10.1109/ICCV.1995.466858. +[95] +Eric E. Keaveny and Andr´e E. X. Brown. “Predicting path from undulations for C. elegans using linear and +nonlinear resistive force theory”. In: Physical Biology 14.2 (Mar. 2017). Publisher: IOP Publishing, p. 025001. +issn: 1478-3975. doi: 10.1088/1478-3975/aa5ce6. url: https://dx.doi.org/10.1088/1478-3975/aa5ce6 +(visited on 10/20/2022). +[96] +John C Crocker and David G Grier. “Methods of digital video microscopy for colloidal studies”. In: Journal of +colloid and interface science 179.1 (1996). Publisher: Elsevier, pp. 298–310. +[97] +Amirhossein Kazerouni et al. Diffusion Models for Medical Image Analysis: A Comprehensive Survey. Nov. 14, +2022. doi: 10.48550/arXiv.2211.07804. arXiv: 2211.07804[cs,eess]. url: http://arxiv.org/abs/2211. +07804 (visited on 01/06/2023). +22 + diff --git a/E9E3T4oBgHgl3EQfVgpZ/content/tmp_files/load_file.txt b/E9E3T4oBgHgl3EQfVgpZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..577431a233fd4ea37f1a169028a5d9b872b4000e --- /dev/null +++ b/E9E3T4oBgHgl3EQfVgpZ/content/tmp_files/load_file.txt @@ -0,0 +1,1245 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf,len=1244 +page_content='Fast spline detection in high density microscopy data Albert Alonso Julius B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Kirkegaard January 12, 2023 Abstract Computer-aided analysis of biological microscopy data has seen a massive improvement with the utilization of general-purpose deep learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Yet, in microscopy studies of multi-organism systems, the problem of collision and overlap remains challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This is particularly true for systems composed of slender bodies such as crawling nematodes, swimming spermatozoa, or the beating of eukaryotic or prokaryotic flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Here, we develop a novel end-to-end deep learning approach to extract precise shape trajectories of generally motile and overlapping splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Our method works in low resolution settings where feature keypoints are hard to define and detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Detection is fast and we demonstrate the ability to track thousands of overlapping organisms simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' While our approach is agnostic to area of application, we present it in the setting of and exemplify its usability on dense experiments of crawling Caenorhabditis elegans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The model training is achieved purely on synthetic data, utilizing a physics-based model for nematode motility, and we demonstrate the model’s ability to generalize from simulations to experimental videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1 Introduction Large-scale, high-throughput quantification of microscopy data have increasingly become possible with the aid of computer vision [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In particular within the last decade, deep learning techniques [7–9] have improved and enabled accurate image analysis of microscopy data in a broad range of areas including cell counting [10, 11], cell segmentation [12–14], nucleus detection [6, 15], sub-cellular segmentation [16], drug discovery [17], cancer detection [18–20], and the identification of infectious diseases [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Detection models serve as the fundamental operation in tracking procedures, and combined with suitable tracking algorithms, these can achieve morphologically resolved organism tracks that can accurately quantify organism motility [23], the application of which ranges from fundamental neuroscience [24–26] and the circuitry of simple organisms [27–30] to drug discovery [31–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Multi-organism detection can be achieved at increasing levels of fidelity: at the crudest, only center-of-mass locations or bounding boxes are predicted [36] which does enable tracking of organisms but provide little morpho- logical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In contrast, pixel-wise segmentation models [12] and pose estimation using keypoints [37] reveal accurate shape dynamics when employed on high-resolution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' However, these methods rely on high definition objects, as segmentation and prediction is highly sensible to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In particular for organisms that are long and slender, pixel-wise segmentation fails at low resolution as correct predictions require sub-pixel accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Moreover, at high densities, these methods may fail due to their inability to properly handle overlap between organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Here, we consider the problem of studying slender organisms at low resolution and high density with the goal to enable both accurate identity tracking and quantification of shape dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This problem has traditionally been approached by employing pixel-wise segmentation and subsequent skeletonization procedures [38–43], an approach that requires ad-hoc procedures to solve the problem of correctly identifying overlapping organisms [44], the com- binatorial complexity of which blows up at high densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To this end we abandon pixel-wise output and instead construct a neural network architecture that predicts, potentially overlapping, splines directly [45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Our method enables both accurate shape prediction and tracking in dense experiments of slender objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This is applicable to a 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='04460v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='CV] 11 Jan 2023 broad class of systems [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1], including tracking of nematode worms [48–50], spiral or elongated bacteria [51–54], spermatozoa [55, 56], the flagella of both eukaryotes [42, 43] and prokaryotes [57], and freely swimming flagella such those of microgametes [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' a b c d Figure 1: Microscopy images of different microorganisms whose slender structure and frequent overlaps makes them hard to detect using classical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans motility experiment from the dataset of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Motile, flexuous, thin, spiral-shaped B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' pilosicoli bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Still from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Beating flagella of the green alga C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' reinhardtii, provided by Kirsty Wan, University of Exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Swimming human spermatozoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' From dataset in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Our method relies on recent advances in deep learning [59–63] and extends these by few simple ideas: We found that humans are better at correctly resolving overlap between moving bodies when given access to videos rather than still micrographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Thus, to allow the neural network to encode the identity of individual bodies as a function of their motion, the input to our neural network is taken to be short video clips rather than single frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Our network outputs multiple independent predictions, and for each produces (1) the spline representing the centre-line of an object, (2) an estimated confidence score for the prediction, and (3) a latent vector, the space of which we induce a metric on that measures whether two predictions are trying to predict the same body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To train the network, each output quantity is associated with a specific loss term, where, importantly, the spline loss term is permutation- invariant in the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To resolve overlap, we do non-max suppression [36], but rather than measuring distances between spline predictions, we use the latent space output, which allows two predictions to be kept even though they are close in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This enables correct predictions for data in which objects overlap very closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Our method is further tailored to support the subsequent tracking process, which must link uniquely predictions from frame to frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To that end, we not only predict the object location at a single timepoint, but also predict consecutive past and future splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Using these time-resolved predictions in the linking process enables high-precision tracking even through dense regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Our method is principally applicable to all microscopy datasets that involve slender bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In this paper, we focus on its applications for tracking dense experiments of crawling C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans worms, a popular model system 2 业in neuroscience [64], human diseases [65], drug discovery [32], motor control [66], memory [67], and ageing [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Studies of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans often rely on phenotypic assays that measure the motility of the nematode worms as function of some environmental condition or treatment [35, 69–81], the throughput of which can be massively increased if overlap between organisms can be tolerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Likewise, resolving identities of organisms during overlap is crucial for studies of interactions between organisms [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Previous work on tracking C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans have generally employed classical computer vision approaches to accurately track single or a few high-definition worms [39, 83–86], or many low-resolution worms at non-overlapping densities [40, 87, 88], in some cases by utilizing a computational model of the worm motion for hypothesis tracking [39, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Recently, deep learning techniques have been utilized to track C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans worms using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' bounding box predictions [89–91] and fully resolved centre-line splines in the case of isolated worms [92], allowing for detection also during periods of self-overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' With this paper, we publish a dataset of videos of motile C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans worms imaged at a wide range of densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The dataset includes ∼ 1,500 labelled splines that we use to evaluate, but not train, our detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We demonstrate that our model can be trained exclusively using synthetically generated data and yet generalizes well to real videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Our method leverages the parallel capabilities of convolutional neural networks and is thus able to handle thousands of detections in a single pass, resulting in real-time detection at ∼ 90 Hz at 512 × 512 resolution on a single GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The code is open source and available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='com/kirkegaardlab/deeptangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2 Results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 Architecture Figure 2 illustrates the overall structure of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Our model is based on single-stage detection models [36, 59] that output many candidate predictions per target in a single forward pass and rely on a score system to prune until a single candidate is left for each target object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The performance of such single-stage models have been shown to enable accurate real-time bounding box detection [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The backbone of our neural network [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2a] consists of convolutional residual networks [60] with the small modification that we employ average pooling rather than max-pooling to avoid translational invariance in the spline predictions, which need to be accurate to a sub-pixel degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We take the input to our model to be a stack consecutive frames in order to provide the model with a temporal context [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This has previously been shown to improve the detection of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' partially hidden objects [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In particular, in present case of motile slender objects where dynamic crossings and overlap between objects are very common, a temporal context can provide the necessary information to resolve the problem of correct identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Furthermore, the temporal context allows the output of our model to include information on the motion of the splines, which we will further exploit for tracking purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The backbone of our neural network performs a 162-fold reduction in resolution when mapping the input images to feature space, from which the network outputs multiple anchored predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We choose the resulting number of candidates to be considerably larger than the number of objects in the frame, thus ensuring that all objects have suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The anchored approach further means that the only restriction on input size is that its dimensions be divisible by 16, and, in particular, it allows training at a certain resolution H × W and subsequent inference at another H′ × W ′ without loss of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The output of our model is composed of spline predictions, confidence scores and latent vectors: Spline predictions We choose to represent the centre-line of the slender bodies of interest by arrays consisting of k equidistant points [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' These coordinate arrays, which we refer to as splines, become high-precision descriptors even for complex shapes when k is chosen large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To reduce the complexity of predicting k points, we embed the spline representation with a principal component (PCA) transform A, the dimension κ of which can be much smaller than k [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The PCA components λ represent shape, and addition hereto, the network also predicts the offset x0 of 3 Neural Network I [ H , W , T ] fθ (λ, x0) [ M , W T , m ] z [ M , W T , K , 2 ] p [ M , D ] s [ M ] qϕ x = x0+Aλ a Latent Space Emergence of clusters 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Score Prunning rl Latent Space Best spline remaining 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Suppression Latent Space 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Repeat b Splines coordinates Synthetic I− I+ I Real I− I+ I L(lx, ls, lp) ˆz Neural Network Backpropagation \uf8ee \uf8f0 z p s \uf8f9 \uf8fb \uf8ee \uf8f0 x− x x+ \uf8f9 \uf8fb Splines z Filtering x = \uf8ee \uf8ef\uf8f0 (x0,0, y0,0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (x0,k, y0,k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (xn,0, yn,0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (xn,k, yn,k) \uf8f9 \uf8fa\uf8fb Predictions Spline points Visualization Visualization c • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••x • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' ψi (xi, yi) ds d •• direct distance •• flip distance e Figure 2: (a) Structure of the detection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Trainable neural networks are colored in gray, and represent the convolutional neural network f(I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' θ) and the latent space encoder q(λ, x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (b) Procedure to prune unfiltered predictions to final detections with the use of the encoded latent space vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (c) Method overview from the input clip I (we use a stack of 11 frames in this work) to the final matrix of splines x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The target frames [I−, I, I+] (center frames from the clip, orange) are explicitly shown for both the synthetic and real videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Additionally, the training setup is represented using lighter color arrows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' from synthetic data to loss backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' After detection, direct visualization of the predicted splines x is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (d) Diagram with a spline descriptor composed of k equidistant points along the skeleton of the nematode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (e) Visual representation of the two distances used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (1), the minimum of which corresponds to correct head-tail alignment and is the one that will be used in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 4 CCthe spline, the internal calculation of which is done in a local coordinate system defined by the anchor points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Thus, instead of predicting 2k floating point values per spline, the network needs only output κ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The temporal context of the input image stack permits output spline prediction also for the non-central images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In our approach, we predict a set of three splines z = [x−, x, x+] corresponding to the three central frames [I−, I, I+] of the input stack [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We consider the central spline x the main output, whereas the past x− and future x+ splines are considered auxiliary predictions whose main purpose lie on their use during the latent space encoding as well as the tracking procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We define the similarity measure between two splines by the standard Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In the case of splines that look symmetric from either end, we exploit this symmetry and employ the flip-invariant distance defined by d2(x, x′) = min � k � i=1 (xi − x′ i)2, k � i=1 (xi − x′ k−i+1)2� , (1) as illustrated in Figure 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Likewise, we define a distance between two collections of consecutive splines z, z′ by their weighted average d2 s = � t ωt d2(zt, z′ t), where the weights can be adjusted to give focus to central predictions, and for the present case we choose ω = 2ω− = 2ω+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The neural network is trained to minimize the distance d2 s between predictions and labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To do so, we let the independent predictors specialize for different shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This is achieved by using a permutation-invariant loss such that the total loss is computed as a sum over the labels only, each using the predictor that best match the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Thus many spline prediction will not contribute to the spline loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Confidence scores Each independent prediction of the network includes a confidence score s, which is used to filter out bad candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In bounding box or mask detection, intersection over union (IoU) is commonly used to evaluate the accuracy of a prediction, however, this metric does not generalize well to spline predictions when there is overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Instead, we introduce a custom metric to define the goodness of a spline set z by comparing it to its label ˆz, ˆs = exp � −d2 s(z,ˆz)/σ2 s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (2) Here, σs is a parameter that sets the scale over which the score varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The metric is sensitive to perturbations on accurate predictions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' predictions close to labels where ds → 0, but loses sensitivity the worse the predictions is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This is a useful feature as correct scoring for good predictions is crucial for choosing the best one, whereas low-scoring predictions are discarded in any case and their relative scoring therefore unimportant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The score prediction is trained using L2 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To avoid conflicting backwards error propagation between this task and that of spline prediction (as scoring bad predictions is easier), we stop the gradient flow in the computational graph on the last layer of the score-predicting part of fθ [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2a] such that it does not interfere with the accuracy of the predicted splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Latent space for candidates suppression Finally, we need to ensure that there is only one prediction per object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Bounding box detectors let the user decide the fraction of overlap between prediction boxes of the same class that should be considered to be targeting the same object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' As our method must work at high densities, this task is complicated by the fact that two predictions might be very close, even completely overlapping in the central frame, and yet represent different objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The task of choosing a suitable cutoff distance is therefore difficult, and we make this a trainable task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We do so by embedding each prediction in a low-dimensional latent space in which comparison between predictions is cheap, thus allowing efficient and fast candidate suppression also at high densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 5 Our method computes the latent vectors p for predictions using an auxiliary neural network, qφ which acts directly on the eigenvalues λ and offsets x0 rather than the more redundant spline coordinate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We induce a Euclidean metric on the latent space with the interpretation that two predictions i, j are predicting the same object with probability P(i ↔ j) = � exp � −||pi − pj||2� if ||x0i − x0j|| ≤ σl, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (3) Here, σl is a real-space visibility cutoff that prevents far predictions to interact in the encoded space, thus avoiding the need to scale the dimensionality of the latent space with the number of candidates or the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We note that when using the flip-invariant metric ds on splines, we explicitly construct the latent space encoder to likewise be flip-invariant (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To train the latent space, we make the assumption that during training predictors are ‘trying’ to predict the label closest to the prediction spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Combined with the probability interpretation, this allows us to use binary cross entropy as a loss function for the probability defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To avoid wrong clustering between undefined close-by predictions, the loss contribution of each prediction is scaled by the product of their real scores ˆsiˆsj, this ensuring that the network focus its attention of good predictions that will not be filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Finally, since the encoder should not to alter the performance of the spline suggestions, the loss on the latent space representations only updates the weights qφ of the encoder, but is trained concurrently with the main model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We employ non-max suppression to choose the best prediction of each object, but with distances measured in latent space, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Concretely: Once all the predictions whose score is lower than a threshold τs have been discarded, multiple candidates are likely to still remain for each target object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The lack of low score predictions expose clusters in the latent space that correspond to single objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We sort the remaining predictions by their score, automatically accepting the highest scored one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Once a prediction i is accepted, all predictions j that have a high probability P(i ↔ j) > τo of being the same object are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This is equivalent to setting an exclusion radius rl in the latent space as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We keep iterating on the remaining predictions, pruning the latent space until all candidates have been iterated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The final number of accepted predictions should equal the number of objects in the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Detection on dense C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans experiments To evaluate our approach, we study microscopy videos of crawling C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans worms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We are particularly interested in videos captured at much higher densities than those typically used in motility experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Thus we evaluate our model on wide-field videos captured under approximately uniform illumination [40], exemplified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In our dataset, the number of nematode worms vary ranging from from ∼ 400 with a small probability of overlap occurring ( ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='05 average overlaps per worm) to extremely densely packed plates with up to ∼ 6,000 nematodes, where there is, on average, one overlap per worm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This means that in the dense plates, detection methods that stop tracking after contact between worms happens are rendered completely ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Defining worm density ρ as the number of worms in a region per square millimeter, we find, as expected, a linear relation between the average amount of overlap per worm and the density [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 4a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Due to the spatial heterogeneity of the worm distribution inside the plate, higher densities can be observed when considering small regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' On 100 mm2 scales, the highest density in the dataset is ρ ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 mm−1, but this jumps to an extreme ρ ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 mm−1 when considering 10 mm2 regions, where humans begin to struggle to correctly identify worms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' For quantitative evaluation of our model, ∼ 200 random regions of the videos were sampled and hand-labelled resulting in ∼ 1,500 labelled worm splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' A sample of frames are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 3b to provide a sense of the different densities encountered in the evaluation dataset, with the predictions of the model overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 6 2736 x 2192 a b Figure 3: Showcase of the capabilities of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (a) Detected splines predicted on an entire densely populated well plate with a single forward pass through the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Inset shows a zoom-in section to demonstrate the accuracy of detection across the entire plate (except near borders, where the plate interferes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The total plate contains around 6,000 splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (b) Close up evaluation of different experimental clips with different densities of worms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Simulation-based training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To train our network, we implement a physics-based synthetic dataset generator to exploit perfectly defined labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This approach removes the need for a supervised dataset, and also allows labelled videos in situations where manual labeling may not be reliable, or where the subjectivity of the human labellers can result in inconsistent labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Physics-based synthetic datasets have successfully been used to train systems on similar conditions, for instance where manual labelling may introduce unnecessary noise or bias to the model [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Our in-silico data generator has two main components: a physics-based model for the organism and a synthetic frame generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In-silico worms are generated on demand every training step which removes the possibility of overfiting to the generated frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In order to train the model to work effectively with a range of worm densities, we generate batches with different numbers of worms in a uniform manner, without bias towards low or high worm counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This teaches the model to handle a variety of densities without overfitting to any specific case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' And to make the model more robust, training also happens on densities whose manual annotation would be extremely challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The simulation and video synthesis are implemented in a GPU framework which enables fast end-to-end training without the performance penalization of data transferring between the accelerator and the host machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We base the worm simulation on resistive force theory, as it has previously been shown to correctly predict the position of the skeleton for short spans of times [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Since the network only perceives the frames surrounding the target frames, we found the total duration of the clip to be short enough that a linear crawling model approximation fits our needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The physics-based model should encapsulate all types of organism behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This can be achieved by 7 1 cmp= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='19mm-2p= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='79mm-2p= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='91mm-2p= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='57mm-2p= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='97mm-2p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='28mm-2 1 mm• b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 Worm density in clip (mm−2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 Average overlaps on a worm a c ρ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='3mm−2 δadtw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='6 px 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 Worm density in clip (mm−2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 Double prediction artefacts Double prediction artefacts Error distance δadtw dependance with density d 0 1 2 3 4 Worm density in clip (mm−2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='000 True Positive rate e 0 1 2 3 4 Worm density in clip (mm−2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='08 False Negative rate f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 Score threshold τs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='56 Average error distance δadtw τo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 τo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='3 τo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 τo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='7 g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 Score threshold τs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='00 Average TP rate h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 Score threshold τs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 Average FN rate i Figure 4: (a) Average number of overlaps counted on frames of pixel size 512 × 512 with different densities of worms (N = 90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (b) Illustration of the asymmetric dynamic time warping distance error corresponding to the average value of the orange euclidean distances between the prediction (green) and the labelled points (white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (c) Example frame with manually labelled points (white) and models predictions (colored).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The metric is only evaluated in the lighter area of size 100 × 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (d) Quantified accuracy of the detections by showing the distance to the manually labelled splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Distributions for different densities are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The violin plots represent the 99 percentile of the data whereas outliers are plotted individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (e–f) Rates for True Positive and False Negative on the manually annotated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (g–i) Performance of the model with different combinations of score (τs) and overlap (τo) thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' N = 1,420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 8 oversampling the behavior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' by making the simulations more diverse in the behavior than reality and thus hope to include all types of real behavior as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Details on the worm simulation and video synthesis can be found in the methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Despite the potential for physics-based simulations to be used for synthetic training data, discrepancies with real data may lead to inaccuracies when applied to real microscopy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This reality gap can be the result of an overly simplified motility model or physics model, or the result of imprecise video synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The gap may be further increased by the fact that the model relies on the PCA transformation matrix A obtained on synthetic data, where the number of PCA components used have been chosen to accurately reproduce all synthetic patterns, but not necessarily to generalize to out-of-sample videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Thus we find that our model is limited to accurate skeleton predictions only on shapes that resemble those produced by our simulations, and the goal of the simulations is therefore to reproduce a broad spectrum of possible motility patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Likewise, we find that our model is susceptible to the brightness of the videos, and accordingly we adjust the real videos to increase their resemblance to the training data (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Metrics Despite being trained exclusively on synthetic data, the model’s inference performance is very good on real clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' From visual inspection, no immediate discrepancies are observed between detections in low density clips and at high density [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 3b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Likewise, per design, the network accuracy is independent on the input clip dimensions, and the parallel structure of convolutions permits the use of large videos covering thousands of nematodes to be processed simultaneously in a single forward pass [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 3a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We note, however, that even though no quality impact on detections is observed when using large fields-of-view clips, there can be a dependency if non-uniform illumination is used as different sections of the frame may have different requirements for preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' For a quantitative assessment of the method accuracy, we compare to the manually labelled dataset, an example of which alongside the model predictions can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' As the predictions are densely defined splines (here, ∼ 50 points), we introduce a custom metric to suitably evaluate the accuracy of the predictions using labels with lower fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The metric used must be shift-invariant, as having points anywhere along the spline should yield zero error regardless of whether the label points precisely coincide with the prediction points or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Likewise, label points should be monotonically assigned along the spline in order to avoid artificially reducing the error for strongly bent or self-coiling worms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Finally, it must be robust against the subjectivity of the labellers, as manual annotations might miss or avoid spots where visibility is low such as the end-points of the worms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To satisfy all these requirements, we introduce a metric based on the dynamical time warping (DTW) distance used to measure similarity between temporal curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In our modified version, asymmetric DTW, summation only runs over label points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Thus, the metric δadtw is defined as follows: Let d(i, j) be the Euclidean distance between label point i and prediction line segment j, then δadtw = min α 1 N N � i=1 d(i, α(i)), (4) where α : [1, N] → [1, M] is a monotonic (non-decreasing or non-increasing) assignments of the N label points to the M prediction line segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' A visual representation of the metric is shown in Figure 4b, and the O(NM) algorithm for its calculation is detailed in the Methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The results of evaluating the trained model on the labelled dataset are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' For reliable comparisons, we first solve the assignment algorithm for the label-prediction pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This means that in the case of two completely overlapped worms, two predictions need to be present to not count as a miss, and likewise, two predictions cannot be considered to target the same label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We find an average error of δadtw ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='54 px with no strong dependency between accuracy and density of worms [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 4d], with the exception of a slight increase in error for extremely dense clips (∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 mm−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The average error corresponds to less than the width of a worm (≈ 2 px ≈ 50 µm), and part of this can be attributed to the fact that human accuracy is also near the half-pixel level [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 4c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Some outliers can be seen however, which can mostly be attributed to an artefact of the model, where the network mistakes a single long worm 9 for two overlapping shorter predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This effect seems particularly sensitive to incorrect intensity normalization of the videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Let σϵ be a cutoff distance above which we no longer consider the predictions to be targeting the closest label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' For all the figures in Figure 4, this cutoff is assumed to be σϵ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 px, and we observe no significant changes by tuning it within the range of sensible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We define the True Positive (TP) rate as the fraction of predictions that both gets assigned a label and this label is within the the distance σϵ of the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Figure 4e shows that the model rarely predicts a spline where there is nothing with a TP rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Nevertheless, there are some predictions that do not get assigned a label which can be attributed to the double-prediction artefacts just mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The likelihood of this happening decreases with density, but the rate is so low that it is almost negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Similarly, we define the False Negative (FN) rate as the fraction of labels that are not assigned a prediction closer than σϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 4f shows that the model in general manages a low FN rate at around ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='015, but that this increases to a rate of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='06 at extreme densities such as ρ ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 mm−2, where clusters tend to be densely packed and manual labeling likewise becomes challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The filtering part of the model depends on the previously introduced thresholds τs and τo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The score threshold, 0 < τs < 1, is used to prune low score predictions [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2b(1)], while the overlap threshold, 0 < τo < 1, is used to decide the probability of two independent predictions to be targeting the same object [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2b(2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Throughout this paper, we have set these to τs = τo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' However, due to their relevance in modifying the filtering process, we evaluate how different combinations of thresholds may alter the performance results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Figures 4g–i show the average performance obtained across all densities when filtering the predictions with variable thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In spite of some dependency between worm density and TP/FN rates, we consider the average metric to be a good indicative of the performance on each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 4g shows the effect of the thresholds on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' No significant dependency on the thresholds is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This can be explained by the fact that accuracy is determined by the best predictors only, which are not discarded until a high τs is used, and ones those are removed, τo becomes irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Further, the fact that there is no notable difference between different values of τo indicates that the clusters are highly compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In contrast, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 4h shows that the TP rate has a stronger dependency on τo at low τs because low score predictions do not form compact clusters, and therefore a larger exclusion radius is required to discard them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 4i shows that misses only begin to occur once the best predictions are discarded, and a strong dependence on the τs is not observed before that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='2 Tracking from consecutive detections Motility assays require not only accurate detections but also the ability to link these across frames to form time- resolved tracks of individual organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This is challenging at high densities where we have the breakdown of the assumption that the closest detected object to the previous frame corresponds to the same identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In general, greedy approaches to particle tracking such as assigning directly the closest particle in consecutive frames frequently leads to failed tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Instead, the process of tracking can be efficiently formulated as a set of linear assignment problems [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Naturally, here we can expand upon particle tracking by using a metric that measures distances not between center-of-mass of the worms, but between the full splines as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This works well for most predictions, but can fail for fast-moving worms or in dense clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' A separate approach to tracking is Kalman filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This would require separate detection of entry and exit events of worms, as well as a probabilistic model for worm motility, which would most likely have to be highly non- linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Kalman filtering is viable for the tracking of few organisms, but for present large-scale systems we require a more efficient approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' As previously mentioned, splines from adjacent frames are also predicted in order to embed temporal information into the latent vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We propose a directed metric that leverages both past x− and future x+ spline predictions [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 5a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Thus to find a mapping σ from one frame to the next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' we solve 10 xi xj x+ i x+ j xk xl x− k x− l d(x+ i (t1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' xk(t2)) + d(x+ j (t1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' xl(t2)) + d(xi(t1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' x− k (t2)) + d(xj(t1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' x− l (t2)) Minimal assignment distance Forward distance df Backward distance db a No constrains Midpoint cutoff Final assignment t0 t1 t2 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 t0 t1 t2 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 t0 t1 t2 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 b d 1 25 49 Position (k) 0 3 6 9 12 15 18 21 24 27 Time (s) 1 25 49 Position (k) π 0 π e f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 Worm density in clip (mm−2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='00 Average track integrity Directed Normal c Only 135 tracks All tracks (∼ 6000) All tracks y x t h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 Worm density in clip (mm−2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='12 mm/s σM ∼ 1/ √ N SE (σM) on CoM speed (vmm) g Figure 5: (a) Illustration of the directed distance used to assign consecutive detections the same identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The simplified drawing shows two independent predictions at adjacent frames and showcases how the assignment scheme computes the identity by comparing future-present and past-present distances and choosing the assignment that minimizes their sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (b) Diagram showcasing how using a location cutoff simplifies the assignment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Nodes represent independent detections at each frame whereas edge values are given by the directed distance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The assignment happens by minimizing the sum of edges at each timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (c) Comparison of using the straightforward spline distance and the proposed directed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The accuracy is evaluated by measuring the integrity of the tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In contrast to other metrics in this paper, this plot has been obtained using synthetic worms as long- term, accurate tracks are required to evaluate the tracking integrity (See Methods for details on Tracking integrity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (d) Qualitative example of 30 s trajectories of the center of mass of the nematodes in a dense experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The still background image represent the last frame of the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To improve the visualization, a small subset of the trajectories are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In contrast, a corner of the frame is used to display all the trajectories to showcase the density of simultaneous tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (e) Two samples of the spline angle ψ of two randomly sampled nematodes from (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (f) Undulations corresponding to 30 s of the detections relative to the center of mass coordinate of nine randomly sampled nematodes from (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (g) Standard error value of the measurements of the center of mass speed as a function of density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (h) Showcase of the possible throughput of the method, by simultaneously tracking more than 6,000 tracks from a full dense plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' A small window on the tracks is shown to showcase their continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 11 CCσ = arg min σ �� i d(xi(t), x− σi(t′)) + d(x+ i (t), xσi(t′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (5) Identity assignment can be seen as a network flow global optimization where nodes represent detections and edges carry cost of assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To avoid having to perform all possible combinations of assignments, we include a physical distance threshold on the midpoint of the central spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This threshold significantly simplifies the assignment scheme and improves the runtime of the filtering process [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 5b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To quantify the performance of these methods, we define the tracking integrity ι as a scalar that indicates how consistent the assignment of a label to a prediction is along the tracked video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Perfect tracks have ι = 1, whereas labels that gets assigned two different identities for half of the duration of the video have ι = 1 2, and so on (see Methods for a detailed definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We evaluate this on synthetically generated videos of 10 seconds (200 frames) that have perfectly labelled tracks, the results of which are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' On videos with densities up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0 mm−2, we achieve an average integrity of ι ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This a ∼ 30 % improvement of the error over using direct spline assignment defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We observe that the integrity is almost perfect at low densities, but drops to ι ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='93 at the highest densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' When applied to high density videos of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans, the tracking method is able to keep track of individual worms as they pass through clusters of other worms [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 5d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In contrast to pixel-level classification of worms, our approach outputs splines directly, and thus subsequent analysis is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' For instance, one may directly study the worm undulations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 5f] or extract the worm spline angle ψ = arctan(y(s, t) − y0(t), x(s, t) − x0(t)) to provide insight into the movement patterns and kinematics of the worm [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 5e].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' One of the key advantages of our methods is its ability to collect a larger number of samples compared to traditional techniques, while still obtaining reliable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' As the standard error decreases with the number of samples, using our methods allows for metrics to be gathered with less uncertainty while still requiring the same experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' For instance, Figure 5g) shows how the error of estimating the average speed of the center of mass of the nematodes decreases with density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This advantage can be extended to tracking large numbers of nematodes in crowded environments, such as extremely dense petri dishes where more than 6,000 concurrent tracks can be simultaneously computed [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 5h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Thus, with our method, we are able to collect a larger number of samples and obtain more precise and reliable results, even in challenging conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 3 Discussion We have introduced a novel deep learning approach for detecting and tracking slender bodies, such as crawling nematodes, in microscopy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The presented convolutional neural network architecture is capable of accurately detecting a large number of overlapping organisms, a task that can be particularly challenging for standard methods such as bounding boxes and pixel-level classifiers due to the issue of occlusion and overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To address this, we have implemented a latent space encoding which allow us to filter by non-maximum suppression and effectively handle overlapping objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Not only is our method capable of accurately detecting and tracking slender bodies, but it also demonstrates strong scalability, performing well across a range of input frame sizes and densities of bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This makes it an ideal tool for a variety of experimental settings where splines are useful descriptors, including studies of crawling nematodes, swimming spermatozoa and beating eukaryotic or prokaryotic flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Besides a suitable detector model, labeled training data is also needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We have demonstrated that relying on a physics-based model to generate synthetic data is adequate to train our network to perform well on real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This is a key achievement as it means that applications of our system for different experimental studies do not require large datasets to be procured, but rather the implementation of a suitable simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Our approach for synthetic data generation relies on over-sampling the behavior of the worms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This is naturally a trade-off as too extreme behavior can lead to datasets that are too hard for the neural network to replicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' For our model, we found that we slightly 12 undersampled certain worms shapes such as strong coiling, which the model therefore could struggle with identifying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Though we did not look into this here, an interesting avenue for future research would be to bootstrap synthetic motility models on small datasets of real organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In a similar fashion, the frame-generator procedure should oversample the textures, pixel intensities and noise of real videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Here, it could be interesting to study whether style transfer [15] or diffusion models [97] could be used to further reduce the gap between training and inference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' For tracking, we introduced a directed metric that employs past and future spline predictions to link them across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' At very high densities this may still fail, in particular because the directed metric yields little advantage if predictions are missing in some frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' A potential way to improve on this could come from utilizing the latent space encoding as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This would require temporal continuity in the latent space representation, which is achievable by modifying the associated loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This should enhance the integrity of tracking, as it could potentially be used to resolve issues such as switches by leveraging the separation of closely physical predictions with different temporal behaviour that characterises the latent encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We believe that these suggestions might be fruitful avenues for further research for improving deep learning models for dense detection of splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In this paper, we have proposed a new approach for fast and precise detection and tracking of slender bodies in microscopy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Its speed and accurate performance across a range of densities and sizes, combined with the ability to handle overlapping objects, makes it a valuable tool for a variety of experimental settings where precise tracking is essential for obtaining quantitative metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 4 Methods Convolutional neural network Most of the weights of the network are at the feature detection convolutional network whose backbone is made of four ResNet groups consisting of 2, 4, 4, 2 blocks with strides 1, 2, 1, 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We modify the original ResNet architecture by replacing the initial max-pooling layer for an average-pool layer to avoid translational invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The final shape of the feature space is [H/16, W/16, C], with C being the number of candidates each cell proposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We have set C = 8 for this project in order to fulfil the condition of M ≫ N even at high densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' All in all, there will always be C candidates per each cell regardless of input size, which leads to a large number of candidates to be sorted in the filtering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The head of the convolutional neural network is composed of two fully connected layers of 512 and C · (3(m + 2) + 1) cells, respectively, with batch normalization in-between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Due to the orientation invariance of the loss function on the spline predictions, it is possible that the splines in the predicted set x−, x, x+ are not aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To remedy this, we aligned them by comparing with the eigenvalues of the flipped spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In order to get the flipped eigenvalues λf, we use λf = A−1JAλ (6) where A is the PCA transformation matrix and J is the exchange matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Latent space encoder The encoder qφ is composed of two fully connected layers with batch normalization in- between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The input of the encoder is the vector of size 3(m + 2) characterizing the splines predictions and the output is D floating point values, corresponding to the coordinates of p in the D-dimensional latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We have found D = 8 to be a well-performing dimension in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Due to the orientation invariance of the splines predictions, we need to construct the encoder to cluster those splines regardless of orientations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To do so, the input values are expanded to include those of the flipped splines λ → (λ, λf) and both are fed to the same layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To ensure symmetry, the output is then summed before passing through the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In doing so, the encoder becomes independent of spline orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 13 Input clips pre-processing The images used to train the model have dark (small pixel intensity) background, as we employ zero-padded convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This is relevant for real recordings, where a negative flip may be necessary to match the network requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' During training, generated clips are normalized using a 1–99 percentile normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' For real clips, we have found that accuracy is improved if we apply CLAHE (adaptive histogram equalization) before prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Likewise, a simple intensity correction factor µ may need to be applied to the videos in order to match the pixel profile of the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' For our dataset, we use correction factors of µ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='2, to get the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Note that we match real data to the synthetic as this avoids the need to retrain the network for different experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Loss functions Spline descriptors are trained as a regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Thus, the loss contribution is given by the custom distance defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To enforce specialization on the predictors, and due to the number of predictions M being considerable larger than the number of bodies N, only the best predictors are accounted for in the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Nevertheless, there may be labels ˆx completely or partially outside the frame at tc, despite being inside at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' To make sure not to punish bad predictions at the boundaries for not matching invisible splines, instead of using the number of simulated bodies N, the subset of bodies completely inside the frame Nv is used and the final loss expression is given by: lx = 1 Nv Nv � i min m d2 s(zm, ˆzi) (7) The score L2 loss is computed as the difference of the values predicted and the score the spline proposals should have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Thus, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (2), we train the predicted score of all predictions using: ls = 1 M M � i � exp � − min n d2 s(zi, ˆzn) σs � − s �2 (8) Finally, the loss function for the latent space encoder is a modified cross entropy loss scaled by the product of scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Denote Pi,j = P(i ↔ j) as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (3), then the encoder loss is defined as an average over all pairs of predictions ⟨i, j⟩ that are physically within the cutoff σl, lp = 1 S ⟨ˆsiˆsj(tij log (Pi,j) + (1 − ti,j) log (1 − Pi,j))⟩⟨i,j⟩ , (9) where S = � ˆsiˆsj, and ti,j indicates whether i and j are targeting the same label k, and is set by tij = � 1 if ki = kj 0 otherwise (10) with ki, kj being the closest labels to the predictions zi, zj respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Training details Training has been done from scratch, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' without the use of a pretrained backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' During training, the frame size for the input clips used was 256×256, but due to the anchored approach this does not constrain inference to happen at the same resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Synthetic input is generated on demand and on device rather than using a fixed pre-generated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Thus, the network never sees the same frame twice and there is no host- to-device data transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' As mentioned in the main text, all networks are trained simultaneously, despite the weights of each one depending on different cost functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The code has been written in Jax using Haiku and training has been carried on a cluster of 8 × NVIDIA A5000’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 14 Inference Inference happens at any resolution whose dimensions are multiple of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The input frames need to be slightly pre-process as described in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Candidate predictions are chosen using a score threshold, and non-maximum suppression in latent space is used for filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Due to the sequential nature of the filtering process, the implementation is written to use the CPU using numba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Worm simulation Worm trajectories are computed by employing a resisitve force theory crawling model used to predict rigid body motions of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans from the undulations [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Thus, we ensure that from a given set of generated undulations, the produced motions will match those of real worms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' From empirical observations, we propose a simple equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (11)) to generate the undulation of the worms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We define the motions by the spline angle ψk(s) with s ∈ [0, 1] [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2d], and decompose this into a linear combination: ψ(s) = ψu(s, t) + ψs(s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (11) This logically separates the worm undulations into two types of motion: one corresponding to a sinusoidal motion ψs and one in which the whole body bends ψu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' These we define by ψu(s, t) = A cos �2π T t + ρ1 � cos (kusk + ρ2) (12) ψs(s, t) = ˜A cos �2π T t + kssk + ρ3 � (13) where ˜A = 1 2 (1 + | sin (2πt) |) A and the rest of parameters are sampled from random distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Although many improvements for the above equations can be suggested, we prefer to keep the model simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Once the values of the parameters for ψ are generated all for the timesteps of the simulation, the positional coordinates are obtained using ⃗x(s, t) = L � s 0 �cos (ψ(s′, t) + γ) sin (ψ(s′, t) + γ) � ds′ (14) where γ is a random orientation and L is the length of the worm (also sampled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Once the skeleton is defined, the rigid body motions are predicted by solving [95] ⃗F = � L 0 ⃗f ds = 0, (15) ⃗τ = � L 0 (⃗x − ⃗xCoM) × ⃗f ds = 0, (16) where the force ⃗f can be calculated from the spline velocity ⃗U = ∂t⃗x + V + Ω × (⃗x − ⃗xCoM) by ⃗f = αt (ˆt · ⃗U) ˆt + αn (ˆn · ⃗U) ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' (17) Here, V and Ω are the center-of-mass velocity and rotational velocity (that we are solving for), and αt and αn = α αt is the tangential and normal drag coefficients, which is also sampled for (α > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We did not find a need for using a non-linear force theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The simulation is run with Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='9 using the Jax library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 15 Video synthesis Given the labels for the splines positions, synthetic videos are generated to be used as input during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In order to add width to each worm, we vary the local body radius r by a function of the form r(s) = ˜R |sin(arccos(as + b))| (18) The pixel values of those circles are calculated with anti-aliasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Once the worms have been rendered, noise artefacts such as uneven background, blurring, Gaussian noise, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' are added to replicate the observed conditions of real experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' During training, standard augmentation techniques are applied as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In the same manner as the motions simulation and the neural network training, frame generation is also written in Python using the Jax library in order to leverage GPU capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Experimental dataset Videos of crawling C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans were filmed using the protocol described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Manually annotated dataset The evaluation dataset is annotated using a custom tool that can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='com/kirkegaardlab/deeptanglelabel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Around ∼ 1,500 splines have been annotated and this dataset (videos and labels) is included in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Asymmetric dynamic time-warped error distance In order to evaluate the manually labelled dataset, we introduce an error distance that compares the similarity between two curves by calculating a distance between each point on one curve and the nearest segment on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' The error distance used is a variation of the dynamic time warping distance, which is widely used for comparing time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We note that, just as is the case for the dynamic time warping distance, this is not a true distance in the mathematical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Algorithm 1: Algorithm for asymmetric dynamic time warping Data: Label curve defined by N points {pi}, and prediction curve defined by M line segments {sj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Result: The asymmetric dynamically time-warped distance from label to prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Initialize matrices C, D with size [N, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' for i = 1 to N do for j = 1 to M do Di,j ← distance from point to segment(pi, sj) C1,1 ← D1,1 for i = 2 to N do Ci,1 = Ci−1,1 + Di,1 for j = 2 to M do C1,j = min (C1,j−1, D1,j−1) for i = 2 to N do for j = 2 to M do Ci,j = min (Ci,j−1, Ci−1,j + Di,j) return CN,M/N Tracking implementation Tracking is done by sequentially predicting individual frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' For better performance, batching of frames allows for parallel detections and can drastically reduce execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Nevertheless, due to the requirement of including surrounding frames for each detection, a considerable increase in memory usage is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Once a collection of spline detections is obtained, each prediction is adapted in order to make it work with the TrackPy Python library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Due to the peculiarity of our distance metric, we implement a custom neighbor strategy (see Code Availability) that avoids the assumption of a symmetric distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 16 It may happen that some detection artefacts appear during the sequential detection performed on tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We have implemented a quick check on the resulting tracks to make sure not to have stubs, and fix obvious branching of tracks due to these artefact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Slight increase in integrity is observed on dense clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Tracking integrity Given a true label of a track of length N, we associate to this track at each time point i a prediction identity Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' We may then define the integrity of the track as ι = 1/N 2 �N i=1 �N j=1[Ii = Ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' For instance, if a label is given identities I = [1, 1, 1, 5, 5, 5, 3, 3, 3] during the track, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' there have been two identity swaps, we find ι = 1 3, which has the interpretation that the track was correct for a third of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This measure will in general scale like ι ∼ N −1, as longer tracks will have higher likelihood of identity swaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Acknowledgments Video of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans were provided by Celia Raimondi, Sunehera Sarwat and Michele Perni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' This work was supported by the Novo Nordisk Foundation, Grant Agreement NNF20OC0062047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' References [1] Nikhil R Pal and Sankar K Pal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “A review on image segmentation techniques”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Pattern recognition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='9 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1277–1294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [2] Dinesh D Patil and Sonal G Deore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Medical image segmentation: a review”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: International Journal of Computer Science and Mobile Computing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 22–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [3] Dzung L Pham, Chenyang Xu, and Jerry L Prince.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “A survey of current methods in medical image segmenta- tion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Annual review of biomedical engineering 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='3 (2000), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 315–337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [4] Anne E Carpenter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “CellProfiler: image analysis software for identifying and quantifying cell phenotypes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Genome biology 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='10 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [5] Rainer Pepperkok and Jan Ellenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “High-throughput fluorescence microscopy for systems biology”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature reviews Molecular cell biology 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='9 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 690–696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [6] Juan C Caicedo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Data-analysis strategies for image-based cell profiling”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature methods 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='9 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 849–863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [7] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Deep learning”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature 521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='7553 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 436–444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [8] Fuyong Xing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Deep learning in microscopy image analysis: A survey”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: IEEE transactions on neural networks and learning systems 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='10 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: IEEE, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 4550–4568.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [9] Erick Moen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Deep learning for cellular image analysis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature methods 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='12 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1233–1246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [10] David A Van Valen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Deep learning automates the quantitative analysis of individual cells in live-cell imaging experiments”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: PLoS computational biology 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='11 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Public Library of Science San Francisco, CA USA, e1005177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [11] Thorsten Falk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “U-Net: deep learning for cell counting, detection, and morphometry”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature methods 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 67–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [12] Olaf Ronneberger, Philipp Fischer, and Thomas Brox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “U-net: Convolutional networks for biomedical image segmentation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: International Conference on Medical image computing and computer-assisted intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Springer, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 234–241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [13] Carsen Stringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Cellpose: a generalist algorithm for cellular segmentation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature methods 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 100–106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 17 [14] Noah F Greenwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Whole-cell segmentation of tissue images with human-level performance using large- scale data annotation and deep learning”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature biotechnology 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='4 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 555–565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [15] Reka Hollandi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “nucleAIzer: a parameter-free deep learning framework for nucleus segmentation using image style transfer”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Cell Systems 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 453–458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [16] Arif Ahmed Sekh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Physics-based machine learning for subcellular segmentation in living cells”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature Machine Intelligence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='12 (Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Number: 12 Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1071–1080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' issn: 2522-5839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1038/s42256-021-00420-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='com/articles/s42256-021- 00420-0 (visited on 10/28/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [17] Paul Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Cellular imaging in drug discovery”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature Reviews Drug Discovery 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='4 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 343–356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [18] Mitko Veta, Paul J van Diest, and Josien PW Pluim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Cutting out the middleman: measuring nuclear area in histopathology slides without segmentation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: International conference on medical image computing and computer-assisted intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Springer, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 632–639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [19] Andre Esteva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Dermatologist-level classification of skin cancer with deep neural networks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: nature 542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='7639 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 115–118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [20] Nicolas Coudray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Classification and mutation prediction from non–small cell lung cancer histopathology images using deep learning”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature medicine 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='10 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1559– 1567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [21] Mahdieh Poostchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Image analysis and machine learning for detecting malaria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Translational Research 194 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 36–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [22] Vibor Laketa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Microscopy in infectious disease research—Imaging across scales”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Journal of molecular biology 430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='17 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2612–2625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [23] Gordon J Berman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Measuring behavior across scales”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: BMC biology 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [24] John W Krakauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Neuroscience needs behavior: correcting a reductionist bias”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Neuron 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='3 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 480–490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [25] Gopal P Sarma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “OpenWorm: overview and recent advances in integrative biological simulation of Caenorhabditis elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Philosophical Transactions of the Royal Society B 373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1758 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: The Royal Society, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 20170382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [26] Kelsey M Hallinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Decoding locomotion from population neural activity in moving C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Elife 10 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: eLife Sciences Publications Limited, e66135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [27] Linda Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Visualizing flagella while tracking bacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Biophysical journal 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='3 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Pub- lisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 630–639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [28] Marco Polin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Chlamydomonas swims with two “gears” in a eukaryotic version of run-and-tumble locomo- tion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Science 325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5939 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: American Association for the Advancement of Science, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 487– 490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [29] Liang Li, Edward C Cox, and Henrik Flyvbjerg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “‘Dicty dynamics’: Dictyostelium motility as persistent random motion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Physical biology 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='4 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: IOP Publishing, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 046006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [30] Andrew Adamatzky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Neuroscience without neurons”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: AIP Conference Proceedings 2425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' eprint: https://aip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='scitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1063/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0082008, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 390001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1063/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0082008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' url: https:// aip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='scitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='org/doi/abs/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1063/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='0082008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 18 [31] David Kokel and Randall T Peterson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Using the zebrafish photomotor response for psychotropic drug screen- ing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Methods in cell biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Elsevier, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 517–524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [32] Linda P O’Reilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans in high-throughput drug discovery”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Advanced drug delivery reviews 69 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 247–253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [33] Demetrio Raldua and Benjamin Pina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “In vivo zebrafish assays for analyzing drug toxicity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Expert opinion on drug metabolism & toxicology 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Taylor & Francis, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 685–697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [34] Adam Michael Stewart, Robert Gerlai, and Allan V Kalueff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Developing highER-throughput zebrafish screens for in-vivo CNS drug discovery”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Frontiers in behavioral neuroscience 9 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Frontiers Media SA, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [35] Michele Perni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “A natural product inhibits the initiation of α-synuclein aggregation and suppresses its toxicity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the National Academy of Sciences 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='6 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: National Acad Sciences, E1009–E1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [36] Joseph Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “You only look once: Unified, real-time object detection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 779–788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [37] Talmo D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Fast animal pose estimation using deep neural networks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature Methods 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Number: 1 Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 117–125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' issn: 1548-7105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1038/s41592- 018-0234-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='com/articles/s41592-018-0234-5 (visited on 10/10/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [38] Wei Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Automatic tracking, feature extraction and classification of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans phenotypes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: IEEE transactions on biomedical engineering 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='10 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: IEEE, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1811–1820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [39] Nicolas Roussel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “A computational model for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans locomotory behavior: application to multiworm tracking”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: IEEE transactions on biomedical engineering 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='10 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: IEEE, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1786–1797.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [40] Michele Perni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Massively parallel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans tracking provides multi-dimensional fingerprints for pheno- typic discovery”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Journal of neuroscience methods 306 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 57–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [41] Jirapat Likitlersuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans tracking and behavioral measurement”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: JoVE (Journal of Visualized Experiments) 69 (2012), e4094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [42] Veikko F Geyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Cell-body rocking is a dominant mechanism for flagellar synchronization in a swimming alga”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the National Academy of Sciences 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='45 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: National Acad Sciences, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 18058–18063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [43] Kirsty Y Wan, Kyriacos C Leptos, and Raymond E Goldstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Lag, lock, sync, slip: the many ‘phases’ of cou- pled flagella”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Journal of the Royal Society Interface 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='94 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: The Royal Society, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 20131160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [44] Nikzad Babaii Rizvandi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Skeleton analysis of population images for detection of isolated and overlapped nematode C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: 2008 16th European signal processing conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' IEEE, 2008, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [45] Pascal Laube, Matthias O Franz, and Georg Umlauf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Deep learning parametrization for B-spline curve ap- proximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: 2018 International Conference on 3D Vision (3DV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' IEEE, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 691–699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [46] Jun Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Deepspline: Data-driven reconstruction of parametric curves and surfaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: arXiv preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='03781 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [47] Soham Mandal and Virginie Uhlmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Splinedist: Automated cell segmentation with spline curves”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: 2021 IEEE 18th International Symposium on Biomedical Imaging (ISBI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1082–1086.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [48] Greg J Stephens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Dimensionality and dynamics in the behavior of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: PLoS computational biology 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='4 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Public Library of Science San Francisco, USA, e1000028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [49] Andr´e EX Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “A dictionary of behavioral motifs reveals clusters of genes affecting Caenorhabditis elegans locomotion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the National Academy of Sciences 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='2 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: National Acad Sciences, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 791–796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 19 [50] Tosif Ahamed, Antonio C Costa, and Greg J Stephens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Capturing the continuous complexity of behaviour in Caenorhabditis elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature Physics 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='2 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 275–283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [51] Noel R Krieg, Joseph P Tomelty, and J Scott Wells Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Inhibitio of Flagellar Coordination in Spirillum volutans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Journal of Bacteriology 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Am Soc Microbiol, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1431–1436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [52] Dhruv K Vig and Charles W Wolgemuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Swimming dynamics of the Lyme disease spirochete”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Physical review letters 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='21 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: APS, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 218104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [53] David J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Hampson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “The Spirochete Brachyspira pilosicoli, Enteric Pathogen of Animals and Humans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Clinical Microbiology Reviews 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 29, 2017), e00087–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1128/CMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='00087-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' url: https: //journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='asm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1128/CMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='00087-17 (visited on 01/09/2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [54] Navish Wadhwa and Howard C Berg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Bacterial motility: machinery and mechanisms”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature Reviews Microbiology 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='3 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 161–173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [55] David M Woolley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “A study of synchronisation between the flagella of bull spermatozoa, with related observations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Journal of Experimental Biology 212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='14 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Company of Biologists, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2215– 2223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [56] Trine B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Haugen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “VISEM: a multimodal video dataset of human spermatozoa”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the 10th ACM Multimedia Systems Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' MMSys ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' New York, NY, USA: Association for Computing Machinery, June 18, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 261–266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' isbn: 9781450362979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1145/3304109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='3325814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' url: https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1145/3304109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='3325814 (visited on 01/09/2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [57] Linda Turner, William S Ryu, and Howard C Berg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Real-time imaging of fluorescent flagellar filaments”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Journal of bacteriology 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='10 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Am Soc Microbiol, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2793–2801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [58] Laurence G Wilson, Lucy M Carter, and Sarah E Reece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “High-speed holographic microscopy of malaria parasites reveals ambidextrous flagellar waveforms”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the National Academy of Sciences 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='47 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: National Acad Sciences, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 18769–18774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [59] Joseph Redmon and Ali Farhadi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Yolov3: An incremental improvement”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: arXiv preprint arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='02767 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [60] Kaiming He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Deep residual learning for image recognition”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [61] Kaiming He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Mask r-cnn”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the IEEE international conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2961–2969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [62] Tsung-Yi Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Focal loss for dense object detection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the IEEE international conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2980–2988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [63] Jiuxiang Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Recent advances in convolutional neural networks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Pattern recognition 77 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 354–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [64] Piali Sengupta and Aravinthan DT Samuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Caenorhabditis elegans: a model system for systems neuroscience”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Current opinion in neurobiology 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='6 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 637–643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [65] Maria Markaki and Nektarios Tavernarakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Caenorhabditis elegans as a model system for human diseases”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Current opinion in biotechnology 63 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 118–125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [66] Jesse M Gray, Joseph J Hill, and Cornelia I Bargmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “A circuit for navigation in Caenorhabditis elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the National Academy of Sciences 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='9 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: National Acad Sciences, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 3184– 3191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [67] Catherine H Rankin, Christine DO Beck, and Catherine M Chiba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Caenorhabditis elegans: a new model system for the study of learning and memory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Behavioural brain research 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 89–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 20 [68] Michael R Klass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “A method for the isolation of longevity mutants in the nematode Caenorhabditis elegans and initial results”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Mechanisms of ageing and development 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='3 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 279–286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [69] CH Opperman and S Chang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Effects of Aldicarb and Fenamiphos on Acetycholinesterase and Motility of Caenorhabditis elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Journal of Nematology 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Society of Nematologists, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [70] Sheng Fong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Energy crisis precedes global metabolic failure in a novel Caenorhabditis elegans Alzheimer Disease model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Scientific reports 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [71] Amy L Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “A new Caenorhabditis elegans model of human huntingtin 513 aggregation and toxicity in body wall muscles”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: PloS one 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='3 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Public Library of Science San Francisco, CA USA, e0173644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [72] Michele Perni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Multistep inhibition of α-synuclein aggregation and toxicity in vitro and in vivo by trodusquemine”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: ACS chemical biology 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='8 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: ACS Publications, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2308–2319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [73] D Dipon Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans discriminates colors to guide foraging”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Science 371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='6533 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: American Association for the Advancement of Science, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1059–1063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [74] James F Morley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “The threshold for polyglutamine-expansion protein aggregation and cellular toxicity is dynamic and influenced by aging in Caenorhabditis elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the National Academy of Sciences 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='16 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: National Acad Sciences, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 10417–10422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [75] Jesse M Gray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Oxygen sensation and social feeding mediated by a C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans guanylate cyclase homo- logue”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature 430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='6997 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 317–322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [76] Linjiao Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Olfactory behavior of swimming C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans analyzed by measuring motile responses to temporal variations of odorants”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Journal of neurophysiology 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: American Physiological Society, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2617–2625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [77] Terence I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Moy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “High Throughput Screen for Novel Antimicrobials using a Whole Animal Infection Model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: ACS chemical biology 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='7 (July 17, 2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 527–533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' issn: 1554-8929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1021/cb900084v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='ncbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='nlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='gov/pmc/articles/PMC2745594/ (visited on 12/09/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [78] Annelie Persson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Natural variation in a neural globin tunes oxygen sensing in wild Caenorhabditis elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature 458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='7241 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1030–1033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [79] Raphael Sznitman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Multi-environment model estimation for motility analysis of Caenorhabditis elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: PLoS One 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='7 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Public Library of Science San Francisco, USA, e11631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [80] Steven D Buckingham, Frederick A Partridge, and David B Sattelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Automated, high-throughput, motility analysis in Caenorhabditis elegans and parasitic nematodes: Applications in the search for new anthelmintics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: International Journal for Parasitology: Drugs and Drug Resistance 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='3 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 226– 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [81] Jeong-Hoon Hahm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans maximum velocity correlates with healthspan and is maintained in worms with an insulin receptor mutation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature communications 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [82] He-Peng Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Collective motion and density fluctuations in bacterial colonies”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the National Academy of Sciences 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='31 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: National Acad Sciences, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 13626–13630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [83] Nicolas Roussel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Robust tracking and quantification of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans body shape and locomotion through coiling, entanglement, and omega bends”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Worm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Issue: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Taylor & Francis, 2014, e982437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [84] Sijie Jason Wang and Zhao-Wen Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Track-a-worm, an open-source system for quantitative assessment of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans locomotory and bending behavior”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: PloS one 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='7 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Public Library of Science San Francisco, USA, e69653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 21 [85] Zhaoyang Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “An imaging system for standardized quantitative analysis of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans behavior”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: BMC bioinformatics 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [86] Ebraheem Fontaine, Alan H Barr, and Joel W Burdick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Tracking of multiple worms and fish for biological studies”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: ICCV Workshop on Dynamical Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Citeseer, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [87] Daniel Ramot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “The Parallel Worm Tracker: a platform for measuring average speed and drug-induced paralysis in nematodes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: PloS one 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='5 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Public Library of Science San Francisco, USA, e2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [88] Nicholas A Swierczek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “High-throughput behavioral analysis in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Nature methods 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='7 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 592–598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [89] Kathleen Bates, Kim N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Le, and Hang Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Deep learning for robust and flexible tracking in behavioral studies for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: PLOS Computational Biology 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='4 (Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 8, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Public Library of Science, e1009942.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Deep-Worm-Tracker: Deep Learning Methods for Accurate Detection and Tracking for Behavioral Studies in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Pages: 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='504475 Section: New Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 19, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1101/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='504475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='biorxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='org/ content/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1101/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='504475v1 (visited on 10/20/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [91] Anthony D Fouad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “High-throughput imaging of Caenorhabditis elegans aging using collective activity monitoring”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: bioRxiv (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Cold Spring Harbor Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [92] Laetitia Hebert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “WormPose: Image synthesis and convolutional networks for pose estimation in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: PLoS computational biology 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='4 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Public Library of Science San Francisco, CA USA, e1008914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [93] Sara Beery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Context R-CNN: Long Term Temporal Context for Per-Camera Object Detection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 13075– 13085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' url: https : / / openaccess .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' thecvf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' com / content _ CVPR _ 2020 / html / Beery _ Context _ R - CNN _ Long_Term_Temporal_Context_for_Per-Camera_Object_Detection_CVPR_2020_paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='html (visited on 12/05/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [94] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Moghaddam and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Pentland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Probabilistic visual learning for object detection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Proceedings of IEEE International Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Proceedings of IEEE International Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' June 1995, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 786–793.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1109/ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='466858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [95] Eric E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Keaveny and Andr´e E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Predicting path from undulations for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' elegans using linear and nonlinear resistive force theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Physical Biology 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='2 (Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: IOP Publishing, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 025001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' issn: 1478-3975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1088/1478-3975/aa5ce6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' url: https://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1088/1478-3975/aa5ce6 (visited on 10/20/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [96] John C Crocker and David G Grier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' “Methods of digital video microscopy for colloidal studies”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' In: Journal of colloid and interface science 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='1 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Publisher: Elsevier, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 298–310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' [97] Amirhossein Kazerouni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Diffusion Models for Medical Image Analysis: A Comprehensive Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 14, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='07804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' arXiv: 2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='07804[cs,eess].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' url: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 07804 (visited on 01/06/2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E3T4oBgHgl3EQfVgpZ/content/2301.04460v1.pdf'} diff --git a/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf b/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cead413454f326895b5949a1ac0f4261634925f7 --- /dev/null +++ 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place Jussieu, 75005 Paris, France +2Centre de Nanosciences et de Nanotechnologies, CNRS, +Universit´e Paris-Saclay, 91120 Palaiseau, France +3IMEP-LAHC, Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, Grenoble INP, 38000 Grenoble, France +4Universit´e Paris-Saclay, Institut d’Optique Graduate School, +CNRS, Laboratoire Charles Fabry, 91127 Palaiseau, France +(Dated: January 13, 2023) +Novel platforms interfacing trapped cold atoms and guided light in nanoscale waveguides are a +promising route to achieve a regime of strong coupling between light and atoms in single pass, +with applications to quantum non-linear optics and quantum simulation. A strong challenge for the +experimental development of this emerging waveguide-QED field of research is to combine facilitated +optical access for atom transport, atom trapping via guided modes and robustness to inherent +nanofabrication imperfections. In this endeavor, here we propose to interface Rubidium atoms with +a photonic crystal waveguide based on a large-index GaInP slab. With a specifically tailored half- +W1 design, we show that a large coupling to the waveguide can be obtained and guided modes can +be used to form two-color dipole traps for atoms at about 100 nm from the edge of the structure. +This optimized device should greatly improve the level of experimental control and facilitate the +atom integration. +I. +INTRODUCTION +Interfacing cold neutral atoms and photons guided in +nanoscale waveguides has raised a large interest over the +recent years, with a wealth of emerging opportunities [1]. +Arrays of atoms can be trapped in the evanescent field +of guided modes and the strong transverse confinement +enables to increase the individual atom-photon coupling +in single pass. Remarkable experimental advances have +been obtained with optical nanofibers [2–5], exploiting +collective effects and chiral properties to realize various +all-fibered functionalities [6–11]. Beyond nanofibers, tai- +lored dispersion relations that can be obtained in pho- +tonic crystal waveguides (PCW) offer unique features. +While the atom-photon coupling can be strongly en- +hanced near a band edge, where guided modes can prop- +agate slowly, atom-photon bound states can also appear +for an atomic transition within a bandgap, with the ca- +pability to implement tunable long-range atom-atom in- +teractions. These features led to a variety of theoretical +proposals for applications in quantum optics and many- +body physics [12–14]. +Despite the promises of this new waveguide-QED +paradigm, trapping atoms in the vicinity of such photonic +crystal waveguides is still at its infancy. This combina- +tion is a daunting challenge due to stringent requirements +when considering real physical implementations. A first +challenge is to keep the atoms as static as possible close to +∗ Corresponding author: alban.urvoy@sorbonne-universite.fr +the structure, so that they can interact with the evanes- +cent mode. While tweezers can be used to maintain the +atoms at a fixed distance [15–18], it is challenging to make +an array of such atoms at distances on the 100 nm range. +Dipole trapping by the evanescent field of guided modes +is necessary but it has remained an important roadblock. +Up to now, only a corrugated slot waveguide (so-called +alligator waveguide) [19–23] has been implemented and +first pioneering demonstrations obtained, albeit with a +limited number of atoms and without stable trapping in +the evanescent field. Some theoretical proposals on novel +interesting structures supporting atom trapping in the +evanescent field have emerged since, such as a slot [24] or +a comb waveguide [25]. Structures must also provide a +large optical access to bring atoms close to their surface. +Eventually, in order to push experimental development, +great care should be put in ensuring that the structure +is robust against fabrication imperfections. +In this paper, we design a novel platform for interfac- +ing trapped cold atoms and a slow-mode photonic crystal +waveguide. +Building from the promises of W1 waveg- +uides, made of a linear defect in a 2D photonic crystal, +and initial work in Ref. [26], we propose a tailored plat- +form for trapping arrays of Rubidium atoms in the prox- +imity, as sketched in Fig. 1(a). +Waveguides based on +a 2D photonic crystal etched in a large refractive-index +slab have well-known strengths and are widely used in the +telecom range. Many techniques have been developed to +shape their dispersion curve with astounding precision +[27–31]. Strong coupling between a single emitter em- +bedded in a W1 waveguide and the guided light has been +demonstrated [32], and successfully exploited for quan- +arXiv:2301.04675v1 [quant-ph] 11 Jan 2023 + +2 +1D +′ +t +(a) +L +a +r +y1 +r3 +(b) +680 +730 +780 +830 +880 +930 +980 +Wavelength [nm] +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +kx[in units of 2 +a ] +320 +340 +360 +380 +400 +420 +440 +Frequency [THz] +Radiative modes +Bulk modes +Bulk modes +87Rb D2 +(c) +o +x +y +z +FIG. 1. A half-W1 slow-mode photonic crystal waveguide coupled to cold atoms. (a) Sketch of the waveguide with an array of +87Rb atoms trapped in the proximity, along the edge. Γ1D and Γ′ correspond to the decay rates in the guided mode and in the +radiation continuum, respectively. The structure is etched in a GaInP membrane (refractive index n = 3.34) suspended in air, +with a slab thickness t of 150 nm. (b) 2D scheme of the optimized waveguide. The initial unshifted and regularly distributed +holes are shown as dotted lines. For the first three rows the position of the holes can be shifted along y and their radius tuned, +amounting to 6 parameters (δyi,δri), i ∈ {1, 2, 3}. For the sake of clarity, only two parameters (δy1 and δr3) are displayed. +(c) Bandstructure of the optimized structure calculated via FDTD simulation. The bulk modes propagate within the slab but +are not guided on the edge of the PCW while the radiative modes are not guided at all. The 87Rb D2 line transition frequency +is aligned with the linear part of a guided band, defined as the slow mode in the text. +tum operations [33]. The proposed platform, sketched in +Fig. 1(a), can be seen as half a W1 waveguide. As its +W1 counterpart, it enables enables dispersion engineer- +ing but in addition offers a 2π solid-angle optical access to +the edge of the structure, allowing for simpler transport +of atoms close to it [26]. We use a large refractive index +GaInP slab that facilitates the design by offering more +flexibility in the engineering of guided modes, and we +show how to trap atoms in the proximity via additional +guided modes. Our effort focuses at each step on making +the design robust to imperfections and on assessing the +experimental feasibility of the full platform. +This paper is organized as follows. First, in section II +we present the specific platform based on a half-W1 +waveguide realized in a GaInP slab with a high refractive +index. We detail the optimization and the resulting ro- +bustness to nanofabrication imperfections, and then pro- +vide the achievable atom-photon coupling. +Second, in +section III we show that guided modes can be used to +trap atoms in the proximity of the waveguide via a two- +color evanescent dipole trap. Stable traps at about 115 +nm from the surface are obtained with low powers that +are compatible with nanophotonic systems. A summary +and outlook is provided in section IV. +II. +ENGINEERED HALF-W1 WAVEGUIDE FOR +RUBIDIUM ATOMS +In this section we introduce the specific half-W1 slow- +mode waveguide designed in this work, based on GaInP. +We identify the required geometrical parameters and +then present the optimizations performed to increase the +robustness to fabrication imperfections, leading thereby +to linear bands. +Finally, the expected coupling to the +guided mode (i.e. Purcell factor) for atoms in the prox- +imity from the surface is detailed. +A. +Description of the half-W1 GaInP waveguide +A periodic modulation of the refractive index in a +medium has deep consequences on light propagation. +The wavevector can be constrained between −π/a and +π/a, where a is the spatial period along the propaga- +tion direction, and we observe the opening of photonic +bandgaps. At the edge of the Brillouin zone (for k = π/a) +the group velocity vanishes [34] and the Purcell factor di- +verges (see Appendix A). +Motivated by the proposal made in Ref. [26], we study +a similar structure with a different material: GaInP. This +material has been chosen for its advantageous optical +and electronic properties. GaInP has a wide electronic +bandgap below 1.85 eV [35], and as such is transparent +for a wide range of wavelengths (from 670 nm up), mean- +ing it could be used with several alkali. At 780 nm, its + +3 +775 +780 +785 + [nm] +0 +10 +20 +30 +40 +50 +group index ng +(a) + = 9.2 nm + navg =27.7 +775 +780 +785 + [nm] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +1D/ +0 +(b) +F′ = 3, m′F = 3 +F = 2, mF = 2 +FIG. 2. Dispersion and atom-coupling properties for the half- +W1 waveguide with the structure optimization specified in +Table I. (a) Calculated group index ng for the slow mode. The +dotted lines delimit the linear band region where the group +index value is constant up to 15%. +(b) Calculated Purcell +factor Γ1D/Γ0 over the same range, for atoms trapped at 115 +nm from the structure. As it can be seen, ng is not the only +parameter affecting this ratio, i.e., the field structure is also +changing, but it is still critical as it diverges with ng just +outside the plateau. +refractive index is n = 3.34, reaching 3.55 at the elec- +tronic band edge. This large index contrast with the air +gives rise to band gaps that are wider and further away +from the light line [34], allowing for more flexibility in +the design of the trapping modes. Finally, this material +has attracted some attention in the recent years as it is +very convenient to operate in the telecom band due to +its low two-photon absorption [36], and growth and fab- +rication processes have therefore been developed and well +mastered. +As shown in figure 1(b), the holes etched in the GaInP +slab do not go up to the edge, leaving a few hundreds +of nanometers of unperturbed slab where the light can +propagate. Being based on a 2D slab rather than a 1D +structure, this geometry should be quite rigid and pre- +vent detrimental effects from low frequency mechanical +modes. The introduced symmetry breaking in the trans- +verse direction allows for a more precise control on the +dispersion properties of the waveguide since it offers ex- +tra degrees of freedom [37], while significantly improving +the optical access. This asymmetry has been harnessed +in Ref. [38] to create many exotic dispersion bands as +Dirac cones, multivalleys, or flat bands. Arrays of 87Rb +will then be trapped on the edge of the waveguide thanks +to a two-color dipole trap, at around 100 nm from the +surface. For comparison, in tapered nanofiber platforms, +atoms sit at more than 200 nm from the silica fiber. +For a given thickness t, chosen here to be t = 150 nm, +the first step to determine the geometrical parameters +consists in finding the lattice period a and hold radius +r of the 2D photonic crystal that allow for a bandgap +at the Rubidium D2 transition. Indeed the width and +position of the band gap is entirely determined by these +values [34]. The band gap has to be wide enough to allow +for at least two guided modes, one that crosses 780 nm, +and a blue-detuned one for trapping, as described later. +Guided bands appear when introducing the defect at the +edge, and we can align the band of interest with respect +to the D2 line by adjusting the width L. +Given these constraints, the geometrical parameters of +the waveguide are found to be: a = 212 nm, r = 63 nm +and L = 337 nm for t = 150 nm. +The corresponding +band structure, computed with the 3D FDTD software +Lumerical [39] is displayed in figure 1(c). Three guided +bands can be found inside the band gap of the 2D pho- +tonic crystal between 360 and 440 THz. The bulk modes +are guided in the slab (kz imaginary) but can propagate +in any direction in the plane, even inside the 2D array of +holes (kx, ky real). Radiative modes have a real k vector +in all directions and are therefore not guided. +B. +Imperfection-robust band engineering +Nanofabrication inherently leads to imperfections, +even if errors below 2 nm can be reached [40]. A spe- +cific effort has been put in our design process to minimize +the impact of such imperfections, thereby facilitating an +experimental realization. +As the Purcell factor diverges at the edge of the Bril- +louin zone, one naive approach could be to align the D2 +line frequency to any band edge of the band structure. +However, fabrication imperfections, to first order, lead +to a shift of the energy of the band [41]. +The flatter +the band, i.e., the smaller the group velocity, the more +it is vulnerable to a shift in frequency [42]. If the D2 +line is aligned with the band edge, an infinitesimal shift +to a lower energy will bring the atomic transition in the +band gap of the 2D photonic crystal, impeding the prop- +agation of the emitted light. +In addition to a shift in +frequency, the disorder introduced during the fabrication +process can lead to strong localization of light inside the +crystal [41, 43, 44]. +Following [26], two main criteria are to be considered +when assessing the robustness of a structure: the group +velocity has to be as independent of the frequency as +possible at the band edge, i.e. ∂vg/∂ω|ωe ∼ 0, and the +distance of the operation frequency to the band edge +∆ω = |ω − ωe| has to be as large as possible. +Designing slow modes with linear bands (i.e., an almost +constant, large group index ng over the widest range of +ω possible) allows us to fulfill these two criteria. First, +a linear dispersion corresponds to a vanishing group ve- +locity dispersion (GVD) and the atom-photon coupling +Row +Position δy (nm) Radius δr (nm) +1 +42.7 +14.2 +2 +53.8 +-11.2 +3 +-3.7 +-10.8 +TABLE I. Calculated changes in row positions and holes radii +via automatic differentiation optimization. All the rows after +the third one are unperturbed. + +4 +-318 +-106 0 106 +318 +x [nm] +-400 +-200 +0 +200 +400 +y [nm] +(a) +-200 +0 +200 +y [nm] +-300 +-225 +-150 +-75 +0 +75 +150 +225 +300 +z [nm] +(b) +-212 +0 +212 +x [nm] +-500 +-250 +0 +250 +500 +y [nm] +(c) +0 +0.2 +0.4 +0.6 +0.8 +1 +Intensity [norm.] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +|C| +FIG. 3. Slow mode structure at the 87Rb D2-line frequency. (a) Normalized intensity, in the (x, y)-plane at z = 0. (b) Same +in the (y, z)-plane at x = −a/2, i.e., crossing the hole nearest to the slab edge. The mode is strongly expelled into the vacuum +around the edge of the waveguide. (c) Polarization ellipticity z-component Cz in the XY plane at z = 0. The other components +of the ellipticity vector are 0. |Cz| = 0 indicates a linear polarization, while we have |Cz|= 1 for a circularly polarized light. +Close to the edge, the polarization has a large circular component due to the strong longitudinal component that appears when +light is confined at the nanoscale. By taking z as the quantification axis, the polarization will be close to σ+ for atoms trapped +in the proximity (91 to 99% at 115 nm from the surface). +is proportional to the group index. Moreover, as shown +in [28], it is possible to design a slow and linear band +over a wide spectral range. As most fabrication imper- +fections lead to a shift ∆ω of the guided bands, both +these constraints aim at placing the relevant frequency +at a position on the band where a small shift will affect +the dispersion at the given frequency only slightly. +It +has been shown that linear bands can be achieved in at +least two types of asymmetric PCWs [38]. Achieving such +vanishing group velocity dispersion has been extensively +studied in the context of W1 waveguides, by tuning the +position of rows of holes [28, 45, 46], chirping the waveg- +uide properties [47], or changing the size of the holes [29]. +Inspired by these previous optimization strategies, we +set the radius of the first three rows of holes as well as +their position along the y axis as optimization parame- +ters, as depicted in figure 1(b). We then have 6 indepen- +dent optimization parameters (δri, δyi), i ∈ {1, 2, 3}. As +full 3D FDTD simulations are computationally intensive, +we use the approximate method of Guided Mode Expan- +sion (GME) [48] thanks to the legume [49] solver to faster +compute the shape of the guided band. We optimize the +shape of the slow-mode band by iteratively varying the +parameters. At each iteration, a cost function enforcing +the minimization of the group velocity dispersion (aver- +aged over the wave vector interval) while setting a target +ng value is evaluated and the (δri, δyi) varied thanks to +automatic differentiation. After a few hundred iterations +we obtain the optimal shifts for achieving this target ng +value over the widest possible spectral range. Finally the +optimized structure was simulated in full 3D FDTD to +validate the results from the approximate GME method. +In order for this optimization to give relevant results, +ng has to be set to an experimentally realistic value, ide- +ally below 60. Indeed, experiments have shown that it is +extremely challenging to reach higher values for the group +index without losses [50]. The most concluding optimiza- +tion results are obtained for a target around ng = 30. +The shifts in position and radius after optimization are +given in table I and the corresponding band structure is +presented in figure 1(c). Figure 2(a) shows that we engi- +neered a band with a constant group index of 28 over a +9 nm range, and hence reach similar performance than a +previous optimization of a W1 waveguide [28]. This fea- +ture offers a two-fold advantage. In addition to making it +robust to shifts caused by fabrication imperfections, the +optimization enables using the half-W1 waveguide in a +large bandwidth regime (≥ 4 THz) with very little dis- +persion. +Finally, as seen in Appendix A, the Purcell factor is +proportional to the group index. Keeping the group in- +dex constant over a wide range enables to keep the Pur- +cell factor constant in case of a shift, as shown in figure +2(b). If it is necessary to have a constant group index it +is not sufficient as the Purcell factor also depends on the +shape of the mode of the electric field (equation A4). As +we move along the guided band, the mode shape changes +slightly, affecting the value of the Purcell factor. +C. +Strong coupling to the slow mode +Given the optimized design, we now turn to the in- +teraction between the slow mode and 87Rb atoms in the +proximity. Taking into account the multilevel character +of Rubidium, we defined a transition-dependent Purcell +factor in equation (A4) given in Appendix A. The group +velocity vg is evaluated from the simulated band struc- +ture, while the other terms are computed from the field +map of the guided mode. Indeed, the Purcell factor is + +5 +F′ = 3 +m′F = +F = 2, mF = 2 +3 ++ +2 +1 +- +(a) +0 +200 +400 +y [nm] +0 +5 +10 +15 +1D/ +0 +-212 +0 +212 +x [nm] +-400 +-200 +0 +200 +y [nm] +(b) +0 +200 +y [nm] +-225 +-150 +-75 +0 +75 +150 +225 +z [nm] +(c) +10 +2 +10 +1 +100 +101 +1D/ +0 +0 +200 +0.0 +0.1 +FIG. 4. Excitation rates for 87Rb atoms in the waveguide proximity. (a) Allowed transitions on the D2 line for an atom in +|F = 2, mF = +2⟩. Because of the large σ+ component (∼ 91% at the position of the atoms) and the values of the Clebsch- +Gordan coefficients, the excitation probability to the |F′ = 3, mF′ = +3⟩ is 100 times higher than the σ− channel. The inset +provides a zoom. (b) Purcell factor in the XY plane, at z = 0. (c). Purcell factor in the YZ plane, at x = −a/2. The red dots +indicate the position of the atoms at 115 nm from the surface. +proportional to the slow-mode intensity. Figure 3(a-b) +show that in order to have the maximum coupling the +atoms should be trapped close to the edge of the waveg- +uide, aligning them to the holes of the first row. +From figure 4(a) we see that for atoms in state +|F = 2, mF = +2⟩ trapped at 115 nm from the edge, the +Purcell factor reaches a value of 1.6. As shown in fig- +ures 4(b) and 4(c), a small modulation in the x direction +exists and the value of the Purcell factor decays rapidly +when going further from the surface. +To quantify the coupling of the atoms to the guided +mode we also define the β factor, β = Γ1D/Γtot with +Γtot = Γ1D + Γ′, and Γ′ the decay rate in all the other +radiation modes than the guided slow mode. Because of +the complex shape of the local density of states accessible +to the atoms, the behaviour of Γ′ is hard to infer, but its +modulation is expected to be minimal as seen in [25, 26]. +Hence we assume for the following Γ′ ≃ Γ0. +At the position of the trap minimum, i.e at 115 nm +from the surface, we find β = 0.62, very close to the +averaged value ˜β = 0.57 for a thermal distribution (at a +temperature of one tenth of the trap depth). This is at +least 50 times better than the current systems involving +nanofibers (β = 10−2) [9] and a significant improvement +with respect to current PCW-based platforms (β = 0.45) +[20]. +III. +TRAPPING RUBIDIUM ATOMS NEAR A +HALF-W1 WAVEGUIDE +Simulations in the previous section were performed for +atoms at 115 nm from the edge of the waveguide. Indeed, +in the following we show a stable trapping scheme based +on an evanescent two-color dipole trap formed by fast +guided modes, allowing the atoms to be trapped between +100 and 150 nm. This trap has been designed following +the ideas implemented in optical nanofibers [2, 51], with +blue- and red-detuned counter-propagating modes. Find- +ing a stable trapping scheme that keeps the atoms close +enough to the surface so that they can couple to the slow +mode with a large Purcell factor is a critical requirement +for experimental implementations. +A. +Two-color dipole trap structure +In contrast with optical nanofibers, the guided modes +are structured along the propagation direction due to the +Bloch wave structure of the light field. The intensity of +the modes, which is an important quantity when looking +at dipole trapping, is periodic with period a, as shown in +figure 3 for the slow mode. This feature constrains the +position of the trapped atoms to the maxima of intensity +of the red-detuned mode. It makes the search for a blue +detuned mode more challenging as this one will also be +structured, while a uniform one would work perfectly well +to repel the atoms from the surface [3]. A blue-detuned +beam with an intensity pattern completely out of phase +with the red-detuned one is needed. Fortunately, modes +separated by a band gap usually have intensity maxima +shifted by a/2 [52]. We then use the highest guided band +for the blue-detuned trap between 400 and 420 THz (fig- +ure 1(c)). +In order to have a full description of the potential +seen by the atoms, we must add the Casimir-Polder +(CP) interaction [53] between the atoms and the surface. +Ground state fluctuations of the vacuum can polarize the +atoms, even if they are not charged. When put in prox- +imity to structures, the vacuum-induced dipole moments +create mirror charges that act on the original dipole, lead- +ing to an additional light shift. +This CP potential is +only significant at very close distances (≤ 150 nm) but +is crucial as it reduces the local density of states and +acts as an attractive potential close to the surface. For +an atom in the proximity of an infinite dielectric plane + +6 +UCP = −C3/d3 [52], where d is the distance to the sur- +face. +As described in Appendix B, for a ground state +87Rb atom close to a GaInP surface, we computed an +approximate C3 = −9.25 × 10−49J.m3. +B. +Trapping potential simulation +The +trapping +potentials +were +obtained +via +nanotrappy +[54], +a +Python +package +developed +by +our group, to design, calculate and optimize dipole traps +around nanoscale waveguides, making the search process +faster and more systematic. +Figure 5 shows a trapping potential in all 3 directions +for an atom in the |F = 2, mF = +2⟩ hyperfine level. For +this trap, a beam red-detuned from the D2 line of 87Rb at +784.5 nm and a beam blue-detuned at 737 nm are used. +A trap of depth 3 mK is obtained with a minimum at +115 nm from the surface. The total powers are Pblue = +1.65 mW and Pred = 0.1 mW, but a stable trap can be +obtained on a wide range of powers. The main limitation +can be the power handling of the structure which is still +to be determined. In Ref. [36], power densities up to +1 GW/cm2 were coupled to similar GaInP PCWs with +group index 8.8. +For our structure which has a cross +section 10 times smaller and a group index 3 times bigger, +this would be equivalent to coupling ≃ 100 mW into our +waveguide. The proposed powers for the trap fall well +below this bound. +The trapping frequencies are large in the x and y direc- +tions, with ωx = 2π×1.89 MHz and ωy = 2π×1.94 MHz. +In the vertical direction however, an important anhar- +monicity of the trap, discussed below, gives ωz = 2π × +133 kHz. +A critical aspect is the trapping along the z direction. +With small powers for the blue-detuned beam, a sym- +metric double well around z = 0 appears. This can be +detrimental for our platform as the atoms will be trapped +at positions not corresponding to the maxima of the Pur- +cell factor. Because of the fixed CP potential, rescaling +the blue and red powers by the same factor does not lead +to a simple reduction of the trap depth, it also goes with +a shift in position, which might lead to reaching this two- +well regime in the z direction. This explains the impor- +tant depth of the simulated traps: the closer to the sur- +face we want the trap minimum to be, the more we have +to compensate for the CP potential with higher beam +power, leading to higher trap depths. This phenomenon +may come from the complex decay of the evanescent wave +away from the surface pointed out in Ref. [55]. Indeed, +this decay is usually multi-exponential, with decay rates +not easily linked to the wavelength or the wavevector of +the guided mode. +Importantly, we also verified that we can achieve a sta- +ble trap in the three directions for a wide range of wave- +lengths, which is a valuable feature for finding the right +trade-off between heating the atoms with off-resonant +scattering and power handling of the waveguide. If we +-212 +0 +212 +424 +x [nm] +-500 +-250 +0 +250 +500 +y [nm] +(a) +-212 +0 +212 +424 +x [nm] +-3.0 +-2.5 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +U [mK] +(b) +0 +100 +200 +300 +y [nm] +-3.0 +-2.5 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +U [mK] +(c) +-200 +0 +200 +z [nm] +-3.0 +-2.5 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +U [mK] +(d) +0 +20 +40 +60 +80 +100 +120 +140 +U [mK] +FIG. 5. Calculated potential of the two-color dipole trap. (a) +2D total trapping potential in the proximity of the waveguide, +in the XY plane. The trapping potential is given along the +three directions in (b), (c) and (d). A periodic stable trap with +depth of about 3 mK is obtained with powers of 1.65 mW for +each blue beam and 100 µW for each red one. The simulations +are performed with the nanotrappy package. +allow the blue power to go up to 3 mW, we can find a +stable trap for blue wavelengths ranging from 724 nm to +738 nm. Pushing the maximum allowed power to 5 mW +we can find a trap for the full available blue-detuned air +band, i.e. ∆λ = 21 nm. For the red-detuned laser, we +have an available range from 780.5 nm up to 786 nm. +Laser diodes are easily available on these wavelengths, +reinforcing the feasibility of our platform. +Finally, as briefly noted before, we used counter- +propagating beams here instead of simple ones, albeit +standing waves are not needed to get a periodic intensity +modulation. The strong ellipticity of the guided modes +(as shown in figure 3(c)), acts as a fictitious magnetic +field on the atoms, splitting the Zeeman levels [56]. If +we start from atoms evenly distributed in all the mF +states, this effect would lead to a large inhomogeneous +broadening up to a few GHz. It can be mitigated by us- +ing counter-propagating trapping beams slightly detuned +from each other, as used for blue detuned beams in some +compensated nanofiber traps [3]. +Via nanotrappy, we +estimated that adding a red-detuned laser at 280 GHz +from the first one and a blue detuned at 250 GHz from +the other reduces this broadening by 90%. Counterprop- +agation creates a running wave at a velocity given by +δω. This pattern propagates but at a speed so large the +atoms only see the average of the potential. + +7 +IV. +CONCLUSION +Many experimental and technological challenges have +yet to be overcome to enable further neutral-atom +waveguide-QED protocols. +As such, experimental ro- +bustness of the targeted waveguide platforms is a critical +requirement, as is evanescent trapping of atoms. In our +work, we proposed and engineered a bona fide platform +for trapping cold Rubidium atoms close to a half-W1 +photonic crystal waveguide based on high-index mate- +rial GaInP. Atoms can be trapped between 100 and 150 +nm from the surface with compatible low-power incident +light. At 115 nm, the slow mode couples to the atoms +with a Purcell factor as high as 1.6 when the group index +is taken around 30. This study has been carried out for +conservative parameters and a strong focus on robustness +against fabrication imperfections has been done by engi- +neering the band structure for a large bandwidth, facili- +tating first implementations. Future generations should +support higher group index, albeit with narrower band- +widths [28]. +This novel platform – tailor-designed for +atom integration, robustness and large optical access – +offers unique advantages for studying coherent and dissi- +pative dynamics in the waveguide-QED framework. +V. +ACKNOWLEDGEMENTS +This work was supported by the French National +Research Agency (NanoStrong Project ANR-18-CE47- +0008), by the R´egion Ile-de-France (DIM SIRTEQ), and +by the European Union’s Horizon 2020 research and in- +novation program under Grant Agreement No.899275 +(DAALI project). A.U. was supported by the European +Union (Marie Curie Fellowship SinglePass 101030421). +J.L. is a member of the Institut Universitaire de France. +Appendix A: Theoretical framework: Reaching +strong Purcell factor +The coupling of the atoms to the guided mode of a +waveguide can be characterized by the Purcell factor +Γ1D/Γ0, which relates the decay rate of the atoms into +the guided mode to the one into free space. For a mul- +tilevel atom we can define the excitation rate ΓF,F ′,mF,q +from the hyperfine level |F, mF⟩ to |F′, mF + q⟩ [57]: +ΓF,F ′,mF,q = 2µ0| ⟨F||ˆd||F⟩|2 +¯h +× +ω2 +q|CmF ,q|2ˆeq · Im G(r, r; ωq) · ˆe∗ +q +(A1) +where the ˆeq, q ∈ {−1, 0, 1}, are the normalized dipole +vectors over all the possible excitation channels (σ−, π, +σ+ respectively) and ωq is the transition frequency be- +tween the specified levels. G(r, r; ωq) is the value of the +classical Green’s tensor at r for a dipole at the same po- +sition. +The CmF ,q are the Clebsch-Gordan coefficients +given by: +CmF ,q = (−1)F ′−1+mF √ +2F + 1 +� +F ′ +1 +F +m′ +F q −mF +� +. (A2) +Note that only the Wigner 3j where m′ +F = mF + q are +non zero. +As we are looking at the decay into a defined guided +mode, it is possible to write the Green’s tensor as a sum +G(r, r; ωq) = G1D(r, r; ωq)+G′(r, r; ωq). From [58] (Eq. +2.89) and [59] we have an analytical expression for the +Green’s function of an effective 1D structured waveguide: +G1D(r, r, ω) = i ac +2ω +� c +vg +� +[E(r) ⊗ E∗(r)] +� +cell drϵ(r)|E(r)|2 +(A3) +where E(r) is the electric field of the guided mode, a the +period of the modulation and vg = ∂ω +∂k the group velocity +of the guided mode. +By neglecting the Zeeman splitting of the mF levels +(ωq constant), the excitation probability of a single atom +in |F, mF⟩ into |F′, mF + q⟩ (through a single excitation +channel) is hence given by: +Γ1D,F,F ′,mF,q = 2πc +ϵ0¯h +| ⟨F||ˆd||F⟩|2 +λ0vg +× +|CmF ,q|2 +|ˆeq · E(r)|2 +� +cell drϵ(r)|E(r)|2 . +(A4) +As such, we see that to reach high Purcell factors we +must either decrease vg which can be achieved by dis- +persion design, or maximize the normalized electric field +amplitude at the position of the atom given by the second +half of equation (A4). +Since we are considering excitation probabilities (as in +Figure 4(a)), we only consider the coupling between the +atom and the injected mode, here the guided mode E(r) +propagating along the positive x axis. If we were instead +to consider decay rates we would have to modify slightly +equation (A4) and sum the contributions of the coupling +of the atom to both forward and backward propagating +guided modes. +Appendix B: Casimir-Polder interactions between +GaInP and Rubidium atoms +To the best of our knowledge, there were no previous +computations of the C3 coefficient of the Casimir-Polder +interactions between GaInP and Rubidium atoms. For +a dielectric wall, we have to adapt the formula for C3 in +the form [60] +C3 ≈ ¯h +4π +� +∞ +0 +α(iξ)ϵ(iξ) − 1 +ϵ(iξ) + 1dξ. +(B1) + +8 +We then have to evaluate α and ϵ over the imaginary axis. +α is simply evaluated using the expression for the scalar +polarizability of 87Rb in the ground state |F = 2⟩ with +complex frequencies. As ϵ(ω) = ϵ′(ω) + iϵ′′(ω), we can +get the dependence in ω from experimental data [35]. To +evaluate it over the imaginary axis we use the Kramers- +Kroenig relation that provides [61] +ϵ(iξ) = 1 + 2 +π +� +∞ +0 +ω[ϵ′(ω) − 1] +ω2 + ξ2 +dω. +(B2) +Finally, for a ground state Rubidium atom close to a +GaInP surface we get C3 += +9.25 × 10−49J.m3 = +h × 1391 Hz.µm3. +[1] D. E. Chang, J. S. Douglas, A. Gonz´alez-Tudela, C.-L. +Hung, and H. J. Kimble, Colloquium: Quantum matter +built from nanoscopic lattices of atoms and photons, Rev. +Mod. Phys. 90, 31002 (2018). +[2] E. Vetsch, D. Reitz, G. Sagu´e, R. Schmidt, S. T. +Dawkins, and A. Rauschenbeutel, Optical Interface Cre- +ated by Laser-Cooled Atoms Trapped in the Evanescent +Field Surrounding an Optical Nanofiber, Phys. Rev. Lett. +104, 203603 (2010). +[3] A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroˆute, +M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kim- +ble, Demonstration of a state-insensitive, compensated +nanofiber trap, Phys. Rev. Lett. 109, 033603 (2012). +[4] T. Nieddu, V. Gokhroo, and S. Nic Chormaic, Optical +nanofibres and neutral atoms, J. Opt. 18, 053001 (2016). +[5] P. Solano, J. A. Grover, J. E. Hoffman, S. Ravets, F. K. +Fatemi, L. A. Orozco, and S. L. Rolston, Chapter seven +- optical nanofibers: A new platform for quantum optics +(Academic Press, 2017) pp. 439–505. +[6] B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and +J. Laurat, Demonstration of a memory for tightly guided +light in an optical nanofiber, Phys. Rev. Lett. 114, +180503 (2015). +[7] N. V. Corzo, B. Gouraud, A. Chandra, A. Goban, A. S. +Sheremet, D. V. Kupriyanov, and J. Laurat, Large Bragg +Reflection from One-Dimensional Chains of Trapped +Atoms Near a Nanoscale Waveguide, Phys. Rev. Lett. +117, 133603 (2016). +[8] H. L. Sørensen, J.-B. B´eguin, K. W. Kluge, I. Iakoupov, +A. S. Sørensen, J. H. M¨uller, E. S. Polzik, and J. Ap- +pel, Coherent backscattering of light off one-dimensional +atomic strings, Phys. Rev. Lett. 117, 133604 (2016). +[9] N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, +B. Gouraud, and J. Laurat, Waveguide-coupled single +collective excitation of atomic arrays, Nature 566, 359 +(2019). +[10] R. Mitsch, C. Sayrin, B. Albrecht, P. Schneeweiss, and +A. Rauschenbeutel, Quantum state-controlled directional +spontaneous emission of photons into a nanophotonic +waveguide, Nat. Commun. 5, 1 (2014). +[11] A. S. Prasad, J. Hinney, S. Mahmoodian, K. Hammerer, +S. Rind, P. Schneeweiss, A. S. Sørensen, J. Volz, and +A. Rauschenbeutel, Correlating photons using the collec- +tive nonlinear response of atoms weakly coupled to an +optical mode, Nat. Photon. 14, 719 (2020). +[12] J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, +H. J. Kimble, and D. E. Chang, Quantum many-body +models with cold atoms coupled to photonic crystals, +Nat. Photon. 9, 326 (2015). +[13] A. Gonz´alez-Tudela, C.-L. Hung, D. E. Chang, I. Cirac, +and H. J. Kimble, Subwavelength vacuum lattices and +atom–atom interactions in two-dimensional photonic +crystals, Nat. Photon. 9, 320 (2015). +[14] M. +Bello, +G. +Platero, +and +A. +Gonz´alez-Tudela, +Spin many-body phases in standard- and topological- +waveguide qed simulators, PRX Quantum 3, 010336 +(2022). +[15] J. D. Thompson, T. G. Tiecke, N. P. De Leon, J. Feist, +A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuleti´c, +and M. D. Lukin, Coupling a single trapped atom to a +nanoscale optical cavity, Science 340, 1202 (2013). +[16] M. E. Kim, T.-H. Chang, B. M. Fields, C.-A. Chen, and +C.-L. Hung, Trapping single atoms on a nanophotonic +circuit with configurable tweezer lattices, Nat. Commun. +10, 1647 (2019). +[17] A. Liu, L. Xu, X.-B. Xu, G.-J. Chen, P. Zhang, G.-Y. +Xiang, G.-C. Guo, Q. Wang, and C.-L. Zou, Proposal +for low-power atom trapping on a GaN-on-sapphire chip, +Phys. Rev. A 106, 033104 (2022). +[18] E. Will, L. Masters, A. Rauschenbeutel, M. Scheucher, +and J. Volz, Coupling a Single Trapped Atom to a +Whispering-Gallery-Mode Microresonator, Phys. Rev. +Lett. 126, 233602 (2021). +[19] A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, +J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, +D. E. Chang, O. Painter, and H. J. Kimble, Atom-light +interactions in photonic crystals, Nat. Commun. 5, 3808 +(2014). +[20] A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Mu˜niz, +O. Painter, and H. J. Kimble, Superradiance for Atoms +Trapped along a Photonic Crystal Waveguide, Phys. Rev. +Lett. 115, 063601 (2015). +[21] J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. +Yu, D. E. Chang, and H. J. Kimble, Atom-atom interac- +tions around the band edge of a photonic crystal waveg- +uide, Proc. Natl. Acad. Sci. U.S.A 113, 10507 (2016). +[22] A. P. Burgers, L. S. Peng, J. A. Muniz, A. C. McClung, +M. J. Martin, and H. J. Kimble, Clocked atom delivery +to a photonic crystal waveguide, Proc. Natl. Acad. Sci. +U.S.A 116, 456 (2019). +[23] S.-P. Yu, J. D. Hood, J. A. Muniz, M. J. Martin, +R. Norte, C.-L. Hung, S. M. Meenehan, J. D. Co- +hen, O. Painter, and H. J. Kimble, Nanowire photonic +crystal waveguides for single-atom trapping and strong +light-matter interactions, Appl. Phys. Lett. 104, 111103 +(2014). +[24] S.-P. Yu, Nano-Photonic Platform for Atom-Light Inter- +action, Ph.D. thesis, California Institute of Technology + +9 +(2017). +[25] N. Fayard, A. Bouscal, J. Berroir, A. Urvoy, T. Ray, +S. Mahapatra, M. Kemiche, J. A. Levenson, J.-J. Greffet, +K. Bencheikh, J. Laurat, and C. Sauvan, Asymmetric +comb waveguide for strong interactions between atoms +and light, Opt. Express 30, 45093 (2022). +[26] X. Zang, J. Yang, R. Faggiani, C. Gill, P. G. Petrov, J.-P. +Hugonin, K. Vynck, S. Bernon, P. Bouyer, V. Boyer, and +P. Lalanne, Interaction between Atoms and Slow Light: +A Study in Waveguide Design, Phys. Rev. Applied 5, +024003 (2016). +[27] A. Y. Petrov and M. Eich, Zero dispersion at small group +velocities in photonic crystal waveguides, Appl. Phys. +Lett. 85, 4866 (2004). +[28] J. Li, T. P. White, L. O. Faolain, A. Gomez-Iglesias, and +T. F. Krauss, Systematic design of flat band slow light +in photonic crystal waveguides, Opt. Express 16, 2621 +(2008). +[29] L. H. Frandsen, A. Lavrinenko, J. Fage-Pedersen, and +P. I. Borel, Photonic crystal waveguides with semi-slow +light and tailored dispersion properties, Opt. Express 14, +2440 (2006). +[30] P. Colman, S. Combri´e, G. Lehoucq, and A. De Rossi, +Control of dispersion in photonic crystal waveguides +using group symmetry theory, Opt. Express 20, 1237 +(2012). +[31] K. Vyas, R. Cheriton, D. Liu, H. Ahmed, S. Schulz, and +K. Dolgaleva, Simulation and fabrication of slow light +suspended air-bridge AlGaAs photonic crystal waveg- +uide, in Proc. SPIE 12004, Integrated Optics: Devices, +Materials, and Technologies XXVI (2022) p. 42. +[32] M. Arcari, I. S¨ollner, A. Javadi, S. Lindskov Hansen, +S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. +Song, S. Stobbe, and P. Lodahl, Near-Unity Coupling +Efficiency of a Quantum Emitter to a Photonic Crystal +Waveguide, Phys. Rev. Lett. 113, 093603 (2014). +[33] H. Le Jeannic, A. Tiranov, J. Carolan, T. Ramos, +Y. Wang, M. H. Appel, S. Scholz, A. D. Wieck, A. Lud- +wig, N. Rotenberg, L. Midolo, J. J. Garc´ıa-Ripoll, A. S. +Sørensen, and P. Lodahl, Dynamical photon–photon in- +teraction mediated by a quantum emitter, Nat. Phys. 18, +1191 (2022). +[34] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. +Meade, Photonic Crystals : Molding the Flow of Light +(Princeton University Press, 2008). +[35] M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yoa, +P. G. Snyder, and J. A. Woollam, Optical constants +of GaxIn1 xP lattice matched to GaAs, J. Appl. Phys. +3416, 372 (2005). +[36] S. Combri´e, Q. V. Tran, A. D. Rossi, and C. Husko, High +quality GaInP nonlinear photonic crystals with mini- +mized nonlinear absorption, Appl. Phys. Lett. 95, 221108 +(2009). +[37] S. L¨u, J. Zhao, and D. Zhang, Flat band slow light +in asymmetric photonic crystal waveguide based on mi- +crofluidic infiltration, Appl. Opt. 49, 3930 (2010). +[38] H. S. Nguyen, F. Dubois, T. Deschamps, S. Cueff, A. Par- +don, J. L. Leclercq, C. Seassal, X. Letartre, and P. Vik- +torovitch, Symmetry Breaking in Photonic Crystals: On- +Demand Dispersion from Flatband to Dirac Cones, Phys. +Rev. Lett. 120, 66102 (2018). +[39] Ansys Lumerical FDTD (2022), simulation software +based on the finite-difference time-domain method. +[40] T. Asano, B.-S. Song, and S. Noda, Analysis of the ex- +perimental Q factors (∼ 1 million) of photonic crystal +nanocavities, Opt. Express 14, 1996 (2006). +[41] R. Faggiani, A. Baron, X. Zang, L. Lalouat, S. A. Schulz, +B. O’Regan, K. Vynck, B. Cluzel, F. De Fornel, T. F. +Krauss, and P. Lalanne, Lower bound for the spatial ex- +tent of localized modes in photonic-crystal waveguides +with small random imperfections, Sci. Rep. 6, 27037 +(2016). +[42] M. Soljaˇci´c and J. D. Joannopoulos, Enhancement of +nonlinear effects using photonic crystals, Nat. Mater. 3, +211 (2004). +[43] S. Mazoyer, J.-P. Hugonin, and P. Lalanne, Disorder- +induced multiple scattering in photonic-crystal waveg- +uides, Phys. Rev. Lett. 103, 063903 (2009). +[44] M. Patterson, S. Hughes, S. Combri´e, N. V. Tran, A. De +Rossi, R. Gabet, and Y. Jaou¨en, Disorder-induced coher- +ent scattering in slow-light photonic crystal waveguides, +Phys. Rev. Lett. 102, 253903 (2009). +[45] J. Wu, Y. Li, C. Peng, and Z. Wang, Wideband and low +dispersion slow light in slotted photonic crystal waveg- +uide, Opt. Commun. 283, 2815 (2010). +[46] J. Liang, L. Y. Ren, M. J. Yun, X. Han, and X. J. Wang, +Wideband ultraflat slow light with large group index in +a W1 photonic crystal waveguide, J. Appl. Phys. 110, +063103 (2011). +[47] D. Mori and T. Baba, Wideband and low dispersion slow +light by chirped photonic crystal coupled waveguide, Opt. +Express 13, 9398 (2005). +[48] L. C. Andreani and D. Gerace, Photonic-crystal slabs +with a triangular lattice of triangular holes investigated +using a guided-mode expansion method, Phys. Rev. B +73, 235114 (2006). +[49] M. Minkov, I. A. Williamson, L. C. Andreani, D. Gerace, +B. Lou, A. Y. Song, T. W. Hughes, and S. Fan, Inverse +Design of Photonic Crystals through Automatic Differ- +entiation, ACS Photonics 7, 1729 (2020). +[50] S. Mazoyer, P. Lalanne, J. C. Rodier, J. P. Hugonin, +M. Spasenovi, L. Kuipers, D. M. Beggs, and T. F. Krauss, +Statistical fluctuations of transmission in slow light +photonic-crystal waveguides, Opt. Express 18, 14654 +(2010). +[51] F. Le Kien, V. I. Balykin, and K. Hakuta, Atom trap and +waveguide using a two-color evanescent light field around +a subwavelength-diameter optical fiber, Phys. Rev. A 70, +063403 (2004). +[52] W. Johnson, V. Dzuba, U. Safronova, and M. Safronova, +Finite-field evaluation of the Lennard-Jones atom-wall +interaction constant C3 for alkali-metal atoms, Phys. +Rev. A 69, 022508 (2004). +[53] H. B. Casimir and D. Polder, The influence of retardation +on the London-van der Waals forces, Phys. Rev. 73, 360 +(1948). +[54] J. Berroir, A. Bouscal, A. Urvoy, T. Ray, and J. Lau- +rat, Nanotrappy: An open-source versatile package for +cold-atom trapping close to nanostructures, Phys. Rev. +Research 4, 013079 (2022). +[55] R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Sub- +wavelength structure of the evanescent field of an optical +Bloch wave, Phys. Rev. Lett. 102, 023902 (2009). +[56] C. Cohen-Tannoudji and J. Dupont-Roc, Experimental +Study of Zeeman Light Shifts in Weak Magnetic Fields, +Phys. Rev. A 5, 968 (1972). + +10 +[57] X. Luan, Towards atom assembly on nanophotonic struc- +tures with optical tweezers, Ph.D. thesis, California Insti- +tute of Technology (2020). +[58] J. D. Hood, Atom-light interactions in a photonic crystal +waveguide, Ph.D. thesis, California Institute of Technol- +ogy (2017). +[59] A. Asenjo-Garcia, J. D. Hood, D. E. Chang, and H. J. +Kimble, Atom-light interactions in quasi-one-dimensional +nanostructures: +A Green’s-function perspective, Phys. +Rev. A 95, 033818 (2017). +[60] A. O. Caride, G. L. Klimchitskaya, V. M. Mostepanenko, +and S. I. Zanette, Dependences of the van der Waals +atom-wall interaction on atomic and material properties, +Phys. Rev. A 71, 042901 (2005). +[61] M. Antezza, L. P. Pitaevskii, and S. Stringari, Effect of +the casimir-polder force on the collective oscillations of +a trapped bose-einstein condensate, Phys. Rev. A 70, +053619 (2004). + diff --git a/F9E3T4oBgHgl3EQftQvk/content/tmp_files/load_file.txt b/F9E3T4oBgHgl3EQftQvk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a8eb3461cfd8a7607819c37cdbae6269e74cdd66 --- /dev/null +++ b/F9E3T4oBgHgl3EQftQvk/content/tmp_files/load_file.txt @@ -0,0 +1,1054 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf,len=1053 +page_content='Systematic design of a robust half-W1 photonic crystal waveguide for interfacing slow light and trapped cold atoms Adrien Bouscal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='1 Malik Kemiche,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 3 Sukanya Mahapatra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='2 Nikos Fayard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='4 J´er´emy Berroir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='1 Tridib Ray,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='1 Jean-Jacques Greffet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='4 Fabrice Raineri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='2 Ariel Levenson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='2 Kamel Bencheikh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='2 Christophe Sauvan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='4 Alban Urvoy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' ∗ and Julien Laurat1 1Laboratoire Kastler Brossel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Sorbonne Universit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' ENS-Universit´e PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Coll`ege de France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 4 place Jussieu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' France 2Centre de Nanosciences et de Nanotechnologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 91120 Palaiseau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' France 3IMEP-LAHC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Grenoble Alpes, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Savoie Mont Blanc, CNRS, Grenoble INP, 38000 Grenoble, France 4Universit´e Paris-Saclay, Institut d’Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127 Palaiseau, France (Dated: January 13, 2023) Novel platforms interfacing trapped cold atoms and guided light in nanoscale waveguides are a promising route to achieve a regime of strong coupling between light and atoms in single pass, with applications to quantum non-linear optics and quantum simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A strong challenge for the experimental development of this emerging waveguide-QED field of research is to combine facilitated optical access for atom transport, atom trapping via guided modes and robustness to inherent nanofabrication imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' In this endeavor, here we propose to interface Rubidium atoms with a photonic crystal waveguide based on a large-index GaInP slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' With a specifically tailored half- W1 design, we show that a large coupling to the waveguide can be obtained and guided modes can be used to form two-color dipole traps for atoms at about 100 nm from the edge of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This optimized device should greatly improve the level of experimental control and facilitate the atom integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' INTRODUCTION Interfacing cold neutral atoms and photons guided in nanoscale waveguides has raised a large interest over the recent years, with a wealth of emerging opportunities [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Arrays of atoms can be trapped in the evanescent field of guided modes and the strong transverse confinement enables to increase the individual atom-photon coupling in single pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Remarkable experimental advances have been obtained with optical nanofibers [2–5], exploiting collective effects and chiral properties to realize various all-fibered functionalities [6–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Beyond nanofibers, tai- lored dispersion relations that can be obtained in pho- tonic crystal waveguides (PCW) offer unique features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' While the atom-photon coupling can be strongly en- hanced near a band edge, where guided modes can prop- agate slowly, atom-photon bound states can also appear for an atomic transition within a bandgap, with the ca- pability to implement tunable long-range atom-atom in- teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' These features led to a variety of theoretical proposals for applications in quantum optics and many- body physics [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Despite the promises of this new waveguide-QED paradigm, trapping atoms in the vicinity of such photonic crystal waveguides is still at its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This combina- tion is a daunting challenge due to stringent requirements when considering real physical implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A first challenge is to keep the atoms as static as possible close to ∗ Corresponding author: alban.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='urvoy@sorbonne-universite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='fr the structure, so that they can interact with the evanes- cent mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' While tweezers can be used to maintain the atoms at a fixed distance [15–18], it is challenging to make an array of such atoms at distances on the 100 nm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Dipole trapping by the evanescent field of guided modes is necessary but it has remained an important roadblock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Up to now, only a corrugated slot waveguide (so-called alligator waveguide) [19–23] has been implemented and first pioneering demonstrations obtained, albeit with a limited number of atoms and without stable trapping in the evanescent field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Some theoretical proposals on novel interesting structures supporting atom trapping in the evanescent field have emerged since, such as a slot [24] or a comb waveguide [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Structures must also provide a large optical access to bring atoms close to their surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Eventually, in order to push experimental development, great care should be put in ensuring that the structure is robust against fabrication imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' In this paper, we design a novel platform for interfac- ing trapped cold atoms and a slow-mode photonic crystal waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Building from the promises of W1 waveg- uides, made of a linear defect in a 2D photonic crystal, and initial work in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [26], we propose a tailored plat- form for trapping arrays of Rubidium atoms in the prox- imity, as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Waveguides based on a 2D photonic crystal etched in a large refractive-index slab have well-known strengths and are widely used in the telecom range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Many techniques have been developed to shape their dispersion curve with astounding precision [27–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Strong coupling between a single emitter em- bedded in a W1 waveguide and the guided light has been demonstrated [32], and successfully exploited for quan- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='04675v1 [quant-ph] 11 Jan 2023 2 1D ′ t (a) L a r y1 r3 (b) 680 730 780 830 880 930 980 Wavelength [nm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='50 kx[in units of 2 a ] 320 340 360 380 400 420 440 Frequency [THz] Radiative modes Bulk modes Bulk modes 87Rb D2 (c) o x y z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A half-W1 slow-mode photonic crystal waveguide coupled to cold atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (a) Sketch of the waveguide with an array of 87Rb atoms trapped in the proximity, along the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Γ1D and Γ′ correspond to the decay rates in the guided mode and in the radiation continuum, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The structure is etched in a GaInP membrane (refractive index n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='34) suspended in air, with a slab thickness t of 150 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (b) 2D scheme of the optimized waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The initial unshifted and regularly distributed holes are shown as dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' For the first three rows the position of the holes can be shifted along y and their radius tuned, amounting to 6 parameters (δyi,δri), i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' For the sake of clarity, only two parameters (δy1 and δr3) are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (c) Bandstructure of the optimized structure calculated via FDTD simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The bulk modes propagate within the slab but are not guided on the edge of the PCW while the radiative modes are not guided at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The 87Rb D2 line transition frequency is aligned with the linear part of a guided band, defined as the slow mode in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' tum operations [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The proposed platform, sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 1(a), can be seen as half a W1 waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' As its W1 counterpart, it enables enables dispersion engineer- ing but in addition offers a 2π solid-angle optical access to the edge of the structure, allowing for simpler transport of atoms close to it [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' We use a large refractive index GaInP slab that facilitates the design by offering more flexibility in the engineering of guided modes, and we show how to trap atoms in the proximity via additional guided modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Our effort focuses at each step on making the design robust to imperfections and on assessing the experimental feasibility of the full platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' First, in section II we present the specific platform based on a half-W1 waveguide realized in a GaInP slab with a high refractive index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' We detail the optimization and the resulting ro- bustness to nanofabrication imperfections, and then pro- vide the achievable atom-photon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Second, in section III we show that guided modes can be used to trap atoms in the proximity of the waveguide via a two- color evanescent dipole trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Stable traps at about 115 nm from the surface are obtained with low powers that are compatible with nanophotonic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A summary and outlook is provided in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' ENGINEERED HALF-W1 WAVEGUIDE FOR RUBIDIUM ATOMS In this section we introduce the specific half-W1 slow- mode waveguide designed in this work, based on GaInP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' We identify the required geometrical parameters and then present the optimizations performed to increase the robustness to fabrication imperfections, leading thereby to linear bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Finally, the expected coupling to the guided mode (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Purcell factor) for atoms in the prox- imity from the surface is detailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Description of the half-W1 GaInP waveguide A periodic modulation of the refractive index in a medium has deep consequences on light propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The wavevector can be constrained between −π/a and π/a, where a is the spatial period along the propaga- tion direction, and we observe the opening of photonic bandgaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' At the edge of the Brillouin zone (for k = π/a) the group velocity vanishes [34] and the Purcell factor di- verges (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Motivated by the proposal made in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [26], we study a similar structure with a different material: GaInP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This material has been chosen for its advantageous optical and electronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' GaInP has a wide electronic bandgap below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='85 eV [35], and as such is transparent for a wide range of wavelengths (from 670 nm up), mean- ing it could be used with several alkali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' At 780 nm, its 3 775 780 785 [nm] 0 10 20 30 40 50 group index ng (a) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='2 nm navg =27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='7 775 780 785 [nm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 1D/ 0 (b) F′ = 3, m′F = 3 F = 2, mF = 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Dispersion and atom-coupling properties for the half- W1 waveguide with the structure optimization specified in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (a) Calculated group index ng for the slow mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The dotted lines delimit the linear band region where the group index value is constant up to 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (b) Calculated Purcell factor Γ1D/Γ0 over the same range, for atoms trapped at 115 nm from the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' As it can be seen, ng is not the only parameter affecting this ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=', the field structure is also changing, but it is still critical as it diverges with ng just outside the plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' refractive index is n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='34, reaching 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='55 at the elec- tronic band edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This large index contrast with the air gives rise to band gaps that are wider and further away from the light line [34], allowing for more flexibility in the design of the trapping modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Finally, this material has attracted some attention in the recent years as it is very convenient to operate in the telecom band due to its low two-photon absorption [36], and growth and fab- rication processes have therefore been developed and well mastered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' As shown in figure 1(b), the holes etched in the GaInP slab do not go up to the edge, leaving a few hundreds of nanometers of unperturbed slab where the light can propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Being based on a 2D slab rather than a 1D structure, this geometry should be quite rigid and pre- vent detrimental effects from low frequency mechanical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The introduced symmetry breaking in the trans- verse direction allows for a more precise control on the dispersion properties of the waveguide since it offers ex- tra degrees of freedom [37], while significantly improving the optical access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This asymmetry has been harnessed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [38] to create many exotic dispersion bands as Dirac cones, multivalleys, or flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Arrays of 87Rb will then be trapped on the edge of the waveguide thanks to a two-color dipole trap, at around 100 nm from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' For comparison, in tapered nanofiber platforms, atoms sit at more than 200 nm from the silica fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' For a given thickness t, chosen here to be t = 150 nm, the first step to determine the geometrical parameters consists in finding the lattice period a and hold radius r of the 2D photonic crystal that allow for a bandgap at the Rubidium D2 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Indeed the width and position of the band gap is entirely determined by these values [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The band gap has to be wide enough to allow for at least two guided modes, one that crosses 780 nm, and a blue-detuned one for trapping, as described later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Guided bands appear when introducing the defect at the edge, and we can align the band of interest with respect to the D2 line by adjusting the width L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Given these constraints, the geometrical parameters of the waveguide are found to be: a = 212 nm, r = 63 nm and L = 337 nm for t = 150 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The corresponding band structure, computed with the 3D FDTD software Lumerical [39] is displayed in figure 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Three guided bands can be found inside the band gap of the 2D pho- tonic crystal between 360 and 440 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The bulk modes are guided in the slab (kz imaginary) but can propagate in any direction in the plane, even inside the 2D array of holes (kx, ky real).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Radiative modes have a real k vector in all directions and are therefore not guided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Imperfection-robust band engineering Nanofabrication inherently leads to imperfections, even if errors below 2 nm can be reached [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A spe- cific effort has been put in our design process to minimize the impact of such imperfections, thereby facilitating an experimental realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' As the Purcell factor diverges at the edge of the Bril- louin zone, one naive approach could be to align the D2 line frequency to any band edge of the band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' However, fabrication imperfections, to first order, lead to a shift of the energy of the band [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The flatter the band, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=', the smaller the group velocity, the more it is vulnerable to a shift in frequency [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' If the D2 line is aligned with the band edge, an infinitesimal shift to a lower energy will bring the atomic transition in the band gap of the 2D photonic crystal, impeding the prop- agation of the emitted light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' In addition to a shift in frequency, the disorder introduced during the fabrication process can lead to strong localization of light inside the crystal [41, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Following [26], two main criteria are to be considered when assessing the robustness of a structure: the group velocity has to be as independent of the frequency as possible at the band edge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' ∂vg/∂ω|ωe ∼ 0, and the distance of the operation frequency to the band edge ∆ω = |ω − ωe| has to be as large as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Designing slow modes with linear bands (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=', an almost constant, large group index ng over the widest range of ω possible) allows us to fulfill these two criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' First, a linear dispersion corresponds to a vanishing group ve- locity dispersion (GVD) and the atom-photon coupling Row Position δy (nm) Radius δr (nm) 1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='2 2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='2 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='8 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Calculated changes in row positions and holes radii via automatic differentiation optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' All the rows after the third one are unperturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 4 318 106 0 106 318 x [nm] 400 200 0 200 400 y [nm] (a) 200 0 200 y [nm] 300 225 150 75 0 75 150 225 300 z [nm] (b) 212 0 212 x [nm] 500 250 0 250 500 y [nm] (c) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='8 1 Intensity [norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 |C| FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Slow mode structure at the 87Rb D2-line frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (a) Normalized intensity, in the (x, y)-plane at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (b) Same in the (y, z)-plane at x = −a/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=', crossing the hole nearest to the slab edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The mode is strongly expelled into the vacuum around the edge of the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (c) Polarization ellipticity z-component Cz in the XY plane at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The other components of the ellipticity vector are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' |Cz| = 0 indicates a linear polarization, while we have |Cz|= 1 for a circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Close to the edge, the polarization has a large circular component due to the strong longitudinal component that appears when light is confined at the nanoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' By taking z as the quantification axis, the polarization will be close to σ+ for atoms trapped in the proximity (91 to 99% at 115 nm from the surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' is proportional to the group index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Moreover, as shown in [28], it is possible to design a slow and linear band over a wide spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' As most fabrication imper- fections lead to a shift ∆ω of the guided bands, both these constraints aim at placing the relevant frequency at a position on the band where a small shift will affect the dispersion at the given frequency only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' It has been shown that linear bands can be achieved in at least two types of asymmetric PCWs [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Achieving such vanishing group velocity dispersion has been extensively studied in the context of W1 waveguides, by tuning the position of rows of holes [28, 45, 46], chirping the waveg- uide properties [47], or changing the size of the holes [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Inspired by these previous optimization strategies, we set the radius of the first three rows of holes as well as their position along the y axis as optimization parame- ters, as depicted in figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' We then have 6 indepen- dent optimization parameters (δri, δyi), i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' As full 3D FDTD simulations are computationally intensive, we use the approximate method of Guided Mode Expan- sion (GME) [48] thanks to the legume [49] solver to faster compute the shape of the guided band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' We optimize the shape of the slow-mode band by iteratively varying the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' At each iteration, a cost function enforcing the minimization of the group velocity dispersion (aver- aged over the wave vector interval) while setting a target ng value is evaluated and the (δri, δyi) varied thanks to automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' After a few hundred iterations we obtain the optimal shifts for achieving this target ng value over the widest possible spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Finally the optimized structure was simulated in full 3D FDTD to validate the results from the approximate GME method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' In order for this optimization to give relevant results, ng has to be set to an experimentally realistic value, ide- ally below 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Indeed, experiments have shown that it is extremely challenging to reach higher values for the group index without losses [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The most concluding optimiza- tion results are obtained for a target around ng = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The shifts in position and radius after optimization are given in table I and the corresponding band structure is presented in figure 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Figure 2(a) shows that we engi- neered a band with a constant group index of 28 over a 9 nm range, and hence reach similar performance than a previous optimization of a W1 waveguide [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This fea- ture offers a two-fold advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' In addition to making it robust to shifts caused by fabrication imperfections, the optimization enables using the half-W1 waveguide in a large bandwidth regime (≥ 4 THz) with very little dis- persion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Finally, as seen in Appendix A, the Purcell factor is proportional to the group index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Keeping the group in- dex constant over a wide range enables to keep the Pur- cell factor constant in case of a shift, as shown in figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' If it is necessary to have a constant group index it is not sufficient as the Purcell factor also depends on the shape of the mode of the electric field (equation A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' As we move along the guided band, the mode shape changes slightly, affecting the value of the Purcell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Strong coupling to the slow mode Given the optimized design, we now turn to the in- teraction between the slow mode and 87Rb atoms in the proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Taking into account the multilevel character of Rubidium, we defined a transition-dependent Purcell factor in equation (A4) given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The group velocity vg is evaluated from the simulated band struc- ture, while the other terms are computed from the field map of the guided mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Indeed, the Purcell factor is 5 F′ = 3 m′F = F = 2, mF = 2 3 + 2 1 (a) 0 200 400 y [nm] 0 5 10 15 1D/ 0 212 0 212 x [nm] 400 200 0 200 y [nm] (b) 0 200 y [nm] 225 150 75 0 75 150 225 z [nm] (c) 10 2 10 1 100 101 1D/ 0 0 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Excitation rates for 87Rb atoms in the waveguide proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (a) Allowed transitions on the D2 line for an atom in |F = 2, mF = +2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Because of the large σ+ component (∼ 91% at the position of the atoms) and the values of the Clebsch- Gordan coefficients, the excitation probability to the |F′ = 3, mF′ = +3⟩ is 100 times higher than the σ− channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The inset provides a zoom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (b) Purcell factor in the XY plane, at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Purcell factor in the YZ plane, at x = −a/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The red dots indicate the position of the atoms at 115 nm from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' proportional to the slow-mode intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Figure 3(a-b) show that in order to have the maximum coupling the atoms should be trapped close to the edge of the waveg- uide, aligning them to the holes of the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' From figure 4(a) we see that for atoms in state |F = 2, mF = +2⟩ trapped at 115 nm from the edge, the Purcell factor reaches a value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' As shown in fig- ures 4(b) and 4(c), a small modulation in the x direction exists and the value of the Purcell factor decays rapidly when going further from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' To quantify the coupling of the atoms to the guided mode we also define the β factor, β = Γ1D/Γtot with Γtot = Γ1D + Γ′, and Γ′ the decay rate in all the other radiation modes than the guided slow mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Because of the complex shape of the local density of states accessible to the atoms, the behaviour of Γ′ is hard to infer, but its modulation is expected to be minimal as seen in [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hence we assume for the following Γ′ ≃ Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' At the position of the trap minimum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='e at 115 nm from the surface, we find β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='62, very close to the averaged value ˜β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='57 for a thermal distribution (at a temperature of one tenth of the trap depth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This is at least 50 times better than the current systems involving nanofibers (β = 10−2) [9] and a significant improvement with respect to current PCW-based platforms (β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='45) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' TRAPPING RUBIDIUM ATOMS NEAR A HALF-W1 WAVEGUIDE Simulations in the previous section were performed for atoms at 115 nm from the edge of the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Indeed, in the following we show a stable trapping scheme based on an evanescent two-color dipole trap formed by fast guided modes, allowing the atoms to be trapped between 100 and 150 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This trap has been designed following the ideas implemented in optical nanofibers [2, 51], with blue- and red-detuned counter-propagating modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Find- ing a stable trapping scheme that keeps the atoms close enough to the surface so that they can couple to the slow mode with a large Purcell factor is a critical requirement for experimental implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Two-color dipole trap structure In contrast with optical nanofibers, the guided modes are structured along the propagation direction due to the Bloch wave structure of the light field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The intensity of the modes, which is an important quantity when looking at dipole trapping, is periodic with period a, as shown in figure 3 for the slow mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This feature constrains the position of the trapped atoms to the maxima of intensity of the red-detuned mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' It makes the search for a blue detuned mode more challenging as this one will also be structured, while a uniform one would work perfectly well to repel the atoms from the surface [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A blue-detuned beam with an intensity pattern completely out of phase with the red-detuned one is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Fortunately, modes separated by a band gap usually have intensity maxima shifted by a/2 [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' We then use the highest guided band for the blue-detuned trap between 400 and 420 THz (fig- ure 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' In order to have a full description of the potential seen by the atoms, we must add the Casimir-Polder (CP) interaction [53] between the atoms and the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Ground state fluctuations of the vacuum can polarize the atoms, even if they are not charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' When put in prox- imity to structures, the vacuum-induced dipole moments create mirror charges that act on the original dipole, lead- ing to an additional light shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This CP potential is only significant at very close distances (≤ 150 nm) but is crucial as it reduces the local density of states and acts as an attractive potential close to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' For an atom in the proximity of an infinite dielectric plane 6 UCP = −C3/d3 [52], where d is the distance to the sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' As described in Appendix B, for a ground state 87Rb atom close to a GaInP surface, we computed an approximate C3 = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='25 × 10−49J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Trapping potential simulation The trapping potentials were obtained via nanotrappy [54], a Python package developed by our group, to design, calculate and optimize dipole traps around nanoscale waveguides, making the search process faster and more systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Figure 5 shows a trapping potential in all 3 directions for an atom in the |F = 2, mF = +2⟩ hyperfine level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' For this trap, a beam red-detuned from the D2 line of 87Rb at 784.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 nm and a beam blue-detuned at 737 nm are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A trap of depth 3 mK is obtained with a minimum at 115 nm from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The total powers are Pblue = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='65 mW and Pred = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='1 mW, but a stable trap can be obtained on a wide range of powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The main limitation can be the power handling of the structure which is still to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [36], power densities up to 1 GW/cm2 were coupled to similar GaInP PCWs with group index 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' For our structure which has a cross section 10 times smaller and a group index 3 times bigger, this would be equivalent to coupling ≃ 100 mW into our waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The proposed powers for the trap fall well below this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The trapping frequencies are large in the x and y direc- tions, with ωx = 2π×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='89 MHz and ωy = 2π×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='94 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' In the vertical direction however, an important anhar- monicity of the trap, discussed below, gives ωz = 2π × 133 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A critical aspect is the trapping along the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' With small powers for the blue-detuned beam, a sym- metric double well around z = 0 appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This can be detrimental for our platform as the atoms will be trapped at positions not corresponding to the maxima of the Pur- cell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Because of the fixed CP potential, rescaling the blue and red powers by the same factor does not lead to a simple reduction of the trap depth, it also goes with a shift in position, which might lead to reaching this two- well regime in the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This explains the impor- tant depth of the simulated traps: the closer to the sur- face we want the trap minimum to be, the more we have to compensate for the CP potential with higher beam power, leading to higher trap depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This phenomenon may come from the complex decay of the evanescent wave away from the surface pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Indeed, this decay is usually multi-exponential, with decay rates not easily linked to the wavelength or the wavevector of the guided mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Importantly, we also verified that we can achieve a sta- ble trap in the three directions for a wide range of wave- lengths, which is a valuable feature for finding the right trade-off between heating the atoms with off-resonant scattering and power handling of the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' If we 212 0 212 424 x [nm] 500 250 0 250 500 y [nm] (a) 212 0 212 424 x [nm] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 U [mK] (b) 0 100 200 300 y [nm] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 U [mK] (c) 200 0 200 z [nm] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='0 U [mK] (d) 0 20 40 60 80 100 120 140 U [mK] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Calculated potential of the two-color dipole trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (a) 2D total trapping potential in the proximity of the waveguide, in the XY plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The trapping potential is given along the three directions in (b), (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A periodic stable trap with depth of about 3 mK is obtained with powers of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='65 mW for each blue beam and 100 µW for each red one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The simulations are performed with the nanotrappy package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' allow the blue power to go up to 3 mW, we can find a stable trap for blue wavelengths ranging from 724 nm to 738 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Pushing the maximum allowed power to 5 mW we can find a trap for the full available blue-detuned air band, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' ∆λ = 21 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' For the red-detuned laser, we have an available range from 780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='5 nm up to 786 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Laser diodes are easily available on these wavelengths, reinforcing the feasibility of our platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Finally, as briefly noted before, we used counter- propagating beams here instead of simple ones, albeit standing waves are not needed to get a periodic intensity modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The strong ellipticity of the guided modes (as shown in figure 3(c)), acts as a fictitious magnetic field on the atoms, splitting the Zeeman levels [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' If we start from atoms evenly distributed in all the mF states, this effect would lead to a large inhomogeneous broadening up to a few GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' It can be mitigated by us- ing counter-propagating trapping beams slightly detuned from each other, as used for blue detuned beams in some compensated nanofiber traps [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Via nanotrappy, we estimated that adding a red-detuned laser at 280 GHz from the first one and a blue detuned at 250 GHz from the other reduces this broadening by 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Counterprop- agation creates a running wave at a velocity given by δω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This pattern propagates but at a speed so large the atoms only see the average of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 7 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' CONCLUSION Many experimental and technological challenges have yet to be overcome to enable further neutral-atom waveguide-QED protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' As such, experimental ro- bustness of the targeted waveguide platforms is a critical requirement, as is evanescent trapping of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' In our work, we proposed and engineered a bona fide platform for trapping cold Rubidium atoms close to a half-W1 photonic crystal waveguide based on high-index mate- rial GaInP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Atoms can be trapped between 100 and 150 nm from the surface with compatible low-power incident light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' At 115 nm, the slow mode couples to the atoms with a Purcell factor as high as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='6 when the group index is taken around 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This study has been carried out for conservative parameters and a strong focus on robustness against fabrication imperfections has been done by engi- neering the band structure for a large bandwidth, facili- tating first implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Future generations should support higher group index, albeit with narrower band- widths [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' This novel platform – tailor-designed for atom integration, robustness and large optical access – offers unique advantages for studying coherent and dissi- pative dynamics in the waveguide-QED framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work was supported by the French National Research Agency (NanoStrong Project ANR-18-CE47- 0008), by the R´egion Ile-de-France (DIM SIRTEQ), and by the European Union’s Horizon 2020 research and in- novation program under Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='899275 (DAALI project).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' was supported by the European Union (Marie Curie Fellowship SinglePass 101030421).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' is a member of the Institut Universitaire de France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Appendix A: Theoretical framework: Reaching strong Purcell factor The coupling of the atoms to the guided mode of a waveguide can be characterized by the Purcell factor Γ1D/Γ0, which relates the decay rate of the atoms into the guided mode to the one into free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' For a mul- tilevel atom we can define the excitation rate ΓF,F ′,mF,q from the hyperfine level |F, mF⟩ to |F′, mF + q⟩ [57]: ΓF,F ′,mF,q = 2µ0| ⟨F||ˆd||F⟩|2 ¯h × ω2 q|CmF ,q|2ˆeq · Im G(r, r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' ωq) · ˆe∗ q (A1) where the ˆeq, q ∈ {−1, 0, 1}, are the normalized dipole vectors over all the possible excitation channels (σ−, π, σ+ respectively) and ωq is the transition frequency be- tween the specified levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' G(r, r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' ωq) is the value of the classical Green’s tensor at r for a dipole at the same po- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' The CmF ,q are the Clebsch-Gordan coefficients given by: CmF ,q = (−1)F ′−1+mF √ 2F + 1 � F ′ 1 F m′ F q −mF � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (A2) Note that only the Wigner 3j where m′ F = mF + q are non zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' As we are looking at the decay into a defined guided mode, it is possible to write the Green’s tensor as a sum G(r, r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' ωq) = G1D(r, r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' ωq)+G′(r, r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' ωq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' From [58] (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='89) and [59] we have an analytical expression for the Green’s function of an effective 1D structured waveguide: G1D(r, r, ω) = i ac 2ω � c vg � [E(r) ⊗ E∗(r)] � cell drϵ(r)|E(r)|2 (A3) where E(r) is the electric field of the guided mode, a the period of the modulation and vg = ∂ω ∂k the group velocity of the guided mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' By neglecting the Zeeman splitting of the mF levels (ωq constant), the excitation probability of a single atom in |F, mF⟩ into |F′, mF + q⟩ (through a single excitation channel) is hence given by: Γ1D,F,F ′,mF,q = 2πc ϵ0¯h | ⟨F||ˆd||F⟩|2 λ0vg × |CmF ,q|2 |ˆeq · E(r)|2 � cell drϵ(r)|E(r)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (A4) As such, we see that to reach high Purcell factors we must either decrease vg which can be achieved by dis- persion design, or maximize the normalized electric field amplitude at the position of the atom given by the second half of equation (A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Since we are considering excitation probabilities (as in Figure 4(a)), we only consider the coupling between the atom and the injected mode, here the guided mode E(r) propagating along the positive x axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' If we were instead to consider decay rates we would have to modify slightly equation (A4) and sum the contributions of the coupling of the atom to both forward and backward propagating guided modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Appendix B: Casimir-Polder interactions between GaInP and Rubidium atoms To the best of our knowledge, there were no previous computations of the C3 coefficient of the Casimir-Polder interactions between GaInP and Rubidium atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' For a dielectric wall, we have to adapt the formula for C3 in the form [60] C3 ≈ ¯h 4π � +∞ 0 α(iξ)ϵ(iξ) − 1 ϵ(iξ) + 1dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (B1) 8 We then have to evaluate α and ϵ over the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' α is simply evaluated using the expression for the scalar polarizability of 87Rb in the ground state |F = 2⟩ with complex frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' As ϵ(ω) = ϵ′(ω) + iϵ′′(ω), we can get the dependence in ω from experimental data [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' To evaluate it over the imaginary axis we use the Kramers- Kroenig relation that provides [61] ϵ(iξ) = 1 + 2 π � +∞ 0 ω[ϵ′(ω) − 1] ω2 + ξ2 dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' (B2) Finally, for a ground state Rubidium atom close to a GaInP surface we get C3 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='25 × 10−49J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='m3 = h × 1391 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='µm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Chang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Douglas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gonz´alez-Tudela, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hung, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kimble, Colloquium: Quantum matter built from nanoscopic lattices of atoms and photons, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 90, 31002 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Vetsch, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Reitz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Sagu´e, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Schmidt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Dawkins, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rauschenbeutel, Optical Interface Cre- ated by Laser-Cooled Atoms Trapped in the Evanescent Field Surrounding an Optical Nanofiber, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 104, 203603 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Goban, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Choi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Alton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Ding, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lacroˆute, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Pototschnig, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Thiele, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Stern, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kim- ble, Demonstration of a state-insensitive, compensated nanofiber trap, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 109, 033603 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Nieddu, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gokhroo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Nic Chormaic, Optical nanofibres and neutral atoms, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 18, 053001 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Solano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Grover, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hoffman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Ravets, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Fatemi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Orozco, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rolston, Chapter seven optical nanofibers: A new platform for quantum optics (Academic Press, 2017) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 439–505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [6] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gouraud, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Maxein, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Nicolas, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Morin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Laurat, Demonstration of a memory for tightly guided light in an optical nanofiber, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 114, 180503 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [7] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Corzo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gouraud, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Chandra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Goban, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Sheremet, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kupriyanov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Laurat, Large Bragg Reflection from One-Dimensional Chains of Trapped Atoms Near a Nanoscale Waveguide, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 117, 133603 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Sørensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' B´eguin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kluge, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Iakoupov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Sørensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' M¨uller, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Polzik, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Ap- pel, Coherent backscattering of light off one-dimensional atomic strings, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 117, 133604 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [9] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Corzo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Raskop, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Chandra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Sheremet, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gouraud, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Laurat, Waveguide-coupled single collective excitation of atomic arrays, Nature 566, 359 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Mitsch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Sayrin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Albrecht, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Schneeweiss, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rauschenbeutel, Quantum state-controlled directional spontaneous emission of photons into a nanophotonic waveguide, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 5, 1 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Prasad, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hinney, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Mahmoodian, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hammerer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rind, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Schneeweiss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Sørensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Volz, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rauschenbeutel, Correlating photons using the collec- tive nonlinear response of atoms weakly coupled to an optical mode, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 14, 719 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Douglas, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Habibian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hung, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gorshkov, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kimble, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Chang, Quantum many-body models with cold atoms coupled to photonic crystals, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 9, 326 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gonz´alez-Tudela, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hung, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Chang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Cirac, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kimble, Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 9, 320 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Bello, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Platero, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gonz´alez-Tudela, Spin many-body phases in standard- and topological- waveguide qed simulators, PRX Quantum 3, 010336 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Thompson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Tiecke, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' De Leon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Feist, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Akimov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gullans, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Zibrov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Vuleti´c, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lukin, Coupling a single trapped atom to a nanoscale optical cavity, Science 340, 1202 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kim, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Chang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Fields, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Chen, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hung, Trapping single atoms on a nanophotonic circuit with configurable tweezer lattices, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 10, 1647 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Xu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Xiang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Guo, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Wang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Zou, Proposal for low-power atom trapping on a GaN-on-sapphire chip, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A 106, 033104 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [18] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Will, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Masters, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rauschenbeutel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Scheucher, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Volz, Coupling a Single Trapped Atom to a Whispering-Gallery-Mode Microresonator, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 126, 233602 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Goban, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hood, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Muniz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Martin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' McClung, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Choi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Chang, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Painter, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kimble, Atom-light interactions in photonic crystals, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 5, 3808 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Goban, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hood, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Mu˜niz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Painter, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kimble, Superradiance for Atoms Trapped along a Photonic Crystal Waveguide, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 115, 063601 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hood, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Goban, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Asenjo-Garcia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Yu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Chang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kimble, Atom-atom interac- tions around the band edge of a photonic crystal waveg- uide, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='A 113, 10507 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Burgers, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Peng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Muniz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' McClung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Martin, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kimble, Clocked atom delivery to a photonic crystal waveguide, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='A 116, 456 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hood, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Muniz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Martin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Norte, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Meenehan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Co- hen, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Painter, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kimble, Nanowire photonic crystal waveguides for single-atom trapping and strong light-matter interactions, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 104, 111103 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Yu, Nano-Photonic Platform for Atom-Light Inter- action, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' thesis, California Institute of Technology 9 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [25] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Fayard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Bouscal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Berroir, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Urvoy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Ray, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Mahapatra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kemiche, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Levenson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Greffet, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Bencheikh, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Laurat, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Sauvan, Asymmetric comb waveguide for strong interactions between atoms and light, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Express 30, 45093 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [26] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Zang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Yang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Faggiani, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Petrov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hugonin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Vynck, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Bernon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Bouyer, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Boyer, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lalanne, Interaction between Atoms and Slow Light: A Study in Waveguide Design, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Applied 5, 024003 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [27] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Petrov and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Eich, Zero dispersion at small group velocities in photonic crystal waveguides, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 85, 4866 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' White, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Faolain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gomez-Iglesias, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Krauss, Systematic design of flat band slow light in photonic crystal waveguides, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Express 16, 2621 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [29] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Frandsen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lavrinenko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Fage-Pedersen, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Borel, Photonic crystal waveguides with semi-slow light and tailored dispersion properties, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Express 14, 2440 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [30] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Colman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Combri´e, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lehoucq, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' De Rossi, Control of dispersion in photonic crystal waveguides using group symmetry theory, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Express 20, 1237 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [31] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Vyas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Cheriton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Ahmed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Schulz, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Dolgaleva, Simulation and fabrication of slow light suspended air-bridge AlGaAs photonic crystal waveg- uide, in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' SPIE 12004, Integrated Optics: Devices, Materials, and Technologies XXVI (2022) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Arcari, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S¨ollner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Javadi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lindskov Hansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Mahmoodian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Thyrrestrup, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Stobbe, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lodahl, Near-Unity Coupling Efficiency of a Quantum Emitter to a Photonic Crystal Waveguide, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 113, 093603 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [33] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Le Jeannic, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Tiranov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Carolan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Ramos, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Appel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Scholz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Wieck, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lud- wig, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rotenberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Midolo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Garc´ıa-Ripoll, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Sørensen, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lodahl, Dynamical photon–photon in- teraction mediated by a quantum emitter, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 18, 1191 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [34] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Joannopoulos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Johnson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Winn, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Meade, Photonic Crystals : Molding the Flow of Light (Princeton University Press, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Schubert, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gottschalch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Herzinger, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Yoa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Snyder, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Woollam, Optical constants of GaxIn1 xP lattice matched to GaAs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 3416, 372 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [36] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Combri´e, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Tran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rossi, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Husko, High quality GaInP nonlinear photonic crystals with mini- mized nonlinear absorption, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 95, 221108 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [37] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' L¨u, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Zhao, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Zhang, Flat band slow light in asymmetric photonic crystal waveguide based on mi- crofluidic infiltration, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 49, 3930 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [38] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Nguyen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Dubois, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Deschamps, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Cueff, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Par- don, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Leclercq, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Seassal, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Letartre, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Vik- torovitch, Symmetry Breaking in Photonic Crystals: On- Demand Dispersion from Flatband to Dirac Cones, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 120, 66102 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [39] Ansys Lumerical FDTD (2022), simulation software based on the finite-difference time-domain method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [40] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Asano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Song, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Noda, Analysis of the ex- perimental Q factors (∼ 1 million) of photonic crystal nanocavities, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Express 14, 1996 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [41] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Faggiani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Baron, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Zang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lalouat, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Schulz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' O’Regan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Vynck, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Cluzel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' De Fornel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Krauss, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lalanne, Lower bound for the spatial ex- tent of localized modes in photonic-crystal waveguides with small random imperfections, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 6, 27037 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Soljaˇci´c and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Joannopoulos, Enhancement of nonlinear effects using photonic crystals, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 3, 211 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [43] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Mazoyer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hugonin, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lalanne, Disorder- induced multiple scattering in photonic-crystal waveg- uides, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 103, 063903 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Patterson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hughes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Combri´e, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Tran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' De Rossi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gabet, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Jaou¨en, Disorder-induced coher- ent scattering in slow-light photonic crystal waveguides, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 102, 253903 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [45] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Peng, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Wang, Wideband and low dispersion slow light in slotted photonic crystal waveg- uide, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 283, 2815 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Liang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Ren, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Yun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Han, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Wang, Wideband ultraflat slow light with large group index in a W1 photonic crystal waveguide, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 110, 063103 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [47] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Mori and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Baba, Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Express 13, 9398 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [48] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Andreani and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gerace, Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' B 73, 235114 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Minkov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Williamson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Andreani, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Gerace, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Song, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hughes, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Fan, Inverse Design of Photonic Crystals through Automatic Differ- entiation, ACS Photonics 7, 1729 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [50] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Mazoyer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lalanne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rodier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hugonin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Spasenovi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kuipers, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Beggs, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Krauss, Statistical fluctuations of transmission in slow light photonic-crystal waveguides, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Express 18, 14654 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [51] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Le Kien, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Balykin, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hakuta, Atom trap and waveguide using a two-color evanescent light field around a subwavelength-diameter optical fiber, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A 70, 063403 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [52] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Johnson, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Dzuba, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Safronova, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Safronova, Finite-field evaluation of the Lennard-Jones atom-wall interaction constant C3 for alkali-metal atoms, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A 69, 022508 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [53] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Casimir and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Polder, The influence of retardation on the London-van der Waals forces, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 73, 360 (1948).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Berroir, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Bouscal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Urvoy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Ray, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lau- rat, Nanotrappy: An open-source versatile package for cold-atom trapping close to nanostructures, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Research 4, 013079 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [55] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Engelen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Mori, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Baba, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kuipers, Sub- wavelength structure of the evanescent field of an optical Bloch wave, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 102, 023902 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [56] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Cohen-Tannoudji and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Dupont-Roc, Experimental Study of Zeeman Light Shifts in Weak Magnetic Fields, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A 5, 968 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' 10 [57] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Luan, Towards atom assembly on nanophotonic struc- tures with optical tweezers, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' thesis, California Insti- tute of Technology (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [58] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hood, Atom-light interactions in a photonic crystal waveguide, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' thesis, California Institute of Technol- ogy (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [59] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Asenjo-Garcia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Hood, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Chang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Kimble, Atom-light interactions in quasi-one-dimensional nanostructures: A Green’s-function perspective, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A 95, 033818 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [60] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Caride, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Klimchitskaya, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Mostepanenko, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Zanette, Dependences of the van der Waals atom-wall interaction on atomic and material properties, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A 71, 042901 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' [61] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Antezza, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Pitaevskii, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Stringari, Effect of the casimir-polder force on the collective oscillations of a trapped bose-einstein condensate, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} +page_content=' A 70, 053619 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQftQvk/content/2301.04675v1.pdf'} diff --git a/GtE0T4oBgHgl3EQfhgEK/content/tmp_files/2301.02431v1.pdf.txt b/GtE0T4oBgHgl3EQfhgEK/content/tmp_files/2301.02431v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f2487dc5b41e0c44223786ccbe2fad39e595e88f --- /dev/null +++ b/GtE0T4oBgHgl3EQfhgEK/content/tmp_files/2301.02431v1.pdf.txt @@ -0,0 +1,1645 @@ +Equilibrium Spacetime Correlations of the Toda Lattice on +the Hydrodynamic Scale +Guido Mazzuca∗, Tamara Grava†, Thomas Kriecherbauer‡, Kenneth T-R +McLaughlin §, Christian B. Mendl¶, Herbert Spohn‖ +January 9, 2023 +Abstract +We report on molecular dynamics simulations of spacetime correlations of the Toda lattice +in thermal equilibrium. The correlations of stretch, momentum, and energy are computed +numerically over a wide range of pressure and temperature. Our numerical results are com- +pared with the predictions from linearized generalized hydrodynamics on the Euler scale. The +system size is N = 3000, 4000 and time t = 600, at which ballistic scaling is well confirmed. +With no adjustable parameters, the numerically obtained scaling functions agree with the +theory within a precision of less than 3.5%. +1 +Introduction +A central goal of Statistical Mechanics is to explore the structure of equilibrium correlations for +observables of physical interest. These could be static correlations, but more ambitiously also +correlations in spacetime. An interesting, but very fine-tuned class of hamiltonians are integrable +many-body systems, either classical or quantum. This choice restricts us to systems in one dimen- +sion. Then, generically, static correlations have exponential decay whether the model is integrable +or not. However, the dynamics of correlations is entirely different. In nonintegrable chains correla- +tions propagate as a few narrow peaks at constant speed which then show characteristic sub-ballistic +broadening. On the other hand for integrable models correlations still spread ballistically but now +∗Department of Mathematics, The Royal Institute of Technology, Stockholm, Sweden. +Email: mazzuca@kth.se +†International School for Advanced Studies (SISSA), Trieste, Italy, School of Mathematics, University of Bristol, +UK and INFN sezione di Trieste, +Email: grava@sissa.it +‡Department of Mathematics, Universit¨at Bayreuth, Germany +Email: thomas.kriecherbauer@uni-bayreuth.de +§Tulane University, New Orleans, United States +Email: kmclaughlin@tulane.edu +¶Technische Universit¨at M¨unchen Department of Informatics, Boltzmannstraße 3, 85748, Garching, Germany +Email: christian.mendl@tum.de +‖Technische Universit¨at M¨unchen Department of Mathematics and Physics, Boltzmannstraße 3 and James- +Franck-Str. 1, 85748 Garching, Germany +Email: spohn@ma.tum.de +1 +arXiv:2301.02431v1 [cond-mat.stat-mech] 6 Jan 2023 + +with a broad spectrum of velocities. Such behaviour was confirmed through a molecular dynamics +(MD) simulation of the Ablowitz-Ladik model [32], an integrable discretization of the nonlinear +Schr¨odinger equation. A further confirmation came from the simulation of the Toda chain [22]. On +the theoretical side, the 2016 construction of generalized hydrodynamics (GHD) was an important +breakthrough [3, 6]. This theory provides a powerful tool through which, at least in principle, +the precise form of the spectrum of correlations can be predicted. With such a development MD +simulations can also be viewed as probing the validity of GHD. +From the side of condensed matter physics, integrable quantum models have received consider- +able attention. Because of size limitations, the simulation of macroscopic profiles are preferred. But +time correlations have also been studied through DMRG simulations [4,5,8,34]. In recent years, +attention has been given to the spacetime spin-spin correlation of the XXZ model at half-filling +and at the isotropic point [10,20,25]. The same quantity has also been investigated for a discrete +classical chain with 3-spins of unit length and interactions such that the model is integrable [7]. A +comparable situation occurs for the classical sinh-Gordon equation, which is integrable as a nonlin- +ear continuum wave equation and possesses an integrable discretization, see [2] for MD simulations +for equilibrium time correlations of the discrete model. +In our contribution we study the correlations of the Toda chain in thermal equilibrium through +MD simulations and compare with predictions from GHD. We will comment on the connection +to [22] in the last section. To make our article reasonably self-contained we first discuss the Landau- +Lifshitz theory for nonintegrable chains. This theory provides the connection between spacetime +correlations and linearized hydrodynamics. For the Toda chain, the theory has to be extended so +as to accommodate an infinite number of conserved fields. We report on MD simulations of the +Toda chain and compare with linearized GHD. +2 +Landau-Lifshitz theory +The dynamics of the Toda chain is governed by the Hamiltonian +H = +� +j∈Z +� 1 +2p2 +j + exp(−(qj+1 − qj)) +� +, +(1) +where (qj, pj) ∈ R2 are position and momentum of the j-th particle [43,44]. Introducing the j-th +stretch (free volume) through rj = qj+1 − qj, the equations of motion read +d +dtrj = pj+1 − pj , +d +dtpj = −e−rj + e−rj−1, +j ∈ Z. +(2) +By tradition, one introduces coefficients for the range and strength of the interaction potential +through (g/γ) exp(−γ(qj+1 − qj)). However, by a suitable change of spacetime scales, the form (2) +can be regained, see the discussion in Section 5. The Toda hamiltonian has no free parameters. +Since the equilibrium measure for (1) is of product form, static correlations are easily accessible. +Time correlations are more challenging, see [36,37] for early attempts. A novel approach has been +developed, now known as GHD. The guiding idea is to first identify the hydrodynamic equations +for the Toda chain, which by necessity are a set of nonlinear coupled hyperbolic conservation laws. +Given such an input one can construct the corresponding Landau-Lifshitz theory [13,24], as based +on linearized GHD. +Before entering into details, it will be useful to first recall the Landau-Lifshitz theory for a +chain with a generic interaction potential, denoted by V (for the Toda lattice V (x) = e−x), see [38] +2 + +and references listed therein. Thus in (1) the interaction term reads V (qj+1 −qj) and the equations +of motion become +d +dtrj = pj+1 − pj , +d +dtpj = V ′(rj) − V ′(rj−1). +To define spacetime correlations we first have to specify the random initial data modelling thermal +equilibrium. By Galileian invariance one restricts to the case of zero average momentum. Then +the Gibbs states are characterized by the inverse temperature β > 0 and a parameter P such +that the physical pressure equals P/β. For simplicity, we will also refer to P as pressure. The +allowed range of P depends on V . If V diverges faster than |x| for |x| → ∞, then P ∈ R. For the +Toda lattice P > 0 because of the one-sided divergence of the exponential. In thermal equilibrium +{(rj, pj), j ∈ Z} are a collection of i.i.d. random variables with single site probability density +Z0(P, β)−1 exp +� +− β +� 1 +2p2 +0 + V (r0) +� +− Pr0 +� +. +(3) +Here Z0(P, β) is the normalizing partition function. Note that, with our convention, P and β appear +linearly in the exponent. Expectations with respect to such i.i.d. random variables are denoted +by ⟨·⟩P,β. +We also shorten the notation for the covariance through ⟨X1X2⟩c +P,β = ⟨X1X2⟩P,β − +⟨X1⟩P,β⟨X2⟩P,β, where the particular random variables X1, X2 will be obvious from the context. +For general V , the conserved fields are stretch, momentum, and energy with densities +⃗Q(j) = +� +rj, pj, ej +� +, +ej = 1 +2p2 +j + Vj, +(4) +using as shorthand Vj = V (rj). ⃗Q is a three-vector with components labeled by n = 0, 1, 2. The +static space correlator is defined through +Cm,n(j) = ⟨Qm(j)Qn(0)⟩c +P,β +(5) +and the static susceptibility by summing over space, +Cm,n = +� +j∈Z +⟨Qm(j)Qn(0)⟩c +P,β, +m, n = 0, 1, 2. Since the underlying measure is product, only the j = 0 term is nonvanishing and +C = +� +� +� +⟨r0r0⟩c +P,β +0 +⟨r0e0⟩c +P,β +0 +⟨p0p0⟩c +P,β +0 +⟨r0e0⟩c +P,β +0 +⟨e0e0⟩c +P,β +� +� +� , +the zero entries resulting from ⟨p0⟩P,β = 0, ⟨p3 +0⟩P,β = 0, and r0, p0 being independent random +variables. Later on we will need the statistics of the conserved fields on the hydrodynamic scale. +More precisely, for smooth test functions f, we consider the random field +⃗ξϵ(f) = √ϵ +� +j∈Z +f(ϵj) +�⃗Q(j) − ⟨⃗Q(0)⟩P,β +� +. +Then, by the central limit theorem for independent random variables, +lim +ϵ→0 +⃗ξϵ(f) = +� +R +dxf(x)⃗u(x), +3 + +where the limit field ⃗u(x) is a Gaussian random field on R with mean zero, E(⃗u(x)) = 0, and +covariance +E(um(x)un(x′)) = Cm,nδ(x − x′), +(6) +in other words, ⃗u(x) is Gaussian white noise with correlated components. +Microscopically, spacetime correlations are defined by evolving one of the observables to time +t which yields +Sm,n(j, t) = ⟨Qm(j, t)Qn(0, 0)⟩c +P,β. +(7) +Note that the Gibbs measure is spacetime stationary and thus without loss of generality both +arguments in Qn in (7) can be taken as (0, 0). To understand the structure of Sm,n one has to rely +on approximations. For the long time ballistic regime a standard scheme is the Landau-Lifshitz +theory, which views Qn(0, 0) as a small perturbation of the initial Gibbs measure at the origin. +This perturbation will propagate and is then probed by the average of Qm at the spacetime point +(j, t). For large (j, t) the microscopic dynamics is approximated by the Euler equations, but only +in their linearized version since the perturbation is small. More concretely, the approximate theory +will be a continuum field ⃗u(x, t) over R × R, which is governed by +∂t⃗u(x, t) + A∂x⃗u(x, t) = 0 , +(8) +with random initial conditions as specified in (6). The 3×3 matrix A is constant, i.e. independent +of (x, t). To explain the structure of A requires some further efforts. We refer to [38] for more +details and proofs of the key identities. +From the equations of motion one infers that to each density Qn(j, t) there is a current density +Jn(j, t) such that +d +dtQn(j, t) + Jn(j + 1, t) − Jn(j, t) = 0. +Explicitly, the current densities are +⃗J(j) = −(pj, V ′ +j−1, pjV ′ +j−1), +(9) +where we adopted the convention that omission of time argument t means time 0 fields. One then +defines the static current-conserved field correlator +Bm,n(j) = ⟨Jm(j)Qn(0)⟩c +P,β, +(10) +and the corresponding susceptibility +Bm,n = +� +j∈Z +⟨Jm(j)Qn(0)⟩c +P,β. +Despite its asymmetric looking definition, +Bm,n = Bn,m. +(11) +As a general property, Euler equations are built on thermally averaged currents. Linearizing +them with respect to the average fields yields +A = BC −1. +4 + +Here B appears when differentiating the average currents with respect to the chemical potentials +and C −1 when switching from intensive to extensive variables. By construction C = C T and C > 0, +in addition B = BT according to (11). Hence +A = C 1/2C −1/2BC −1/2C −1/2, +which ensures that A has real eigenvalues and a complete set of left-right eigenvectors. Anharmonic +lattices are symmetric under time reversal, which implies the eigenvalues ⃗c = (−c, 0, c), with c > 0 +the isentropic speed of sound. We denote the right, resp. left eigenvectors of A by |ψα⟩ and ⟨ ˜ψα|, +α = 0, 1, 2. With this input the solution to (8) with initial conditions (6) reads +SLL +m,n(x, t) = E +� +um(x, t)un(0, 0) +� += (δ(x − At)C)m,n = +2 +� +α=0 +δ(x − cαt)(|ψα⟩⟨ ˜ψα|C)m,n +with m, n = 0, 1, 2. There are three δ-peaks, the heat peak standing still and two sound peaks +propagating in opposite directions with speed c. Specifying m, n, each peak has a signed weight +which depends on C and the left-right eigenvectors of A. +The Landau-Lifshitz theory asserts that the microscopic correlator +Sm,n(j, t) ≃ SLL +m,n(x, t) +for j = ⌊xt⌋, ⌊·⌋ denoting integer part, with t sufficiently large. +The reader might be disap- +pointed by the conclusion. But with such basic information the fine-structure of the peaks can be +investigated, in particular their specific sub-ballistic broadening and corresponding scaling func- +tions [31,38,39]. +When turning to the Toda lattice, the conservation laws are now labeled by n = 0, 1, ... and +thus A, B, C become infinite dimensional matrices. The corresponding Landau-Lifshitz theory has +been worked out in [40]. As to be discussed in the following section, with appropriate adjustments +Eq. (12) is still valid. +3 +Toda lattice, linearized generalized hydrodynamics +The conservation laws of the Toda lattice are obtained from a Lax matrix [11,26]. For this purpose, +we first introduce the Flaschka variables +aj = e−rj/2. +Then the equations of motion become +d +dtaj = 1 +2aj(pj − pj+1), +d +dtpj = a2 +j−1 − a2 +j. +(13) +The Lax matrix, L, is defined by +Lj,j = pj, +Lj,j+1 = Lj+1,j = aj, +j ∈ Z, and Li,j = 0 otherwise. Clearly L = LT. The conserved fields are labelled by nonnegative +integers and have densities given by +Q0(j) = rj, +Qn(j) = (Ln)j,j , +(14) +5 + +with n ≥ 1. Note that Qn(j) is local in the sense that it depends only on the variables with indices +in the interval [j − n, j + n]. An explicit expression for these quantities is given in [15]. For the +current densities one obtains +J0(j) = −pj, +Jn(j) = (LnL +↓)j,j, +n = 1, 2, ... , +(15) +where L↓ is the lower triangular part of L. Then under the Toda dynamics +d +dtQn(j, t) + Jn(j + 1, t) − Jn(j, t) = 0, +which is the n-th conservation law in local form. +The first items in the list are stretch and momentum for which our current definitions agree +with those in (4), (9). However, for n = 2 one obtains (L2)0,0 = p2 +0 + a2 +−1 + a2 +0 and (L2L↓)0,0 = +a2 +−1(p−1 + p0), which differs from (4), (9) on two accounts. First there is the trivial factor of 2. +In our numerical plots we use the physical energy density ej. The second point is more subtle. +Densities are not uniquely defined, since one can add a difference of some local function and its +shift by one. When summing a particular choice for the density over some spatial interval, the +result differs from another choice of the density by a boundary term only. Thus the bulk term will +have a correction of order 1/(length of interval), which does not affect the hydrodynamic equations. +For the currents the difference can be written as a total time derivative which is again a boundary +term when integrating over some time interval. In this section we adopt the conventions (14) and +(15), since the analysis heavily relies on the Lax matrix. Beyond n = 2, while the fields no longer +have a name, they still have to be taken into account in a hydrodynamic theory. +The infinite volume static field-field correlator is defined as in (5) and the current-field correlator +as in (10). In particularly, B = BT. Of course, C, B are now matrices in the Hilbert space of +sequences indexed by N0, i.e. the space ℓ2(N0). To distinguish 3 × 3 matrices from their infinite +dimensional counterparts, for the latter we use standard italic symbols. The spacetime correlator +of the Toda lattice is defined by +Sm,n(j, t) = ⟨Qm(j, t)Qn(0, 0)⟩c +P,β. +(16) +and we plan to construct its Landau-Litshitz approximation. In essence this amounts to an analysis +of +� +eAtC +� +m,n, +A = BC−1. +(17) +While we are mainly interested in the physical fields corresponding to the indices m, n = 0, 1, 2, +for the operator in (17) an understanding of the infinite dimensional matrices is required. +Starting from the basics, the free energy of the Toda lattice is given by +Feq(P, β) = log +� +β/2π + P log β − log Γ(P). +In particular, the average stretch, ν, is determined through +ν(P, β) = ∂PFeq(P, β) = ⟨Q0(0)⟩P,β = log β − ψ(P), +(18) +with ψ the digamma function. Expectations of higher order fields can be written as moments of a +probability measure denoted by νρp, +κn = ⟨Qn(0)⟩P,β = +� +R +dwνρp(w)wn, +(19) +6 + +n ≥ 1. ρp is called particle density. To determine this density one first has to solve the thermody- +namic Bethe equations (TBA). For this purpose we introduce the integral operator +Tf(w) = 2 +� +R +dw′ log |w − w′|f(w′), +w ∈ R, considered as an operator on L2(R, dw) and define the number density +ρn(w) = e−ε(w), +(20) +with quasi-energies ε. The quasi-energies satisfy the TBA equation +ε(w) = 1 +2βw2 − µ − (Te−ε)(w), +(21) +where the chemical potential µ has to be adjusted such that +� +R +dwρn(w) = P. +(22) +Thereby the number density depends on the parameters P and β. +The TBA equation is closely connected to the β-ensemble of random matrix theory. We rewrite +(21) as +− log ρn(w) = 1 +2αw2 − µ − αP(Tρn)(w). +As α → ∞, the entropy term on the lefthand side can be neglected and one recognizes the defining +equation for the Wigner semi-cirle law on the interval [−2 +√ +P, 2 +√ +P]. The Lax DOS is the P- +derivative of ρn, which diverges as (w ± 2 +√ +P)−1/2 at the two borders. As α is lowered the borders +become smeared to eventually cross over to a Gaussian. +In practice, the TBA equation has to be solved numerically. But for thermal equilibrium an +exact solution is available [1, 12, 35]. Denoting the solution of (21) for β = 1 and the constraint +(22) by ρ∗ +n one has +ρ∗ +n(w) = +e−w2/2 +√ +2π| ˆfP(w)|2, +ˆfP(w) = +� ∞ +0 +dtfP(t)eiwt, +fP(t) = +√ +2π−1Γ(P)−1/2tP−1e− 1 +2 t2. +(23) +In our numerical simulations it is of advantage to use the exact solution. +The TBA equation is a standard tool from GHD as one way to write the Euler-Lagrange +equations for the variational principle associated with the generalized free energy. For the Toda +lattice such a variational formula was obtained in [9,42]. Proofs using methods from the theory of +large deviations and transfer operator method have also become available [16,27,29,30]. +Next we introduce the dressing transformation of some function f by +f dr = +� +1 − Tρn +�−1f +with ρn regarded as a multiplication operator. Then number and particle density are related as +ρn(w) = +ρp(w) +1 + Tρp(w) +(24) +with inverse +ρp = (1 − ρnT)−1ρn = ρnςdr +0 , +(25) +7 + +using the convention ςn(w) = wn. +For the average currents similar identities are available. +The central novel quantity is the +effective velocity +veff = ςdr +1 +ςdr +0 +, +(26) +see [3,6,41,45]. Then +⟨J0(0)⟩P,β = −κ1, +and, for n ≥ 1, +⟨Jn(0)⟩P,β = +� +R +dwρp(w)(veff(w) − κ1)wn. +In thermal equilibrium we have κ1 = 0. +Since in the following there will be many integrals over R, let us first introduce the abbreviation +⟨f⟩ = +� +R +dwf(w). +With this notation the C matrix turns out to be of the form +C0,0 = ν3⟨ρpςdr +0 ςdr +0 ⟩, +C0,n = Cn,0 = −ν2⟨ρpςdr +0 (ςn − κnς0)dr⟩, +Cm,n = ν⟨ρp(ςm − κmς0)dr(ςn − κnς0)dr⟩, +m, n ≥ 1. Note that the matrix C has the block structure +C = +�C0,0 +C0,n +Cm,0 +Cm,n +� +, +in the sense that Cm,n for m, n ≥ 1 follows a simple pattern. This structure will reappear for B +and eAtC. +The field-current correlator B can be computed in a similar fashion with the result +B0,0 = ν2⟨ρp(veff − κ1)ςdr +0 ςdr +0 ⟩, +B0,n = Bn,0 = −ν⟨ρp(veff − κ1)ςdr +0 (ςn − κnς0)dr⟩, +Bm,n = ⟨ρp(veff − κ1)(ςm − κmς0)dr(ςn − κnς0)dr⟩. +As in (12), we want to determine the propagator of the Landau-Lifshitz theory, denoted by +SLL +m,n(x, t). In principle, all pieces have been assembled. However to figure out the exponential of A +requires its diagonalization. Details can be found in [40] and we only mention that one constructs +a linear similarity transformation, R, such that R−1AR is multiplication by +ν−1(veff(w) − κ1) +(30) +in L2(R, dw). Here veff is the effective velocity defined in (26). Using the block convention as in +(28), the spacetime correlator in the Landau-Lifshitz approximation is given by +SLL(x, t) = +� +R +dwδ +� +x − tν−1(veff(w) − κ1) +� +νρp(w) +× +� +ν2ςdr +0 (w)2 +νςdr +0 (w)(ςn − κnς0)dr(w) +νςdr +0 (w)(ςm − κmς0)dr(w) +(ςm − κmς0)dr(w)(ςn − κnς0)dr(w) +� +. +8 + +Note that SLL(x, 0) = δ(x)C. As a property of the Euler equations, the expression (31) possesses +exact ballistic scaling, +SLL +m,n(x, t) = 1 +t SLL +m,n(x/t, 1). +(32) +The correlator Sm,n(j, t) is computed in our MD simulations which will then be compared with +SLL +m,n(x, t). +4 +Numerical simulations +For a molecular dynamics simulation one has to first specify a finite ring [1, . . . , N] with suitable +boundary conditions. For the dynamics of positions qj and momenta pj one imposes +qN+1 = q1 + νN. +(33) +The parameter ν fixes the free volume per particle and can have either sign. In our simulation, +we actually allow for a fluctuating free volume by choosing random initial conditions such that +{r1, p1, . . . , rN, pN} are i.i.d. random variables with a single site distribution as specified in (3). +Then the deterministic time evolution is governed by (13) with boundary conditions +r0 = rN, +pN+1 = p1. +In fact, the boundary condition in (33) amounts to the micro-canonical constraint +N +� +j=1 +rj = νN. +If one sets ν = ⟨Q0(0)⟩P,β, then for large N, by the equivalence of ensembles, the two schemes +for sampling the correlator Sm,n(j, t) should differ by the size of statistical fluctuations. For a +few representative examples we checked that indeed the equivalence of ensembles holds for the +particular observables under study. +Returning to the choice of system size there is an important physical constraint. In all sim- +ulations one observes a sharp right and left front, which travel with constant speed and beyond +which spatial correlations are exponentially small. On a ring necessarily the two fronts will collide +after some time. Such an encounter has a noticeable effect on the molecular dynamics which is not +captured by the linearized GHD analysis. Therefore the simulation time is limited by the time of +first collision. Indeed, we note in Figures 1-3 that both linearized GHD and MD clearly display +maximal speeds of at most ∆j/∆t = 2 for the entire range of (P, β, m, n) displayed in these figures. +Taking into account that the initial correlations are proportional to δ0j, we conclude that for a +ring of size N = 3000 there will be no collision of the two fronts up to time t = 750 which is larger +than t = 600 used in our simulations. +Before displaying and discussing our results, we provide more details on numerically solving +the TBA equations and on the actual scheme used for MD. +4.1 +Details of the numerical implementation +4.1.1 +Solving linearized GHD +To numerically solve the linearized GHD equations, we use a numerical method similar to the one +from [33]. First, Eq. (23) can be expressed in terms of the parabolic cylinder function Dν(z), which +is readily available in Mathematica. This provides the solution to the TBA equations (21), (22). +9 + +Then, we use a simple finite element discretization of the w-dependent functions by hat func- +tions, resulting in piecewise linear functions on a uniform grid. After precomputing the integral +operator T in (20) for such hat functions, the dressing transformation (24) becomes a linear sys- +tem of equations, which can be solved numerically. This procedure yields ςdr +n , and subsequently +ρp via (25) and veff via (26). The moments can be computed from κn = +� +R dwνρn(w)ςdr +n (w), or +(equivalently) Eq. (19). +To evaluate the correlator in (31), we note that the delta-function in the integrand results in a +parametrized curve, with the first coordinate (corresponding to x/t) equal to ˜veff from (30), and +the second coordinate equal to the remaining terms in the integrand divided by the Jacobi factor +| d +dw ˜veff(w)| resulting from the delta-function. +4.1.2 +Molecular dynamics simulations +We approximate the expectation value that is contained in the MD-definition of the correlations +Sm,n in equation (16) by the following numerical scheme, whose implementation program is written +in Python, and can be found at [28]. First, we generate the random initial conditions distributed +according to the Gibbs measure, as given by (3) for the i.i.d. random variables (rj, pj)1≤j≤N. +Specifically, the variables pj are distributed according to a standard normal random variable, that +we generate with Numpy v1.23’s native function random.default rng().normal [18], times 1/√β. +It takes a brief calculation to see that rj can be chosen to be − ln(X/(2β)) where X is chi-square +distributed with shape parameter 2P. We obtain the random variable X using Numpy v1.23’s +native function random.default rng().chisquare. Having chosen the initial conditions in such +a manner, we solve equation (2). +For the evolution, we adapt the classical St¨ormer–Verlet algorithm [17] of order 2 to work with +the variables (p, r). Specifically, we used a time step equal to δ = 0.05, and, given the solution +(r(t), p(t)) at time t, we approximate the solution at time t + δ through the following scheme, +pj +� +t + δ +2 +� += pj(t) − δ +2 +� +e−rj(t) − erj−1(t)� +, +rj(t + δ) = rj(t) + δ +� +pj+1 +� +t + δ +2 +� +− pj +� +t + δ +2 +�� +, +pj(t + δ) = pj +� +t + δ +2 +� +− δ +2 +� +e−rj(t+δ) − erj−1(t+δ)� +, +for all j = 1, . . . , N. In this part of the implementation, we extensively used the library Numba [23] +to speed up the computations. +Our approximation for the expectation Sm,n is then extracted from 3×106 trials with indepen- +dent initial conditions. Here we take the empirical mean of all trials where for each trial we also +take the mean of the N = 3000 sets of data that are generated by choosing each site on the ring +for j = 0. +To evaluate the quality of our numerical simulations, we have repeated the numerical experi- +ments up to five times including variations for the length of the ring and evaluating the solutions at +more intermediate time steps than displayed in the figures below. Furthermore, we have compared +the results with the corresponding outcomes obtained by a MATLAB program that has been devel- +oped independently from the Python program, and that follows a different numerical scheme. It +uses MATLAB’s random number generators randn for initial momenta and rand combined with the +10 + +rejection method to produce initial stretches. The dynamics is then evaluated by the solver ode45, +which exploits the Runge–Kutta method to numerically solve the Hamiltonian system associated +with (1) on the ring. We found that the deviations between different experiments are comparable +to the size of the amplitudes of the high frequency oscillations that are present in figures 4-5. These +oscillations are due to the random fluctuations of the empirical means around their expectation +values Sm,n. Agreement of different experiments up to the order of these oscillations therefore +shows the consistency of the corresponding numerical results. +We also want to mention that all the pictures that appeared in this paper are made using the +library matplotlib [19]. +4.2 +Comparison of linearized GHD with MD at time t = 600 +We compare the GHD predictions with MD simulations for three different temperatures that +correspond to β = 0.5 (Fig. 1), β = 1 (Fig. 2), and β = 2 (Fig. 3). For each β we choose three +different values for the pressure parameter P in such a way that the corresponding mean stretches, +given by (18), are positive (≈ 2.57) for low pressure, negative (≈ −0.42) for high pressure and +approximately zero for medium pressure. We summarize their values in Table 1. +pressure +β = 0.5 +β = 1 +β = 2 +low +P = 0.32, ⟨r⟩ ≈ +2.58 +P = 0.4, ⟨r⟩ ≈ +2.56 +P = 0.52, ⟨r⟩ ≈ +2.56 +medium +P = 0.95, ⟨r⟩ ≈ −0.03 +P = 1.5, ⟨r⟩ ≈ −0.04 +P = 2.55, ⟨r⟩ ≈ −0.03 +high +P = 1.21, ⟨r⟩ ≈ −0.42 +P = 2.0, ⟨r⟩ ≈ −0.42 +P = 3.53, ⟨r⟩ ≈ −0.42 +Table 1: Values for β and P and the corresponding mean stretches used in experiments +In each of the nine cases we have evaluated the Landau-Lifshitz approximations SLL +m,n(·, 1), see +(31), of the correlators for all 0 ≤ n ≤ m ≤ 2 using the numerical scheme described in Section 4.1.1. +Their graphs are displayed in Figures 1-3 as dashed lines. Note that the speeds of the sound peaks +depend significantly on both pressure and temperature. Moreover, the predicted fine-structure of +both the heat and the sound peaks are quite different for low pressure when compared to medium +and high pressure. +The colored lines in Figures 1-3 show our numerical results for the corresponding molecular +dynamics. According to the predicted ballistic scaling (32) we plot tSm,n(j, t) as a function of +j/t for t = 600. Here the values of Sm,n(j, t) are approximated using the numerics explained in +Section 4.1.2. +The agreement between linearized GHD and MD is striking, in particular since there are no +adjustable parameters. In all of the 54 comparisons shown in Figures 1-3 the GHD predictions +for the fine-structure of heat and sound peaks are in excellent agreement with the ones observed +from molecular dynamics at time t = 600. As we show in more detail in the next subsection +the largest deviations occur mostly near the sound peaks and do not exceed 3.5% of the peaks’ +maximal values. +4.3 +Deviation of linearized GHD from MD at times t = 150 and t = 600 +The purpose of this subsection is twofold. On the one hand we have a look at the small differences +between GHD predictions and molecular dynamics simulations that can hardly be detected in +11 + +1.0 +0.5 +0.0 +0.5 +1.0 +0 +2 +4 +6 +8 +S00, S11, S22, += 0.5, P = 0.32 +S00 +S11 +S22 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0 +1 +2 +3 +4 +5 +6 +7 +S00, S11, S22, += 0.5, P = 0.95 +S00 +S11 +S22 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +2 +4 +6 +8 +S00, S11, S22, += 0.5, P = 1.21 +S00 +S11 +S22 +1.0 +0.5 +0.0 +0.5 +1.0 +4 +2 +0 +2 +4 +S21, S20, S10, += 0.5, P = 0.32 +S21 +S20 +S10 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +4 +3 +2 +1 +0 +1 +2 +3 +4 +S21, S20, S10, += 0.5, P = 0.95 +S21 +S20 +S10 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +4 +3 +2 +1 +0 +1 +2 +3 +4 +S21, S20, S10, += 0.5, P = 1.21 +S21 +S20 +S10 +Figure 1: Toda correlation functions: GHD predictions y �→ SLL +m,n(y, 1) vs. numerical simulations +of the molecular dynamics y �→ tSm,n(yt, t) at t = 600 for β = 0.5 with low pressure (top), medium +pressure (middle) and high pressure (bottom). Numerical simulations are colored according to +the legend, the corresponding GHD predictions are displayed by dashed lines. Number of trials: +3 × 106. +12 + +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +0 +2 +4 +6 +8 +S00, S11, S22, += 1.0, P = 0.40 +S00 +S11 +S22 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +S00, S11, S22, += 1.0, P = 1.50 +S00 +S11 +S22 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +1 +2 +3 +4 +S00, S11, S22, += 1.0, P = 2.00 +S00 +S11 +S22 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +3 +2 +1 +0 +1 +2 +3 +S21, S20, S10, += 1.0, P = 0.40 +S21 +S20 +S10 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2 +1 +0 +1 +2 +S21, S20, S10, += 1.0, P = 1.50 +S21 +S20 +S10 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2 +1 +0 +1 +2 +S21, S20, S10, += 1.0, P = 2.00 +S21 +S20 +S10 +Figure 2: Toda correlation functions: GHD predictions y �→ SLL +m,n(y, 1) vs. numerical simulations +of the molecular dynamics y �→ tSm,n(yt, t) at t = 600 for β = 1.0 with low pressure (top), medium +pressure (middle) and high pressure (bottom). Numerical simulations are colored according to +the legend, the corresponding GHD predictions are displayed by dashed lines. Number of trials: +3 × 106. +13 + +1.0 +0.5 +0.0 +0.5 +1.0 +0 +1 +2 +3 +4 +5 +6 +7 +S00, S11, S22, += 2.0, P = 0.52 +S00 +S11 +S22 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +S00, S11, S22, += 2.0, P = 2.55 +S00 +S11 +S22 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +S00, S11, S22, += 2.0, P = 3.53 +S00 +S11 +S22 +1.0 +0.5 +0.0 +0.5 +1.0 +2 +1 +0 +1 +2 +S21, S20, S10, += 2.0, P = 0.52 +S21 +S20 +S10 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +S21, S20, S10, += 2.0, P = 2.55 +S21 +S20 +S10 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +S21, S20, S10, += 2.0, P = 3.53 +S21 +S20 +S10 +Figure 3: Toda correlation functions: GHD predictions y �→ SLL +m,n(y, 1) vs. numerical simulations +of the molecular dynamics y �→ tSm,n(yt, t) at t = 600 for β = 2.0 with low pressure (top), medium +pressure (middle) and high pressure (bottom). Numerical simulations are colored according to +the legend, the corresponding GHD predictions are displayed by dashed lines. Number of trials: +3 × 106. +14 + +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.00 +0.25 +0.50 +0.75 +1.00 +t: 150 +t: 600 +GHD +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.02 +0.00 +0.02 += 2.00, P = 2.55, S1, 1 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +t: 150 +t: 600 +GHD +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.02 +0.00 +0.02 +0.04 += 2.00, P = 2.55, S1, 0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +t: 150 +t: 600 +GHD +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.02 +0.00 +0.02 += 1.00, P = 1.50, S1, 1 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +1 +0 +1 +t: 150 +t: 600 +GHD +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.02 +0.00 +0.02 +0.04 += 1.00, P = 1.50, S1, 0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +2.0 +t: 150 +t: 600 +GHD +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.05 +0.00 +0.05 += 0.50, P = 0.95, S1, 1 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +1 +0 +1 +t: 150 +t: 600 +GHD +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.050 +0.025 +0.000 +0.025 +0.050 += 0.50, P = 0.95, S1, 0 +Figure 4: Toda correlation functions S1,1 (left) and S1,0 (right) for medium pressure and increasing +temperatures (top to bottom). For each value of β and P the top panels show GHD prediction +vs. numerical simulations as in Figures 1-3 but with the the molecular dynamics evaluated at two +times t = 150 and t = 600. The bottom panels display the differences between the GHD prediction +and numerical simulations at time t = 150 (red) and at time t = 600 (green). Number of trials: +3 × 106. +15 + +1.00 +0.75 +0.50 +0.25 0.00 +0.25 +0.50 +0.75 +1.00 +0 +2 +4 +6 +8 +t: 150 +t: 600 +GHD +1.00 +0.75 +0.50 +0.25 0.00 +0.25 +0.50 +0.75 +1.00 +0.1 +0.0 +0.1 +0.2 += 1.00, P = 0.40, S0, 0 +1.00 +0.75 +0.50 +0.25 0.00 +0.25 +0.50 +0.75 +1.00 +3 +2 +1 +0 +1 +t: 150 +t: 600 +GHD +1.00 +0.75 +0.50 +0.25 0.00 +0.25 +0.50 +0.75 +1.00 +0.1 +0.0 +0.1 += 1.00, P = 0.40, S2, 0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.00 +0.25 +0.50 +0.75 +1.00 +t: 150 +t: 600 +GHD +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.02 +0.00 +0.02 += 1.00, P = 1.50, S0, 0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +1.5 +1.0 +0.5 +0.0 +t: 150 +t: 600 +GHD +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.050 +0.025 +0.000 +0.025 +0.050 += 1.00, P = 1.50, S2, 0 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +t: 150 +t: 600 +GHD +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0.02 +0.01 +0.00 +0.01 +0.02 += 1.00, P = 2.00, S0, 0 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +1.5 +1.0 +0.5 +0.0 +t: 150 +t: 600 +GHD +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0.050 +0.025 +0.000 +0.025 +0.050 += 1.00, P = 2.00, S2, 0 +Figure 5: Toda correlation functions S0,0 (left) and S2,0 (right) for β = 1 and increasing pressure +(top to bottom). For each value of β and P the top panels show GHD prediction vs. numerical +simulations as in Figure 2 but with the the molecular dynamics evaluated at two times t = 150 and +t = 600. The bottom panels display the differences between the GHD prediction and numerical +simulations at time t = 150 (red) and at time t = 600 (green).Number of trials: 3 × 106. +16 + +Figures 1-3. On the other hand we indicate how these differences evolve in time by including time +t = 150 for the molecular dynamics. Recall that the GHD predictions are time-invariant in the +scaling y �→ tSm,n(yt, t) we have chosen, see (32). +From the 54 comparisons that are displayed in Figures 1-3 we select 12 cases that are repre- +sentative and show all the phenomena that we have observed. In Figure 4 we consider correlations +S1,1 and S1,0 at medium pressure (cf. Table 1) for all three values of β. The small scale fluctuations +displayed in the bottom panels are due to the approximation of expectation values by empirical +averages. Their amplitudes become smaller if one increases the number of trials. Note that the +difference in amplitudes of these fluctions between t = 150 and t = 600 is mostly due to the scaling +y �→ tSm,n(yt, t) that we use. This implies that the values of the correlations are multiplied by a +factor that is 4 times larger at the later time. The same holds for the plots in Figure 5 where the +correlations S0,0 and S2,0 are shown for fixed β = 1 and our three different choices for pressure. +We now summarize our main findings: +1. The deviations occur mostly near the sound peaks and amount to 1.5%-3.5% of the peaks’ +maximal values at time t = 600. +2. There appear to be small but systematic deviations concerning the shape of the sound peak +in all cases. One would need to conduct experiments with a higher resolution, i.e. more sites +and consequently larger times and more trials, to determine whether there is indeed such +a systematic deviation. With the resolution present in our experiments the question of a +systematic deviation with respect to the shape of the peak cannot be decided. +3. In some of the experiments the maximal deviations would be significantly smaller if a constant +only depending on the values of β, P, m, n is added to all values of Sm,n(j, t), see e.g. +correlations S0,0 and S2,0 for β = 1, P = 0.4 in Figure 5. This seems to be related to the +approximation errors for the means ⟨r⟩, ⟨p⟩, and ⟨e⟩, that appear to be less pronounced in +the case of momentum p. We have observed that these deviations decrease as the number +of trials is increased and we do not expect a systematic deviation between GHD and MD in +this respect. +4. For (β; P) ∈ {(0.5; 0.95), (0.5; 1.21)} we observe that the size of the deviations is essentially +the same for times t = 150 and t = 600 whereas for (β; P) ∈ {(0.5; 0.32), (1; 0.4), (2; 0.52), +(2; 2.55), (2; 3.53)} these deviations are significantly larger at the smaller time. The remain- +ing two cases (β; P) ∈ {(1; 1.5), (1; 2)} are somewhat in between, also depending on the +correlation function that is considered, see Figure 5. This is an indication that the speed +of convergence of tSm,n(yt, t) to the GHD prediction SLL +m,n(y, 1) as t → ∞ depends on the +values of β and P. As a rule we have observed that both increasing temperature or increasing +pressure leads to a faster speed of convergence. +5 +Conclusions and outlook +As can be seen from Table 1, we picked the intermediate pressure such that ν ≃ 0. In the particle +picture ν = 0 corresponds to the boundary condition q1 = qN. In thermal equilibrium the positions +then perform an unbiased random walk with typical excursions of order +√ +N. Thus the free volume +is of order 1/ +√ +N. The particles are extremely dense and the picture of successive pair collisions +breaks down completely. So one might wonder whether GHD is still valid under such extreme +conditions. ν = 0 poses no particular difficulties for MD simulations. In GHD the factor 1/ν +17 + +appears in the expression for veff, see Eq. (31). This makes the numerical scheme slow and only +values close to ν = 0 are accessible. However the correlator S changes smoothly through ν = 0. +GHD also covers this seemingly singular point. +Simultaneously A. Kundu [21] posted a somewhat puzzling note. He considers the parameter +values β = 1, P = 1. When cutting the matrices Cm,n and Am,n at low orders, the resulting +Sm,n consists of a few δ-peaks which move at constant velocity. After ballistic scaling, with high +precission they turn out to lie on the curve obtained from GHD. A theoretical explanations seems +to be missing. +In [22] the molecular dynamics of Toda lattice correlations are simulated for the potential +Vkd(x) = g +γ e−γx +with arbitrary γ, g > 0. To distinguish their parameters from ours, the variables in [22] are here +denoted by ¯t, ¯r, ¯P, ¯β. +¯P is the physical pressure and, comparing the Gibbs weights, one obtains +the relations +β = g +γ +¯β, +P = 1 +γ +¯P ¯β. +From the equations of motions one deduces +¯t = +1 +√γgt, +r(t) = γ¯r(¯t), +p(t) = g +γ ¯p(¯t). +Thus, translating to our units, the MD simulations reported in [22] are (i) P = 0.01, β = 0.01, +N = 1024, t = 400, (ii) P = 1, β = 1, N = 1024, t = 200, 300, and (iii) P = 400, β = 400, +N = 256, t = 80. In fact, in all three cases the time scales are identical, t = ¯t. Since GHD was not +available yet, no comparison could have been attempted. +Case (i) is a very dilute chain. In this limit νρp is a unit Gaussian. The dressed functions +become polynomials as ςdr +0 (w) = a0, ςdr +1 (w) = a1w, and ςdr +2 (w) = a2w2 + a3 with coefficients +a0, ..., a3 depending on (P, β). Note that for a noninteracting fluid a3 would vanish. As a result +S0,0 is Gaussian, S1,1 has two peaks, and S2,2 has either two or three peaks. +This is in good +agreement with [22] and explains our motivation not to venture into the low density regime. Case +(ii) interpolates between our β = 1, P = 0.40 and β = 1, P = 1.5. Note that now S0,0 has a local +minimum at w = 0, which is very different from the structure in the dilute regime. On the other +hand, S2,2 has a local maximum at w = 0, as is the case for low density/high temperature. +The most interesting parameter value is (iii), which deserves more detailed studies. The issue +is the behavior of the Toda chain at very low temperatures. Simply letting β → ∞ will freeze +any motion. But the simultaneous limit β → ∞ with P = ¯Pβ at fixed physical pressure ¯P is +meaningful, at least statistically. In this limit ν > 0 always. Also the density of states converges +to the arcsine distribution, +lim +β→∞ νρp(w) = +1 +π +√ +4 ¯P − w2, +|w| ≤ 2 +� +¯P. +To understand the dynamical behavior, the effective potential is expanded as +e−r + ¯Pr ≃ 1 +2 ¯P(r − r0)2 + c0 +at its minimum r0. Since β is large, the initial fluctuations are of order 1/√β. Therefore the dy- +namics can be approximated by a harmonic chain with ω2 = ¯P. The equilibrium time correlations +18 + +of the harmonic chain have intricate oscillatory behavior [14], which in the large β limit should +match with the Toda lattice, as partially evidenced through case (iii). Clearly, GHD cannot repro- +duce such fine details. Still, when averaged on suitable scales, the gross behavior of the harmonic +chain oscillations might be visible. +Acknowledgements +This material is based upon work supported by the National Science Foundation under Grant +No. 1440140, while five of the authors were in residence at the Mathematical Sciences Research +Institute in Berkeley, California, during the fall semester of 2021. +The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cam- +bridge, for support and hospitality during the programme “Dispersive hydrodynamics: mathemat- +ics, simulation and experiments, with applications in nonlinear waves” where some work on this +paper was undertaken. This work was supported by EPSRC grant no EP/R014604/1. TG ac- +knowledges the support of the European Union’s H2020 Marie Sk�lodowska–Curie grant No. 778010 +IPaDEGAN, of INdAM/GNFM and of the research project Mathematical Methods in NonLinear +Physics (MMNLP), Gruppo 4-Fisica Teorica of INFN. GM is financed by the KAM grant number +2018.0344. KTRM was supported by a Visiting Wolfson research fellowship from the Royal Society. +References +[1] R. Allez, J. P. Bouchaud, and A. Guionnet, Invariant β ensembles and the Gauss- +Wigner crossover, Phys. Rev. Lett., 109 (2012), pp. 1–5. +[2] A. Bastianello, B. Doyon, G. Watts, and T. Yoshimura, Generalized hydrodynamics +of classical integrable field theory: the sinh-Gordon model, SciPost Physics, 4 (2018). +[3] B. Bertini, M. Collura, J. De Nardis, and M. Fagotti, Transport in out-of- +equilibrium XXZ chains: exact profiles of charges and currents, Phys. Rev. Lett., 117 (2016), +p. 207201. +[4] V. B. Bulchandani, X. Cao, and J. E. Moore, Kinetic theory of quantum and classical +Toda lattices, Journal of Physics A: Mathematical and Theoretical, 52 (2019), p. 33LT01. +[5] V. B. Bulchandani, R. Vasseur, C. Karrasch, and J. E. Moore, Solvable hydrody- +namics of quantum integrable systems, Physical Review Letters, 119 (2017). +[6] O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura, Emergent hydrodynamics in +integrable quantum systems out of equilibrium, Phys. Rev. X, 6 (2016), p. 041065. +[7] A. Das, M. Kulkarni, H. Spohn, and A. Dhar, Kardar-Parisi-Zhang scaling for an +integrable lattice Landau-Lifshitz spin chain, Physical Review E, 100 (2019). +[8] B. Doyon, Exact large-scale correlations in integrable systems out of equilibrium, SciPost +Physics, 5 (2018). +[9] B. Doyon, Generalized hydrodynamics of the classical Toda system, Journal of Mathematical +Physics, 60 (2019), p. 073302. +19 + +[10] M. Dupont and J. E. Moore, Universal spin dynamics in infinite-temperature one- +dimensional quantum magnets, Physical Review B, 101 (2020). +[11] H. Flaschka, The Toda lattice. I. Existence of integrals, Phys. Rev. B (3), 9 (1974), +pp. 1924–1925. +[12] P. J. Forrester and G. Mazzuca, The classical β-ensembles with β proportional to 1/N: +from loop equations to Dyson’s disordered chain, J. Math. Phys., 62 (2021), pp. Paper No. +073505, 22. +[13] D. Forster, Hydrodynamic fluctuations, broken symmetry, and correlation functions, 1975. +[14] T. Grava, T. Kriecherbauer, G. Mazzuca, and K. D. T.-R. McLaughlin, Corre- +lation functions for a chain of short range oscillators, J. Stat. Phys., 183 (2021), pp. Paper +No. 1, 31. +[15] T. Grava, A. Maspero, G. Mazzuca, and A. Ponno, Adiabatic invariants for the FPUT +and Toda chain in the thermodynamic limit, Comm. Math. Phys., 380 (2020), pp. 811–851. +[16] A. Guionnet and R. Memin, Large deviations for Gibbs ensembles of the classical Toda +chain, Electronic Journal of Probability, 27 (2022), pp. 1 – 29. +[17] E. Hairer, G. Wanner, and C. Lubich, Symplectic Integration of Hamiltonian Systems, +Springer Berlin Heidelberg, Berlin, Heidelberg, 2006, pp. 179–236. +[18] C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, +D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, +S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del R´ıo, M. Wiebe, +P. Peterson, P. G´erard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, +H. Abbasi, C. Gohlke, and T. E. Oliphant, Array programming with NumPy, Nature, +585 (2020), pp. 357–362. +[19] J. D. Hunter, Matplotlib: A 2d graphics environment, Computing in Science & Engineering, +9 (2007), pp. 90–95. +[20] E. Ilievski, J. D. Nardis, M. Medenjak, and T. Prosen, Superdiffusion in one- +dimensional quantum lattice models, Physical Review Letters, 121 (2018). +[21] A. Kundu, Integrable hydrodynamics of Toda chain: case of small systems, 2022. +[22] A. Kundu and A. Dhar, Equilibrium dynamical correlations in the Toda chain and other +integrable models, Phys. Rev. E, 94 (2016), pp. 062130, 13. +[23] S. K. Lam, A. Pitrou, and S. Seibert, Numba: A LLVM-based Python JIT compiler, in +Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC, LLVM +’15, New York, NY, USA, 2015, Association for Computing Machinery. +[24] L. Landau and E. Lifshitz, Fluid Mechanics: Volume 6, no. v. 6, Elsevier Science, 2013. +[25] M. Ljubotina, M. ˇZ nidariˇc, and T. Prosen, Kardar-Parisi-Zhang physics in the quan- +tum Heisenberg magnet, Physical Review Letters, 122 (2019). +20 + +[26] S. V. Manakov, Complete integrability and stochastization of discrete dynamical systems, +ˇZ. `Eksper. Teoret. Fiz., 67 (1974), pp. 543–555. +[27] G. Mazzuca, On the mean density of states of some matrices related to the beta ensembles +and an application to the Toda lattice, J. Math. Phys., 63 (2022), pp. Paper No. 043501, 13. +[28] G. Mazzuca, Toda correlation functions, 2022. +available at https://github.com/ +gmazzuca/TodaCorrelation. +[29] G. Mazzuca and T. Grava, Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, +Circular β-ensemble and double confluent Heun equation. +[30] G. Mazzuca and R. Memin, Large deviations for Ablowitz-Ladik lattice, and the Schur +flow, 2022. +[31] C. B. Mendl and H. Spohn, Equilibrium time-correlation functions for one-dimensional +hard-point systems, Phys. Rev. E, 90 (2014), p. 012147. +[32] C. B. Mendl and H. Spohn, Low temperature dynamics of the one-dimensional discrete +nonlinear Schr¨odinger equation, Journal of Statistical Mechanics: Theory and Experiment, +2015 (2015), p. P08028. +[33] C. B. Mendl and H. Spohn, High-low pressure domain wall for the classical Toda lattice, +SciPost Phys. Core, 5 (2022), p. 002. +[34] F. S. Møller, G. Perfetto, B. Doyon, and J. Schmiedmayer, Euler-scale dynamical +correlations in integrable systems with fluid motion, SciPost Physics Core, 3 (2020). +[35] M. Opper, Analytical solution of the classical Bethe ansatz equation for the Toda chain, +Phys. Lett. A, 112 (1985), pp. 201–203. +[36] T. Schneider, Classical statistical mechanics of lattice dynamic model systems: transfer +integral and molecular-dynamics studies, (1983), pp. 212–241. +[37] T. Schneider and E. Stoll, Excitation spectrum of the Toda lattice: a molecular-dynamics +study, Phys. Rev. Lett., 45 (1980), pp. 997 – 1002. +[38] H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys., 154 +(2014), pp. 1191–1227. +[39] +, The Kardar-Parisi-Zhang equation: a statistical physics perspective, in Stochastic pro- +cesses and random matrices: Lecture notes of the Les Houches Summer School July 2015, +G. Schehr, A. Altland, Y. V. Fyodorov, N. O’Connell, and L. F. Cugliandolo, eds., vol. 104, +Oxford University Press, 2017, pp. 177–227. +[40] +, Ballistic space-time correlators of the classical Toda lattice, J. Phys. A, 53 (2020), +pp. 265004, 17. +[41] +, Collision rate ansatz for the classical Toda lattice, Phys. Rev. E, 101 (2020), +pp. 060103(R), 4. +21 + +[42] +, Generalized Gibbs ensembles of the classical Toda chain, J. Stat. Phys., 180 (2020), +pp. 4–22. +[43] M. Toda, Vibration of a chain with nonlinear interaction, J. Phys. Soc. Jpn., 22 (1967), +pp. 431 – 436. +[44] +, Theory of Nonlinear Lattices, Springer Berlin, Heidelberg, 1989. +[45] T. Yoshimura and H. Spohn, Collision rate ansatz for quantum integrable systems, SciPost +Phys., 9 (2020), pp. Paper No. 040, 14. +22 + diff --git a/GtE0T4oBgHgl3EQfhgEK/content/tmp_files/load_file.txt b/GtE0T4oBgHgl3EQfhgEK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8de942814e1afa73016b0c964b0b8a8d8eba8d4d --- /dev/null +++ b/GtE0T4oBgHgl3EQfhgEK/content/tmp_files/load_file.txt @@ -0,0 +1,1235 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf,len=1234 +page_content='Equilibrium Spacetime Correlations of the Toda Lattice on the Hydrodynamic Scale Guido Mazzuca∗, Tamara Grava†, Thomas Kriecherbauer‡, Kenneth T-R McLaughlin §, Christian B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Mendl¶, Herbert Spohn‖ January 9, 2023 Abstract We report on molecular dynamics simulations of spacetime correlations of the Toda lattice in thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The correlations of stretch, momentum, and energy are computed numerically over a wide range of pressure and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Our numerical results are com- pared with the predictions from linearized generalized hydrodynamics on the Euler scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The system size is N = 3000, 4000 and time t = 600, at which ballistic scaling is well confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' With no adjustable parameters, the numerically obtained scaling functions agree with the theory within a precision of less than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 1 Introduction A central goal of Statistical Mechanics is to explore the structure of equilibrium correlations for observables of physical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' These could be static correlations, but more ambitiously also correlations in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' An interesting, but very fine-tuned class of hamiltonians are integrable many-body systems, either classical or quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This choice restricts us to systems in one dimen- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Then, generically, static correlations have exponential decay whether the model is integrable or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' However, the dynamics of correlations is entirely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In nonintegrable chains correla- tions propagate as a few narrow peaks at constant speed which then show characteristic sub-ballistic broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' On the other hand for integrable models correlations still spread ballistically but now ∗Department of Mathematics, The Royal Institute of Technology, Stockholm, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Email: mazzuca@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='se †International School for Advanced Studies (SISSA), Trieste, Italy, School of Mathematics, University of Bristol, UK and INFN sezione di Trieste, Email: grava@sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='it ‡Department of Mathematics, Universit¨at Bayreuth, Germany Email: thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='kriecherbauer@uni-bayreuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='de §Tulane University, New Orleans, United States Email: kmclaughlin@tulane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='edu ¶Technische Universit¨at M¨unchen Department of Informatics, Boltzmannstraße 3, 85748, Garching, Germany Email: christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='mendl@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='de ‖Technische Universit¨at M¨unchen Department of Mathematics and Physics, Boltzmannstraße 3 and James- Franck-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 1, 85748 Garching, Germany Email: spohn@ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='de 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='02431v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='stat-mech] 6 Jan 2023 with a broad spectrum of velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Such behaviour was confirmed through a molecular dynamics (MD) simulation of the Ablowitz-Ladik model [32], an integrable discretization of the nonlinear Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' A further confirmation came from the simulation of the Toda chain [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' On the theoretical side, the 2016 construction of generalized hydrodynamics (GHD) was an important breakthrough [3, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This theory provides a powerful tool through which, at least in principle, the precise form of the spectrum of correlations can be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' With such a development MD simulations can also be viewed as probing the validity of GHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' From the side of condensed matter physics, integrable quantum models have received consider- able attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Because of size limitations, the simulation of macroscopic profiles are preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' But time correlations have also been studied through DMRG simulations [4,5,8,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In recent years, attention has been given to the spacetime spin-spin correlation of the XXZ model at half-filling and at the isotropic point [10,20,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The same quantity has also been investigated for a discrete classical chain with 3-spins of unit length and interactions such that the model is integrable [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' A comparable situation occurs for the classical sinh-Gordon equation, which is integrable as a nonlin- ear continuum wave equation and possesses an integrable discretization, see [2] for MD simulations for equilibrium time correlations of the discrete model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In our contribution we study the correlations of the Toda chain in thermal equilibrium through MD simulations and compare with predictions from GHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' We will comment on the connection to [22] in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' To make our article reasonably self-contained we first discuss the Landau- Lifshitz theory for nonintegrable chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This theory provides the connection between spacetime correlations and linearized hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For the Toda chain, the theory has to be extended so as to accommodate an infinite number of conserved fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' We report on MD simulations of the Toda chain and compare with linearized GHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 2 Landau-Lifshitz theory The dynamics of the Toda chain is governed by the Hamiltonian H = � j∈Z � 1 2p2 j + exp(−(qj+1 − qj)) � , (1) where (qj, pj) ∈ R2 are position and momentum of the j-th particle [43,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Introducing the j-th stretch (free volume) through rj = qj+1 − qj, the equations of motion read d dtrj = pj+1 − pj , d dtpj = −e−rj + e−rj−1, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (2) By tradition, one introduces coefficients for the range and strength of the interaction potential through (g/γ) exp(−γ(qj+1 − qj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' However, by a suitable change of spacetime scales, the form (2) can be regained, see the discussion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The Toda hamiltonian has no free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Since the equilibrium measure for (1) is of product form, static correlations are easily accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Time correlations are more challenging, see [36,37] for early attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' A novel approach has been developed, now known as GHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The guiding idea is to first identify the hydrodynamic equations for the Toda chain, which by necessity are a set of nonlinear coupled hyperbolic conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Given such an input one can construct the corresponding Landau-Lifshitz theory [13,24], as based on linearized GHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Before entering into details, it will be useful to first recall the Landau-Lifshitz theory for a chain with a generic interaction potential, denoted by V (for the Toda lattice V (x) = e−x), see [38] 2 and references listed therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Thus in (1) the interaction term reads V (qj+1 −qj) and the equations of motion become d dtrj = pj+1 − pj , d dtpj = V ′(rj) − V ′(rj−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' To define spacetime correlations we first have to specify the random initial data modelling thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' By Galileian invariance one restricts to the case of zero average momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Then the Gibbs states are characterized by the inverse temperature β > 0 and a parameter P such that the physical pressure equals P/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For simplicity, we will also refer to P as pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The allowed range of P depends on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' If V diverges faster than |x| for |x| → ∞, then P ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For the Toda lattice P > 0 because of the one-sided divergence of the exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In thermal equilibrium {(rj, pj), j ∈ Z} are a collection of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' random variables with single site probability density Z0(P, β)−1 exp � − β � 1 2p2 0 + V (r0) � − Pr0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (3) Here Z0(P, β) is the normalizing partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Note that, with our convention, P and β appear linearly in the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Expectations with respect to such i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' random variables are denoted by ⟨·⟩P,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' We also shorten the notation for the covariance through ⟨X1X2⟩c P,β = ⟨X1X2⟩P,β − ⟨X1⟩P,β⟨X2⟩P,β, where the particular random variables X1, X2 will be obvious from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For general V , the conserved fields are stretch, momentum, and energy with densities ⃗Q(j) = � rj, pj, ej � , ej = 1 2p2 j + Vj, (4) using as shorthand Vj = V (rj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' ⃗Q is a three-vector with components labeled by n = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The static space correlator is defined through Cm,n(j) = ⟨Qm(j)Qn(0)⟩c P,β (5) and the static susceptibility by summing over space, Cm,n = � j∈Z ⟨Qm(j)Qn(0)⟩c P,β, m, n = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Since the underlying measure is product, only the j = 0 term is nonvanishing and C = � � � ⟨r0r0⟩c P,β 0 ⟨r0e0⟩c P,β 0 ⟨p0p0⟩c P,β 0 ⟨r0e0⟩c P,β 0 ⟨e0e0⟩c P,β � � � , the zero entries resulting from ⟨p0⟩P,β = 0, ⟨p3 0⟩P,β = 0, and r0, p0 being independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Later on we will need the statistics of the conserved fields on the hydrodynamic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' More precisely, for smooth test functions f, we consider the random field ⃗ξϵ(f) = √ϵ � j∈Z f(ϵj) �⃗Q(j) − ⟨⃗Q(0)⟩P,β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Then, by the central limit theorem for independent random variables, lim ϵ→0 ⃗ξϵ(f) = � R dxf(x)⃗u(x), 3 where the limit field ⃗u(x) is a Gaussian random field on R with mean zero, E(⃗u(x)) = 0, and covariance E(um(x)un(x′)) = Cm,nδ(x − x′), (6) in other words, ⃗u(x) is Gaussian white noise with correlated components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Microscopically, spacetime correlations are defined by evolving one of the observables to time t which yields Sm,n(j, t) = ⟨Qm(j, t)Qn(0, 0)⟩c P,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (7) Note that the Gibbs measure is spacetime stationary and thus without loss of generality both arguments in Qn in (7) can be taken as (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' To understand the structure of Sm,n one has to rely on approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For the long time ballistic regime a standard scheme is the Landau-Lifshitz theory, which views Qn(0, 0) as a small perturbation of the initial Gibbs measure at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This perturbation will propagate and is then probed by the average of Qm at the spacetime point (j, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For large (j, t) the microscopic dynamics is approximated by the Euler equations, but only in their linearized version since the perturbation is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' More concretely, the approximate theory will be a continuum field ⃗u(x, t) over R × R, which is governed by ∂t⃗u(x, t) + A∂x⃗u(x, t) = 0 , (8) with random initial conditions as specified in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The 3×3 matrix A is constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' independent of (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' To explain the structure of A requires some further efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' We refer to [38] for more details and proofs of the key identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' From the equations of motion one infers that to each density Qn(j, t) there is a current density Jn(j, t) such that d dtQn(j, t) + Jn(j + 1, t) − Jn(j, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Explicitly, the current densities are ⃗J(j) = −(pj, V ′ j−1, pjV ′ j−1), (9) where we adopted the convention that omission of time argument t means time 0 fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' One then defines the static current-conserved field correlator Bm,n(j) = ⟨Jm(j)Qn(0)⟩c P,β, (10) and the corresponding susceptibility Bm,n = � j∈Z ⟨Jm(j)Qn(0)⟩c P,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Despite its asymmetric looking definition, Bm,n = Bn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (11) As a general property, Euler equations are built on thermally averaged currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Linearizing them with respect to the average fields yields A = BC −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 4 Here B appears when differentiating the average currents with respect to the chemical potentials and C −1 when switching from intensive to extensive variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' By construction C = C T and C > 0, in addition B = BT according to (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Hence A = C 1/2C −1/2BC −1/2C −1/2, which ensures that A has real eigenvalues and a complete set of left-right eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Anharmonic lattices are symmetric under time reversal, which implies the eigenvalues ⃗c = (−c, 0, c), with c > 0 the isentropic speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' We denote the right, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' left eigenvectors of A by |ψα⟩ and ⟨ ˜ψα|, α = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' With this input the solution to (8) with initial conditions (6) reads SLL m,n(x, t) = E � um(x, t)un(0, 0) � = (δ(x − At)C)m,n = 2 � α=0 δ(x − cαt)(|ψα⟩⟨ ˜ψα|C)m,n with m, n = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' There are three δ-peaks, the heat peak standing still and two sound peaks propagating in opposite directions with speed c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Specifying m, n, each peak has a signed weight which depends on C and the left-right eigenvectors of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The Landau-Lifshitz theory asserts that the microscopic correlator Sm,n(j, t) ≃ SLL m,n(x, t) for j = ⌊xt⌋, ⌊·⌋ denoting integer part, with t sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The reader might be disap- pointed by the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' But with such basic information the fine-structure of the peaks can be investigated, in particular their specific sub-ballistic broadening and corresponding scaling func- tions [31,38,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' When turning to the Toda lattice, the conservation laws are now labeled by n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' and thus A, B, C become infinite dimensional matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The corresponding Landau-Lifshitz theory has been worked out in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' As to be discussed in the following section, with appropriate adjustments Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (12) is still valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 3 Toda lattice, linearized generalized hydrodynamics The conservation laws of the Toda lattice are obtained from a Lax matrix [11,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For this purpose, we first introduce the Flaschka variables aj = e−rj/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Then the equations of motion become d dtaj = 1 2aj(pj − pj+1), d dtpj = a2 j−1 − a2 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (13) The Lax matrix, L, is defined by Lj,j = pj, Lj,j+1 = Lj+1,j = aj, j ∈ Z, and Li,j = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Clearly L = LT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The conserved fields are labelled by nonnegative integers and have densities given by Q0(j) = rj, Qn(j) = (Ln)j,j , (14) 5 with n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Note that Qn(j) is local in the sense that it depends only on the variables with indices in the interval [j − n, j + n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' An explicit expression for these quantities is given in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For the current densities one obtains J0(j) = −pj, Jn(j) = (LnL ↓)j,j, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' , (15) where L↓ is the lower triangular part of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Then under the Toda dynamics d dtQn(j, t) + Jn(j + 1, t) − Jn(j, t) = 0, which is the n-th conservation law in local form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The first items in the list are stretch and momentum for which our current definitions agree with those in (4), (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' However, for n = 2 one obtains (L2)0,0 = p2 0 + a2 −1 + a2 0 and (L2L↓)0,0 = a2 −1(p−1 + p0), which differs from (4), (9) on two accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' First there is the trivial factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In our numerical plots we use the physical energy density ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The second point is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Densities are not uniquely defined, since one can add a difference of some local function and its shift by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' When summing a particular choice for the density over some spatial interval, the result differs from another choice of the density by a boundary term only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Thus the bulk term will have a correction of order 1/(length of interval), which does not affect the hydrodynamic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For the currents the difference can be written as a total time derivative which is again a boundary term when integrating over some time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In this section we adopt the conventions (14) and (15), since the analysis heavily relies on the Lax matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Beyond n = 2, while the fields no longer have a name, they still have to be taken into account in a hydrodynamic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The infinite volume static field-field correlator is defined as in (5) and the current-field correlator as in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In particularly, B = BT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Of course, C, B are now matrices in the Hilbert space of sequences indexed by N0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' the space ℓ2(N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' To distinguish 3 × 3 matrices from their infinite dimensional counterparts, for the latter we use standard italic symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The spacetime correlator of the Toda lattice is defined by Sm,n(j, t) = ⟨Qm(j, t)Qn(0, 0)⟩c P,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (16) and we plan to construct its Landau-Litshitz approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In essence this amounts to an analysis of � eAtC � m,n, A = BC−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (17) While we are mainly interested in the physical fields corresponding to the indices m, n = 0, 1, 2, for the operator in (17) an understanding of the infinite dimensional matrices is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Starting from the basics, the free energy of the Toda lattice is given by Feq(P, β) = log � β/2π + P log β − log Γ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In particular, the average stretch, ν, is determined through ν(P, β) = ∂PFeq(P, β) = ⟨Q0(0)⟩P,β = log β − ψ(P), (18) with ψ the digamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Expectations of higher order fields can be written as moments of a probability measure denoted by νρp, κn = ⟨Qn(0)⟩P,β = � R dwνρp(w)wn, (19) 6 n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' ρp is called particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' To determine this density one first has to solve the thermody- namic Bethe equations (TBA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For this purpose we introduce the integral operator Tf(w) = 2 � R dw′ log |w − w′|f(w′), w ∈ R, considered as an operator on L2(R, dw) and define the number density ρn(w) = e−ε(w), (20) with quasi-energies ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The quasi-energies satisfy the TBA equation ε(w) = 1 2βw2 − µ − (Te−ε)(w), (21) where the chemical potential µ has to be adjusted such that � R dwρn(w) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (22) Thereby the number density depends on the parameters P and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The TBA equation is closely connected to the β-ensemble of random matrix theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' We rewrite (21) as − log ρn(w) = 1 2αw2 − µ − αP(Tρn)(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' As α → ∞, the entropy term on the lefthand side can be neglected and one recognizes the defining equation for the Wigner semi-cirle law on the interval [−2 √ P, 2 √ P].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The Lax DOS is the P- derivative of ρn, which diverges as (w ± 2 √ P)−1/2 at the two borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' As α is lowered the borders become smeared to eventually cross over to a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In practice, the TBA equation has to be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' But for thermal equilibrium an exact solution is available [1, 12, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Denoting the solution of (21) for β = 1 and the constraint (22) by ρ∗ n one has ρ∗ n(w) = e−w2/2 √ 2π| ˆfP(w)|2, ˆfP(w) = � ∞ 0 dtfP(t)eiwt, fP(t) = √ 2π−1Γ(P)−1/2tP−1e− 1 2 t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (23) In our numerical simulations it is of advantage to use the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The TBA equation is a standard tool from GHD as one way to write the Euler-Lagrange equations for the variational principle associated with the generalized free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For the Toda lattice such a variational formula was obtained in [9,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Proofs using methods from the theory of large deviations and transfer operator method have also become available [16,27,29,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Next we introduce the dressing transformation of some function f by f dr = � 1 − Tρn �−1f with ρn regarded as a multiplication operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Then number and particle density are related as ρn(w) = ρp(w) 1 + Tρp(w) (24) with inverse ρp = (1 − ρnT)−1ρn = ρnςdr 0 , (25) 7 using the convention ςn(w) = wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For the average currents similar identities are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The central novel quantity is the effective velocity veff = ςdr 1 ςdr 0 , (26) see [3,6,41,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Then ⟨J0(0)⟩P,β = −κ1, and, for n ≥ 1, ⟨Jn(0)⟩P,β = � R dwρp(w)(veff(w) − κ1)wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In thermal equilibrium we have κ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Since in the following there will be many integrals over R, let us first introduce the abbreviation ⟨f⟩ = � R dwf(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' With this notation the C matrix turns out to be of the form C0,0 = ν3⟨ρpςdr 0 ςdr 0 ⟩, C0,n = Cn,0 = −ν2⟨ρpςdr 0 (ςn − κnς0)dr⟩, Cm,n = ν⟨ρp(ςm − κmς0)dr(ςn − κnς0)dr⟩, m, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Note that the matrix C has the block structure C = �C0,0 C0,n Cm,0 Cm,n � , in the sense that Cm,n for m, n ≥ 1 follows a simple pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This structure will reappear for B and eAtC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The field-current correlator B can be computed in a similar fashion with the result B0,0 = ν2⟨ρp(veff − κ1)ςdr 0 ςdr 0 ⟩, B0,n = Bn,0 = −ν⟨ρp(veff − κ1)ςdr 0 (ςn − κnς0)dr⟩, Bm,n = ⟨ρp(veff − κ1)(ςm − κmς0)dr(ςn − κnς0)dr⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' As in (12), we want to determine the propagator of the Landau-Lifshitz theory, denoted by SLL m,n(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In principle, all pieces have been assembled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' However to figure out the exponential of A requires its diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Details can be found in [40] and we only mention that one constructs a linear similarity transformation, R, such that R−1AR is multiplication by ν−1(veff(w) − κ1) (30) in L2(R, dw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Here veff is the effective velocity defined in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Using the block convention as in (28), the spacetime correlator in the Landau-Lifshitz approximation is given by SLL(x, t) = � R dwδ � x − tν−1(veff(w) − κ1) � νρp(w) × � ν2ςdr 0 (w)2 νςdr 0 (w)(ςn − κnς0)dr(w) νςdr 0 (w)(ςm − κmς0)dr(w) (ςm − κmς0)dr(w)(ςn − κnς0)dr(w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 8 Note that SLL(x, 0) = δ(x)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' As a property of the Euler equations, the expression (31) possesses exact ballistic scaling, SLL m,n(x, t) = 1 t SLL m,n(x/t, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (32) The correlator Sm,n(j, t) is computed in our MD simulations which will then be compared with SLL m,n(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 4 Numerical simulations For a molecular dynamics simulation one has to first specify a finite ring [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' , N] with suitable boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For the dynamics of positions qj and momenta pj one imposes qN+1 = q1 + νN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (33) The parameter ν fixes the free volume per particle and can have either sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In our simulation, we actually allow for a fluctuating free volume by choosing random initial conditions such that {r1, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' , rN, pN} are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' random variables with a single site distribution as specified in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Then the deterministic time evolution is governed by (13) with boundary conditions r0 = rN, pN+1 = p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In fact, the boundary condition in (33) amounts to the micro-canonical constraint N � j=1 rj = νN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' If one sets ν = ⟨Q0(0)⟩P,β, then for large N, by the equivalence of ensembles, the two schemes for sampling the correlator Sm,n(j, t) should differ by the size of statistical fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For a few representative examples we checked that indeed the equivalence of ensembles holds for the particular observables under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Returning to the choice of system size there is an important physical constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In all sim- ulations one observes a sharp right and left front, which travel with constant speed and beyond which spatial correlations are exponentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' On a ring necessarily the two fronts will collide after some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Such an encounter has a noticeable effect on the molecular dynamics which is not captured by the linearized GHD analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Therefore the simulation time is limited by the time of first collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Indeed, we note in Figures 1-3 that both linearized GHD and MD clearly display maximal speeds of at most ∆j/∆t = 2 for the entire range of (P, β, m, n) displayed in these figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Taking into account that the initial correlations are proportional to δ0j, we conclude that for a ring of size N = 3000 there will be no collision of the two fronts up to time t = 750 which is larger than t = 600 used in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Before displaying and discussing our results, we provide more details on numerically solving the TBA equations and on the actual scheme used for MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='1 Details of the numerical implementation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='1 Solving linearized GHD To numerically solve the linearized GHD equations, we use a numerical method similar to the one from [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' First, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (23) can be expressed in terms of the parabolic cylinder function Dν(z), which is readily available in Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This provides the solution to the TBA equations (21), (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 9 Then, we use a simple finite element discretization of the w-dependent functions by hat func- tions, resulting in piecewise linear functions on a uniform grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' After precomputing the integral operator T in (20) for such hat functions, the dressing transformation (24) becomes a linear sys- tem of equations, which can be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This procedure yields ςdr n , and subsequently ρp via (25) and veff via (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The moments can be computed from κn = � R dwνρn(w)ςdr n (w), or (equivalently) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' To evaluate the correlator in (31), we note that the delta-function in the integrand results in a parametrized curve, with the first coordinate (corresponding to x/t) equal to ˜veff from (30), and the second coordinate equal to the remaining terms in the integrand divided by the Jacobi factor | d dw ˜veff(w)| resulting from the delta-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='2 Molecular dynamics simulations We approximate the expectation value that is contained in the MD-definition of the correlations Sm,n in equation (16) by the following numerical scheme, whose implementation program is written in Python, and can be found at [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' First, we generate the random initial conditions distributed according to the Gibbs measure, as given by (3) for the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' random variables (rj, pj)1≤j≤N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Specifically, the variables pj are distributed according to a standard normal random variable, that we generate with Numpy v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='23’s native function random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='default rng().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='normal [18], times 1/√β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' It takes a brief calculation to see that rj can be chosen to be − ln(X/(2β)) where X is chi-square distributed with shape parameter 2P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' We obtain the random variable X using Numpy v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='23’s native function random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='default rng().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='chisquare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Having chosen the initial conditions in such a manner, we solve equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For the evolution, we adapt the classical St¨ormer–Verlet algorithm [17] of order 2 to work with the variables (p, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Specifically, we used a time step equal to δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='05, and, given the solution (r(t), p(t)) at time t, we approximate the solution at time t + δ through the following scheme, pj � t + δ 2 � = pj(t) − δ 2 � e−rj(t) − erj−1(t)� , rj(t + δ) = rj(t) + δ � pj+1 � t + δ 2 � − pj � t + δ 2 �� , pj(t + δ) = pj � t + δ 2 � − δ 2 � e−rj(t+δ) − erj−1(t+δ)� , for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In this part of the implementation, we extensively used the library Numba [23] to speed up the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Our approximation for the expectation Sm,n is then extracted from 3×106 trials with indepen- dent initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Here we take the empirical mean of all trials where for each trial we also take the mean of the N = 3000 sets of data that are generated by choosing each site on the ring for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' To evaluate the quality of our numerical simulations, we have repeated the numerical experi- ments up to five times including variations for the length of the ring and evaluating the solutions at more intermediate time steps than displayed in the figures below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Furthermore, we have compared the results with the corresponding outcomes obtained by a MATLAB program that has been devel- oped independently from the Python program, and that follows a different numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' It uses MATLAB’s random number generators randn for initial momenta and rand combined with the 10 rejection method to produce initial stretches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The dynamics is then evaluated by the solver ode45, which exploits the Runge–Kutta method to numerically solve the Hamiltonian system associated with (1) on the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' We found that the deviations between different experiments are comparable to the size of the amplitudes of the high frequency oscillations that are present in figures 4-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' These oscillations are due to the random fluctuations of the empirical means around their expectation values Sm,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Agreement of different experiments up to the order of these oscillations therefore shows the consistency of the corresponding numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' We also want to mention that all the pictures that appeared in this paper are made using the library matplotlib [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='2 Comparison of linearized GHD with MD at time t = 600 We compare the GHD predictions with MD simulations for three different temperatures that correspond to β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 1), β = 1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 2), and β = 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For each β we choose three different values for the pressure parameter P in such a way that the corresponding mean stretches, given by (18), are positive (≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='57) for low pressure, negative (≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='42) for high pressure and approximately zero for medium pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' We summarize their values in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' pressure β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 β = 1 β = 2 low P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='32, ⟨r⟩ ≈ +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='58 P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='4, ⟨r⟩ ≈ +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='56 P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='52, ⟨r⟩ ≈ +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='56 medium P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='95, ⟨r⟩ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='03 P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5, ⟨r⟩ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='04 P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='55, ⟨r⟩ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='03 high P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='21, ⟨r⟩ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='42 P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0, ⟨r⟩ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='42 P = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='53, ⟨r⟩ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='42 Table 1: Values for β and P and the corresponding mean stretches used in experiments In each of the nine cases we have evaluated the Landau-Lifshitz approximations SLL m,n(·, 1), see (31), of the correlators for all 0 ≤ n ≤ m ≤ 2 using the numerical scheme described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Their graphs are displayed in Figures 1-3 as dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Note that the speeds of the sound peaks depend significantly on both pressure and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Moreover, the predicted fine-structure of both the heat and the sound peaks are quite different for low pressure when compared to medium and high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The colored lines in Figures 1-3 show our numerical results for the corresponding molecular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' According to the predicted ballistic scaling (32) we plot tSm,n(j, t) as a function of j/t for t = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Here the values of Sm,n(j, t) are approximated using the numerics explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The agreement between linearized GHD and MD is striking, in particular since there are no adjustable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In all of the 54 comparisons shown in Figures 1-3 the GHD predictions for the fine-structure of heat and sound peaks are in excellent agreement with the ones observed from molecular dynamics at time t = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' As we show in more detail in the next subsection the largest deviations occur mostly near the sound peaks and do not exceed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5% of the peaks’ maximal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='3 Deviation of linearized GHD from MD at times t = 150 and t = 600 The purpose of this subsection is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' On the one hand we have a look at the small differences between GHD predictions and molecular dynamics simulations that can hardly be detected in 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0 2 4 6 8 S00, S11, S22, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5, P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='32 S00 S11 S22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0 1 2 3 4 5 6 7 S00, S11, S22, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5, P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='95 S00 S11 S22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0 2 4 6 8 S00, S11, S22, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5, P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='21 S00 S11 S22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 4 2 0 2 4 S21, S20, S10, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5, P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='32 S21 S20 S10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': 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+page_content='5 with low pressure (top), medium pressure (middle) and high pressure (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Numerical simulations are colored according to the legend, the corresponding GHD predictions are displayed by dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Number of trials: 3 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='75 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 2 1 0 1 2 S21, S20, S10, = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0, P = 1.' 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for β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 with low pressure (top), medium pressure (middle) and high pressure (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Numerical simulations are colored according to the legend, the corresponding GHD predictions are displayed by dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Number of trials: 3 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0 1 2 3 4 5 6 7 S00, S11, S22, = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0, P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='52 S00 S11 S22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 S00, S11, S22, = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0, P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='55 S00 S11 S22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 S00, S11, S22, = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0, P = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='53 S00 S11 S22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 2 1 0 1 2 S21, S20, S10, = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0, P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='52 S21 S20 S10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 S21, S20, S10, = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0, P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='55 S21 S20 S10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 S21, S20, S10, = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0, P = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='53 S21 S20 S10 Figure 3: Toda correlation functions: GHD predictions y �→ SLL m,n(y, 1) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' numerical simulations of the molecular dynamics y �→ tSm,n(yt, t) at t = 600 for β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 with low pressure (top), medium pressure (middle) and high pressure (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Numerical simulations are colored according to the legend, the corresponding GHD predictions are displayed by dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Number of trials: 3 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 t: 150 t: 600 GHD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='02 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00, P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='55, S1, 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 t: 150 t: 600 GHD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='04 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00, P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='55, S1, 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 t: 150 t: 600 GHD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='02 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00, P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='50, S1, 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1 0 1 t: 150 t: 600 GHD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='04 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00, P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='50, S1, 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 t: 150 t: 600 GHD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='05 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='50, P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='95, S1, 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1 0 1 t: 150 t: 600 GHD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='050 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='50, P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='95, S1, 0 Figure 4: Toda correlation functions S1,1 (left) and S1,0 (right) for medium pressure and increasing temperatures (top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For each value of β and P the top panels show GHD prediction vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' numerical simulations as in Figures 1-3 but with the the molecular dynamics evaluated at two times t = 150 and t = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The bottom panels display the differences between the GHD prediction and numerical simulations at time t = 150 (red) and at time t = 600 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Number of trials: 3 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='50 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='8 t: 150 t: 600 GHD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='02 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00, P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00, S0, 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 t: 150 t: 600 GHD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='050 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00, P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='00, S2, 0 Figure 5: Toda correlation functions S0,0 (left) and S2,0 (right) for β = 1 and increasing pressure (top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For each value of β and P the top panels show GHD prediction vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' numerical simulations as in Figure 2 but with the the molecular dynamics evaluated at two times t = 150 and t = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The bottom panels display the differences between the GHD prediction and numerical simulations at time t = 150 (red) and at time t = 600 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='Number of trials: 3 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 16 Figures 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' On the other hand we indicate how these differences evolve in time by including time t = 150 for the molecular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Recall that the GHD predictions are time-invariant in the scaling y �→ tSm,n(yt, t) we have chosen, see (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' From the 54 comparisons that are displayed in Figures 1-3 we select 12 cases that are repre- sentative and show all the phenomena that we have observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In Figure 4 we consider correlations S1,1 and S1,0 at medium pressure (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Table 1) for all three values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The small scale fluctuations displayed in the bottom panels are due to the approximation of expectation values by empirical averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Their amplitudes become smaller if one increases the number of trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Note that the difference in amplitudes of these fluctions between t = 150 and t = 600 is mostly due to the scaling y �→ tSm,n(yt, t) that we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This implies that the values of the correlations are multiplied by a factor that is 4 times larger at the later time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The same holds for the plots in Figure 5 where the correlations S0,0 and S2,0 are shown for fixed β = 1 and our three different choices for pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' We now summarize our main findings: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The deviations occur mostly near the sound peaks and amount to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5%-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5% of the peaks’ maximal values at time t = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' There appear to be small but systematic deviations concerning the shape of the sound peak in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' One would need to conduct experiments with a higher resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' more sites and consequently larger times and more trials, to determine whether there is indeed such a systematic deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' With the resolution present in our experiments the question of a systematic deviation with respect to the shape of the peak cannot be decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In some of the experiments the maximal deviations would be significantly smaller if a constant only depending on the values of β, P, m, n is added to all values of Sm,n(j, t), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' correlations S0,0 and S2,0 for β = 1, P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='4 in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This seems to be related to the approximation errors for the means ⟨r⟩, ⟨p⟩, and ⟨e⟩, that appear to be less pronounced in the case of momentum p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' We have observed that these deviations decrease as the number of trials is increased and we do not expect a systematic deviation between GHD and MD in this respect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' For (β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' P) ∈ {(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='95), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='21)} we observe that the size of the deviations is essentially the same for times t = 150 and t = 600 whereas for (β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' P) ∈ {(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='32), (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='4), (2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='52), (2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='55), (2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='53)} these deviations are significantly larger at the smaller time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The remain- ing two cases (β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' P) ∈ {(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5), (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 2)} are somewhat in between, also depending on the correlation function that is considered, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This is an indication that the speed of convergence of tSm,n(yt, t) to the GHD prediction SLL m,n(y, 1) as t → ∞ depends on the values of β and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' As a rule we have observed that both increasing temperature or increasing pressure leads to a faster speed of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 5 Conclusions and outlook As can be seen from Table 1, we picked the intermediate pressure such that ν ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In the particle picture ν = 0 corresponds to the boundary condition q1 = qN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In thermal equilibrium the positions then perform an unbiased random walk with typical excursions of order √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Thus the free volume is of order 1/ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The particles are extremely dense and the picture of successive pair collisions breaks down completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' So one might wonder whether GHD is still valid under such extreme conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' ν = 0 poses no particular difficulties for MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In GHD the factor 1/ν 17 appears in the expression for veff, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This makes the numerical scheme slow and only values close to ν = 0 are accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' However the correlator S changes smoothly through ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' GHD also covers this seemingly singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Simultaneously A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Kundu [21] posted a somewhat puzzling note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' He considers the parameter values β = 1, P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' When cutting the matrices Cm,n and Am,n at low orders, the resulting Sm,n consists of a few δ-peaks which move at constant velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' After ballistic scaling, with high precission they turn out to lie on the curve obtained from GHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' A theoretical explanations seems to be missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In [22] the molecular dynamics of Toda lattice correlations are simulated for the potential Vkd(x) = g γ e−γx with arbitrary γ, g > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' To distinguish their parameters from ours, the variables in [22] are here denoted by ¯t, ¯r, ¯P, ¯β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' ¯P is the physical pressure and, comparing the Gibbs weights, one obtains the relations β = g γ ¯β, P = 1 γ ¯P ¯β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' From the equations of motions one deduces ¯t = 1 √γgt, r(t) = γ¯r(¯t), p(t) = g γ ¯p(¯t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Thus, translating to our units, the MD simulations reported in [22] are (i) P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='01, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='01, N = 1024, t = 400, (ii) P = 1, β = 1, N = 1024, t = 200, 300, and (iii) P = 400, β = 400, N = 256, t = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In fact, in all three cases the time scales are identical, t = ¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Since GHD was not available yet, no comparison could have been attempted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Case (i) is a very dilute chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In this limit νρp is a unit Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The dressed functions become polynomials as ςdr 0 (w) = a0, ςdr 1 (w) = a1w, and ςdr 2 (w) = a2w2 + a3 with coefficients a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', a3 depending on (P, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Note that for a noninteracting fluid a3 would vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' As a result S0,0 is Gaussian, S1,1 has two peaks, and S2,2 has either two or three peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This is in good agreement with [22] and explains our motivation not to venture into the low density regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Case (ii) interpolates between our β = 1, P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='40 and β = 1, P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Note that now S0,0 has a local minimum at w = 0, which is very different from the structure in the dilute regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' On the other hand, S2,2 has a local maximum at w = 0, as is the case for low density/high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The most interesting parameter value is (iii), which deserves more detailed studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The issue is the behavior of the Toda chain at very low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Simply letting β → ∞ will freeze any motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' But the simultaneous limit β → ∞ with P = ¯Pβ at fixed physical pressure ¯P is meaningful, at least statistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' In this limit ν > 0 always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Also the density of states converges to the arcsine distribution, lim β→∞ νρp(w) = 1 π √ 4 ¯P − w2, |w| ≤ 2 � ¯P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' To understand the dynamical behavior, the effective potential is expanded as e−r + ¯Pr ≃ 1 2 ¯P(r − r0)2 + c0 at its minimum r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Since β is large, the initial fluctuations are of order 1/√β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Therefore the dy- namics can be approximated by a harmonic chain with ω2 = ¯P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The equilibrium time correlations 18 of the harmonic chain have intricate oscillatory behavior [14], which in the large β limit should match with the Toda lattice, as partially evidenced through case (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Clearly, GHD cannot repro- duce such fine details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Still, when averaged on suitable scales, the gross behavior of the harmonic chain oscillations might be visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 1440140, while five of the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the fall semester of 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cam- bridge, for support and hospitality during the programme “Dispersive hydrodynamics: mathemat- ics, simulation and experiments, with applications in nonlinear waves” where some work on this paper was undertaken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' This work was supported by EPSRC grant no EP/R014604/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' TG ac- knowledges the support of the European Union’s H2020 Marie Sk�lodowska–Curie grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 778010 IPaDEGAN, of INdAM/GNFM and of the research project Mathematical Methods in NonLinear Physics (MMNLP), Gruppo 4-Fisica Teorica of INFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' GM is financed by the KAM grant number 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='0344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' KTRM was supported by a Visiting Wolfson research fellowship from the Royal Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Allez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Bouchaud, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Guionnet, Invariant β ensembles and the Gauss- Wigner crossover, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', 109 (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Bastianello, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Doyon, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Watts, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Yoshimura, Generalized hydrodynamics of classical integrable field theory: the sinh-Gordon model, SciPost Physics, 4 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Bertini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Collura, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' De Nardis, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Fagotti, Transport in out-of- equilibrium XXZ chains: exact profiles of charges and currents, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', 117 (2016), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 207201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Bulchandani, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Cao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Moore, Kinetic theory of quantum and classical Toda lattices, Journal of Physics A: Mathematical and Theoretical, 52 (2019), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 33LT01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [5] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Bulchandani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Vasseur, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Karrasch, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Moore, Solvable hydrody- namics of quantum integrable systems, Physical Review Letters, 119 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [6] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Castro-Alvaredo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Doyon, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' X, 6 (2016), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 041065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Das, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Kulkarni, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Spohn, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Dhar, Kardar-Parisi-Zhang scaling for an integrable lattice Landau-Lifshitz spin chain, Physical Review E, 100 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [8] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Doyon, Exact large-scale correlations in integrable systems out of equilibrium, SciPost Physics, 5 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [9] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Doyon, Generalized hydrodynamics of the classical Toda system, Journal of Mathematical Physics, 60 (2019), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 073302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 19 [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Dupont and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Moore, Universal spin dynamics in infinite-temperature one- dimensional quantum magnets, Physical Review B, 101 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Flaschka, The Toda lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Existence of integrals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' B (3), 9 (1974), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 1924–1925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Forrester and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Mazzuca, The classical β-ensembles with β proportional to 1/N: from loop equations to Dyson’s disordered chain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', 62 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 073505, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Forster, Hydrodynamic fluctuations, broken symmetry, and correlation functions, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [14] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Grava, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Kriecherbauer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Mazzuca, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' McLaughlin, Corre- lation functions for a chain of short range oscillators, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', 183 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 1, 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Grava, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Maspero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Mazzuca, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Ponno, Adiabatic invariants for the FPUT and Toda chain in the thermodynamic limit, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', 380 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 811–851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Guionnet and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Memin, Large deviations for Gibbs ensembles of the classical Toda chain, Electronic Journal of Probability, 27 (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 1 – 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [17] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Hairer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Wanner, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Lubich, Symplectic Integration of Hamiltonian Systems, Springer Berlin Heidelberg, Berlin, Heidelberg, 2006, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 179–236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Harris, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Millman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' van der Walt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Gommers, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Virtanen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Cournapeau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Wieser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Taylor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Berg, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Smith, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Kern, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Picus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Hoyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' van Kerkwijk, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Brett, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Haldane, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' del R´ıo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Wiebe, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Peterson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' G´erard-Marchant, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Sheppard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Reddy, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Weckesser, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Abbasi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Gohlke, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Oliphant, Array programming with NumPy, Nature, 585 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 357–362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Hunter, Matplotlib: A 2d graphics environment, Computing in Science & Engineering, 9 (2007), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 90–95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [20] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Ilievski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Nardis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Medenjak, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Prosen, Superdiffusion in one- dimensional quantum lattice models, Physical Review Letters, 121 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Kundu, Integrable hydrodynamics of Toda chain: case of small systems, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Kundu and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Dhar, Equilibrium dynamical correlations in the Toda chain and other integrable models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' E, 94 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 062130, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Lam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Pitrou, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Seibert, Numba: A LLVM-based Python JIT compiler, in Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC, LLVM ’15, New York, NY, USA, 2015, Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [24] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Landau and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Lifshitz, Fluid Mechanics: Volume 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 6, Elsevier Science, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Ljubotina, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' ˇZ nidariˇc, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Prosen, Kardar-Parisi-Zhang physics in the quan- tum Heisenberg magnet, Physical Review Letters, 122 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 20 [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Manakov, Complete integrability and stochastization of discrete dynamical systems, ˇZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' `Eksper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Teoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', 67 (1974), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 543–555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [27] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Mazzuca, On the mean density of states of some matrices related to the beta ensembles and an application to the Toda lattice, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', 63 (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 043501, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [28] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Mazzuca, Toda correlation functions, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content='com/ gmazzuca/TodaCorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [29] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Mazzuca and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Grava, Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, Circular β-ensemble and double confluent Heun equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [30] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Mazzuca and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Memin, Large deviations for Ablowitz-Ladik lattice, and the Schur flow, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [31] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Mendl and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Spohn, Equilibrium time-correlation functions for one-dimensional hard-point systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' E, 90 (2014), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 012147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [32] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Mendl and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Spohn, Low temperature dynamics of the one-dimensional discrete nonlinear Schr¨odinger equation, Journal of Statistical Mechanics: Theory and Experiment, 2015 (2015), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' P08028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [33] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Mendl and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Spohn, High-low pressure domain wall for the classical Toda lattice, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Core, 5 (2022), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [34] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Møller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Perfetto, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Doyon, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Schmiedmayer, Euler-scale dynamical correlations in integrable systems with fluid motion, SciPost Physics Core, 3 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Opper, Analytical solution of the classical Bethe ansatz equation for the Toda chain, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' A, 112 (1985), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 201–203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [36] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Schneider, Classical statistical mechanics of lattice dynamic model systems: transfer integral and molecular-dynamics studies, (1983), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 212–241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [37] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Schneider and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Stoll, Excitation spectrum of the Toda lattice: a molecular-dynamics study, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', 45 (1980), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 997 – 1002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [38] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', 154 (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 1191–1227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [39] , The Kardar-Parisi-Zhang equation: a statistical physics perspective, in Stochastic pro- cesses and random matrices: Lecture notes of the Les Houches Summer School July 2015, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Schehr, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Altland, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Fyodorov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' O’Connell, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Cugliandolo, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 104, Oxford University Press, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 177–227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [40] , Ballistic space-time correlators of the classical Toda lattice, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' A, 53 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 265004, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [41] , Collision rate ansatz for the classical Toda lattice, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' E, 101 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 060103(R), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 21 [42] , Generalized Gibbs ensembles of the classical Toda chain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', 180 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 4–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Toda, Vibration of a chain with nonlinear interaction, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', 22 (1967), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 431 – 436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [44] , Theory of Nonlinear Lattices, Springer Berlin, Heidelberg, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' [45] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Yoshimura and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Spohn, Collision rate ansatz for quantum integrable systems, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=', 9 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 040, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfhgEK/content/2301.02431v1.pdf'} diff --git a/I9FLT4oBgHgl3EQfJi8x/content/tmp_files/2301.12004v1.pdf.txt b/I9FLT4oBgHgl3EQfJi8x/content/tmp_files/2301.12004v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a438e24fb8b0af4b6080b31313c153c3429f9b1 --- /dev/null +++ b/I9FLT4oBgHgl3EQfJi8x/content/tmp_files/2301.12004v1.pdf.txt @@ -0,0 +1,1626 @@ +Understanding the Effectiveness of Very Large +Language Models on Dialog Evaluation +Jessica Huynh, Cathy Jiao, Prakhar Gupta, Shikib Mehri, Payal Bajaj, Vishrav +Chaudhary, Maxine Eskenazi +Abstract Language models have steadily increased in size over the past few years. +They achieve a high level of performance on various natural language processing +(NLP) tasks such as question answering and summarization. Large language models +(LLMs) have been used for generation and can now output human-like text. Due to +this, there are other downstream tasks in the realm of dialog that can now harness the +LLMs’ language understanding capabilities. Dialog evaluation is one task that this +paper will explore. It concentrates on prompting with LLMs: BLOOM, OPT, GPT- +3, Flan-T5, InstructDial and TNLGv2. The paper shows that the choice of datasets +used for training a model contributes to how well it performs on a task as well as on +how the prompt should be structured. Specifically, the more diverse and relevant the +group of datasets that a model is trained on, the better dialog evaluation performs. +This paper also investigates how the number of examples in the prompt and the type +of example selection used affect the model’s performance. +Jessica Huynh +Carnegie Mellon University, e-mail: jhuynh@cs.cmu.edu +Cathy Jiao +Carnegie Mellon University, e-mail: cljiao@cs.cmu.edu +Prakhar Gupta +Carnegie Mellon University, e-mail: prakharg@cs.cmu.edu +Shikib Mehri +Amazon, e-mail: asmehri@amazon.com (work done while at Carnegie Mellon University) +Payal Bajaj +Microsoft Turing, e-mail: pabajaj@microsoft.com +Vishrav Chaudhary +Microsoft Turing, e-mail: vchaudhary@microsoft.com +Maxine Eskenazi +Carnegie Mellon University, e-mail: max@cs.cmu.edu +1 +arXiv:2301.12004v1 [cs.CL] 27 Jan 2023 + +2 +J. Huynh et al. +1 Introduction +In recent years, language models such as GPT-3 [5] have grown larger, and their per- +formance on downstream natural language processing (NLP) tasks has significantly +improved in low-resource settings where only a few instances per task are available +(few-shot). The larger these models are, the higher their performances trend on tasks +such as language generation and evaluation [39]. They can generate coherent, fluent +and interesting responses. However, they can also produce responses that are repet- +itive and un-engaging [29], in addition to being hard to control. Dialog evaluation is +the task of assessing the quality of responses generated by dialog models in terms +of properties like those mentioned above. However, one significant impediment for +open-domain dialog generation research is the lack of meaningful automatic metrics +for open-domain dialog evaluation. Standard language generation metrics have been +shown to be ineffective for dialog evaluation [11], a large part of which is because +conversations can be followed by multiple valid responses. Standard automatic met- +rics (e.g. BLEU [24]), which use references for evaluation, cannot deal with this +quality, known as the one-to-many response problem. Many recently introduced au- +tomatic metrics for dialog evaluation [21, 12] have attained increasingly stronger +correlations with human judgment. Since human dialog evaluation typically mea- +sures multiple fine-grained properties (e.g. appropriate, interesting, consistent), au- +tomatic evaluation metrics should be expected to do the same. This paper explores +several fine-grained metrics that are measured both at turn-level (i.e. relevance and +fluency), and dialog-level (i.e. consistency and coherence). +Automatic dialog evaluation continues to be an evolving topic, but with fine- +grained metrics and definitions varying across different human-annotated datasets +[22, 46], it is important to be able to create reasonable automatic metrics with lim- +ited data. Large language models (LLMs) that have been pre-trained on large-scale +datasets are able to perform zero and few-shot inference [26, 32], and they have ex- +hibited good reasoning skills [5, 39] in addition to having implicitly learned some +notion of dialog quality [21]. This makes them suitable for open-domain dialog +evaluation in zero-shot and extreme few-shot settings. While there have been a few +attempts to use LLMs for dialog evaluation [36], there has not, to our knowledge, +been a systematic study of LLMs for this task. This paper explores several aspects +of LLM use in dialog evaluation: the effect of model type and size and the choice +of training data as well as the use of in-context examples for dialog evaluation (the +number and quality of the examples used). The experiments herein employ bench- +marks to test both how well LLMs can be used for fine-grained evaluation, and how +generalizable the models’ performance is across multiple domains and datasets. + +Very Large Language Models for Dialog Evaluation +3 +2 Related Work +2.1 LLMs +Several LLMs have been released recently: T5 [27], GPT-3 [5], BLOOM [4], OPT +[42], and TNLGv2 [34]. The following models, the sizes of which are shown in +Figure 1, are explored here: +• +T5, trained on the 750B Colossal Clean Crawled Corpus (C4) contains heuristi- +cally cleaned natural language English text from the web. Specific models con- +sidered are: +– Flan-T5 [8], T5 fine-tuned on 1836 tasks, including dialog tasks and data. +– InstructDial [13], T5 fine-tuned specifically on 48 dialog tasks. +• +GPT-3 includes a 570B filtered CommonCrawl corpus [27] in addition to Web- +Text [26], Books1, Books2, and Wikipedia [16]. +– InstructGPT (text-davinci-002) [23], GPT-3 fine-tuned with a prompting +dataset and 175B parameters. +• +BLOOM was trained on 46 languages and 13 programming languages with a +multilingual focus. +• +OPT contains data from the RoBERTa corpus [18], the Pile [9], and PushShift.io +Reddit [2, 29]. +• +TNLGv2 is trained on a subset of the Pile (notably excluding corpora classified +as having natural dialog), two CommonCrawl snapshots [27], RealNews [40], +and CC-Stories [37]. +Fig. 1 Large Language Models, comparison of select approximate sizes + +530B +I size +Model +175B +30B +7B +！ +0.5B +BLOOM +OPT +InstructGPT Flan-T5 InstructDial TNLGv2 +LLM +Fine-tuned LLM +LLM4 +J. Huynh et al. +As the number of parameters in these models increases, performance also in- +creases: TNLGv2 530B, with around three times the number of parameters, out- +performs the original GPT-3 on a variety of NLP tasks [34]. LLMs are also gener- +alizable; they perform well on many NLP tasks in few-shot settings and zero-shot +settings [38, 32]. However, several drawbacks and areas for exploration remain for +LLMs that should be noted. Recent work has shown that performance on certain +zero-shot tasks plateaus as model parameter size grows exponentially [5]. LLMs +also struggle with parsing social situations [33] and correctly using context [1], +which are important in dialog settings. This raises questions on the performance of +LLMs for dialog evaluation, and how an LLM’s performance changes as it increases +in size. +The data that a model is trained on also influences the performance of down- +stream tasks. T5 is fine-tuned on various subtasks, but pre-trained with C4. When +pre-trained with domain-specific data, T5 performs better on tasks in that domain +[3, 27]. Furthermore, adding several domains of data during pre-training makes the +model likely to perform better [18, 42, 7]. Notably, BLOOM, OPT, Flan-T5, In- +structGPT, and InstructDial are partially trained or fine-tuned on dialog datasets. +Details on the content of these datasets can be found in Appendix A. This is impor- +tant because natural dialog data is difficult to obtain, so either scripted conversations +or Reddit threads are used since they are the most readily available. This dearth of +data is the reason that few-shot prompting is of interest. While work such as [39] +acknowledges emergent abilities in larger language models in few-shot prompting +settings, this paper explores discrepancies in performance specifically for dialog +evaluation. +2.2 Dialog Evaluation +Dialog evaluation presents a unique combination of challenges; it must consider +multiple speakers [44], context that informs the current dialog turn, and the one-to- +many aspect mentioned above [45]. +Metrics such as USR [22] and FED [21] were created to address some of +these challenges; they are reference-free, capture complex aspects of dialog, and +have good correlation with human evaluation. These metrics use models such as +RoBERTa (125 million parameters) [18] and DialoGPT (345 or 762 million param- +eters) [43] respectively. However, the best performing versions of these models are +smaller than most models examined in this paper, and are fine-tuned on dialog data +or on a specific dialog task. Other automatic evaluation metrics include GRADE +[14] and DEB [31]. With current LLMs’ large increase in hyperparameters, their +plethora of training data, and their promising generalizable performance on NLP +tasks, these model-based metrics should improve as well. + +Very Large Language Models for Dialog Evaluation +5 +2.3 Example selection for few-shot learning +The example selection process for prompting LLMs is of great interest. Prompting +an LLM with a task and a few examples enables the model to adapt to a new task +without completely fine-tuning it. In particular, in-context examples can provide im- +portant cues to help LLMs make predictions on tasks. Recent work has used a vari- +ety of methods to examine example selection. Common methods measure semantic +similarity between example embeddings [17, 35]. Alternatively, retrieval methods +(e.g. BM25 [28]) have been used directly, or as a precursor to training a selection +retriever [30]. +These example selection methods have shown promise in few-shot NLP tasks. +In [35], the two-step framework for annotating and selecting in-context examples +from large unlabeled data showed competitive performance across 10 tasks such as +classification, commonsense reasoning, dialog state tracking, and code generation. +[17] showed that selecting examples with similar sentence embeddings yields higher +GPT-3 performance than random selection. However, the authors acknowledge that +further investigation is required to find more efficient in-context example retrieval +methods. +Moreover, the wording and order of examples presented in prompts can also +affect model performance [10, 17, 15]. Lu et al [19] observed order sensitivity across +0.1B to 175B parameter GPT-2 and GPT-3 models when the models were probed +with different text classification tasks and several in-context examples. Also, the +wording of the in-context examples depends on the data used for model training; +for unfamiliar prompt formats, model performance may decrease [15]. Increasing +the size of the model and the amount of data does not resolve the issue since the +same instability is still prevalent [47]. Thus this paper studies the effect of example +selection on dialog evaluation. +3 Evaluation Settings +Two settings for dialog evaluation are explored: fine-grained evaluation and multi- +domain evaluation. In-context examples are explored in both. +3.1 Fine-Grained Evaluation +Fine-grained metrics can be measured at both the turn level (e.g. informativeness +and relevance), and the dialog level (e.g. coherence and diversity). The FED dataset +[21] is used. It consists of 124 open-domain dialogs of humans with humans or +with machines, for which each dialog has 3 responses that are chosen for annotation +(8 turn-level and 10 dialog-level qualities along with overall turn- and dialog-level +quality). This dataset was chosen due to the large number of previously studied fine- + +6 +J. Huynh et al. +grained qualities as listed in Section 4.1, with the exception of correctness and error +recovery, which are only specifically present in FED. +In the experiments, the LM is prompted to output a rating (an integer value - see +Appendix B) to evaluate each fine-grained quality in a response. The final rating +for each fine-grained quality is a weighted sum of the K-top ratings outputted from +the LM. Formally, given the K-top predicted ratings r1,r2,...,rK along with their +corresponding log probabilities, p1,..., pK, the weight, wi, of each rating ri is derived +as: +wi = +pi +∑K +j=1 pj +The final rating, r, is calculated as: +r = +K +∑ +i=1 +ri ∗wi +In order to provide a more accurate view of the LM’s performance, K = 3 in the +following experiments. Additionally, this scoring mechanism converts the LM pre- +dictions onto a continuous scale, which more closely mirrors the average of human +ratings. Results are reported with the Spearman correlations to the average human +ratings for each fine-grained quality. +3.2 Multi-domain Evaluation +This task tests automatic dialog evaluation metrics for robustness across multiple +dialog domains. The analysis uses only the overall quality metric since many of +the domain datasets do not have fine-grained annotations. The Spearman correla- +tion is used between human ratings and model predictions on the evaluation sets +released by DSTC 10 Track 5 [6] “Automatic Evaluation and Moderation of Open- +domain Dialogue Systems”. These sets contain human judgement ratings for dialog +responses. In this setting, a model is shown a dialog context and a response, and it +outputs “yes” if the response is a good response to that context, otherwise it outputs +“no”. An example can be seen in Appendix C. The probability of the “goodness” of +the response (i.e., the rating), g, is calculated as: +g = +pmodel(yes) +pmodel(yes)+ pmodel(no) +where pmodel(yes) and pmodel(no) are the log probilities of the model outputs +for “yes” and “no”. Evaluation is carried out on 8 representative evaluation sets out +of the 14 DSTC10 evaluation sets [6]. This subset was chosen because it covers +multiple domains and datasets, such as persona, topic and chitchat-based responses. +A robust dialog metric should perform well across all the domains and evaluation +sets considered. + +Very Large Language Models for Dialog Evaluation +7 +The evaluation sets used for fine-grained evaluation, FED-Turn (FT) and FED- +Dial (FD) [21], are included as two of the eight datasets. The other datasets include: +TopicalChat-USR (TU, knowledge-grounded open-domain conversations rated for +six different dialog qualities) [22]; PersonaChat-USR (PU, persona-conditioned +conversations annotated with the USR schema) [22]; DailyDialog-Zhao (DZ, more +formal language conversations rated for appropriateness) [46]; DailyDialog-Gupta +(DGU, rated for appropriateness) [11]; DailyDialog-GRADE (DGR; annotated for +coherence) [14]; and Empathetic-GRADE (EG, emotionally grounded conversa- +tions annotated for coherence) [14]. Although some of these datasets are not directly +annotated for whether a response is good, the metric they use remains a component +for overall quality, and thus it is treated as the indicator of the overall quality of the +response in the experiments. +3.3 In-Context Examples +This paper uses two methods for example selection: random selection, and algorith- +mic selection using BM25 [20] which calculates document similarity. The examples +remain consistent for each evaluation test point. The random selection experiment is +run three times, and the mean and standard deviation of the runs are reported. There +are three configurations for BM25 between the test point and each possible example +point - comparing the context only (BM25C), the response only (BM25R), and the +concatenated context and response together (BM25C+R). +With the FED dataset, an additional method, manual selection, is added for ex- +ample selection. For each fine-grained dialog quality, a set of three dialogs which +span a wide range of ratings is chosen that remains constant over every test point. In +theory, the model should be able to show increased performance if it sees examples +of very good, good and bad responses for fine-grained metrics. For the DSTC10 +datasets, an additional experiment tested how the number of examples used affects +model performance. +4 Experiments and Results +The in-context example experiments are carried out on the largest available model, +530B TNLGv2, to explore the ceiling of model performance on the dialog evalua- +tion task. 6.7B TNLGv2 is used for a direct comparison of how much performance +gain is provided by using more parameters. +BLOOM and OPT are examined up to 7B and 30B respectively for the fine- +grained metric evaluation task. 1 Smaller LLMs do not perform as well with in- +1 Due to limitations in compute power, larger BLOOM and OPT models were not explored. How- +ever, as the largest available GPT-3 model is explored, the comparisons appear sufficient to show +the performance of a variety of LLMs. + +8 +J. Huynh et al. +context examples unless they have been specifically tuned for the task, so only +the 7B and 6.7B models for BLOOM and OPT respectively are explored for the +DSTC10 datasets. Flan-T5 and InstructDial are analyzed in the 3B setting for con- +sistency. Lastly, InstructGPT (text-davinci-002) is used, which has 175B parame- +ters. +4.1 Fine-grained Metric Evaluation +FED is separated into turn-level and dialog-level metrics. The dataset has anno- +tations for 8 different turn-level metrics, consisting of interestingness, engaging- +ness, specificity, relevance, correctness, semantic appropriateness, understandabil- +ity, and fluency, with the addition of overall quality. FED annotates three different +responses for each dialog context; one FED dialog is treated as one example. The +corresponding rating is inserted after the response statement in the prompt, an exam- +ple of which can be seen in Appendix B. FED also looks at 10 different dialog-level +metrics for a system’s responses: coherence, error recovery, consistency, diversity, +topic depth, likeability, understandingness, flexibility, informativeness, and inquisi- +tiveness, with overall quality included. The model is prompted with the full dialog +context with the rating. +The FED metric was previously evaluated with both fine-tuned (ft) and from- +scratch 345M and 762M DialoGPT [43] models. In the following experiments on +FED, 3 in-context examples were used for prompting in Tables 1, 2, 3 and 4 and +Appendix D and E. +4.1.1 In-Context Example Selection +This setting evaluates 2 versions of the TNLGv2 model: 6.7B and 530B. These +models are compared to the 762M ft DialoGPT model and the results are shown in +Tables 1 and 2 and Appendix D. +First, the performances of these models are compared over the three example +selection methods: manual, random, and algorithmic. With manually chosen in- +context examples, the 530B TNLGv2 model outperforms the DialoGPT model on +almost all turn-level metrics except for understandability and fluency. There are +significant gains in all of the dialog-level metrics as well. Since DialoGPT is fine- +tuned on Reddit threads, more casual language is expected, compared to models +like TNLGv2 where many of the training datasets consist of more formal language. +Since the wording of conversational responses tends to be more casual, it is not sur- +prising that the fine-tuned DialoGPT model outperforms even the largest TNLGv2 +model for fluency and understandability. However, the TNLGv2 models show large +improvement on predicting turn- and dialog-level quality. This suggests that the +TNLGv2 models have a strong grasp on overall quality, which may be due to train- +ing on more formal language. + +Very Large Language Models for Dialog Evaluation +9 +BM25C+R generally outperforms BM25C and BM25R. However, when choos- +ing examples with BM25C+R, the correlation of understandability with human +annotations increases significantly when using the 6.7B TNLGv2 model. 6.7B +TNLGv2 consistently outperforms 530B TNLGv2 in this aspect with any BM25 +method. It appears that the smaller model is more influenced by the similarity of +language in the examples than the larger one. +Even when given random examples, the TNLGv2 models outperform the 762M +ft DialoGPT model on a majority of the fine-grained metrics. This shows that larger +models can better detect what constitutes a good response based on these metrics +even if they are not given hand-picked examples. However, they generally do not +outperform the manually or algorithmically chosen examples as expected. +An additional observation is that there are certain factors that cause models to +perform better or worse on specific metrics: number of parameters the model has, +the type of training data, and the difficulty of the task. LLMs are able to provide +increases in performance of over 50% for 15 out of 20 turn- and dialog-level met- +rics compared to DialoGPT with 530B TNLGv2 and manually-chosen examples. +However, if the 530B TNLGv2 model is compared to the 6.7B TNLGv2 model, +this increase is only observed for 2 out of the 20 metrics: correctness and under- +standability. LLMs can achieve high correlations with human judgement, but there +is a limit to how much more performance gains can increase with extremely large +models. +Specificity, relevance, and correctness all relate to the context of the conversation +while the other metrics are more turn-specific. It follows that relevance and correct- +ness with BM25C+R on the 6.7B TNLGv2 model outperform the 530B TNLGv2 +model with manual examples. However, specificity performs worse. Choosing both +diverse ratings and similar example points are important. This finding further sup- +ports the idea that the nature of the data used to train these LLMs is important. Had +the training data been more similar to conversational language, an increase could +have been observed in the correlations for these metrics without choosing algorith- +mically similar examples. +TNLGv2 struggles with understandability; it performs the worst at the highest +correlation of 0.193. It also has unstable performance; performing at significance +with random examples and with algorithmically chosen examples on 6.7B, but not +with manually chosen ones. This shows that choosing examples with diverse ratings +helps a model less for metrics that it already performs poorly on; it would better +benefit from examples that are similar. +In general, even with the difference in training data, it is easier to obtain an overall +sense of the conversation than a metric for a single turn for the larger models due +to the large amount of parameters and variety of data that they have seen. When +choosing examples based on context, the larger models generally perform worse; it +appears that having different examples is more important for dialog-level metrics +than for turn-level metrics. + +10 +J. Huynh et al. +manual +random +BM25C+R +Quality +762M ft +6.7B +530B +6.7B +530B +6.7B +530B +Interesting +0.408 +0.455 +0.474 +0.293 ± 0.03 +0.398± 0.02 +0.358 +0.383 +Engaging +0.318 +0.459 +0.484 +0.235± 0.04 +0.352± 0.02 +0.378 +0.383 +Specific +0.267 +0.305 +0.450 +0.188± 0.02 +0.289± 0.01 +0.268 +0.322 +Relevant +0.152 +0.214 +0.300 +0.179± 0.04 +0.299± 0.03 +0.392 +0.357 +Correct +0.133 +0.195 +0.393 +0.171± 0.04 +0.338± 0.04 +0.399 +0.377 +Sem. Approp. +0.155 +0.292 +0.395 +0.163± 0.03 +0.270± 0.01 +0.291 +0.294 +Understandable +0.111 +0.021* +0.036* +0.146± 0.02 +0.129± 0.02 +0.193 +0.062* +Fluent +0.224 +0.164 +0.195 +0.052*± 0.03 +0.112*± 0.01 +0.096* +0.178 +Overall +0.209 +0.371 +0.475 +0.256± 0.02 +0.380± 0.01 +0.474 +0.514 +Table 1 Turn-level fine-grained metrics on the FED dataset for manually, randomly, and BM25 +chosen examples over the TNLGv2 6.7B and 530B models. BM25C+R stands for examples chosen +by BM25 considering both the context and the response of the test point. Bold values indicate the +best value for the metric and * values indicate correlations that are not statistically significant. +manual +random +BM25C +Quality +762M ft +6.7B +530B +6.7B +530B +6.7B +530B +Coherent +0.251 +0.599 +0.727 +0.443± 0.03 +0.533± 0.02 +0.618 +0.512 +Error Recovery +0.165* +0.474 +0.578 +0.348± 0.04 +0.463± 0.06 +0.492 +0.419 +Consistent +0.116* +0.276 +0.382 +0.270± 0.02 +0.205* ± 0.04 +0.238 +0.046* +Diverse +0.420 +0.625 +0.620 +0.434± 0.06 +0.490± 0.02 +0.496 +0.548 +Topic Depth +0.476 +0.640 +0.659 +0.361± 0.03 +0.531± 0.04 +0.559 +0.472 +Likeable +0.262 +0.619 +0.686 +0.511± 0.03 +0.580± 0.01 +0.568 +0.515 +Understanding +0.306 +0.517 +0.638 +0.479± 0.06 +0.496± 0.02 +0.567 +0.428 +Flexible +0.293 +0.617 +0.656 +0.491± 0.05 +0.553± 0.03 +0.614 +0.451 +Informative +0.288 +0.569 +0.547 +0.391± 0.04 +0.452± 0.04 +0.523 +0.419 +Inquisitive +0.163 +0.537 +0.527 +0.436± 0.05 +0.444± 0.02 +0.334 +0.252 +Overall +0.443 +0.630 +0.688 +0.479± 0.05 +0.570± 0.02 +0.607 +0.531 +Table 2 Dialog-level fine-grained metrics on the FED dataset for manually, randomly, and BM25 +chosen examples over the TNLGv2 6.7B and 530B models. BM25C stands for examples chosen +by BM25 considering only the context of the test point. +4.1.2 Comparisons Across LLMs +These model comparisons are performed using manually chosen in-context exam- +ples, since that is what generally performed the best in both turn-level and dialog- +level metrics in Tables 3 and 4. Comparisons across smaller versions of BLOOM +and OPT can be found in Appendix E. +Even though the large versions of BLOOM and OPT could not be run, it is ap- +parent that both of these models outperform TNLGv2 on understandability, and +that OPT 6.7B can outperform TNLGv2 530B on fluency. Data dissimilarities were +noted above in Section 4.1.1 between the TNLGv2 model and the FED data. Al- +though BLOOM was only trained on some English data, it has still seen some ca- +sual language, while OPT was partially trained on Reddit data. Thus the language +appearing in the BLOOM and OPT training sets more closely matches that of the +conversations used here. This explains the increase in performance. +BLOOM 7B outperforms 6.7B TNLGv2 on correctness, while OPT 6.7B out- +performs 6.7B TNLGv2 on relevance, correctness, semantic appropriateness and + +Very Large Language Models for Dialog Evaluation +11 +TNLG +BLOOM +OPT +Flan-T5 +InstructGPT +Quality +6.7B +530B +7B +6.7B +30B +3B +175B +Interesting +0.455 +0.474 +0.291 +0.429 +0.399 +0.519 +0.551 +Engaging +0.459 +0.484 +0.435 +0.446 +0.349 +0.425 +0.489 +Specific +0.305 +0.450 +0.296 +0.275 +0.207 +0.433 +0.421 +Relevant +0.214 +0.300 +0.109 +0.272 +0.289 +0.435 +0.471 +Correct +0.195 +0.393 +0.235 +0.342 +0.354 +0.378 +0.376 +Sem. Approp. +0.292 +0.395 +0.258 +0.371 +0.382 +0.277 +0.374 +Understandable +0.021* +0.036* +0.159 +0.131 +0.073* +0.297 +0.382 +Fluent +0.164 +0.195 +0.111 +0.201 +0.188 +0.200 +0.204 +Overall +0.371 +0.475 +0.274 +0.368 +0.433 +0.445 +0.536 +Table 3 Turn-level fine-grained metrics on the FED dataset for manually chosen examples over +the TNLGv2, BLOOM, OPT, Flan-T5, and InstructGPT models. +TNLG +BLOOM +OPT +FLAN-T5 +InstructGPT +Quality +6.7B +530B +7B +6.7B +30B +3B +175B +Coherent +0.599 +0.727 +0.613 +0.558 +0.584 +0.730 +0.707 +Error Recovery +0.474 +0.578 +0.474 +0.377 +0.479 +0.398 +0.560 +Consistent +0.276 +0.382 +0.323 +0.237 +0.309 +0.410 +0.517 +Diverse +0.625 +0.620 +0.498 +0.454 +0.607 +0.544 +0.628 +Topic Depth +0.640 +0.659 +0.637 +0.544 +0.609 +0.650 +0.680 +Likeable +0.619 +0.686 +0.566 +0.544 +0.571 +0.659 +0.672 +Understanding +0.517 +0.638 +0.484 +0.505 +0.483 +0.637 +0.694 +Flexible +0.617 +0.656 +0.499 +0.528 +0.592 +0.595 +0.688 +Informative +0.569 +0.547 +0.462 +0.497 +0.522 +0.662 +0.647 +Inquisitive +0.537 +0.527 +0.539 +0.461 +0.537 +0.487 +0.578 +Overall +0.630 +0.688 +0.531 +0.374 +0.530 +0.585 +0.690 +Table 4 Dialog-level fine-grained metrics on the FED dataset for manually chosen examples over +the TNLGv2, BLOOM, OPT, Flan-T5, and InstructGPT models. +fluency in addition. As previously noted, relevance and correctness are turn-level +metrics that take more of the context into account, so with training data that is more +similar to casual language, these models perform better. It should be noted that +the overall turn- and dialog-level quality results were not surpassed by any smaller +model, thus the very large models will have an advantage for overall metrics. +Flan-T5 outperforms the largest model, TNLGv2 530B, on interestingness, rele- +vance, and understandability at turn level and coherence, consistency, and informa- +tiveness at dialog level. There is a larger performance drop for the semantic appro- +priateness, error recovery, and overall dialog-level quality metrics. Error recovery +is a relatively new metric [21]. Even though Flan-T5 was fine-tuned on many di- +alog tasks, it may not have seen data that addresses this specific metric. Flan-T5 +only has 3B parameters, and the fact that it outperforms 530B TNLGv2 shows the +importance of use of dialog data during pre-training or fine-tuning. +InstructGPT, being fine-tuned with prompting at 175B parameters, is more suit- +able for the present experiments. It performs very well on both turn- and dialog-level +metrics, outperforming 530B TNLGv2 on almost all metrics. Since InstructGPT has +already seen prompting, the model can better understand a task through only instruc- +tions or combinations of instructions and in-context examples. + +12 +J. Huynh et al. +Model +TU +DZ +PU +DGU +DGR +FT +EG +FD +Experiments with Random Examples +4ex +0.112 ± 0.03 0.428 ± 0.01 0.403 ± 0.02 0.542 ± 0.00 0.338 ± 0.01 0.318 ± 0.02 0.248 ± 0.04 0.290 ± 0.05 +8ex +0.169 ± 0.03 0.430 ± 0.03 0.331 ± 0.03 0.570 ± 0.01 0.429 ± 0.05 0.337 ± 0.01 0.200 ± 0.04 0.339 ± 0.18 +12ex +0.148 ± 0.03 0.453 ± 0.02 0.384 ± 0.02 0.565 ± 0.01 0.410 ± 0.06 0.412 ± 0.03 0.160 ± 0.02 0.351 ± 0.08 +Experiments with Algorithmically Retrieved Examples +4ex BM25R +0.247 +0.424 +0.252 +0.482 +0.342 +0.364 +0.144 +0.264 +4ex BM25C +0.129 +0.424 +0.339 +0.510 +0.370 +0.172 +0.192 +0.549 +4ex BM25C+R +0.213 +0.441 +0.432 +0.479 +0.371 +0.137 +0.211 +0.479 +8ex BM25R +0.309 +0.487 +0.275 +0.536 +0.304 +0.426 +0.121 +0.419 +8ex BM25C +0.227 +0.564 +0.460 +0.627 +0.387 +0.323 +0.123 +0.518 +8ex BM25C+R +0.185 +0.458 +0.439 +0.526 +0.308 +0.377 +0.171 +0.530 +12ex BM25R +0.300 +0.474 +0.358 +0.570 +0.337 +0.393 +0.095* +0.414 +12ex BM25C +0.278 +0.688 +0.449 +0.674 +0.397 +0.377 +0.106* +0.492 +12ex BM25C+R +0.202 +0.491 +0.452 +0.465 +0.349 +0.358 +0.148 +0.493 +Best of DSTC10 baselines +0.319 +0.532 +0.493 +0.596 +0.363 +0.247 +0.395 +0.555 +Table 5 Spearman correlation of model predictions for overall quality with human ratings for +TNLGv2 530B model with algorithmically chosen examples. TU, PU, PZ, DZ, CG, DGU, DGR, +EG, FT and FD are abbreviations for TopicalChat-USR, PersonaChat-USR [22], PersonaChat- +Zhao [46], DailyDialog-Zhao [46], ConvAI2-GRADE [14], DailyDialog-Gupta [11], DailyDialog- +GRADE [14], Empathetic-GRADE [14], FED-Turn and FED-Dial [21]. +Model +TU +DZ +PU +DGU +DGR +FT +EG +FD +Few-shot in-context Experiments +BLOOM-7B-4ex +0.027* +0.075 +0.123 +0.127 +0.131 +0.117 +0.012 +0.289 +OPT-6.7B-4ex +0.115 +0.258 +0.444 +0.228 +0.091* +0.486 +0.044* +0.657 +TNLG-6.7B-4ex +0.124 +0.198 +0.237 +0.209 +0.214 +0.296 +0.057* +0.314 +TNLG-530B-4ex +0.129 +0.424 +0.339 +0.510 +0.370 +0.172 +0.192 +0.549 +Flan-T5-3B-4ex +0.447 +0.657 +0.578 +0.714 +0.379 +0.442 +0.396 +0.492 +InstructGPT-175B-4ex +0.616 +0.716 +0.687 +0.746 +0.472 +0.506 +0.305 +0.412 +Zero-shot Experiments +Flan-T5-3B-0ex +0.357 +0.599 +0.533 +0.677 +0.351 +0.380 +0.418 +0.444 +InstructDial-3B-0ex +0.446 +0.601 +0.376 +0.634 +0.286 +0.263 +0.475 +0.228 +Best of DSTC10 baselines +0.319 +0.532 +0.493 +0.596 +0.363 +0.247 +0.395 +0.555 +Best TNLGv2 value +0.309 +0.688 +0.460 +0.678 +0.429 +0.426 +0.248 +0.549 +Table 6 Spearman correlation of model predictions for overall quality with human ratings with 4 +examples chosen with BM25 using context. Macro average scores are also shown. +4.2 DSTC10 Datasets +The same set of experiments were carried out on the 8 datasets in the DSTC10 chal- +lenge in Tables 5 and 6, and Appendix F. The previous best performing metrics on +DSTC10 are compiled from [13], which include both reference-free and fine-tuned +metrics (see Appendix G). Quality is evaluated in terms of how good a response is +to the context. +4.2.1 In-Context Example Selection +Experiments are performed with randomly chosen examples and examples that were +chosen by BM25 over 4, 8, and 12 examples in Table 5 and Appendix F. Higher +correlation results are obtained on 4 datasets (DZ, DGU, DGR, and FT) with com- +parable results on 3 datasets (TU, PU, and FD), as compared to the best DSTC10 +baselines. Most of the best results are on the 530B TNLGv2 model, which will be + +Very Large Language Models for Dialog Evaluation +13 +discussed in this section, as compared to the 6.7B TNLGv2 model. Several factors +are relevant here: the language of the dataset, the way the dataset was created, and +how the dataset was annotated. +DailyDialog contains more formal language, thus TNLGv2 should perform well +since its training dataset includes data sources with formal language. DZ, DGU, and +DGR almost always perform the best when examples are chosen from looking at +the context; adding the response generally leads to poorer performance. Since these +datasets are annotated for appropriateness and coherence, context is more important +than a more turn-specific metric. +TopicalChat was created through knowledge-grounding. The conversations could +thus have more substance than a purely open-domain un-prompted conversation. It +thus follows that response selection will work the best when choosing examples. +PersonaChat has conversations that are persona-conditioned, so the quality of the +conversation should take into account the entire conversation for each persona. It +performs better with examples chosen for context and response or with just context. +FED is split into turn- and dialog-level annotations, thus, for turn-level annota- +tions choosing examples based on responses should work best, and for dialog-level +annotations choosing examples based on either the context or the context and re- +sponse should perform the best. Choosing examples with context and response per- +forms the best for EG, but randomly choosing examples outperforms that result. It +may be that with emotionally grounded conversations, the model needs more, or +more diverse examples due to the different ways emotion can be expressed. +In general, choosing examples algorithmically improves performance over ran- +domly choosing examples. This is consistent with previous experiments above. +However, randomly-chosen examples perform better on the DGR and EG datasets +on the 530B TNLGv2 model. This may be because these two datasets were rated +for coherence. Algorithmically, choosing examples based on context and response +performs the best on EG, as was seen for coherence in FED in Section 4.1.1. +4.2.2 Comparisons Across LLMs +Table 6 compares the evaluation results across various LLMs. Due to model input +length restrictions, the following experiments were carried out using 4 in-context ex- +amples or in a zero-shot setting. BM25 is only used with the context as the example +selection strategy, since it performed well with the TNLGv2 models. +In the few-shot setting, models that were not fine-tuned or trained with prompting +(BLOOM, OPT) did not have consistent results across the datasets. However, those +that were fine-tuned or prompted (Flan-T5, InstructGPT, InstructDial) had results +that were close to or surpassed the previous best DSTC10 baselines. InstructGPT +performed the best. Even in the zero-shot setting, Flan-T5 outperforms the baseline +in 6 of the datasets, and InstructDial in 5. +These results clearly show that for dialog evaluation, it is insufficient to simply +train on large amounts of general internet data. Specialized approaches such as in- +struction tuning on multiple tasks improve the generalization capabilities of models + +14 +J. Huynh et al. +in zero- and few-shot settings. It is not surprising that InstructGPT performs the best +since it fine-tunes a very large language model with instructions. +5 Conclusion +LLMs have the potential to significantly contribute to dialog evaluation. Current +LLMs perform well for this task in a few-shot setting. However, this performance +varies greatly depending on the content of and number of examples in the prompt. +Models prefer more similar examples for metrics that they struggle to evaluate, +while preferring examples with more diverse ratings for metrics that they can evalu- +ate well. Very large language models also still afford performance gains, especially +for overall quality evaluation at the turn and dialog level. Even though large lan- +guage models perform better at dialog-level fine-grained metrics, there are still pre- +viously shown issues with how these models understand social situations and use +context that may hinder further improvement if not addressed. +Performance is also affected by the model’s training data. Smaller language mod- +els that are fine-tuned on instructions, trained on dialog data, and/or trained on mul- +tiple dialog tasks outperform larger language models. These smaller models also +perform more consistently over different domains. This indicates that LLMs should +have more diverse pre-training data in order to be able to handle a larger variety of +tasks in few or zero-shot settings. +More work needs to be done on understanding how a large language model mod- +els different types of tasks. In-context example selection and example wording still +remains unstable across large language models in many tasks, and the performance +variation over different dialog domains in this paper demonstrates that as well. +Presently, the LLMs explored in this paper have their own strengths. Smaller +models such as BLOOM and OPT could share more training data similarity with +dialog tasks based on their objective. TNLGv2 530B provides a very large lan- +guage model that has shown improvement in dialog evaluation along with other +NLP tasks. Flan-T5 and InstructDial show the efficacy of fine-tuning a LLM on +dialog tasks, and InstructGPT shows the importance of training a model to better +recognize prompts. The evaluations of these models provide suggestions for the +characteristics of the best LLMs to use for dialog evaluation. Future work in using +LLMs for other NLP tasks can benefit from such comprehensive analyses. Once a +better understanding of LLMs is realized, the capabilities of large language models +for zero- and few-shot tasks will increase greatly. +6 Acknowledgements +We would like to thank Microsoft for allowing us to use TNLGv2. J.H. was sup- +ported by the NSF Graduate Research Fellowship under Grant Nos. DGE1745016 + +Very Large Language Models for Dialog Evaluation +15 +and DGE2140739. The opinions expressed in this paper do not necessarily reflect +those of that funding agency. +References +[1] Agarwal O, Yang Y, Wallace BC, Nenkova A (2021) Interpretability analysis for named en- +tity recognition to understand system predictions and how they can improve. Computational +Linguistics 47(1):117–140 +[2] Baumgartner J, Zannettou S, Keegan B, Squire M, Blackburn J (2020) The pushshift reddit +dataset. In: Proceedings of the international AAAI conference on web and social media, +vol 14, pp 830–839 +[3] Beltagy I, Lo K, Cohan A (2019) Scibert: A pretrained language model for scientific text. +arXiv preprint arXiv:190310676 +[4] BigScience Workshop (2022) Bloom (revision 4ab0472). DOI 10.57967/hf/0003, URL +https://huggingface.co/bigscience/bloom +[5] Brown T, Mann B, Ryder N, Subbiah M, Kaplan JD, Dhariwal P, Neelakantan A, Shyam P, +Sastry G, Askell A, et al (2020) Language models are few-shot learners. Advances in neural +information processing systems 33:1877–1901 +[6] Chen Z, Sadoc J, D’Haro LF, Banchs R, Rudnicky A (2021) Automatic evaluation and mod- +eration of open-domain dialogue systems. arXiv preprint arXiv:211102110 +[7] Chowdhery A, Narang S, Devlin J, Bosma M, Mishra G, Roberts A, Barham P, Chung HW, +Sutton C, Gehrmann S, et al (2022) Palm: Scaling language modeling with pathways. arXiv +preprint arXiv:220402311 +[8] Chung HW, Hou L, Longpre S, Zoph B, Tay Y, Fedus W, Li E, Wang X, Dehghani +M, Brahma S, et al (2022) Scaling instruction-finetuned language models. arXiv preprint +arXiv:221011416 +[9] Gao L, Biderman S, Black S, Golding L, Hoppe T, Foster C, Phang J, He H, Thite A, +Nabeshima N, Presser S, Leahy C (2020) The Pile: An 800gb dataset of diverse text for +language modeling. arXiv preprint arXiv:210100027 +[10] Gao T, Fisch A, Chen D (2020) Making pre-trained language models better few-shot learners. +arXiv preprint arXiv:201215723 +[11] Gupta P, Mehri S, Zhao T, Pavel A, Eskenazi M, Bigham J (2019) Investigating evaluation of +open-domain dialogue systems with human generated multiple references. In: Proceedings of +the 20th Annual SIGdial Meeting on Discourse and Dialogue, Association for Computational +Linguistics, Stockholm, Sweden, pp 379–391, DOI 10.18653/v1/W19-5944, URL https: +//aclanthology.org/W19-5944 +[12] Gupta P, Tsvetkov Y, Bigham J (2021) Synthesizing adversarial negative responses for robust +response ranking and evaluation. In: Findings of the Association for Computational Linguis- +tics: ACL-IJCNLP 2021, Association for Computational Linguistics, Online, pp 3867–3883, +DOI 10.18653/v1/2021.findings-acl.338, URL https://aclanthology.org/2021. +findings-acl.338 +[13] Gupta P, Jiao C, Yeh YT, Mehri S, Eskenazi M, Bigham JP (2022) Improving zero and few- +shot generalization in dialogue through instruction tuning. arXiv preprint arXiv:220512673 +[14] Huang L, Ye Z, Qin J, Lin L, Liang X (2020) GRADE: Automatic graph-enhanced coherence +metric for evaluating open-domain dialogue systems. In: Proceedings of the 2020 Conference +on Empirical Methods in Natural Language Processing (EMNLP), Association for Compu- +tational Linguistics, Online, pp 9230–9240, DOI 10.18653/v1/2020.emnlp-main.742, URL +https://aclanthology.org/2020.emnlp-main.742 +[15] Jiang Z, Xu FF, Araki J, Neubig G (2020) How can we know what language models know? +Transactions of the Association for Computational Linguistics 8:423–438 + +16 +J. Huynh et al. +[16] Kaplan J, McCandlish S, Henighan T, Brown TB, Chess B, Child R, Gray S, Radford A, Wu +J, Amodei D (2020) Scaling laws for neural language models. DOI 10.48550/ARXIV.2001. +08361, URL https://arxiv.org/abs/2001.08361 +[17] Liu J, Shen D, Zhang Y, Dolan B, Carin L, Chen W (2021) What makes good in-context +examples for gpt-3? DOI 10.48550/ARXIV.2101.06804, URL https://arxiv.org/ +abs/2101.06804 +[18] Liu Y, Ott M, Goyal N, Du J, Joshi M, Chen D, Levy O, Lewis M, Zettlemoyer L, Stoy- +anov V (2019) Roberta: A robustly optimized bert pretraining approach. arXiv preprint +arXiv:190711692 +[19] Lu Y, Bartolo M, Moore A, Riedel S, Stenetorp P (2021) Fantastically ordered prompts and +where to find them: Overcoming few-shot prompt order sensitivity. DOI 10.48550/ARXIV. +2104.08786, URL https://arxiv.org/abs/2104.08786 +[20] Manning CD (2008) Introduction to information retrieval. Syngress Publishing, +[21] Mehri S, Eskenazi M (2020) Unsupervised evaluation of interactive dialog with DialoGPT. +In: Proceedings of the 21th Annual Meeting of the Special Interest Group on Discourse and +Dialogue, Association for Computational Linguistics, 1st virtual meeting, pp 225–235, URL +https://aclanthology.org/2020.sigdial-1.28 +[22] Mehri S, Eskenazi M (2020) USR: An unsupervised and reference free evaluation met- +ric for dialog generation. In: Proceedings of the 58th Annual Meeting of the Association +for Computational Linguistics, Association for Computational Linguistics, Online, pp 681– +707, DOI 10.18653/v1/2020.acl-main.64, URL https://aclanthology.org/2020. +acl-main.64 +[23] Ouyang L, Wu J, Jiang X, Almeida D, Wainwright CL, Mishkin P, Zhang C, Agarwal S, +Slama K, Ray A, et al (2022) Training language models to follow instructions with human +feedback. arXiv preprint arXiv:220302155 +[24] Papineni K, Roukos S, Ward T, Zhu WJ (2002) Bleu: a method for automatic evaluation +of machine translation. In: Proceedings of the 40th annual meeting of the Association for +Computational Linguistics, pp 311–318 +[25] Phy V, Zhao Y, Aizawa A (2020) Deconstruct to reconstruct a configurable evaluation met- +ric for open-domain dialogue systems. In: Proceedings of the 28th International Confer- +ence on Computational Linguistics, International Committee on Computational Linguistics, +Barcelona, Spain (Online), pp 4164–4178, DOI 10.18653/v1/2020.coling-main.368, URL +https://aclanthology.org/2020.coling-main.368 +[26] Radford A, Wu J, Child R, Luan D, Amodei D, Sutskever I, et al (2019) Language models +are unsupervised multitask learners. OpenAI blog 1(8):9 +[27] Raffel C, Shazeer N, Roberts A, Lee K, Narang S, Matena M, Zhou Y, Li W, Liu PJ (2020) +Exploring the limits of transfer learning with a unified text-to-text transformer. Journal of +Machine Learning Research 21(140):1–67, URL http://jmlr.org/papers/v21/ +20-074.html +[28] Robertson S, Zaragoza H (2009) The probabilistic relevance framework: Bm25 and beyond. +Found Trends Inf Retr 3(4):333–389, DOI 10.1561/1500000019, URL https://doi. +org/10.1561/1500000019 +[29] Roller S, Dinan E, Goyal N, Ju D, Williamson M, Liu Y, Xu J, Ott M, Smith EM, Boureau YL, +Weston J (2021) Recipes for building an open-domain chatbot. In: Proceedings of the 16th +Conference of the European Chapter of the Association for Computational Linguistics: Main +Volume, Association for Computational Linguistics, Online, pp 300–325, DOI 10.18653/v1/ +2021.eacl-main.24, URL https://aclanthology.org/2021.eacl-main.24 +[30] Rubin O, Herzig J, Berant J (2021) Learning to retrieve prompts for in-context learning. +DOI 10.48550/ARXIV.2112.08633, URL https://arxiv.org/abs/2112.08633 +[31] Sai AB, Mohankumar AK, Arora S, Khapra MM (2020) Improving dialog evaluation with a +multi-reference adversarial dataset and large scale pretraining. Transactions of the Associa- +tion for Computational Linguistics 8:810–827 + +Very Large Language Models for Dialog Evaluation +17 +[32] Sanh V, Webson A, Raffel C, Bach SH, Sutawika L, Alyafeai Z, Chaffin A, Stiegler A, Scao +TL, Raja A, et al (2021) Multitask prompted training enables zero-shot task generalization. +arXiv preprint arXiv:211008207 +[33] Sap M, LeBras R, Fried D, Choi Y (2022) Neural theory-of-mind? on the limits of social +intelligence in large lms. arXiv preprint arXiv:221013312 +[34] Smith S, Patwary M, Norick B, LeGresley P, Rajbhandari S, Casper J, Liu Z, Prabhumoye +S, Zerveas G, Korthikanti V, et al (2022) Using deepspeed and megatron to train megatron- +turing nlg 530b, a large-scale generative language model. arXiv preprint arXiv:220111990 +[35] Su H, Kasai J, Wu CH, Shi W, Wang T, Xin J, Zhang R, Ostendorf M, Zettlemoyer L, Smith +NA, Yu T (2022) Selective annotation makes language models better few-shot learners. DOI +10.48550/ARXIV.2209.01975, URL https://arxiv.org/abs/2209.01975 +[36] Thoppilan R, De Freitas D, Hall J, Shazeer N, Kulshreshtha A, Cheng HT, Jin A, Bos T, +Baker L, Du Y, et al (2022) Lamda: Language models for dialog applications. arXiv preprint +arXiv:220108239 +[37] Trinh TH, Le QV (2018) A simple method for commonsense reasoning. DOI 10.48550/ +ARXIV.1806.02847, URL https://arxiv.org/abs/1806.02847 +[38] Wei J, Bosma M, Zhao VY, Guu K, Yu AW, Lester B, Du N, Dai AM, Le QV (2021) Fine- +tuned language models are zero-shot learners. arXiv preprint arXiv:210901652 +[39] Wei J, Tay Y, Bommasani R, Raffel C, Zoph B, Borgeaud S, Yogatama D, Bosma M, Zhou +D, Metzler D, et al (2022) Emergent abilities of large language models. arXiv preprint +arXiv:220607682 +[40] Zellers R, Holtzman A, Rashkin H, Bisk Y, Farhadi A, Roesner F, Choi Y (2019) Defend- +ing against neural fake news. DOI 10.48550/ARXIV.1905.12616, URL https://arxiv. +org/abs/1905.12616 +[41] Zhang C, Chen Y, D’Haro LF, Zhang Y, Friedrichs T, Lee G, Li H (2021) DynaEval: +Unifying turn and dialogue level evaluation. In: Proceedings of the 59th Annual Meet- +ing of the Association for Computational Linguistics and the 11th International Joint Con- +ference on Natural Language Processing (Volume 1: Long Papers), Association for Com- +putational Linguistics, Online, pp 5676–5689, DOI 10.18653/v1/2021.acl-long.441, URL +https://aclanthology.org/2021.acl-long.441 +[42] Zhang S, Roller S, Goyal N, Artetxe M, Chen M, Chen S, Dewan C, Diab M, Li X, +Lin XV, et al (2022) Opt: Open pre-trained transformer language models. arXiv preprint +arXiv:220501068 +[43] Zhang Y, Sun S, Galley M, Chen YC, Brockett C, Gao X, Gao J, Liu J, Dolan B (2019) +Dialogpt: Large-scale generative pre-training for conversational response generation. arXiv +preprint arXiv:191100536 +[44] Zhang Z, Guo T, Chen M (2021) Dialoguebert: A self-supervised learning based dialogue +pre-training encoder. In: Proceedings of the 30th ACM International Conference on Informa- +tion & Knowledge Management, pp 3647–3651 +[45] Zhao T, Zhao R, Eskenazi M (2017) Learning discourse-level diversity for neural dialog +models using conditional variational autoencoders. arXiv preprint arXiv:170310960 +[46] Zhao T, Lala D, Kawahara T (2020) Designing precise and robust dialogue response eval- +uators. In: Proceedings of the 58th Annual Meeting of the Association for Computational +Linguistics, Association for Computational Linguistics, Online, pp 26–33, DOI 10.18653/ +v1/2020.acl-main.4, URL https://aclanthology.org/2020.acl-main.4 +[47] Zhao Z, Wallace E, Feng S, Klein D, Singh S (2021) Calibrate before use: Improving few- +shot performance of language models. In: International Conference on Machine Learning, +PMLR, pp 12,697–12,706 + +18 +J. Huynh et al. +A LLMs and Their Training/Fine-tuning Data +Seen Dialog Fine-tuned +Flan-T5 +✓ +✓ +InstructDial +✓ +✓ +InstructGPT +✓ +✓ +BLOOM +✓ +× +OPT +✓ +× +TNLGv2 +× +× +Table 7 LLMs with the datasets they were trained on. During training or fine-tuning: “Seen Dia- +log” indicates that the model has explicitly seen dialog datasets and therefore elements of casual +language, and “fine-tuned” indicates that the model was fine-tuned on dialog data. TNLGv2 has +not seen datasets explicitly categorized as having dialog, but elements of casual language may be +included in the Common Crawl snapshots and other internet-based corpora. Symbols: ✓means that +the category is included and × means that the category is not included. +B Prompt format examples FED +Task: Given a dialog history and a response, rate how interesting the response is with regards +to the dialog history. +== Example 1 == +A: Hi! +B: Hi. This is a pleasant surprise. +A: Haha...thanks! how did you like the gift? +Response: Currently unpacking it I guess. How’s your morning? +Rating: 1/2 +A: Hope you like it! Morning is good. Busy finishing up stuff before the holidays. +B: I think I traveled too much the last couple of months so no holiday for me. But I’m okay +with that. Going anywhere exciting? +A: Yes +Response: Where to? +Rating: 1/2 +A: Hawaii... looking forward to warm beaches. +Response: WOW. Which island? I like Hawaii. +Rating: 2/2 +Table 8 An example of a prompt with one example from FED [21]. Interestingness was rated in +FED over three values corresponding to 0/2, 1/2, and 2/2. The resulting output is truncated to the +integer value of 0, 1, or 2 to be used in evaluation. + +Very Large Language Models for Dialog Evaluation +19 +C Prompt format examples DSTC10 +Instruction: Given a conversation and a response, choose if the response is a good response +to the context +Example +Background info: none +Conversation: +Person A: did your meal meet with your approval ? +Response: yes , i did . it was a good meal . +Question: Is the above response a good response to the conversation? +Answer: Yes +Background info: none +Conversation: +Person B: i really do hate public transportation. +Person A: i agree , it ’s just never on time. +Response : you ’re right. +Question: Is the above response a good response to the conversation? +Answer: +Table 9 An example of a prompt with examples from DSTC 10. +D Additional algorithmically chosen FED examples +BM25C +BM25R +Quality +7B +530B +7B +530B +Interesting +0.336 0.389 +0.355 +0.385 +Engaging +0.308 0.332 +0.328 +0.389 +Specific +0.217 0.224 +0.297 +0.329 +Relevant +0.338 0.314 +0.311 +0.356 +Correct +0.333 0.341 +0.300 +0.383 +Sem. Approp. +0.261 0.270 +0.287 +0.337 +Understandable 0.141 0.028* 0.169 0.029* +Fluent +0.106 0.147 0.096* 0.121 +Overall +0.435 0.438 +0.360 +0.407 +Table 10 Turn-level fine-grained metrics on the FED dataset for algorithmically chosen examples +over the TNLGv2 6.7B and 530B models. BM25C stands for examples chosen by BM25 consid- +ering the context and BM25R stands for examples chosen by BM25 considering the response. + +20 +J. Huynh et al. +E Additional LLM sizes on FED +BLOOM +OPT +Quality +560M +1.1B +1.7B +3B +125M 350M 1.3B 2.7B +Interesting +0.282 +0.331 +0.336 +0.328 +0.187 +0.186 0.388 0.245 +Engaging +0.217 +0.320 +0.278 +0.418 +0.121 +0.252 0.398 0.292 +Specific +0.030* 0.065* 0.204 +0.353 +0.197 0.004* 0.217 0.222 +Relevant +0.076* 0.056* 0.072* 0.091* 0.146 +0.105 0.231 0.177 +Correct +0.106 +0.146 +0.124 +0.173 +0.119 +0.152 0.327 0.270 +Sem. Approp. +0.048* 0.228 +0.205 +0.265 +0.148 +0.278 0.274 0.296 +Understandable -0.017* 0.043* -0.005* 0.087* 0.058* 0.021* 0.189 0.205 +Fluent +0.158 +0.223 0.097* 0.091* 0.109 0.087* 0.158 0.163 +Overall +0.086* 0.179 0.076* 0.285 +0.134 +0.219 0.338 0.197 +Table 11 Turn-level fine-grained metrics on the FED dataset for manually chosen examples over +the smaller sizes of BLOOM and OPT. +BLOOM +OPT +Quality +560M +1.1B +1.7B +3B +125M 350M 1.3B 2.7B +Coherent +0.499 +0.533 +0.531 0.531 0.490 0.514 0.528 0.435 +Error Recovery 0.293 +0.298 +0.322 0.448 0.168 0.380 0.342 0.348 +Consistent +0.217 +0.238 0.129* 0.264 0.193 0.191 0.250 0.268 +Diverse +0.345 +0.430 +0.461 0.518 0.451 0.304 0.491 0.531 +Topic Depth +0.418 +0.414 +0.519 0.462 0.228 0.302 0.462 0.454 +Likeable +0.310 +0.374 +0.421 0.476 0.467 0.395 0.462 0.535 +Understanding +0.276 +0.312 +0.257 0.371 0.389 0.283 0.414 0.494 +Flexible +0.269 +0.432 +0.400 0.441 0.458 0.377 0.460 0.432 +Informative +0.149* 0.384 +0.372 0.537 0.378 0.402 0.381 0.544 +Inquisitive +0.198 +0.350 +0.318 0.339 0.489 0.300 0.439 0.413 +Overall +0.262 0.146* 0.207 0.261 -0.000* 0.319 0.452 0.437 +Table 12 Dialog-level fine-grained metrics on the FED dataset for manually chosen examples over +the smaller sizes of BLOOM and OPT. + +Very Large Language Models for Dialog Evaluation +21 +F DSTC10 Results For TNLGv2 6.7B +Model +TU +DZ +PU +DGU +DGR +FT +EG +FD +Experiments with Random Examples +4ex +0.034* ± 0.05 0.117 ± 0.02 0.206 ± 0.02 0.080* ± 0.05 0.121± 0.05 0.191 ± 0.06 0.005* ± 0.04 0.228 ± 0.03 +8ex +0.054* ± 0.05 0.160 ± 0.02 0.206 ± 0.03 0.109* ± 0.03 0.139 ± 0.08 0.178 ± 0.02 0.060* ± 0.06 0.238 ± 0.11 +12ex +0.063* ± 0.03 0.149 ± 0.00 0.225 ± 0.01 0.114 ± 0.05 0.143 ± 0.06 0.210 ± 0.03 0.052* ± 0.02 0.127 ± 0.04 +Experiments with Algorithmically Retrieved Examples +4ex BM25R +0.148 +0.218 +0.223 +0.202 +0.094* +0.273 +-0.012* +0.335 +4ex BM25C +0.124 +0.198 +0.237 +0.209 +0.214 +0.296 +0.057* +0.314 +4ex BM25C+R +0.05* +0.142 +0.169 +0.167 +0.083* +0.274 +0.038* +0.339 +8ex BM25R +0.077* +0.270 +0.203 +0.222 +0.128 +0.199 +0.042* +0.335 +8ex BM25C +0.184 +0.328 +0.343 +0.526 +0.176 +0.363 +0.073* +0.387 +8ex BM25C+R +0.029* +0.152 +0.020* +0.092 +0.022* +0.348 +0.024* +0.440 +12ex BM25R +0.069* +0.338 +0.153 +0.213 +0.110* +0.250 +0.026* +0.401 +12ex BM25C +0.285 +0.544 +0.325 +0.678 +0.208 +0.330 +0.042* +0.365 +12ex BM25C+R +0.035* +0.168 +0.088* +0.086* +0.100* +0.407 +0.092* +0.343 +Table 13 Spearman correlation of model predictions with human ratings for TNLGv2 6.7B +model with algorithmically chosen examples. TU, PU, PZ, DZ, CG, DGU, DGR, EG, +FT and FD are abbreviations for TopicalChat-USR, PersonaChat-USR +[22], PersonaChat- +Zhao [46], DailyDialog-Zhao [46], ConvAI2-GRADE [14], DailyDialog-Gupta [11], DailyDialog- +GRADE [14], Empathetic-GRADE [14], FED-Turn and FED-Dial [21]. +G DSTC10 Baseline Results +Model +Fine-Tuned on +TU +DZ +PU DGU DGR +FT +EG +FD +DTSC 10 datasets +USL-H [25] +✓ +0.319 +0.385 0.493 0.481 +0.09 0.115 0.237 0.202 +GRADE [14] +✓ +0.176 +0.532 0.329 0.596 0.254 0.048 0.300 0.106 +DynaEval [41] +✓ +-0.013 0.169 0.148 0.038 0.122 0.247 0.159 0.555 +USR [22] +× +0.291 +0.363 0.140 0.353 0.066 0.055 0.268 0.084 +FED [21] +× +-0.090 -0.080 -0.004 0.025 -0.009 0.173 0.005 0.178 +DEB [31] +× +0.123 +0.486 0.351 0.579 0.363 0.044 0.395 0.141 +Best +0.319 +0.532 0.493 0.596 0.363 0.247 0.395 0.555 +Table 14 Spearman correlation of model predictions with human ratings. The models fine-tuned +on DSTC 10 datasets tend to perform better on the DSTC 10 datasets. TU, PU, PZ, DZ, CG, +DGU, DGR, EG, FT and FD are abbreviations for TopicalChat-USR, PersonaChat-USR [22], +PersonaChat-Zhao [46], DailyDialog-Zhao [46], ConvAI2-GRADE [14], DailyDialog-Gupta [11], +DailyDialog-GRADE [14], Empathetic-GRADE [14], FED-Turn and FED-Dial [21]. + diff --git a/I9FLT4oBgHgl3EQfJi8x/content/tmp_files/load_file.txt b/I9FLT4oBgHgl3EQfJi8x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..323800f8fc3ee21f62f1f3807f6efc2064e19b8e --- /dev/null +++ b/I9FLT4oBgHgl3EQfJi8x/content/tmp_files/load_file.txt @@ -0,0 +1,1465 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf,len=1464 +page_content='Understanding the Effectiveness of Very Large Language Models on Dialog Evaluation Jessica Huynh, Cathy Jiao, Prakhar Gupta, Shikib Mehri, Payal Bajaj, Vishrav Chaudhary, Maxine Eskenazi Abstract Language models have steadily increased in size over the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' They achieve a high level of performance on various natural language processing (NLP) tasks such as question answering and summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Large language models (LLMs) have been used for generation and can now output human-like text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Due to this, there are other downstream tasks in the realm of dialog that can now harness the LLMs’ language understanding capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Dialog evaluation is one task that this paper will explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' It concentrates on prompting with LLMs: BLOOM, OPT, GPT- 3, Flan-T5, InstructDial and TNLGv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The paper shows that the choice of datasets used for training a model contributes to how well it performs on a task as well as on how the prompt should be structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Specifically, the more diverse and relevant the group of datasets that a model is trained on, the better dialog evaluation performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This paper also investigates how the number of examples in the prompt and the type of example selection used affect the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Jessica Huynh Carnegie Mellon University, e-mail: jhuynh@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='edu Cathy Jiao Carnegie Mellon University, e-mail: cljiao@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='edu Prakhar Gupta Carnegie Mellon University, e-mail: prakharg@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='edu Shikib Mehri Amazon, e-mail: asmehri@amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='com (work done while at Carnegie Mellon University) Payal Bajaj Microsoft Turing, e-mail: pabajaj@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='com Vishrav Chaudhary Microsoft Turing, e-mail: vchaudhary@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='com Maxine Eskenazi Carnegie Mellon University, e-mail: max@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='12004v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='CL] 27 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Huynh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 1 Introduction In recent years, language models such as GPT-3 [5] have grown larger, and their per- formance on downstream natural language processing (NLP) tasks has significantly improved in low-resource settings where only a few instances per task are available (few-shot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The larger these models are, the higher their performances trend on tasks such as language generation and evaluation [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' They can generate coherent, fluent and interesting responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, they can also produce responses that are repet- itive and un-engaging [29], in addition to being hard to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Dialog evaluation is the task of assessing the quality of responses generated by dialog models in terms of properties like those mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, one significant impediment for open-domain dialog generation research is the lack of meaningful automatic metrics for open-domain dialog evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Standard language generation metrics have been shown to be ineffective for dialog evaluation [11], a large part of which is because conversations can be followed by multiple valid responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Standard automatic met- rics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' BLEU [24]), which use references for evaluation, cannot deal with this quality, known as the one-to-many response problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Many recently introduced au- tomatic metrics for dialog evaluation [21, 12] have attained increasingly stronger correlations with human judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Since human dialog evaluation typically mea- sures multiple fine-grained properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' appropriate, interesting, consistent), au- tomatic evaluation metrics should be expected to do the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This paper explores several fine-grained metrics that are measured both at turn-level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' relevance and fluency), and dialog-level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' consistency and coherence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Automatic dialog evaluation continues to be an evolving topic, but with fine- grained metrics and definitions varying across different human-annotated datasets [22, 46], it is important to be able to create reasonable automatic metrics with lim- ited data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Large language models (LLMs) that have been pre-trained on large-scale datasets are able to perform zero and few-shot inference [26, 32], and they have ex- hibited good reasoning skills [5, 39] in addition to having implicitly learned some notion of dialog quality [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This makes them suitable for open-domain dialog evaluation in zero-shot and extreme few-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' While there have been a few attempts to use LLMs for dialog evaluation [36], there has not, to our knowledge, been a systematic study of LLMs for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This paper explores several aspects of LLM use in dialog evaluation: the effect of model type and size and the choice of training data as well as the use of in-context examples for dialog evaluation (the number and quality of the examples used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The experiments herein employ bench- marks to test both how well LLMs can be used for fine-grained evaluation, and how generalizable the models’ performance is across multiple domains and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Very Large Language Models for Dialog Evaluation 3 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1 LLMs Several LLMs have been released recently: T5 [27], GPT-3 [5], BLOOM [4], OPT [42], and TNLGv2 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The following models, the sizes of which are shown in Figure 1, are explored here: T5, trained on the 750B Colossal Clean Crawled Corpus (C4) contains heuristi- cally cleaned natural language English text from the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Specific models con- sidered are: – Flan-T5 [8], T5 fine-tuned on 1836 tasks, including dialog tasks and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' – InstructDial [13], T5 fine-tuned specifically on 48 dialog tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' GPT-3 includes a 570B filtered CommonCrawl corpus [27] in addition to Web- Text [26], Books1, Books2, and Wikipedia [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' – InstructGPT (text-davinci-002) [23], GPT-3 fine-tuned with a prompting dataset and 175B parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' BLOOM was trained on 46 languages and 13 programming languages with a multilingual focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' OPT contains data from the RoBERTa corpus [18], the Pile [9], and PushShift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='io Reddit [2, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' TNLGv2 is trained on a subset of the Pile (notably excluding corpora classified as having natural dialog), two CommonCrawl snapshots [27], RealNews [40], and CC-Stories [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 1 Large Language Models, comparison of select approximate sizes 530B I size Model 175B 30B 7B ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='5B BLOOM OPT InstructGPT Flan-T5 InstructDial TNLGv2 LLM Fine-tuned LLM LLM4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Huynh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' As the number of parameters in these models increases, performance also in- creases: TNLGv2 530B, with around three times the number of parameters, out- performs the original GPT-3 on a variety of NLP tasks [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' LLMs are also gener- alizable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' they perform well on many NLP tasks in few-shot settings and zero-shot settings [38, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, several drawbacks and areas for exploration remain for LLMs that should be noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Recent work has shown that performance on certain zero-shot tasks plateaus as model parameter size grows exponentially [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' LLMs also struggle with parsing social situations [33] and correctly using context [1], which are important in dialog settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This raises questions on the performance of LLMs for dialog evaluation, and how an LLM’s performance changes as it increases in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The data that a model is trained on also influences the performance of down- stream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' T5 is fine-tuned on various subtasks, but pre-trained with C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' When pre-trained with domain-specific data, T5 performs better on tasks in that domain [3, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Furthermore, adding several domains of data during pre-training makes the model likely to perform better [18, 42, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Notably, BLOOM, OPT, Flan-T5, In- structGPT, and InstructDial are partially trained or fine-tuned on dialog datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Details on the content of these datasets can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This is impor- tant because natural dialog data is difficult to obtain, so either scripted conversations or Reddit threads are used since they are the most readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This dearth of data is the reason that few-shot prompting is of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' While work such as [39] acknowledges emergent abilities in larger language models in few-shot prompting settings, this paper explores discrepancies in performance specifically for dialog evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='2 Dialog Evaluation Dialog evaluation presents a unique combination of challenges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' it must consider multiple speakers [44], context that informs the current dialog turn, and the one-to- many aspect mentioned above [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Metrics such as USR [22] and FED [21] were created to address some of these challenges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' they are reference-free, capture complex aspects of dialog, and have good correlation with human evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' These metrics use models such as RoBERTa (125 million parameters) [18] and DialoGPT (345 or 762 million param- eters) [43] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, the best performing versions of these models are smaller than most models examined in this paper, and are fine-tuned on dialog data or on a specific dialog task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Other automatic evaluation metrics include GRADE [14] and DEB [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' With current LLMs’ large increase in hyperparameters, their plethora of training data, and their promising generalizable performance on NLP tasks, these model-based metrics should improve as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Very Large Language Models for Dialog Evaluation 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='3 Example selection for few-shot learning The example selection process for prompting LLMs is of great interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Prompting an LLM with a task and a few examples enables the model to adapt to a new task without completely fine-tuning it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In particular, in-context examples can provide im- portant cues to help LLMs make predictions on tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Recent work has used a vari- ety of methods to examine example selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Common methods measure semantic similarity between example embeddings [17, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Alternatively, retrieval methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' BM25 [28]) have been used directly, or as a precursor to training a selection retriever [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' These example selection methods have shown promise in few-shot NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In [35], the two-step framework for annotating and selecting in-context examples from large unlabeled data showed competitive performance across 10 tasks such as classification, commonsense reasoning, dialog state tracking, and code generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' [17] showed that selecting examples with similar sentence embeddings yields higher GPT-3 performance than random selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, the authors acknowledge that further investigation is required to find more efficient in-context example retrieval methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Moreover, the wording and order of examples presented in prompts can also affect model performance [10, 17, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Lu et al [19] observed order sensitivity across 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1B to 175B parameter GPT-2 and GPT-3 models when the models were probed with different text classification tasks and several in-context examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Also, the wording of the in-context examples depends on the data used for model training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' for unfamiliar prompt formats, model performance may decrease [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Increasing the size of the model and the amount of data does not resolve the issue since the same instability is still prevalent [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Thus this paper studies the effect of example selection on dialog evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 3 Evaluation Settings Two settings for dialog evaluation are explored: fine-grained evaluation and multi- domain evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In-context examples are explored in both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1 Fine-Grained Evaluation Fine-grained metrics can be measured at both the turn level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' informativeness and relevance), and the dialog level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' coherence and diversity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The FED dataset [21] is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' It consists of 124 open-domain dialogs of humans with humans or with machines, for which each dialog has 3 responses that are chosen for annotation (8 turn-level and 10 dialog-level qualities along with overall turn- and dialog-level quality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This dataset was chosen due to the large number of previously studied fine- 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Huynh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' grained qualities as listed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1, with the exception of correctness and error recovery, which are only specifically present in FED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In the experiments, the LM is prompted to output a rating (an integer value - see Appendix B) to evaluate each fine-grained quality in a response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The final rating for each fine-grained quality is a weighted sum of the K-top ratings outputted from the LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Formally, given the K-top predicted ratings r1,r2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=',rK along with their corresponding log probabilities, p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=', pK, the weight, wi, of each rating ri is derived as: wi = pi ∑K j=1 pj The final rating, r, is calculated as: r = K ∑ i=1 ri ∗wi In order to provide a more accurate view of the LM’s performance, K = 3 in the following experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Additionally, this scoring mechanism converts the LM pre- dictions onto a continuous scale, which more closely mirrors the average of human ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Results are reported with the Spearman correlations to the average human ratings for each fine-grained quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='2 Multi-domain Evaluation This task tests automatic dialog evaluation metrics for robustness across multiple dialog domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The analysis uses only the overall quality metric since many of the domain datasets do not have fine-grained annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The Spearman correla- tion is used between human ratings and model predictions on the evaluation sets released by DSTC 10 Track 5 [6] “Automatic Evaluation and Moderation of Open- domain Dialogue Systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' These sets contain human judgement ratings for dialog responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In this setting, a model is shown a dialog context and a response, and it outputs “yes” if the response is a good response to that context, otherwise it outputs “no”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' An example can be seen in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The probability of the “goodness” of the response (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=', the rating), g, is calculated as: g = pmodel(yes) pmodel(yes)+ pmodel(no) where pmodel(yes) and pmodel(no) are the log probilities of the model outputs for “yes” and “no”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Evaluation is carried out on 8 representative evaluation sets out of the 14 DSTC10 evaluation sets [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This subset was chosen because it covers multiple domains and datasets, such as persona, topic and chitchat-based responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' A robust dialog metric should perform well across all the domains and evaluation sets considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Very Large Language Models for Dialog Evaluation 7 The evaluation sets used for fine-grained evaluation, FED-Turn (FT) and FED- Dial (FD) [21], are included as two of the eight datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The other datasets include: TopicalChat-USR (TU, knowledge-grounded open-domain conversations rated for six different dialog qualities) [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' PersonaChat-USR (PU, persona-conditioned conversations annotated with the USR schema) [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DailyDialog-Zhao (DZ, more formal language conversations rated for appropriateness) [46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DailyDialog-Gupta (DGU, rated for appropriateness) [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DailyDialog-GRADE (DGR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' annotated for coherence) [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' and Empathetic-GRADE (EG, emotionally grounded conversa- tions annotated for coherence) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Although some of these datasets are not directly annotated for whether a response is good, the metric they use remains a component for overall quality, and thus it is treated as the indicator of the overall quality of the response in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='3 In-Context Examples This paper uses two methods for example selection: random selection, and algorith- mic selection using BM25 [20] which calculates document similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The examples remain consistent for each evaluation test point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The random selection experiment is run three times, and the mean and standard deviation of the runs are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' There are three configurations for BM25 between the test point and each possible example point - comparing the context only (BM25C), the response only (BM25R), and the concatenated context and response together (BM25C+R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' With the FED dataset, an additional method, manual selection, is added for ex- ample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' For each fine-grained dialog quality, a set of three dialogs which span a wide range of ratings is chosen that remains constant over every test point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In theory, the model should be able to show increased performance if it sees examples of very good, good and bad responses for fine-grained metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' For the DSTC10 datasets, an additional experiment tested how the number of examples used affects model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 4 Experiments and Results The in-context example experiments are carried out on the largest available model, 530B TNLGv2, to explore the ceiling of model performance on the dialog evalua- tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='7B TNLGv2 is used for a direct comparison of how much performance gain is provided by using more parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' BLOOM and OPT are examined up to 7B and 30B respectively for the fine- grained metric evaluation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 1 Smaller LLMs do not perform as well with in- 1 Due to limitations in compute power, larger BLOOM and OPT models were not explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' How- ever, as the largest available GPT-3 model is explored, the comparisons appear sufficient to show the performance of a variety of LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Huynh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' context examples unless they have been specifically tuned for the task, so only the 7B and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='7B models for BLOOM and OPT respectively are explored for the DSTC10 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Flan-T5 and InstructDial are analyzed in the 3B setting for con- sistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Lastly, InstructGPT (text-davinci-002) is used, which has 175B parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1 Fine-grained Metric Evaluation FED is separated into turn-level and dialog-level metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The dataset has anno- tations for 8 different turn-level metrics, consisting of interestingness, engaging- ness, specificity, relevance, correctness, semantic appropriateness, understandabil- ity, and fluency, with the addition of overall quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' FED annotates three different responses for each dialog context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' one FED dialog is treated as one example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The corresponding rating is inserted after the response statement in the prompt, an exam- ple of which can be seen in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' FED also looks at 10 different dialog-level metrics for a system’s responses: coherence, error recovery, consistency, diversity, topic depth, likeability, understandingness, flexibility, informativeness, and inquisi- tiveness, with overall quality included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The model is prompted with the full dialog context with the rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The FED metric was previously evaluated with both fine-tuned (ft) and from- scratch 345M and 762M DialoGPT [43] models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In the following experiments on FED, 3 in-context examples were used for prompting in Tables 1, 2, 3 and 4 and Appendix D and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1 In-Context Example Selection This setting evaluates 2 versions of the TNLGv2 model: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='7B and 530B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' These models are compared to the 762M ft DialoGPT model and the results are shown in Tables 1 and 2 and Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' First, the performances of these models are compared over the three example selection methods: manual, random, and algorithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' With manually chosen in- context examples, the 530B TNLGv2 model outperforms the DialoGPT model on almost all turn-level metrics except for understandability and fluency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' There are significant gains in all of the dialog-level metrics as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Since DialoGPT is fine- tuned on Reddit threads, more casual language is expected, compared to models like TNLGv2 where many of the training datasets consist of more formal language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Since the wording of conversational responses tends to be more casual, it is not sur- prising that the fine-tuned DialoGPT model outperforms even the largest TNLGv2 model for fluency and understandability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, the TNLGv2 models show large improvement on predicting turn- and dialog-level quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This suggests that the TNLGv2 models have a strong grasp on overall quality, which may be due to train- ing on more formal language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Very Large Language Models for Dialog Evaluation 9 BM25C+R generally outperforms BM25C and BM25R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, when choos- ing examples with BM25C+R, the correlation of understandability with human annotations increases significantly when using the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='7B TNLGv2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='7B TNLGv2 consistently outperforms 530B TNLGv2 in this aspect with any BM25 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' It appears that the smaller model is more influenced by the similarity of language in the examples than the larger one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Even when given random examples, the TNLGv2 models outperform the 762M ft DialoGPT model on a majority of the fine-grained metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This shows that larger models can better detect what constitutes a good response based on these metrics even if they are not given hand-picked examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, they generally do not outperform the manually or algorithmically chosen examples as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' An additional observation is that there are certain factors that cause models to perform better or worse on specific metrics: number of parameters the model has, the type of training data, and the difficulty of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' LLMs are able to provide increases in performance of over 50% for 15 out of 20 turn- and dialog-level met- rics compared to DialoGPT with 530B TNLGv2 and manually-chosen examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, if the 530B TNLGv2 model is compared to the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='7B TNLGv2 model, this increase is only observed for 2 out of the 20 metrics: correctness and under- standability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' LLMs can achieve high correlations with human judgement, but there is a limit to how much more performance gains can increase with extremely large models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Specificity, relevance, and correctness all relate to the context of the conversation while the other metrics are more turn-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' It follows that relevance and correct- ness with BM25C+R on the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='7B TNLGv2 model outperform the 530B TNLGv2 model with manual examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, specificity performs worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Choosing both diverse ratings and similar example points are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This finding further sup- ports the idea that the nature of the data used to train these LLMs is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Had the training data been more similar to conversational language, an increase could have been observed in the correlations for these metrics without choosing algorith- mically similar examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' TNLGv2 struggles with understandability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' it performs the worst at the highest correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' It also has unstable performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' performing at significance with random examples and with algorithmically chosen examples on 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='7B, but not with manually chosen ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This shows that choosing examples with diverse ratings helps a model less for metrics that it already performs poorly on;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' it would better benefit from examples that are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In general, even with the difference in training data, it is easier to obtain an overall sense of the conversation than a metric for a single turn for the larger models due to the large amount of parameters and variety of data that they have seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' When choosing examples based on context, the larger models generally perform worse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' it appears that having different examples is more important for dialog-level metrics than for 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+page_content='531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='530 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='585 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='690 Table 4 Dialog-level fine-grained metrics on the FED dataset for manually chosen examples over the TNLGv2, BLOOM, OPT, Flan-T5, and InstructGPT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' fluency in addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' As previously noted, relevance and correctness are turn-level metrics that take more of the context into account, so with training data that is more similar to casual language, these models perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' It should be noted that the overall turn- and dialog-level quality results were not surpassed by any smaller model, thus the very large models will have an advantage for overall metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Flan-T5 outperforms the largest model, TNLGv2 530B, on interestingness, rele- vance, and understandability at turn level and coherence, consistency, and informa- tiveness at dialog level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' There is a larger performance drop for the semantic appro- priateness, error recovery, and overall dialog-level quality metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Error recovery is a relatively new metric [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Even though Flan-T5 was fine-tuned on many di- alog tasks, it may not have seen data that addresses this specific metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Flan-T5 only has 3B parameters, and the fact that it outperforms 530B TNLGv2 shows the importance of use of dialog data during pre-training or fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' InstructGPT, being fine-tuned with prompting at 175B parameters, is more suit- able for the present experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' It performs very well on both turn- and dialog-level metrics, outperforming 530B TNLGv2 on almost all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Since InstructGPT has already seen prompting, the model can better understand a task through only instruc- tions or combinations of instructions and in-context examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 12 J.' metadata={'source': 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CG, DGU, DGR, EG, FT and FD are abbreviations for TopicalChat-USR, PersonaChat-USR [22], PersonaChat- Zhao [46], DailyDialog-Zhao [46], ConvAI2-GRADE [14], DailyDialog-Gupta [11], DailyDialog- GRADE [14], Empathetic-GRADE [14], FED-Turn and FED-Dial [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Model TU DZ PU DGU DGR FT EG FD Few-shot in-context Experiments BLOOM-7B-4ex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='027* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='117 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='289 OPT-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='7B-4ex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='091* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='044* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='657 TNLG-6.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='376 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='634 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='228 Best of DSTC10 baselines 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='532 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='555 Best TNLGv2 value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='549 Table 6 Spearman correlation of model predictions for overall quality with human ratings with 4 examples chosen with BM25 using context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Macro average scores are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='2 DSTC10 Datasets The same set of experiments were carried out on the 8 datasets in the DSTC10 chal- lenge in Tables 5 and 6, and Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The previous best performing metrics on DSTC10 are compiled from [13], which include both reference-free and fine-tuned metrics (see Appendix G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Quality is evaluated in terms of how good a response is to the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1 In-Context Example Selection Experiments are performed with randomly chosen examples and examples that were chosen by BM25 over 4, 8, and 12 examples in Table 5 and Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Higher correlation results are obtained on 4 datasets (DZ, DGU, DGR, and FT) with com- parable results on 3 datasets (TU, PU, and FD), as compared to the best DSTC10 baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Most of the best results are on the 530B TNLGv2 model, which will be Very Large Language Models for Dialog Evaluation 13 discussed in this section, as compared to the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='7B TNLGv2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Several factors are relevant here: the language of the dataset, the way the dataset was created, and how the dataset was annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DailyDialog contains more formal language, thus TNLGv2 should perform well since its training dataset includes data sources with formal language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DZ, DGU, and DGR almost always perform the best when examples are chosen from looking at the context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' adding the response generally leads to poorer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Since these datasets are annotated for appropriateness and coherence, context is more important than a more turn-specific metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' TopicalChat was created through knowledge-grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The conversations could thus have more substance than a purely open-domain un-prompted conversation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' It thus follows that response selection will work the best when choosing examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' PersonaChat has conversations that are persona-conditioned, so the quality of the conversation should take into account the entire conversation for each persona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' It performs better with examples chosen for context and response or with just context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' FED is split into turn- and dialog-level annotations, thus, for turn-level annota- tions choosing examples based on responses should work best, and for dialog-level annotations choosing examples based on either the context or the context and re- sponse should perform the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Choosing examples with context and response per- forms the best for EG, but randomly choosing examples outperforms that result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' It may be that with emotionally grounded conversations, the model needs more, or more diverse examples due to the different ways emotion can be expressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In general, choosing examples algorithmically improves performance over ran- domly choosing examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This is consistent with previous experiments above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, randomly-chosen examples perform better on the DGR and EG datasets on the 530B TNLGv2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This may be because these two datasets were rated for coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Algorithmically, choosing examples based on context and response performs the best on EG, as was seen for coherence in FED in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='2 Comparisons Across LLMs Table 6 compares the evaluation results across various LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Due to model input length restrictions, the following experiments were carried out using 4 in-context ex- amples or in a zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' BM25 is only used with the context as the example selection strategy, since it performed well with the TNLGv2 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In the few-shot setting, models that were not fine-tuned or trained with prompting (BLOOM, OPT) did not have consistent results across the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, those that were fine-tuned or prompted (Flan-T5, InstructGPT, InstructDial) had results that were close to or surpassed the previous best DSTC10 baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' InstructGPT performed the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Even in the zero-shot setting, Flan-T5 outperforms the baseline in 6 of the datasets, and InstructDial in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' These results clearly show that for dialog evaluation, it is insufficient to simply train on large amounts of general internet data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Specialized approaches such as in- struction tuning on multiple tasks improve the generalization capabilities of models 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Huynh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' in zero- and few-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' It is not surprising that InstructGPT performs the best since it fine-tunes a very large language model with instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 5 Conclusion LLMs have the potential to significantly contribute to dialog evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Current LLMs perform well for this task in a few-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' However, this performance varies greatly depending on the content of and number of examples in the prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Models prefer more similar examples for metrics that they struggle to evaluate, while preferring examples with more diverse ratings for metrics that they can evalu- ate well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Very large language models also still afford performance gains, especially for overall quality evaluation at the turn and dialog level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Even though large lan- guage models perform better at dialog-level fine-grained metrics, there are still pre- viously shown issues with how these models understand social situations and use context that may hinder further improvement if not addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Performance is also affected by the model’s training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Smaller language mod- els that are fine-tuned on instructions, trained on dialog data, and/or trained on mul- tiple dialog tasks outperform larger language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' These smaller models also perform more consistently over different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This indicates that LLMs should have more diverse pre-training data in order to be able to handle a larger variety of tasks in few or zero-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' More work needs to be done on understanding how a large language model mod- els different types of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In-context example selection and example wording still remains unstable across large language models in many tasks, and the performance variation over different dialog domains in this paper demonstrates that as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Presently, the LLMs explored in this paper have their own strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Smaller models such as BLOOM and OPT could share more training data similarity with dialog tasks based on their objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' TNLGv2 530B provides a very large lan- guage model that has shown improvement in dialog evaluation along with other NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Flan-T5 and InstructDial show the efficacy of fine-tuning a LLM on dialog tasks, and InstructGPT shows the importance of training a model to better recognize prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The evaluations of these models provide suggestions for the characteristics of the best LLMs to use for dialog evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Future work in using LLMs for other NLP tasks can benefit from such comprehensive analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Once a better understanding of LLMs is realized, the capabilities of large language models for zero- and few-shot tasks will increase greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 6 Acknowledgements We would like to thank Microsoft for allowing us to use TNLGv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' was sup- ported by the NSF Graduate Research Fellowship under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DGE1745016 Very Large Language Models for Dialog Evaluation 15 and DGE2140739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The opinions expressed in this paper do not necessarily reflect those of that funding agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' References [1] Agarwal O, Yang Y, Wallace BC, Nenkova A (2021) Interpretability analysis for named en- tity recognition to understand system predictions and how they can improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Computational Linguistics 47(1):117–140 [2] Baumgartner J, Zannettou S, Keegan B, Squire M, Blackburn J (2020) The pushshift reddit dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: Proceedings of the international AAAI conference on web and social media, vol 14, pp 830–839 [3] Beltagy I, Lo K, Cohan A (2019) Scibert: A pretrained language model for scientific text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:190310676 [4] BigScience Workshop (2022) Bloom (revision 4ab0472).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='57967/hf/0003, URL https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='co/bigscience/bloom [5] Brown T, Mann B, Ryder N, Subbiah M, Kaplan JD, Dhariwal P, Neelakantan A, Shyam P, Sastry G, Askell A, et al (2020) Language models are few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Advances in neural information processing systems 33:1877–1901 [6] Chen Z, Sadoc J, D’Haro LF, Banchs R, Rudnicky A (2021) Automatic evaluation and mod- eration of open-domain dialogue systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:211102110 [7] Chowdhery A, Narang S, Devlin J, Bosma M, Mishra G, Roberts A, Barham P, Chung HW, Sutton C, Gehrmann S, et al (2022) Palm: Scaling language modeling with pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:220402311 [8] Chung HW, Hou L, Longpre S, Zoph B, Tay Y, Fedus W, Li E, Wang X, Dehghani M, Brahma S, et al (2022) Scaling instruction-finetuned language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:221011416 [9] Gao L, Biderman S, Black S, Golding L, Hoppe T, Foster C, Phang J, He H, Thite A, Nabeshima N, Presser S, Leahy C (2020) The Pile: An 800gb dataset of diverse text for language modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:210100027 [10] Gao T, Fisch A, Chen D (2020) Making pre-trained language models better few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:201215723 [11] Gupta P, Mehri S, Zhao T, Pavel A, Eskenazi M, Bigham J (2019) Investigating evaluation of open-domain dialogue systems with human generated multiple references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: Proceedings of the 20th Annual SIGdial Meeting on Discourse and Dialogue, Association for Computational Linguistics, Stockholm, Sweden, pp 379–391, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='18653/v1/W19-5944, URL https: //aclanthology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/W19-5944 [12] Gupta P, Tsvetkov Y, Bigham J (2021) Synthesizing adversarial negative responses for robust response ranking and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: Findings of the Association for Computational Linguis- tics: ACL-IJCNLP 2021, Association for Computational Linguistics, Online, pp 3867–3883, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='18653/v1/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='findings-acl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='338, URL https://aclanthology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' findings-acl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='338 [13] Gupta P, Jiao C, Yeh YT, Mehri S, Eskenazi M, Bigham JP (2022) Improving zero and few- shot generalization in dialogue through instruction tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:220512673 [14] Huang L, Ye Z, Qin J, Lin L, Liang X (2020) GRADE: Automatic graph-enhanced coherence metric for evaluating open-domain dialogue systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), Association for Compu- tational Linguistics, Online, pp 9230–9240, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='18653/v1/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='emnlp-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='742, URL https://aclanthology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='emnlp-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='742 [15] Jiang Z, Xu FF, Araki J, Neubig G (2020) How can we know what language models know?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Transactions of the Association for Computational Linguistics 8:423–438 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Huynh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' [16] Kaplan J, McCandlish S, Henighan T, Brown TB, Chess B, Child R, Gray S, Radford A, Wu J, Amodei D (2020) Scaling laws for neural language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 08361, URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/abs/2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='08361 [17] Liu J, Shen D, Zhang Y, Dolan B, Carin L, Chen W (2021) What makes good in-context examples for gpt-3?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='06804, URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/ abs/2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='06804 [18] Liu Y, Ott M, Goyal N, Du J, Joshi M, Chen D, Levy O, Lewis M, Zettlemoyer L, Stoy- anov V (2019) Roberta: A robustly optimized bert pretraining approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:190711692 [19] Lu Y, Bartolo M, Moore A, Riedel S, Stenetorp P (2021) Fantastically ordered prompts and where to find them: Overcoming few-shot prompt order sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' 2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='08786, URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/abs/2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='08786 [20] Manning CD (2008) Introduction to information retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Syngress Publishing, [21] Mehri S, Eskenazi M (2020) Unsupervised evaluation of interactive dialog with DialoGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: Proceedings of the 21th Annual Meeting of the Special Interest Group on Discourse and Dialogue, Association for Computational Linguistics, 1st virtual meeting, pp 225–235, URL https://aclanthology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='sigdial-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='28 [22] Mehri S, Eskenazi M (2020) USR: An unsupervised and reference free evaluation met- ric for dialog generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, Association for Computational Linguistics, Online, pp 681– 707, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='18653/v1/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='acl-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='64, URL https://aclanthology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' acl-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='64 [23] Ouyang L, Wu J, Jiang X, Almeida D, Wainwright CL, Mishkin P, Zhang C, Agarwal S, Slama K, Ray A, et al (2022) Training language models to follow instructions with human feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:220302155 [24] Papineni K, Roukos S, Ward T, Zhu WJ (2002) Bleu: a method for automatic evaluation of machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: Proceedings of the 40th annual meeting of the Association for Computational Linguistics, pp 311–318 [25] Phy V, Zhao Y, Aizawa A (2020) Deconstruct to reconstruct a configurable evaluation met- ric for open-domain dialogue systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: Proceedings of the 28th International Confer- ence on Computational Linguistics, International Committee on Computational Linguistics, Barcelona, Spain (Online), pp 4164–4178, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='18653/v1/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='coling-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='368, URL https://aclanthology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='coling-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='368 [26] Radford A, Wu J, Child R, Luan D, Amodei D, Sutskever I, et al (2019) Language models are unsupervised multitask learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' OpenAI blog 1(8):9 [27] Raffel C, Shazeer N, Roberts A, Lee K, Narang S, Matena M, Zhou Y, Li W, Liu PJ (2020) Exploring the limits of transfer learning with a unified text-to-text transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Journal of Machine Learning Research 21(140):1–67, URL http://jmlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/papers/v21/ 20-074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='html [28] Robertson S, Zaragoza H (2009) The probabilistic relevance framework: Bm25 and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Found Trends Inf Retr 3(4):333–389, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1561/1500000019, URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1561/1500000019 [29] Roller S, Dinan E, Goyal N, Ju D, Williamson M, Liu Y, Xu J, Ott M, Smith EM, Boureau YL, Weston J (2021) Recipes for building an open-domain chatbot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume, Association for Computational Linguistics, Online, pp 300–325, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='18653/v1/ 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='eacl-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='24, URL https://aclanthology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='eacl-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='24 [30] Rubin O, Herzig J, Berant J (2021) Learning to retrieve prompts for in-context learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='08633, URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/abs/2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='08633 [31] Sai AB, Mohankumar AK, Arora S, Khapra MM (2020) Improving dialog evaluation with a multi-reference adversarial dataset and large scale pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Transactions of the Associa- tion for Computational Linguistics 8:810–827 Very Large Language Models for Dialog Evaluation 17 [32] Sanh V, Webson A, Raffel C, Bach SH, Sutawika L, Alyafeai Z, Chaffin A, Stiegler A, Scao TL, Raja A, et al (2021) Multitask prompted training enables zero-shot task generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:211008207 [33] Sap M, LeBras R, Fried D, Choi Y (2022) Neural theory-of-mind?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' on the limits of social intelligence in large lms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:221013312 [34] Smith S, Patwary M, Norick B, LeGresley P, Rajbhandari S, Casper J, Liu Z, Prabhumoye S, Zerveas G, Korthikanti V, et al (2022) Using deepspeed and megatron to train megatron- turing nlg 530b, a large-scale generative language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:220111990 [35] Su H, Kasai J, Wu CH, Shi W, Wang T, Xin J, Zhang R, Ostendorf M, Zettlemoyer L, Smith NA, Yu T (2022) Selective annotation makes language models better few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='01975, URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/abs/2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='01975 [36] Thoppilan R, De Freitas D, Hall J, Shazeer N, Kulshreshtha A, Cheng HT, Jin A, Bos T, Baker L, Du Y, et al (2022) Lamda: Language models for dialog applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:220108239 [37] Trinh TH, Le QV (2018) A simple method for commonsense reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='48550/ ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='02847, URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/abs/1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='02847 [38] Wei J, Bosma M, Zhao VY, Guu K, Yu AW, Lester B, Du N, Dai AM, Le QV (2021) Fine- tuned language models are zero-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:210901652 [39] Wei J, Tay Y, Bommasani R, Raffel C, Zoph B, Borgeaud S, Yogatama D, Bosma M, Zhou D, Metzler D, et al (2022) Emergent abilities of large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:220607682 [40] Zellers R, Holtzman A, Rashkin H, Bisk Y, Farhadi A, Roesner F, Choi Y (2019) Defend- ing against neural fake news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='12616, URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' org/abs/1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='12616 [41] Zhang C, Chen Y, D’Haro LF, Zhang Y, Friedrichs T, Lee G, Li H (2021) DynaEval: Unifying turn and dialogue level evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: Proceedings of the 59th Annual Meet- ing of the Association for Computational Linguistics and the 11th International Joint Con- ference on Natural Language Processing (Volume 1: Long Papers), Association for Com- putational Linguistics, Online, pp 5676–5689, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='18653/v1/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='acl-long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='441, URL https://aclanthology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='acl-long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='441 [42] Zhang S, Roller S, Goyal N, Artetxe M, Chen M, Chen S, Dewan C, Diab M, Li X, Lin XV, et al (2022) Opt: Open pre-trained transformer language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:220501068 [43] Zhang Y, Sun S, Galley M, Chen YC, Brockett C, Gao X, Gao J, Liu J, Dolan B (2019) Dialogpt: Large-scale generative pre-training for conversational response generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:191100536 [44] Zhang Z, Guo T, Chen M (2021) Dialoguebert: A self-supervised learning based dialogue pre-training encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: Proceedings of the 30th ACM International Conference on Informa- tion & Knowledge Management, pp 3647–3651 [45] Zhao T, Zhao R, Eskenazi M (2017) Learning discourse-level diversity for neural dialog models using conditional variational autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' arXiv preprint arXiv:170310960 [46] Zhao T, Lala D, Kawahara T (2020) Designing precise and robust dialogue response eval- uators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, Association for Computational Linguistics, Online, pp 26–33, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='18653/ v1/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='acl-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='4, URL https://aclanthology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='org/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='acl-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='4 [47] Zhao Z, Wallace E, Feng S, Klein D, Singh S (2021) Calibrate before use: Improving few- shot performance of language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' In: International Conference on Machine Learning, PMLR, pp 12,697–12,706 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Huynh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' A LLMs and Their Training/Fine-tuning Data Seen Dialog Fine-tuned Flan-T5 ✓ ✓ InstructDial ✓ ✓ InstructGPT ✓ ✓ BLOOM ✓ × OPT ✓ × TNLGv2 × × Table 7 LLMs with the datasets they were trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' During training or fine-tuning: “Seen Dia- log” indicates that the model has explicitly seen dialog datasets and therefore elements of casual language, and “fine-tuned” indicates that the model was fine-tuned on dialog data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' TNLGv2 has not seen datasets explicitly categorized as having dialog, but elements of casual language may be included in the Common Crawl snapshots and other internet-based corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Symbols: ✓means that the category is included and × means that the category is not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' B Prompt format examples FED Task: Given a dialog history and a response, rate how interesting the response is with regards to the dialog history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' == Example 1 == A: Hi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' B: Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' This is a pleasant surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' A: Haha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='thanks!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' how did you like the gift?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Response: Currently unpacking it I guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' How’s your morning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Rating: 1/2 A: Hope you like it!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Morning is good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Busy finishing up stuff before the holidays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' B: I think I traveled too much the last couple of months so no holiday for me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' But I’m okay with that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Going anywhere exciting?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' A: Yes Response: Where to?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Rating: 1/2 A: Hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' looking forward to warm beaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Response: WOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Which island?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' I like Hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Rating: 2/2 Table 8 An example of a prompt with one example from FED [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Interestingness was rated in FED over three values corresponding to 0/2, 1/2, and 2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The resulting output is truncated to the integer value of 0, 1, or 2 to be used in evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Very Large Language Models for Dialog Evaluation 19 C Prompt format examples DSTC10 Instruction: Given a conversation and a response, choose if the response is a good response to the context Example Background info: none Conversation: Person A: did your meal meet with your approval ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Response: yes , i did .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' it was a good meal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Question: Is the above response a good response to the conversation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Answer: Yes Background info: none Conversation: Person B: i really do hate public transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Person A: i agree , it ’s just never on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Response : you ’re right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Question: Is the above response a good response to the conversation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' Answer: Table 9 An example of a prompt with examples from DSTC 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' D Additional algorithmically chosen FED examples BM25C BM25R Quality 7B 530B 7B 530B Interesting 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='385 Engaging 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='308 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='028* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='029* Fluent 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='096* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='121 Overall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='435 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='438 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='407 Table 10 Turn-level fine-grained metrics on the FED dataset for algorithmically chosen examples over the TNLGv2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='7B and 530B models.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content='555 Table 14 Spearman correlation of model predictions with human ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' The models fine-tuned on DSTC 10 datasets tend to perform better on the DSTC 10 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} +page_content=' TU, PU, PZ, DZ, CG, DGU, DGR, EG, FT and FD are abbreviations for TopicalChat-USR, PersonaChat-USR [22], PersonaChat-Zhao [46], DailyDialog-Zhao [46], ConvAI2-GRADE [14], DailyDialog-Gupta [11], DailyDialog-GRADE [14], Empathetic-GRADE [14], FED-Turn and FED-Dial [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FLT4oBgHgl3EQfJi8x/content/2301.12004v1.pdf'} diff --git a/ItE1T4oBgHgl3EQfYARl/vector_store/index.faiss b/ItE1T4oBgHgl3EQfYARl/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..12992be25134cb5fabf610e1c0debe71a791f0b7 --- /dev/null +++ b/ItE1T4oBgHgl3EQfYARl/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4b06f08498fded9c426fe44f04a280d43caba7a32dc0ac96b39973047394f0a1 +size 1835053 diff --git a/JNAzT4oBgHgl3EQfjv3G/content/tmp_files/2301.01522v1.pdf.txt b/JNAzT4oBgHgl3EQfjv3G/content/tmp_files/2301.01522v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..608fe1a00d25f0f850c5874c8387e8c6cc3add5d --- /dev/null +++ b/JNAzT4oBgHgl3EQfjv3G/content/tmp_files/2301.01522v1.pdf.txt @@ -0,0 +1,2150 @@ +B − L model in light of the CDF II result +Sanjoy Mandal,1, ∗ Hemant Prajapati,2, † and Rahul Srivastava2, ‡ +1Korea Institute for Advanced Study, Seoul 02455, Korea +2Department of Physics, Indian Institute of Science Education and Research - Bhopal, +Bhopal Bypass Road, Bhauri, Bhopal 462066, India +(Dated: January 5, 2023) +Recent CDF II collaboration results on W mass measurements contradict Standard +Model (SM) prediction, requiring new physics to explain this anomaly. To explain +this issue, in this paper we investigate the idea of using the U(1)B−L gauged SM +extension. We demonstrate that B −L extended models can explain the revised best +fit values for S, T, and U following the CDF II results. We studied the parameter +space of models with and without mixing between neutral gauge bosons. We also +reviewed the dark matter constraints and demonstrated that there is parameter +space which is compatible with current W boson mass, relic abundance, and direct +detection experiments. +I. +INTRODUCTION +The Standard model (SM) of electroweak theory with SU(2) ⊗ U(1) gauge symmetry +is highly successful in explaining most of the observations in particle physics experiments. +With the recent finding of the Higgs like boson with mass 125 GeV at the LHC [1, 2], seems +to complete the SM. Despite its ability to explain most observable phenomena, the SM can- +not be considered the final theory of particle physics. There is an ever-increasing number of +observations such as the discovery of neutrino oscillations [3] and the existence of dark mat- +ter (DM) at cosmic scales [4] that put serious questions on the SM predictions. In addition +to this, the CDF-II collaboration recently published their high precision measurement of the +W boson mass M CDF +W += 80.4335 ± 0.0094 GeV [5], which reveals a 7-σ difference from the +SM expectation M SM +W += 80.354 ± 0.007 GeV [6]. This leads us to investigate the extension +of SM, which can account for the aforementioned problems with SM. +In the absence of right-handed neutrino (RHN), the neutrinos are massless in the SM. +RHNs have been a prevalent feature of many extensions of SM such as various seesaw +mechanisms [7–10] to generate neutrino masses. In recent years, a number of models have +been proposed that combine neutrino mass generations and the existence of DM into a +∗ smandal@kias.re.kr +† hemant19@iiserb.ac.in +‡ rahul@iiserb.ac.in +arXiv:2301.01522v1 [hep-ph] 4 Jan 2023 + +2 +single framework. +Motivated by this, people have studied extensively beyond standard +model (BSM) framework based on the gauged U(1)B−L model [11–17]. The most intriguing +aspect of this model is that it includes three RHNs to cancel gauge and mixed gauge-gravity +anomalies and generate tiny neutrino masses through the seesaw mechanism. This type of +model predicts the existence of a new neutral gauge boson Z′ that can mix with the SM +neutral gauge boson Z [18]. One also has the possibility of explaining the DM in these type +of model with an additional scalar field, χd, that is a SM singlet but charged under U(1)B−L. +An advantage of this scenario is that one does not need to impose any ad hoc Z2 symmetry +to stabilise the DM. Instead, stability of χd can be guaranteed by appropriately choosing its +B − L charge. +In this work we show that, despite its simplicity, in addition to neutrino mass generation +and DM, B − L model can also explain the recent CDF-II W boson mass measurements. +The new boson associated with U(1)B−L symmetry mixes with the SM neural Z boson to +provide S, T, U corrections that are compatible with current W boson mass measurements. +Specifically we investigate two distinct scenarios: one with no mass mixing between two +neutral bosons and one with mass mixing. We study the difference in parameter space in +both cases. We show that the parameter space consistent with the best fit S, T, U values +following the CDF II results is also consistent with the DM physics constraints in the model +we proposed. +The paper is organised as follows: in Sec. II, we briefly discuss the possibility of having +kinetic mixing between two field strength tensors corresponding to U(1)Y and U(1)B−L. We +investigate in detail whether or not addressing simply the impacts of kinetic mixing at the +tree level may resolve the W mass anomaly. In Sec. III, we study two B − L gauged models +without taking into account the effects of kinetic mixing. The first model is a minimal +B − L extension of the SM with no mass mixing between the SM neutral gauge boson +Z and U(1)B−L neutral gauge boson Z′. In the second model, we introduce mass mixing +between these neutral gauge bosons and also introduce a scalar DM candidate. In Sec. IV, +we discuss how one can parametrise the new physics contributions to W mass in terms of +oblique parameters S, T and U. In Sec. V, we described the effective lagrangian approach +to parameterise this novel physics, as well as the parameter space that is compatible with +the S, T, and U parameters following the CDF-II data. We focused our attention on the +chiral B − L model. Following that, we reviewed the DM constraints derived from Planck’s +measurement of the relic density, as well as the constraints derived from the direct detection +experiments. Finally, we demonstrated that, in the chiral B −L model, the parameter space +we found agrees with the current measurements of the W boson mass, relic abundance, and +direct detection experiments. + +3 +II. +KINETIC-MIXING AND W MASS +We note that a kinetic mixing can occur provided there are two or more field strength +tensors Bµν and Xµν which are neutral under some gauge symmetry. Thus in our case with +the gauge group SU(3)c ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1)B−L , the kinetic terms can be expressed +as follows +LKinetic = −1 +4BµνBµν − 1 +4XµνXµν − κ +2BµνXµν, +(1) +where Bµν and Xµν are the filed strength tensors of the gauge groups U(1)Y and U(1)X, +respectively. The requirement of positive kinetic energy implies that kinetic coefficient |κ| < +1. One can diagonalize the kinetic mixing term as follow +� ˜B +˜X +� += +� +1 +κ +0 +√ +1 − κ2 +� � +B +X +� +. +(2) +Let’s first determine the gauge boson mass spectrum setting the kinetic mixing κ = 0 to fix +our notation. With the kinetic mixing κ = 0, the covariant derivative can be defined as +Dµ = ∂µ − igsT aGa +µ − igT aW a +µ − ig′Y Bµ − igXYXXµ, +(3) +where gauge coupling gX is a free parameter. In addition to SM Higgs doublet Φ, one adds +a scalar, χ, singlet of the SM but charged under U(1)B−L, that spontaneously breaks the +B − L symmetry. In minimal case, the U(1)B−L charge of χ is qχ = 2. To determine the +gauge boson mass spectrum, we have to expand the following scalar kinetic terms +Ls = (DµΦ)†(DµΦ) + (Dµχ)†(Dµχ), +(4) +and have to replace the fields Φ and χ by the following expressions such as +Φ = 1 +√ +2 +� +0 +vΦ + R1 +� +, ⟨χ⟩ = 1 +√ +2(vχ + R2). +(5) +With this above replacement we can expand the scalar kinetic terms (DµΦ)†(DµΦ) and +(Dµχ)†(Dµχ) as follows +(DµΦ)†(DµΦ) ≡ 1 +2∂µR1∂µR1 + 1 +8(R1 + vΦ)2� +g2|W µ +1 − iW µ +2 |2 + (gW µ +3 − g′Bµ)2� +, +(6) +(Dµχ)†(Dµχ) ≡ 1 +2∂µR2∂µR2 + 1 +2(R2 + vχ)2(g +′ +1Xµ)2, +(7) +where we have defined g +′ +1 = gXqχ. With this, the mass matrix of the neutral gauge bosons +is given by +LM = 1 +2V T +0 M 2 +GV0, +(8) + +4 +where +V T +0 = +� +Bµ W3µ Xµ +� +and M 2 +G = +� +� +� +1 +4g′2v2 +Φ +− 1 +4gg′v2 +Φ +0 +− 1 +4gg′v2 +Φ +1 +4g2v2 +Φ +0 +0 +0 +g +′2 +1 v2 +χ +� +� +� . +(9) +In the kinetic term diagonalized basis ˜V T +0 = ( ˜Bµ W3µ ˜Xµ), the mass matrix of the neutral +gauge boson can be written as +LM = 1 +2 +˜V T +0 STM 2 +GS ˜V0 = 1 +2 +˜V T +0 ˜ +M 2 +G ˜V0, +(10) +where +S = +� +� +� +1 0 − +κ +√ +1−κ2 +0 1 +0 +0 0 +1 +√ +1−κ2 +� +� +� , ˜ +M 2 +G = STM 2 +GS = +� +� +� +1 +4g′2v2 +Φ +− 1 +4gg′v2 +Φ +1 +4g′˜gtv2 +Φ +− 1 +4gg′v2 +Φ +1 +4g2v2 +Φ +− 1 +4g˜gtv2 +Φ +1 +4g′˜gtv2 +Φ − 1 +4g˜gtv2 +Φ +1 +4˜g2 +t v2 +Φ + g +′′2 +1 v2 +χ +� +� +� , +(11) +with ˜gt = − +g′κ +√ +1−κ2. Following linear combination of ˜Bµ, W µ +3 and ˜Xµ gives definite mass +eigenstates Aµ, Zµ and Z +′µ, +� +� +� +˜Bµ +W µ +3 +˜Xµ +� +� +� = +� +� +� +cos θw − sin θw cos θ +sin θw sin θ +sin θw +cos θw cos θ +− cos θw sin θ +0 +sin θ +cos θ +� +� +� +� +� +� +Aµ +Zµ +Z +′µ +� +� +� , +(12) +where +tan2θ = +2˜gt +� +g2 + g′2 +˜g2 +t + 16 +� +vχ +2vΦ +�2 +g +′′2 +1 − g2 − g′2 +with ˜gt = − +g′κ +√ +1 − κ2 and g +′′ +1 = +g′ +1 +√ +1 − κ2. +(13) +Masses of physical gauge bosons A, Z and Z +′ are given by, +MA = 0, M 2 +Z,Z′ = 1 +8 +� +Cv2 +Φ ∓ +� +−D + v4 +ΦC2 +� +, +(14) +where, +C = g2 + g′2 + ˜g2 +t + 16 +� vχ +2vΦ +�2 +g +′′2 +1 , +D = 16v2 +Φv2 +χ(g2 + g′2)g +′′2 +1 . +(15) +The covariant derivative with the kinetic mixing can be expressed in terms of the orthogonal +fields ˜B and ˜X as +Dµ = ∂µ − igsT aGa +µ − igT aW a +µ − ig′Y ˜Bµ − i +� +gXYX +1 +√ +1 − κ2 − g′Y +κ +√ +1 − κ2 +� +˜Xµ. +(16) + +5 +W mass: Now let’s try to see whether one can explain the CDF-II anomaly considering +only kinetic mixing and ignoring any other loop corrections due to new neutral gauge boson +Z′. +Specifically, we consider that the shift in W boson mass measured by CDF-II also +modifies the Z boson mass at the tree level as the ρ parameter should be equal to one at +tree-level. Further, we investigate whether new physics contribution through kinetic mixing +is sufficient to reduce this change in Z mass to the experimental value of MZ = 91.1876 +GeV [6]. The tree-level formula for the W and Z mass is given as follows +M 2 +W = v2 +Φg2 +4 +, +M 2 +Z|κ=0 = v2 +Φ +4 +� +g2 + g′2� +. +(17) +Taking the CDF II measured W mass, MW = 80.4335 ± 0.0094 GeV [5] and using the PDG +values for other input parameters, sin2 θw = 0.23121 ± 0.00004, Gf = 1.1663787(6) × 10−5 +(GeV)−2 [6], we calculated weak couplings that is consistent with CDF-II measured W mass +and then calculated the Z mass. With this, the central value of the theoretically computed +Z mass is given as +MZ|κ=0 = 91.7345 GeV, +(18) +which is of course larger than the experimental value of MZ = 91.1876 GeV [6]. Note that +in B − L model, Z mass is affected by the presence of the kinetic mixing parameter κ, see +Eq. (14): +MZ = f (gx, MZ′, κ) . +(19) +As a result, the new physics contribution from kinetic mixing κ can reduce the Z mass to +the experimental value. In Fig. 1, we show how Z mass depends on the kinetic mixing κ, gx +and MZ′. The various lines in each panel of Fig. 1 correspond to different values of gx and +MZ′ while the ratio MZ′ +gx +remains constant. The ratio is kept at 6 TeV, 8 TeV and 10 TeV +in the top left, top right and bottom panel, respectively. It is clear from Fig. 1 that when +only the central values are considered, the change in Z mass that touches the experimental +value occurs only at low Z′ mass. Also comparing top left, top right and bottom panel of +Fig. 1, we see that at high MZ′ +gx +ratio, the kinetic mixing is not sufficient to reduce Z mass +to the experimental value. +The mass of Z′ and the gauge coupling gx can be constrained with collider data. From +LEP II data the bound +MZ′ +gx +≳ 6 − 7 TeV, +(20) +was derived in Refs. [19, 20, 23]. Current ATLAS and CMS searches for dilepton resonances +at the LHC can also be used to constrain MZ′ via the Drell-Yan process, pp → Z′ → +ℓ¯ℓ, with ℓ = e, µ [21–23]. From Fig. 2, we see that the LHC dilepton constraints are the +most stringent up to MZ = 6 TeV, beyond which the resonant Z′ production is kinematically +limited at √s = 13 TeV. Hence comparing Fig. 1 and Fig. 2, we can conclude that in view + +6 +gx=0.1 , MZ' = 600 +gx=0.2 , MZ' = 1200 +gx=0.3 , MZ' =1800 +gx=0.4 , MZ' = 2400 +0.2 +0.4 +0.6 +0.8 +1.0 +90.50 +90.75 +91.00 +91.25 +91.50 +91.75 +92.00 +κ +MZ (GeV) +MZ(exp) +gx=0.1 , MZ' = 800 +gx=0.2 , MZ' = 1600 +gx=0.3 , MZ' =2400 +gx=0.4 , MZ' = 3200 +0.2 +0.4 +0.6 +0.8 +1.0 +90.50 +90.75 +91.00 +91.25 +91.50 +91.75 +92.00 +κ +MZ (GeV) +MZ(exp) +gx=0.1 , MZ' = 1000 +gx=0.2 , MZ' = 2000 +gx=0.3 , MZ' = 3000 +gx=0.4 , MZ' = 4000 +0.2 +0.4 +0.6 +0.8 +1.0 +90.50 +90.75 +91.00 +91.25 +91.50 +91.75 +92.00 +κ +MZ (GeV) +MZ(exp) +FIG. 1: Z mass versus kinetic mixing κ. The coloured lines in each panel correspond to +different gx and MZ values while keeping the ratio MZ′ +gx +constant. +of current experimental constraints on gx − MZ′, it is not possible to explain CDF-II W +anomaly at tree-level with help of kinetic mixing κ. +III. +MINIMAL AND CHIRAL B − L MODEL +We saw in the previous section that kinetic mixing alone is insufficient to explain the W +mass anomaly. From this point forward, we will ignore kinetic mixing and concentrate on the +loop contribution from the U(1)B−L Z′ gauge sector in order to explain the W mass anomaly. +In this section, we study two U(1)B−L gauged SM extensions: minimal and chiral extensions. +Under U(1)B−L the SM quarks and leptons have charge 1/3 and −1 respectively. As a result, + +7 +2 +4 +6 +8 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +MZ '[ TeV ] +gx +CMS13(2l) +ATLAS13(2l) +LEP-II +FIG. 2: Constraint on gx as a function of MZ′. The shaded regions are ruled out from +LEP-II [19, 20], ATLAS and CMS dilepton searches [21, 22]. +B −L is an anomalous symmetry that requires the inclusion of additional fermions to gauge +it consistently. The gauge group U(1)B−L has the potential to cause the following triangle +gauge anomalies: +[SU(3)c]2[U(1)B−L] = +� +q +XqL − +� +q +XqR, +(21a) +[SU(2)L]2[U(1)B−L] = +� +l +XlL + 3 +� +q +XqL, +(21b) +[U(1)Y ]2[U(1)B−L] = +� +lq +(Y 2 +lLXlL + 3Y 2 +qLXqL) − +� +lq +(Y 2 +lRXlR + 3Y 2 +qRXqR), +(21c) +[U(1)Y ][U(1)B−L]2 = +� +lq +(YlLX2 +lL + 3YqLX2 +qL) − +� +lq +(YlRX2 +lR + 3YqRX2 +qR). +(21d) +In addition to this we have two more equations +[U(1)B−L]3 = +� +lq +(X3 +lL + 3X3 +qL) − +� +lq +(X3 +lR + 3X3 +qR), +(22a) +[G]2[U(1)B−L] = +� +lq +(XlL + 3XqL) − +� +lq +(XlR + 3XqR), +(22b) +where, X is the U(1)B−L charge and Y is the hyper charge. Anomalies from the first four +equations of Eq. (21), cancel within the SM particle content. To cancel anomalies arising +from Eq. (22), we add three generations of RHNs(νi +R, i = 1, 2, 3) with U(1)B−L charges +(x1, x2, x3). This gives us the following two conditions: +x1 + x2 + x3 = −3, +(23a) +x3 +1 + x3 +2 + x3 +3 = −3. +(23b) + +8 +We will discuss two charge assignment for the νi +R that cancel anomalies. The first is the +vector solution (also sometime called minimal B − L extension), in which the RHNs has the +same charge as the left-handed neutrino: (−1, −1, −1). Second assignment makes neutrinos +chiral under U(1)B−L with RHNs charges: (5, −4, −4). +Fields +( SU(3)c ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1)B−L ) +LL +(1, 2, − 1 +2, −1) +QL +(3, 2, 1 +6, 1 +3) +eR +(1, 1, −1, −1) +νR +(1, 1, 0, −1) +uR +(3, 1, 2 +3, 1 +3) +dR +(3, 1, − 1 +3, 1 +3) +Φ +(1, 2, 1 +2, 0) +χ +(1, 1, 0, 2) +TABLE I: Matter content and charge assignment of the vector B − L model. For brevity, +the generation index is suppressed. +A. +Vector B − L Model +This model is a simple extension of the SM. The particle contents and their charges +under the gauge group SU(3)c ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1)B−L are given in Table. I. The new +particles are three RHNs with B − L charge −1 to cancel the gauge anomalies and a new +scalar filed χ, singlet of the SM but charged under U(1)B−L, that spontaneously breaks the +B − L symmetry. We assign B − L charge +2 for scalar field χ so that νi +R gets Majorana +mass after B − L breaking which further gives rise to light neutrino mass through seesaw +mechanism. We begin by writing down the Lagrangian of the scalar sector. The most general +renormalizable and SU(3)c ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1)B−L gauge invariant scalar sector is +given by +Ls = (DµΦ)†(DµΦ) + (Dµχ)†(Dµχ) − V(Φ, χ), +(24) +where the covariant derivative is defined in Eq. (3). The scalar potential V(Φ, χ) is given by +V(Φ, χ) = m2 +χ(χ∗χ) + 1 +2λχ(χ∗χ)2 + m2 +Φ(Φ†Φ) + 1 +2λΦ(Φ†Φ)2 + λΦχ(χ∗χ)(Φ†Φ). +(25) +The breaking of the electroweak and the U(1)B−L gauge symmetries are driven by the +vacuum expectation values(vev) of the scalar fields Φ and χ. Denoting the vevs of field + +9 +Φ and χ as vΦ and vχ, the fields Φ and χ after symmetry breaking can be written in the +following form: +Φ = 1 +√ +2 +� +√ +2G+ +vΦ + R1 + iI1 +� +, +χ = 1 +√ +2(vχ + R2 + iI2). +(26) +G± are the Goldstone boson corresponding to W ±. I1 and I2 will mix and give rise to the +Goldstone bosons corresponding to the neutral gauge bosons Z and Z′. The mass matrix of +CP-even Higgs scalars in the basis (R1, R2) reads as +M2 +R = +� +A C +C B +� += +� +v2 +ΦλΦ +vΦvχλΦχ +vΦvχλΦχ +v2 +χλχ +� +. +(27) +The mass eigenvalues of light and heavy mass eigenstates as +m2 +h = 1 +2 +� +A + B − +� +(A − B)2 + 4C2 +� +, +(28) +m2 +H = 1 +2 +� +A + B + +� +(A − B)2 + 4C2 +� +. +(29) +We follow the convention m2 +h ≤ m2 +H and have identified h as the SM Higgs discovered at +LHC, with mass mh = 125 GeV. The two mass eigenstates h, H are related with the (R1, R2) +fields through the following rotation matrix as +� +h +H +� += U +� +R1 +R2 +� += +� +cos θ − sin θ +sin θ +cos θ +� � +R1 +R2 +� +, with tan 2θ = +2C +B − A. +(30) +In the absence of kinetic mixing, neutral bosons cannot have mass mixing because the +scalar doublet Φ does not carry any B − L charge. The gauge boson masses are given as +M 2 +Z = v2 +Φ +4 +� +g2 + g′2� +, +M 2 +W = v2 +Φg2 +4 +, +MZ′ = 2vχgx, +(31) +where g and g′ are SU(2) and hypercharge coupling respectively. +In B − L model neutrino masses are generated by seesaw mechanism. Apart from the SM +L +⟨Φ⟩ +νR +νR +⟨χ⟩ +⟨Φ⟩ +L +1 +FIG. 3: Neutrino mass generation in B − L model through type-I seesaw mechanism. + +10 +part, the Yukawa sector of the model can be written in a gauge-invariant way as +−LY ⊃ Y ij +ν L +i ˜Φνj +R + yij +M +2 νc +RiνRjχ + H.c., +(32) +The first and second terms will give the Dirac and Majorana contributions to the neutrino +mass generation. We assume without loss of any generality a basis in which yij +M is diagonal. +After the breaking of electroweak and U(1)B−L symmetry, we can write the mass term as +−LM ⊃ νLmDνR + 1 +2νc +RMRνR + H.c., +(33) +where mD = +yνvΦ +√ +2 +and MR = +yMvχ +√ +2 . Now using the fact that Majorana mass terms are +symmetric and νc +RmT +ν νc +L = νLmννR, we can write the LM in the following matrix form +−LM ⊃ +1 +2 +� +νL (νR)c +� � +0 +mD +mT +D MR +� � +(νL)c +νR +� +. +(34) +From the above mass matrix, one can easily recover the seesaw formula for light Majorana +neutrinos as, Mν ≈ mDM −1 +R mT +D and the heavy neutrino mass as MN ≈ MR with the +assumption mD ≪ MR. +Fields +( SU(3)C ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1)B−L ) +LL +(1, 2, −1/2, −1) +QL +(3, 2, 1/6, 1/3) +eR +(1, 1, −1, −1) +uR +(3, 1, 2/3, 1/3) +dR +(3, 1, −1/3, 1/3) +ν1 +R +(1, 1, 0, 5) +ν2,3 +R +(1, 1, 0, −4) +Φ +(1, 2, 1/2, 0) +ϕ +(1, 2, 1/2, −3) +σ +(1, 1, 0, 3) +χd +(1, 1, 0, 1/2) +TABLE II: Matter content and charge assignment of the chiral B − L model. For brevity, +the generation index is suppressed. +B. +Chiral B − L Model +Another U(1)B−L gauged model will be discussed in this section. For RHNs, we use a +chiral anomaly cancellation solution (5, −4, −4). Apart from the SM particle content and + +11 +RHNs, in scalar sector we add one more SU(2)L doublet ϕ and a scalar singlet σ. ϕ is with +hypercharge +1/2 and U(1)B−L charge −3, whereas scalar σ has U(1)B−L charge +3. We +also include a scalar DM χd with a charge of U(1)B−L of +1/2. The advantage is that one +does not need to impose any ad hoc Z2 symmetry to stabilise the DM. Instead, the stability +of χd can be guaranteed by this nontrivial B − L charge. The most general renormalizable +and SU(3)c ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1)B−L gauge invariant scalar sector is given by +Ls = (DµΦ)†DµΦ + (Dµϕ)†Dµϕ + (Dµσ)†Dµσ + (Dµχd)†Dµχd − V(Φ, ϕ, σ, χd), +(35) +where again the covariant derivative is defined in Eq. (3). The scalar potential V(Φ, ϕ, σ, χd) +is given by +V(Φ, ϕ, σ, χd) = m2 +σ(σ∗σ) + 1 +2λσ(σ∗σ)2 + m2 +Φ(Φ†Φ) + 1 +2λΦ(Φ†Φ)2 + m2 +ϕ(ϕ†ϕ) + 1 +2λϕ(ϕ†ϕ)2 ++ m2 +χd(χ∗ +dχd) + 1 +2λχd(χ∗ +dχd)2 − µ(Φ†ϕ)σ − µ(ϕ†Φ)σ∗ + λΦσ(Φ†Φ)(σσ∗) ++ λϕσ(ϕ†ϕ)(σσ∗) + λΦϕ1(Φ†Φ)(ϕ†ϕ) + λΦϕ2(Φ†ϕ)(ϕ†Φ) + λΦχd(Φ†Φ)(χ∗ +dχd) ++ λϕχd(ϕ†ϕ)(χ∗ +dχd) + λσχd(σσ∗)(χ∗ +dχd). +(36) +Neutral components of Φ and ϕ spontaneously break electroweak symmetry. +A singlet +scalar σ, along with ϕ, breaks the U(1)B−L spontaneously. First we solve the minimization +equations for the mass parameters mΦ, mϕ, mσ in the potential. We get +2m2 +Φ + v2 +ΦλΦ − +√ +2µ +vΦ +vϕvσ + v2 +χλΦσ + v2 +ϕ(λΦϕ1 + λΦϕ2) = 0, +(37a) +2m2 +σ − +√ +2µ +vσ +vΦvϕ + v2 +σλσ + v2 +ΦλΦσ + v2 +ϕλϕσ = 0, +(37b) +2m2 +ϕ − +√ +2µ +vϕ +vΦvσ + v2 +σλϕσ + v2 +ϕλϕ + v2 +Φ(λΦϕ1 + λΦϕ2) = 0. +(37c) +The fields Φ, ϕ and σ can be written in unitary gauge after symmetry breaking in the +following form: +Φ = 1 +√ +2 +� +√ +2G+ +1 +vΦ + R1 + iI1 +� +, +ϕ = 1 +√ +2 +� +√ +2G+ +2 +vϕ + R2 + iI2 +� +, σ = 1 +√ +2(vσ + R3 + iI3). +(38) +G± +1 and G± +2 will mix and give rises to the Goldstone bosons G± corresponding to the W ± +boson. One electrically charged field remains as the physical field. The mass matrix of these +electrically charged fields in the basis (G+ +1 , G+ +2 ) reads as +M2 +± = 1 +2 +� +� +� +√ +2µvσvϕ +vΦ +− v2 +ϕλΦϕ2 +vΦvϕλΦϕ2 − +√ +2µvσ +vΦvϕλΦϕ2 − +√ +2µvσ +√ +2µvσvΦ +vϕ +− v2 +ΦλΦϕ2 +� +� +� . +(39) + +12 +Mass eigen states are given as +M 2 +H± = +v2 +2vΦvϕ +�√ +2µvσ − vΦvϕλΦϕ2 +� +, +(40) +where, v = +� +v2 +Φ + v2 +ϕ. +The two mass eigenstates G±, H± are related with the (G± +1 , G± +2 ) fields through the fol- +lowing rotation matrix as +� +G± +H± +� += U +� +G± +1 +G± +2 +� += +� +cos α +sin α +− sin α cos α +� � +G± +1 +G± +2 +� +, with tan α = vϕ +vΦ +. +(41) +In pseudo-scalar sector I1, I2 and I3 mix together and gives two Goldstone boson G0 +1, G0 +2 +corresponding to the neutral gauge bosons Z and Z′ and one pseudo scalar field remains as +a physical massive field H0. The mass matrix in the basis (I1, I2, I3) can be written as +M2 +I = 1 +√ +2 +� +� +� +µvϕvσ +vΦ +− µvσ +− µvϕ +−µvσ +µvΦvσ +vϕ +µvΦ +−µvϕ +µvΦ +µvΦvϕ +vσ +� +� +� . +(42) +Mass of the physical eigenstate is given as +M 2 +H0 = +µ +√ +2vΦvϕvσ +� +v2 +Φv2 +ϕ + v2 +σv2� +. +(43) +Mass eigenstates G0 +1, G0 +2, H0 are related with the (I1, I2, I3) fields through the following +rotation matrix as +� +�� +G0 +1 +G0 +2 +H0 +� +�� = U +� +�� +I1 +I2 +I3 +� +�� = +� +�� +cos α +sin α +0 +− sin α cos β cos α cos β − sin α +− sin α sin β cos α sin β +cos β +� +�� +� +�� +I1 +I2 +I3 +� +�� , +(44) +Where, +tan α = vϕ +vΦ +, tan β = vσv +vΦvϕ +. +(45) +Three CP-even neutral scalars are mixed together. The mass matrix in the basis (R1, R2, R3) +can be expressed as +M2 +S = 1 +2 +� +������� +2v2 +ΦλΦ + +√ +2µ +vΦ vϕvσ +2vΦvϕλ12 − +√ +2µvσ +2vΦvσλΦσ − +√ +2vϕµ +2vΦvϕλ12 − +√ +2µvσ +2v2 +ϕλϕ + +√ +2µ +vϕ vΦvσ +2vϕvσλϕσ − +√ +2vΦµ +2vΦvσλΦσ − +√ +2vϕµ +2vϕvσλϕσ − +√ +2vΦµ +2v2 +σλσ + +√ +2µ +vσ vΦvϕ, +� +������� +, +(46) + +13 +where, λ12 = λΦϕ1 + λΦϕ2 . The matrix M2 +S can be diagonalized by an orthogonal matrix : +OT +RM 2 +ROR = diag(m2 +H1, m2 +H2, m2 +H3) with +� +� +� +H1 +H2 +H3 +� +� +� = OR +� +� +� +R1 +R2 +R3 +� +� +� . +(47) +We assume the mass eigenstates to be ordered by their masses mH1 ≤ mH2 ≤ mH3. H1 = h +is identified with the SM Higgs of 125 GeV. We will use the standard parameterization +OR = R23R13R12 where +R12 = +� +� +� +c12 −s12 0 +s12 +c12 +0 +0 +0 +1 +� +� +� , +R13 = +� +� +� +c13 0 −s13 +0 +1 +0 +s13 0 +c13 +� +� +� , +R23 = +� +� +� +1 +0 +0 +0 c23 −s23 +0 s23 +c23 +� +� +� +(48) +cij = cos θij, sij = sin θij, where the angles θij can be chosen to lie in the range − π +2 ≤ θij ≤ π +2. +Finally the mass of dark matter χd will be given as +M 2 +DM = 2m2 +χd + v2 +ΦλΦχd + v2 +σλσχd + v2 +ϕλϕχd +2 +. +(49) +Neutrino mass: The Yukawa sector of the model can be written in a gauge-invariant way +as +−LY = Y ij +e L +iΦej +R + Y ij +u Q +i ˜Φuj +R + Y ij +d Q +iΦdj +R + Y ij +ν L +i ˜ϕνj +R + H.c. +(50) +We see from the last term of Eq. (50) that the three RHNs pair up with the three left-handed +L +νR +⟨φ⟩ +⟨Φ⟩ +⟨σ⟩ +1 +FIG. 4: Neutrino mass generation in Chiral B − L model through Dirac type II seesaw +neutrinos of the SM to form Dirac particles. Note the importance of unconventional B − L +charges of ϕ and νi +R to generate the Dirac neutrino mass. This B − L charge assignment is +needed to forbid Majorana mass terms for the νi +R while simultaneously enforcing a Yukawa + +14 +coupling structure in which only ϕ couples to RHNs. After the spontaneous breaking of +electroweak and U(1)B−L symmetry, we can write the neutrino mass term as +− LM = νLmννR + H.c. +(51) +where mν = Y ij +ν vϕ +√ +2 . The smallness of the neutrino masses relative to those of the quarks and +charged leptons is explained by smallness of the second Higgs doublet vev vϕ ∼ eV for large +Yukawa coupling Yν ∼ O(1). In this model, smallness of vϕ arises very naturally and this +can be understood from Eq. (43) with the approximation vσ ≫ vΦ,ϕ: +vϕ ≈ +µvσv2 +√ +2M 2 +H0vΦ +. +(52) +Hence, the vev of the neutral component of the field ϕ is inversely proportional to the mass of +the heavy scalar. This provided a natural explanation for the low vev and thus low neutrino +masses. Note that in our analysis for simplicity, we assumed one neutrino to be massless. +Gauge sector: As field ϕ is charged under both SM and U(1)B−L gauge group, even in +the absence of kinetic mixing this will introduce mixing between SM neutral boson Z with +the new neutral gauge boson Z′ corresponding to U(1)B−L. Again to determine the gauge +boson mass spectrum we have to expand the following kinetic terms: +(DµΦ)†DµΦ + (Dµϕ)†Dµϕ + (Dµσ)†Dµσ, +(53) +and have to replace the fields Φ, ϕ and σ by the expressions given in Eq. (38). The gauge +bosons mass matrix in the basis (Bµ, W µ +3 , Xµ) can be written as +M2 +V = v2 +4 +� +�� +g′2 +− gg′ +− 6u2g′gx +−gg′ +g2 +6u2ggx +−6u2g′gx +6u2ggx +36b2g2 +x +� +�� , where u = vϕ +v , and b2 = u2 + v2 +σ +v2 . +(54) +Mass matrix in Eq. (54) can be diagonalized by the following unitary matrix +� +�� +Aµ +Zµ +Z′µ +� +�� = +� +�� +cos θw +sin θw +0 +− cos α′ cos θw cos α′ cos θw +− sin α′ +− sin α′ sin θw +sin α′ cos θw +cos α′ +� +�� +� +�� +Bµ +W µ +3 +Xµ +� +�� , +(55) +where, tan θw = g′ +g , and tan 2α′ = +C′ +B′. After rotation we get a massless photon and two +heavy bosons: +MA = 0, M 2 +Z = v2 +8 +� +A′ − +√ +B′2 + C′2 +� +and M 2 +Z′ = v2 +8 +� +A′ + +√ +B′2 + C′2 +� +, +(56) +where, A′ = 36b2g2 +x + (g2 + g′2), +B′ = 36b2g2 +x − (g2 + g′2) and C′ = 12gxu2� +g2 + g′2. + +15 +IV. +W MASS AND THE S, T, U PARAMETERS +In the SM, the W boson mass can be calculated very precisely in terms of the precisely +measured input parameters {GF, αem, MZ}. The W boson mass is related with these pa- +rameters in the following way [24–26]: +M 2 +W = M 2 +Z +2 +� +1 + +� +1 − +4παem +√ +2GFM 2 +Z (1 + ∆r) +� +, +(57) +where ∆r represents the quantum corrections. Taking the central values of the input param- +eters, MZ = 91.1876 GeV, α−1 +em = 137.036, GF = 1.1663787 × 10−5 GeV−2 and considering +the SM value of ∆r ≈ 0.038, Eq. (57) gives us the theoretical prediction of W boson mass +80.360 GeV, with the theoretical uncertainty of near about 4 MeV. This theoretical predic- +tion is 7σ away from the recently announced CDF-II results. Note that the new physics +contribution to the parameter ∆r can be reparametrised in terms of the self energy correc- +tions to the gauge bosons. Specifically, dominant BSM effects can be written in terms of +the three gauge boson self-energy parameters known as the oblique parameters S, T and +U provided that the new physics mass scale is greater than the electroweak scale and that +it contributes only through virtual loops to the electroweak precision observables. The W +boson mass in terms of these parameters can be written as [27]: +MW = M SM +W +� +1 − +α +4(cos2 θw − sin2 θw) (S − 1.55T − 1.24U) +� +. +(58) +Recently, Ref. [28] gave the values of these parameters from an analysis of precision elec- +troweak data including the CDF-II new result of the W-mass: +S = 0.06 ± 0.1, T = 0.11 ± 0.12, U = 0.13 ± 0.09. +(59) +with the correlation +ρST = 0.90, ρSU = −0.59 and ρTU = −0.85. +(60) +V. +NEW PHYSICS CONTRIBUTION TO S, T, U +Using six dimensional SU(2)L invariant effective operator we can parametrise new physics +that only couples to SM vector bosons and Higgs. Effects related with dimension 6 operators +can be expressed in the following way [20, 29] +L = LSM + 2 +v2 (cWBOWB + cHOH + cWWOWW + cBBOBB) . +(61) +We presented two generic models with heavy new neutral vector bosons in section. (III). In +the first model the doublet scalar is not charged under the new symmetry and hence there is + +16 +no mass mixing between SM neutral bosons and the new Z′ boson. In the second model we +introduced a doublet which is charged under new symmetry and hence it can mix Z′ boson +with SM neutral bosons. This novel physics is characterized by oblique parameters in an +effective lagrangian approach, as [30] +S = 4 sin2 θw +αZ +2M 2 +Wg2 +x +g2g′2M 2 +Z′ [Ze − Zφ + ZL] +� +g2Ze + g′2(Ze + 2ZL) +� +, +(62a) +T = 1 +αZ +4M 2 +Wg2 +x +g2M 2 +Z′ [Ze − Zφ + ZL]2 , +(62b) +U = 4 sin2 θw +αZ +4M 2 +Wg2 +x +g2M 2 +Z′ [Ze − Zφ + ZL] [Ze + 2ZL] . +(62c) +The U(1)B−L charges of the Lepton singlet, Lepton doublet are Ze, ZL respectively and Zφ +is the total B − L charge of both the scalar doublets. MZ′ is the mass of the new heavy +boson, and gx is the new gauge coupling. +Zϕ = 0 +Zϕ =-3 +20 +40 +60 +80 +100 +-0.3 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +MZ' /gx [TeV] +S +S = 0.16 +S = -0.04 +Zϕ = 0 +Zϕ =-3 +20 +40 +60 +80 +100 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +MZ' /gx [TeV] +T +T = 0.23 +T = -0.01 +Zϕ = 0 +Zϕ =-3 +20 +40 +60 +80 +100 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +MZ' /gx [TeV] +U +U = 0.22 +U = 0.04 +FIG. 5: The S, T, U parameters versus ratio of the mass of new boson and its gauge +coupling. The red line represents vector B − L model, whereas the blue line represents +chiral B − L model. +Taking the central values of the input parameters, M CDF +W += 80.4335 [5], sin2 θw = + +17 +0.23121, αZ = 1/127.935 [6, 31]. +In Fig. 5, we plotted The S, T, U parameters versus +ratio of the mass of new boson and its gauge coupling for two different scenarios. The red +line depicts the vector B − L model, whereas the blue line depicts the chiral B − L model. +The dotted lines correspond to the updated best fit values of the S, T, U parameters after +the CDF results [28]. +S +T +U +5 +10 +15 +20 +0.2 +0.4 +0.6 +0.8 +1.0 +MZ'[TeV] +gx +S +T +U +5 +10 +15 +20 +0.2 +0.4 +0.6 +0.8 +1.0 +MZ'[TeV] +gx +FIG. 6: The new boson’s gauge coupling versus mass. The space permitted by the new +S, T, U (3σ) is shown by a different coloured band. The graph on the left is for vector +B − L, while the graph on the right is for chiral B − L model. +In Fig. 6 the parameter space that permits us to solve the W anomaly is shown. The blue +line represents the maximum permissible value for T, and the region between the blue line +and the MZ′ axis represents the allowable parameter space fulfilled by the best fit T value +in 3σ range. The allowed region for S and U is shown by the colours red and cyan. +The allowed parameter space in Fig. 7 to satisfy CDF-II measurements is the overlap +zone between the S, T and U bands. As previously stated, the left panel depicts the case +with no mass mixing between SM neutral gauge bosons and the new U(1)B−L neutral gauge +boson, whereas the right panel shows the case with mass mixing, as ϕ has a charge of −3 +under B − L symmetry. Notice that the parameter space is improved when scalar doublet +mixes the new heavy boson with SM neutral bosons. As a result, we will now concentrate +our efforts on the chiral B-L model. +Taking the scenario where ϕ has a charge of −3 under B − L symmetry, we showed the +points (dark cyan) that satisfy W mass in Fig. 8. + +18 +5 +10 +15 +20 +0.2 +0.4 +0.6 +0.8 +1.0 +MZ'[TeV] +gx +5 +10 +15 +20 +0.2 +0.4 +0.6 +0.8 +1.0 +MZ'[TeV] +gx +FIG. 7: The green band represents the permitted parameter space that is consistent with +CDF II, W mass measurement. The graph on the left is for Zφ = 0, while the graph on the +right is for Zφ = −3. +5 +10 +15 +20 +0.2 +0.4 +0.6 +0.8 +1.0 +MZ' [TeV] +gx +FIG. 8: For chiral B − L model, we showed the space permitted by the new oblique +parameters (S, T, U) after CDF-II results (3σ) in green colour. Dark Cyan points are +consistent with CDF-II, W mass measurement. +VI. +DARK MATTER CONSTRAINTS +In this section we collect the results of our analysis of DM phenomenology. As previously +stated, in the second model outlined in the Sec. III, χd is the scalar DM. It carries the + +19 +U(1)B−L charge 1/2, which forbids any term in the potential that results in the decay of χd. +We study the χd relic density and its direct detection prospects. Specifically, we determine +the regions in the parameter space of the model where the DM constraints and S, T, U +consistent with CDF-II W mass measurements can be satisfied. The SARAH-4.14.5 [32, 33] +package is used to calculate all of the vertices and mass matrices, among other things. +All the expressions are verified analytically and numerical calculation are performed by +package SPheno-4.0.2 [34, 35]. The relic abundance, on the other hand, is determined using +micrOMEGAS-5.2.13 [36]. +There are several DM annihilation channels present in this model which is shown in +Appendix. A. They involve annilation to quarks, leptons, neutrinos, gauge bosons (Z, Z′), +neutral scalars (Hi, H0) and charged scalar (H±). Altogether, they determine the relic abun- +dance of χd. Note that as χd is charged under B − L, the DM χd has both the gauge and +scalar interactions. The gauge interactions allow the annihilation of the dark matter particle +into fermions mediated by the gauge boson, χdχ∗ +d → Z′∗ → f ¯f. One should also consider +the direct annihilation into two gauge bosons, χdχ∗ +d → Z′Z′, when kinematically accessi- +ble. Hence in pure gauge interaction case there are very few parameters (MDM, gx, MZ′) +which plays the role in determining DM phenomenology. Due to the strong experimental +constraints on MZ′/gx, the annihilation into Z′Z′ is suppressed, hence only annihilation to +fermions turns out to be relevant. Due to the structure of the gauge coupling, the annihila- +tion channel χdχ∗ +d → Z′∗ → f ¯f is velocity suppressed (∝ v2). This is why the relic density +tends to be much higher than the observed value except in a narrow region close to the +resonance (MDM ∼ MZ′/2). And even at the resonance, the relic density can be too large +to be in agreement with the data, as illustrated in Ref. [17]. Specifically the requirements +to have correct relic is to be near resonance and relatively large coupling gx and low MZ′, +which in view of Fig. 2 is ruled out. +Besides the gauge interactions, the DM χd, also has scalar interactions, see Eq. 36. Scalar +Parameter +Range +mχd +[ 0, 104 ] GeV +λΦχd +[ 10−6, 1 ] +λϕχd +[ 10−6, 1 ] +λσχd +[ 10−6, 1 ] +µ +[ 10−6, 1 ] +TABLE III: Ranges of variation of the input parameters used in our numerical scan. +interactions between the DM and the SM scalar give rise to the well-known Higgs-portal +scenario. But there are some differences between this simplistic Higgs-portal scenario and +our B − L case. First of all, the DM field is necessarily complex as it is charged under + +20 +U(1)B−L – rather than real. In addition to this there will be many additional annihilation +channels due to the presence of additional scalars, both neutral and charged, such as H2,3, +H0 and H±. +In the following, instead of separately studying gauge interaction and scalar interaction, +we will focus on the general case. To do this we first need to identify the parameters which +are relevant for DM analysis. Even though the model introduces new free parameters, not +all of them are important to DM analysis. For example, the self quartic couplings and some +mixed quartic couplings such as λΦσ, λϕσ, λΦϕ1,2 does not play any role in DM phenomenology. +Hence, we choose to fix these parameters. The remaining free parameters relevant for DM +analysis can be chosen as: +mχd, λΦχd, λϕχd, λσχd, gx and MZ′. +(63) +We will look at how the DM phenomenology of this model is affected by the above-mentioned +parameter. To carry out the numerical scan, we varied these parameters as listed in the +Table. III. We varied them on the logarithmic scale. The gauge coupling gx and mass MZ′ +are varied according to the allowable parameter space coming from S, T, U restriction +consistent with CDF-II W mass measurements (Dark cyan points ), as illustrated in Fig. 8. +FIG. 9: Relic density vs the mass of the Dark matter. The blue and grey dots show over +and under abundance relic density points, respectively, whilst the magenta points reflect +the 3 σ range for cold dark matter obtained from Planck satellite data. +In Fig. 9, we show the relic density as a function of the mass of the scalar DM χd. The blue + +108 +106 +104 +102 +Qh +100 +10-2 +10-4 +10-6E +102 +103 +10° +MpM [GeV]21 +and grey points represent the over and under-abundance relic density regions, respectively, +whereas the magenta points in the narrow band fall in the 3σ range for cold dark matter +derived from the Planck satellite data [4]: +0.1126 ≤ Ωh2 ≤ 0.1246. +(64) +Various features of the Fig. 9 can be understood from different DM annihilation channels +shown in Appendix. A. Annihilation in the low mass region of DM is dominated via exchange +of SM Higgs (H1 = h) to SM fermionic final states. As DM approaches half of the Higgs mass +( MDM ≈ mh/2), h becomes on-shell and these annihilation channels become very efficient. +Notice that there is no dip at MDM ≈ MZ/2 because the mixing between Z and Z′ is not +strong enough for annihilation through Z exchange to be effective. For MDM ≥ 80 GeV, +annihilation of DM to Z and W final states comes into picture (χdχ∗ +d → ZZ, W +W −) and +hence we get another dip in that region. We see a dip in relic density in the DM mass range +2.5 TeV-4 TeV. This is due to the fact that the combination of gx and MZ′ required to get +correct oblique parameters forces the vev of singlet scalar (vσ) to be around 8 TeV-10 TeV. +This high vev pushes the mass of CP-even scalars H2 and H3 to be in the range of 5-8 TeV +and we get a dip when DM mass is roughly half of this range due to the the H2,3 mediated +s-channel annihilation to SM final states. A sub-dominant role is played by annihilation +into HiHj, H0H0 and ZZ, Z′Z′ via the direct 4-point vertices HiHjχdχ∗ +d, H0H0χdχ∗ +d and +ZZχdχ∗ +d or ZZ′χdχ∗ +d, respectively. +Also there could be additional contribution from χd +exchange in the t-channel. As the annihilation cross section is inversely proportional to the +mass of dark matter, at very high value of DM mass the relic density increases. +Direct Detection: Let us now study the direct detection prospects of our DM candi- +dates χd. A large number of experiments are being conducted to demonstrate the parti- +cle nature of dark matter through direct detection. Various direct detection experiments, +XENON1T [37], LZ[38], XENONnT[39], LUX[40, 41], PandaX-II[42], impose constraints. +These experiments are designed to measure the tiny recoil in the detector target nuclei pro- +duced by the elastic collisions between DM and target nuclei. +The effective lagrangian for nucleon-DM interactions is expressed as +Leff = aNNNχ2 +d, +(65) +Where aN is the effective nucleon-DM coupling. +The spin independent scattering cross +section via the Higgs(H1,2,3) interaction is given by +σSI +N−χd = µ2M 2 +Nf 2 +N +4πM 2 +DMv2 +�λH1χ2 +d +M 2 +H1 +(OR)11 + +λH2χ2 +d +M 2 +H2 +(OR)21 + +λH3χ2 +d +M 2 +H3 +(OR)31 +� +, +(66) +Where (OR)ij is the elements of the mass matrix defined in Eq. (47), fN is the form factor, +which depends on the hadronic matrix elements and µ = +MNMDM +MN+MDM is the reduced mass for + +22 +nucleon-DM system. The trilinear couplings are given as +λHiχ2 +d = 2 [vΦλΦχd(OR)i1 + vϕλϕχd(OR)i2 + vσλσχd(OR)i3] . +(67) +Eq. (66) is an extension of the expression corresponding to the scalar DM case [43]. The +cross section per nucleon for Dark matter-nuclei interaction through Z′ is given as [44] +σ0 = 1 +π +� +MDMMn +MDM + AMn +�2 � g2 +x +2M 2 +Z′ +�2 +, +(68) +Where A is the number of nucleons in the target, we have set it to 131 for Xenon. The +nucleon mass is Mn = 0.938919 GeV. +FIG. 10: WIMP-nucleon spin-independent cross section for the scalar dark matter χd. The +colour code has the same meaning as in Fig. 9. The solid red line denotes the latest upper +bound from the LZ [38] collaboration, the dashed red line corresponds to XENON1T [37] +limit and the dashed brown line corresponds to the “neutrino floor” lower limit [45, 46]. +In Fig. (10) we imposed the direct detection constraints on our scalar dark matter χd. +We performed the numerical scan with the micrOMEGAS-5.2.13 and varied the parameters +as shown in the table (III). The colour code has the same meaning as in Fig. 9. The LZ and +XENON1T experiment imposes the most stringent constraints. As a result, we plotted the +most recent upper bound from the both LZ and XENON1T collaboration [37], as shown by +the solid and dashed red line respectively. The brown line represents the lower limit, which +corresponds to the “neutrino floor” from the coherent elastic neutrino scattering [45, 46]. +In Fig. 11, we demonstrated the parameter space that is compatible with all of the afore- +mentioned requirements. The green bands show the permitted values for oblique parameters + +[cm?] +10-44 +XENONIT +10-48 +10-50 +101 +102 +103 +MpM [GeV]23 +S, T and U. The region with the grey shading is the ATLAS’s most stringent collider con- +straint. The magenta points fulfill M CDF +W +, relic density and limitations from direct detection +experiments all at the same time. +5 +10 +15 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +MZ'[TeV] +gx +ATLAS13(2l) +FIG. 11: We showed the space permitted by the new oblique parameters (S, T, U) after +CDF-II results in green colour. The grey region is ruled out by most stringent collider +constraints from ATLAS. The magenta points satisfy M CDF +W +, relic density and limits from +direct detection experiments simultaneously. +It is worth noting that there is a significant parameter space in the DM high mass region, +which is consistent with recent measurements of W boson mass, relic abundance, collider +constraints, and the direct detection experiments. +VII. +CONCLUSION +The U(1)B−L gauged extension of the SM is very simple in its nature. +It’s minimal +version just needs three right-handed neutrinos to cancel gauge anomalies. +It naturally +explains the small neutrino masses through a seesaw mechanism. Despite their simplicity, +these type of models can explain the recent CDF-II measurement of the W boson mass, +which reveals considerable disagreement with the SM predictions. The new neutral boson +associated with the new U(1)B−L symmetry can provide the loop corrections to gauge boson +two-point functions that are compatible with the most recently revised oblique parameter +values as a consequence of the CDF-II results. +We investigated and find that kinetic mixing alone can not explain the W anomaly +at the tree level. +In addition, we investigated the two models with and without mass +mixing between neutral bosons. +We focused our attention on chiral B − L model. +We + +24 +imposed constraints derived from direct detection of dark matter and relic abundance and +demonstrated that the chiral U(1)B−L model can explain the W anomaly. +ACKNOWLEDGMENTS +Work of S.M. has been supported by KIAS Individual Grants (PG086001) at Korea Insti- +tute for Advanced Study. The work of R.S. is supported by the Government of India,SERB +Startup Grant SRG/2020/002303. The work of H.P. is supported by the Prime Minister +Research Fellowship (ID: 0401969). +Appendix A: Annihilation channels for scalar DM χd +In the chiral B − L model the relic abundance of the DM candidate χd is determined by +the annihilation diagrams shown in Fig. 12. + +25 +H1/H2/H3 +χd +χ∗ +d +q +q +H1/H2/H3 +χd +χ∗ +d +l+ +l− +1 +H1/H2/H3 +χd +χ∗ +d +Z/W + +Z′/W − +Z/Z′ +χd +χ∗ +d +ν +ν +1 +H1/H2/H3 +χd +χ∗ +d +ν +ν +H1/H2/H3 +χd +χ∗ +d +Z/Z′ +Z/Z′ +1 +Hi +χd +χ∗ +d +Hj +Hk +Z/Z′ +χd +χ∗ +d +q/l+ +q/l− +1 +H1/H2/H3 +χd +χ∗ +d +W +/W − +H−/H+ +Z/Z′ +χd +χ∗ +d +W +/W − +H−/H+ +1 +Z/Z′ +χd +χ∗ +d +Z′ +Hi +Z/Z′ +χd +χ∗ +d +Z +Hi +1 +χd +χ∗ +d +Hi +Hj +χd +χ∗ +d +H0 +H0 +χd +χ∗ +d +Z/Z′/Z +Z/Z′/Z′ +H1/H2/H3 +χd +χ∗ +d +Z′/Z +H0 +H1/H2/H3 +χd +χ∗ +d +H0 +H0 +1 +Z/Z′ +χd +χ∗ +d +W + +W − +χd +Hi +χ∗ +d +χd +Z/Z′ +χd +Hj +χ∗ +d +χd +Hi +1 +χd +Z/Z′/Z′ +χ∗ +d +χd +Z/Z/Z′ +1 +FIG. 12: Feynman diagrams that contributes to the relic density of the scalar dark matter +χd. + +26 +[1] ATLAS Collaboration, G. Aad et al., “Observation of a new particle in the search for the +Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B 716 +(2012) 1–29, arXiv:1207.7214 [hep-ex]. +[2] CMS Collaboration, S. Chatrchyan et al., “Observation of a New Boson at a Mass of 125 +GeV with the CMS Experiment at the LHC,” Phys. Lett. B 716 (2012) 30–61, +arXiv:1207.7235 [hep-ex]. +[3] Super-Kamiokande Collaboration, Y. Fukuda et al., “Evidence for oscillation of +atmospheric neutrinos,” Phys. Rev. Lett. 81 (1998) 1562–1567, arXiv:hep-ex/9807003. +[4] Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological +parameters,” Astron. Astrophys. 641 (2020) A6, arXiv:1807.06209 [astro-ph.CO]. +[Erratum: Astron.Astrophys. 652, C4 (2021)]. +[5] CDF Collaboration, T. Aaltonen et al., “High-precision measurement of the W boson mass +with the CDF II detector,” Science 376 no. 6589, (2022) 170–176. +[6] Particle Data Group Collaboration, P. A. Zyla et al., “Review of Particle Physics,” PTEP +2020 no. 8, (2020) 083C01. +[7] J. Schechter and J. W. F. Valle, “Neutrino Masses in SU(2) x U(1) Theories,” Phys. Rev. D +22 (1980) 2227. +[8] J. Schechter and J. W. F. Valle, “Neutrino Decay and Spontaneous Violation of Lepton +Number,” Phys. Rev. D 25 (1982) 774. +[9] R. N. Mohapatra and J. W. F. Valle, “Neutrino Mass and Baryon Number Nonconservation +in Superstring Models,” Phys. Rev. D 34 (1986) 1642. +[10] M. Malinsky, J. C. Romao, and J. W. F. Valle, “Novel supersymmetric SO(10) seesaw +mechanism,” Phys. Rev. Lett. 95 (2005) 161801, arXiv:hep-ph/0506296. +[11] N. Okada and S. Okada, “Z′-portal right-handed neutrino dark matter in the minimal +U(1)X extended Standard Model,” Phys. Rev. D 95 no. 3, (2017) 035025, +arXiv:1611.02672 [hep-ph]. +[12] A. Das, S. Gola, S. Mandal, and N. Sinha, “Two-component scalar and fermionic dark +matter candidates in a generic U(1)X model,” Phys. Lett. B 829 (2022) 137117, +arXiv:2202.01443 [hep-ph]. +[13] A. Das, N. Okada, and D. Raut, “Heavy Majorana neutrino pair productions at the LHC in +minimal U(1) extended Standard Model,” Eur. Phys. J. C 78 no. 9, (2018) 696, +arXiv:1711.09896 [hep-ph]. +[14] L. Basso, Phenomenology of the minimal B-L extension of the Standard Model at the LHC. +PhD thesis, Southampton U., 2011. arXiv:1106.4462 [hep-ph]. + +27 +[15] W. Emam and S. Khalil, “Higgs and Z-prime phenomenology in B-L extension of the +standard model at LHC,” Eur. Phys. J. C 52 (2007) 625–633, arXiv:0704.1395 [hep-ph]. +[16] L. Basso, A. Belyaev, S. Moretti, and G. M. Pruna, “Tree Level Unitarity Bounds for the +Minimal B-L Model,” Phys. Rev. D 81 (2010) 095018, arXiv:1002.1939 [hep-ph]. +[17] W. Rodejohann and C. E. Yaguna, “Scalar dark matter in the B−L model,” JCAP 12 +(2015) 032, arXiv:1509.04036 [hep-ph]. +[18] P. Langacker, “The Physics of Heavy Z′ Gauge Bosons,” Rev. Mod. Phys. 81 (2009) +1199–1228, arXiv:0801.1345 [hep-ph]. +[19] M. Carena, A. Daleo, B. A. Dobrescu, and T. M. P. Tait, “Z′ gauge bosons at the +Tevatron,” Phys. Rev. D 70 (2004) 093009, arXiv:hep-ph/0408098. +[20] G. Cacciapaglia, C. Csaki, G. Marandella, and A. Strumia, “The Minimal Set of Electroweak +Precision Parameters,” Phys. Rev. D 74 (2006) 033011, arXiv:hep-ph/0604111. +[21] ATLAS Collaboration, G. Aad et al., “Search for high-mass dilepton resonances using 139 +fb−1 of pp collision data collected at √s =13 TeV with the ATLAS detector,” Phys. Lett. B +796 (2019) 68–87, arXiv:1903.06248 [hep-ex]. +[22] CMS Collaboration, “Search for a narrow resonance in high-mass dilepton final states in +proton-proton collisions using 140 fb−1 of data at √s = 13 TeV,”. +[23] A. Das, P. S. B. Dev, Y. Hosotani, and S. Mandal, “Probing the minimal U(1)X model at +future electron-positron colliders via the fermion pair-production channel,” +arXiv:2104.10902 [hep-ph]. +[24] D. L´opez-Val and T. Robens, “∆r and the W-boson mass in the singlet extension of the +standard model,” Phys. Rev. D 90 (2014) 114018, arXiv:1406.1043 [hep-ph]. +[25] W. Hollik, “Electroweak theory,” J. Phys. Conf. Ser. 53 (2006) 7–43. +[26] S. Amrith, J. M. Butterworth, F. F. Deppisch, W. Liu, A. Varma, and D. Yallup, “LHC +Constraints on a B − L Gauge Model using Contur,” JHEP 05 (2019) 154, +arXiv:1811.11452 [hep-ph]. +[27] C. Cai, Z.-H. Yu, and H.-H. Zhang, “CEPC Precision of Electroweak Oblique Parameters +and Weakly Interacting Dark Matter: the Fermionic Case,” Nucl. Phys. B 921 (2017) +181–210, arXiv:1611.02186 [hep-ph]. +[28] C.-T. Lu, L. Wu, Y. Wu, and B. Zhu, “Electroweak Precision Fit and New Physics in light +of W Boson Mass,” arXiv:2204.03796 [hep-ph]. +[29] R. Barbieri, A. Pomarol, R. Rattazzi, and A. Strumia, “Electroweak symmetry breaking +after LEP-1 and LEP-2,” Nucl. Phys. B 703 (2004) 127–146, arXiv:hep-ph/0405040. +[30] A. Strumia, “Interpreting electroweak precision data including the W-mass CDF anomaly,” +arXiv:2204.04191 [hep-ph]. +[31] K. S. Babu, S. Jana, and V. P. K., “Correlating W-Boson Mass Shift with Muon g − 2 in the +2HDM,” arXiv:2204.05303 [hep-ph]. + +28 +[32] F. Staub, “Exploring new models in all detail with SARAH,” Adv. High Energy Phys. 2015 +(2015) 840780, arXiv:1503.04200 [hep-ph]. +[33] M. D. Goodsell, S. Liebler, and F. Staub, “Generic calculation of two-body partial decay +widths at the full one-loop level,” Eur. Phys. J. C 77 no. 11, (2017) 758, arXiv:1703.09237 +[hep-ph]. +[34] W. Porod, “SPheno, a program for calculating supersymmetric spectra, SUSY particle +decays and SUSY particle production at e+ e- colliders,” Comput. Phys. Commun. 153 +(2003) 275–315, arXiv:hep-ph/0301101. +[35] W. Porod and F. Staub, “SPheno 3.1: Extensions including flavour, CP-phases and models +beyond the MSSM,” Comput. Phys. Commun. 183 (2012) 2458–2469, arXiv:1104.1573 +[hep-ph]. +[36] G. B´elanger, F. Boudjema, A. Goudelis, A. Pukhov, and B. Zaldivar, “micrOMEGAs5.0 : +Freeze-in,” Comput. Phys. Commun. 231 (2018) 173–186, arXiv:1801.03509 [hep-ph]. +[37] XENON Collaboration, E. Aprile et al., “Dark Matter Search Results from a One Ton-Year +Exposure of XENON1T,” Phys. Rev. Lett. 121 no. 11, (2018) 111302, arXiv:1805.12562 +[astro-ph.CO]. +[38] LUX-ZEPLIN Collaboration, J. Aalbers et al., “First Dark Matter Search Results from +the LUX-ZEPLIN (LZ) Experiment,” arXiv:2207.03764 [hep-ex]. +[39] E. Aprile et al., “Search for New Physics in Electronic Recoil Data from XENONnT,” +arXiv:2207.11330 [hep-ex]. +[40] LUX Collaboration, D. S. Akerib et al., “First results from the LUX dark matter +experiment at the Sanford Underground Research Facility,” Phys. Rev. Lett. 112 (2014) +091303, arXiv:1310.8214 [astro-ph.CO]. +[41] LUX Collaboration, D. S. Akerib et al., “Results from a search for dark matter in the +complete LUX exposure,” Phys. Rev. Lett. 118 no. 2, (2017) 021303, arXiv:1608.07648 +[astro-ph.CO]. +[42] PandaX-II Collaboration, Q. Wang et al., “Results of dark matter search using the full +PandaX-II exposure,” Chin. Phys. C 44 no. 12, (2020) 125001, arXiv:2007.15469 +[astro-ph.CO]. +[43] J. M. Cline, K. Kainulainen, P. Scott, and C. Weniger, “Update on scalar singlet dark +matter,” Phys. Rev. D 88 (2013) 055025, arXiv:1306.4710 [hep-ph]. [Erratum: +Phys.Rev.D 92, 039906 (2015)]. +[44] E. Ma, N. Pollard, R. Srivastava, and M. Zakeri, “Gauge B − L Model with Residual Z3 +Symmetry,” Phys. Lett. B 750 (2015) 135–138, arXiv:1507.03943 [hep-ph]. +[45] J. Billard, L. Strigari, and E. Figueroa-Feliciano, “Implication of neutrino backgrounds on +the reach of next generation dark matter direct detection experiments,” Phys. Rev. D 89 +no. 2, (2014) 023524, arXiv:1307.5458 [hep-ph]. + +29 +[46] J. Billard et al., “Direct Detection of Dark Matter – APPEC Committee Report,” +arXiv:2104.07634 [hep-ex]. + diff --git a/JNAzT4oBgHgl3EQfjv3G/content/tmp_files/load_file.txt b/JNAzT4oBgHgl3EQfjv3G/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c30feb91dfcea8084c738966c85085810664c12 --- /dev/null +++ b/JNAzT4oBgHgl3EQfjv3G/content/tmp_files/load_file.txt @@ -0,0 +1,1000 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf,len=999 +page_content='B − L model in light of the CDF II result Sanjoy Mandal,1, ∗ Hemant Prajapati,2, † and Rahul Srivastava2, ‡ 1Korea Institute for Advanced Study, Seoul 02455, Korea 2Department of Physics, Indian Institute of Science Education and Research - Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462066, India (Dated: January 5, 2023) Recent CDF II collaboration results on W mass measurements contradict Standard Model (SM) prediction, requiring new physics to explain this anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' To explain this issue, in this paper we investigate the idea of using the U(1)B−L gauged SM extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We demonstrate that B −L extended models can explain the revised best fit values for S, T, and U following the CDF II results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We studied the parameter space of models with and without mixing between neutral gauge bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We also reviewed the dark matter constraints and demonstrated that there is parameter space which is compatible with current W boson mass, relic abundance, and direct detection experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' INTRODUCTION The Standard model (SM) of electroweak theory with SU(2) ⊗ U(1) gauge symmetry is highly successful in explaining most of the observations in particle physics experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' With the recent finding of the Higgs like boson with mass 125 GeV at the LHC [1, 2], seems to complete the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Despite its ability to explain most observable phenomena, the SM can- not be considered the final theory of particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' There is an ever-increasing number of observations such as the discovery of neutrino oscillations [3] and the existence of dark mat- ter (DM) at cosmic scales [4] that put serious questions on the SM predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In addition to this, the CDF-II collaboration recently published their high precision measurement of the W boson mass M CDF W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4335 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='0094 GeV [5], which reveals a 7-σ difference from the SM expectation M SM W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='354 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='007 GeV [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' This leads us to investigate the extension of SM, which can account for the aforementioned problems with SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In the absence of right-handed neutrino (RHN), the neutrinos are massless in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' RHNs have been a prevalent feature of many extensions of SM such as various seesaw mechanisms [7–10] to generate neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In recent years, a number of models have been proposed that combine neutrino mass generations and the existence of DM into a ∗ smandal@kias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='kr † hemant19@iiserb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='in ‡ rahul@iiserb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='01522v1 [hep-ph] 4 Jan 2023 2 single framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Motivated by this, people have studied extensively beyond standard model (BSM) framework based on the gauged U(1)B−L model [11–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The most intriguing aspect of this model is that it includes three RHNs to cancel gauge and mixed gauge-gravity anomalies and generate tiny neutrino masses through the seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' This type of model predicts the existence of a new neutral gauge boson Z′ that can mix with the SM neutral gauge boson Z [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' One also has the possibility of explaining the DM in these type of model with an additional scalar field, χd, that is a SM singlet but charged under U(1)B−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' An advantage of this scenario is that one does not need to impose any ad hoc Z2 symmetry to stabilise the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Instead, stability of χd can be guaranteed by appropriately choosing its B − L charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In this work we show that, despite its simplicity, in addition to neutrino mass generation and DM, B − L model can also explain the recent CDF-II W boson mass measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The new boson associated with U(1)B−L symmetry mixes with the SM neural Z boson to provide S, T, U corrections that are compatible with current W boson mass measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Specifically we investigate two distinct scenarios: one with no mass mixing between two neutral bosons and one with mass mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We study the difference in parameter space in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We show that the parameter space consistent with the best fit S, T, U values following the CDF II results is also consistent with the DM physics constraints in the model we proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The paper is organised as follows: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' II, we briefly discuss the possibility of having kinetic mixing between two field strength tensors corresponding to U(1)Y and U(1)B−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We investigate in detail whether or not addressing simply the impacts of kinetic mixing at the tree level may resolve the W mass anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' III, we study two B − L gauged models without taking into account the effects of kinetic mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The first model is a minimal B − L extension of the SM with no mass mixing between the SM neutral gauge boson Z and U(1)B−L neutral gauge boson Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In the second model, we introduce mass mixing between these neutral gauge bosons and also introduce a scalar DM candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' IV, we discuss how one can parametrise the new physics contributions to W mass in terms of oblique parameters S, T and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' V, we described the effective lagrangian approach to parameterise this novel physics, as well as the parameter space that is compatible with the S, T, and U parameters following the CDF-II data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We focused our attention on the chiral B − L model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Following that, we reviewed the DM constraints derived from Planck’s measurement of the relic density, as well as the constraints derived from the direct detection experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Finally, we demonstrated that, in the chiral B −L model, the parameter space we found agrees with the current measurements of the W boson mass, relic abundance, and direct detection experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' KINETIC-MIXING AND W MASS We note that a kinetic mixing can occur provided there are two or more field strength tensors Bµν and Xµν which are neutral under some gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Thus in our case with the gauge group SU(3)c ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1)B−L , the kinetic terms can be expressed as follows LKinetic = −1 4BµνBµν − 1 4XµνXµν − κ 2BµνXµν, (1) where Bµν and Xµν are the filed strength tensors of the gauge groups U(1)Y and U(1)X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The requirement of positive kinetic energy implies that kinetic coefficient |κ| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' One can diagonalize the kinetic mixing term as follow � ˜B ˜X � = � 1 κ 0 √ 1 − κ2 � � B X � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (2) Let’s first determine the gauge boson mass spectrum setting the kinetic mixing κ = 0 to fix our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' With the kinetic mixing κ = 0, the covariant derivative can be defined as Dµ = ∂µ − igsT aGa µ − igT aW a µ − ig′Y Bµ − igXYXXµ, (3) where gauge coupling gX is a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In addition to SM Higgs doublet Φ, one adds a scalar, χ, singlet of the SM but charged under U(1)B−L, that spontaneously breaks the B − L symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In minimal case, the U(1)B−L charge of χ is qχ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' To determine the gauge boson mass spectrum, we have to expand the following scalar kinetic terms Ls = (DµΦ)†(DµΦ) + (Dµχ)†(Dµχ), (4) and have to replace the fields Φ and χ by the following expressions such as Φ = 1 √ 2 � 0 vΦ + R1 � , ⟨χ⟩ = 1 √ 2(vχ + R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (5) With this above replacement we can expand the scalar kinetic terms (DµΦ)†(DµΦ) and (Dµχ)†(Dµχ) as follows (DµΦ)†(DµΦ) ≡ 1 2∂µR1∂µR1 + 1 8(R1 + vΦ)2� g2|W µ 1 − iW µ 2 |2 + (gW µ 3 − g′Bµ)2� , (6) (Dµχ)†(Dµχ) ≡ 1 2∂µR2∂µR2 + 1 2(R2 + vχ)2(g ′ 1Xµ)2, (7) where we have defined g ′ 1 = gXqχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' With this, the mass matrix of the neutral gauge bosons is given by LM = 1 2V T 0 M 2 GV0, (8) 4 where V T 0 = � Bµ W3µ Xµ � and M 2 G = � � � 1 4g′2v2 Φ − 1 4gg′v2 Φ 0 − 1 4gg′v2 Φ 1 4g2v2 Φ 0 0 0 g ′2 1 v2 χ � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (9) In the kinetic term diagonalized basis ˜V T 0 = ( ˜Bµ W3µ ˜Xµ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' the mass matrix of the neutral gauge boson can be written as LM = 1 2 ˜V T 0 STM 2 GS ˜V0 = 1 2 ˜V T 0 ˜ M 2 G ˜V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (10) where S = � � � 1 0 − κ √ 1−κ2 0 1 0 0 0 1 √ 1−κ2 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' ˜ M 2 G = STM 2 GS = � � � 1 4g′2v2 Φ − 1 4gg′v2 Φ 1 4g′˜gtv2 Φ − 1 4gg′v2 Φ 1 4g2v2 Φ − 1 4g˜gtv2 Φ 1 4g′˜gtv2 Φ − 1 4g˜gtv2 Φ 1 4˜g2 t v2 Φ + g ′′2 1 v2 χ � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (11) with ˜gt = − g′κ √ 1−κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Following linear combination of ˜Bµ, W µ 3 and ˜Xµ gives definite mass eigenstates Aµ, Zµ and Z ′µ, � � � ˜Bµ W µ 3 ˜Xµ � � � = � � � cos θw − sin θw cos θ sin θw sin θ sin θw cos θw cos θ − cos θw sin θ 0 sin θ cos θ � � � � � � Aµ Zµ Z ′µ � � � , (12) where tan2θ = 2˜gt � g2 + g′2 ˜g2 t + 16 � vχ 2vΦ �2 g ′′2 1 − g2 − g′2 with ˜gt = − g′κ √ 1 − κ2 and g ′′ 1 = g′ 1 √ 1 − κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (13) Masses of physical gauge bosons A, Z and Z ′ are given by, MA = 0, M 2 Z,Z′ = 1 8 � Cv2 Φ ∓ � −D + v4 ΦC2 � , (14) where, C = g2 + g′2 + ˜g2 t + 16 � vχ 2vΦ �2 g ′′2 1 , D = 16v2 Φv2 χ(g2 + g′2)g ′′2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (15) The covariant derivative with the kinetic mixing can be expressed in terms of the orthogonal fields ˜B and ˜X as Dµ = ∂µ − igsT aGa µ − igT aW a µ − ig′Y ˜Bµ − i � gXYX 1 √ 1 − κ2 − g′Y κ √ 1 − κ2 � ˜Xµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (16) 5 W mass: Now let’s try to see whether one can explain the CDF-II anomaly considering only kinetic mixing and ignoring any other loop corrections due to new neutral gauge boson Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Specifically, we consider that the shift in W boson mass measured by CDF-II also modifies the Z boson mass at the tree level as the ρ parameter should be equal to one at tree-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Further, we investigate whether new physics contribution through kinetic mixing is sufficient to reduce this change in Z mass to the experimental value of MZ = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1876 GeV [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The tree-level formula for the W and Z mass is given as follows M 2 W = v2 Φg2 4 , M 2 Z|κ=0 = v2 Φ 4 � g2 + g′2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (17) Taking the CDF II measured W mass, MW = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4335 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='0094 GeV [5] and using the PDG values for other input parameters, sin2 θw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='23121 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='00004, Gf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1663787(6) × 10−5 (GeV)−2 [6], we calculated weak couplings that is consistent with CDF-II measured W mass and then calculated the Z mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' With this, the central value of the theoretically computed Z mass is given as MZ|κ=0 = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='7345 GeV, (18) which is of course larger than the experimental value of MZ = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1876 GeV [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Note that in B − L model, Z mass is affected by the presence of the kinetic mixing parameter κ, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (14): MZ = f (gx, MZ′, κ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (19) As a result, the new physics contribution from kinetic mixing κ can reduce the Z mass to the experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 1, we show how Z mass depends on the kinetic mixing κ, gx and MZ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The various lines in each panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 1 correspond to different values of gx and MZ′ while the ratio MZ′ gx remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The ratio is kept at 6 TeV, 8 TeV and 10 TeV in the top left, top right and bottom panel, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' It is clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 1 that when only the central values are considered, the change in Z mass that touches the experimental value occurs only at low Z′ mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Also comparing top left, top right and bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 1, we see that at high MZ′ gx ratio, the kinetic mixing is not sufficient to reduce Z mass to the experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The mass of Z′ and the gauge coupling gx can be constrained with collider data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' From LEP II data the bound MZ′ gx ≳ 6 − 7 TeV, (20) was derived in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [19, 20, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Current ATLAS and CMS searches for dilepton resonances at the LHC can also be used to constrain MZ′ via the Drell-Yan process, pp → Z′ → ℓ¯ℓ, with ℓ = e, µ [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 2, we see that the LHC dilepton constraints are the most stringent up to MZ = 6 TeV, beyond which the resonant Z′ production is kinematically limited at √s = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Hence comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 2, we can conclude that in view 6 gx=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="1 , MZ' = 600 gx=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="2 , MZ' = 1200 gx=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="3 , MZ' =1800 gx=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="4 , MZ' = 2400 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='50 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='75 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='00 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='25 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='50 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='75 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='00 κ MZ (GeV) MZ(exp) gx=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="1 , MZ' = 800 gx=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="2 , MZ' = 1600 gx=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="3 , MZ' =2400 gx=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="4 , MZ' = 3200 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='50 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='75 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='00 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='25 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='50 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='75 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='00 κ MZ (GeV) MZ(exp) gx=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="1 , MZ' = 1000 gx=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="2 , MZ' = 2000 gx=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="3 , MZ' = 3000 gx=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="4 , MZ' = 4000 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='50 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='75 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='00 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='25 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='50 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='75 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='00 κ MZ (GeV) MZ(exp) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 1: Z mass versus kinetic mixing κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The coloured lines in each panel correspond to different gx and MZ values while keeping the ratio MZ′ gx constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' of current experimental constraints on gx − MZ′, it is not possible to explain CDF-II W anomaly at tree-level with help of kinetic mixing κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' MINIMAL AND CHIRAL B − L MODEL We saw in the previous section that kinetic mixing alone is insufficient to explain the W mass anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' From this point forward, we will ignore kinetic mixing and concentrate on the loop contribution from the U(1)B−L Z′ gauge sector in order to explain the W mass anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In this section, we study two U(1)B−L gauged SM extensions: minimal and chiral extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Under U(1)B−L the SM quarks and leptons have charge 1/3 and −1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' As a result, 7 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="500 1 MZ '[ TeV ] gx CMS13(2l) ATLAS13(2l) LEP-II FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 2: Constraint on gx as a function of MZ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The shaded regions are ruled out from LEP-II [19, 20], ATLAS and CMS dilepton searches [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' B −L is an anomalous symmetry that requires the inclusion of additional fermions to gauge it consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The gauge group U(1)B−L has the potential to cause the following triangle gauge anomalies: [SU(3)c]2[U(1)B−L] = � q XqL − � q XqR, (21a) [SU(2)L]2[U(1)B−L] = � l XlL + 3 � q XqL, (21b) [U(1)Y ]2[U(1)B−L] = � lq (Y 2 lLXlL + 3Y 2 qLXqL) − � lq (Y 2 lRXlR + 3Y 2 qRXqR), (21c) [U(1)Y ][U(1)B−L]2 = � lq (YlLX2 lL + 3YqLX2 qL) − � lq (YlRX2 lR + 3YqRX2 qR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (21d) In addition to this we have two more equations [U(1)B−L]3 = � lq (X3 lL + 3X3 qL) − � lq (X3 lR + 3X3 qR), (22a) [G]2[U(1)B−L] = � lq (XlL + 3XqL) − � lq (XlR + 3XqR), (22b) where, X is the U(1)B−L charge and Y is the hyper charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Anomalies from the first four equations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (21), cancel within the SM particle content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' To cancel anomalies arising from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (22), we add three generations of RHNs(νi R, i = 1, 2, 3) with U(1)B−L charges (x1, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' This gives us the following two conditions: x1 + x2 + x3 = −3, (23a) x3 1 + x3 2 + x3 3 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (23b) 8 We will discuss two charge assignment for the νi R that cancel anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The first is the vector solution (also sometime called minimal B − L extension), in which the RHNs has the same charge as the left-handed neutrino: (−1, −1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Second assignment makes neutrinos chiral under U(1)B−L with RHNs charges: (5, −4, −4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Fields ( SU(3)c ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1)B−L ) LL (1, 2, − 1 2, −1) QL (3, 2, 1 6, 1 3) eR (1, 1, −1, −1) νR (1, 1, 0, −1) uR (3, 1, 2 3, 1 3) dR (3, 1, − 1 3, 1 3) Φ (1, 2, 1 2, 0) χ (1, 1, 0, 2) TABLE I: Matter content and charge assignment of the vector B − L model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' For brevity, the generation index is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Vector B − L Model This model is a simple extension of the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The particle contents and their charges under the gauge group SU(3)c ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1)B−L are given in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The new particles are three RHNs with B − L charge −1 to cancel the gauge anomalies and a new scalar filed χ, singlet of the SM but charged under U(1)B−L, that spontaneously breaks the B − L symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We assign B − L charge +2 for scalar field χ so that νi R gets Majorana mass after B − L breaking which further gives rise to light neutrino mass through seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We begin by writing down the Lagrangian of the scalar sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The most general renormalizable and SU(3)c ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1)B−L gauge invariant scalar sector is given by Ls = (DµΦ)†(DµΦ) + (Dµχ)†(Dµχ) − V(Φ, χ), (24) where the covariant derivative is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The scalar potential V(Φ, χ) is given by V(Φ, χ) = m2 χ(χ∗χ) + 1 2λχ(χ∗χ)2 + m2 Φ(Φ†Φ) + 1 2λΦ(Φ†Φ)2 + λΦχ(χ∗χ)(Φ†Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (25) The breaking of the electroweak and the U(1)B−L gauge symmetries are driven by the vacuum expectation values(vev) of the scalar fields Φ and χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Denoting the vevs of field 9 Φ and χ as vΦ and vχ, the fields Φ and χ after symmetry breaking can be written in the following form: Φ = 1 √ 2 � √ 2G+ vΦ + R1 + iI1 � , χ = 1 √ 2(vχ + R2 + iI2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (26) G± are the Goldstone boson corresponding to W ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' I1 and I2 will mix and give rise to the Goldstone bosons corresponding to the neutral gauge bosons Z and Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The mass matrix of CP-even Higgs scalars in the basis (R1, R2) reads as M2 R = � A C C B � = � v2 ΦλΦ vΦvχλΦχ vΦvχλΦχ v2 χλχ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (27) The mass eigenvalues of light and heavy mass eigenstates as m2 h = 1 2 � A + B − � (A − B)2 + 4C2 � , (28) m2 H = 1 2 � A + B + � (A − B)2 + 4C2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (29) We follow the convention m2 h ≤ m2 H and have identified h as the SM Higgs discovered at LHC, with mass mh = 125 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The two mass eigenstates h, H are related with the (R1, R2) fields through the following rotation matrix as � h H � = U � R1 R2 � = � cos θ − sin θ sin θ cos θ � � R1 R2 � , with tan 2θ = 2C B − A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (30) In the absence of kinetic mixing, neutral bosons cannot have mass mixing because the scalar doublet Φ does not carry any B − L charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The gauge boson masses are given as M 2 Z = v2 Φ 4 � g2 + g′2� , M 2 W = v2 Φg2 4 , MZ′ = 2vχgx, (31) where g and g′ are SU(2) and hypercharge coupling respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In B − L model neutrino masses are generated by seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Apart from the SM L ⟨Φ⟩ νR νR ⟨χ⟩ ⟨Φ⟩ L 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 3: Neutrino mass generation in B − L model through type-I seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 10 part, the Yukawa sector of the model can be written in a gauge-invariant way as −LY ⊃ Y ij ν L i ˜Φνj R + yij M 2 νc RiνRjχ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', (32) The first and second terms will give the Dirac and Majorana contributions to the neutrino mass generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We assume without loss of any generality a basis in which yij M is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' After the breaking of electroweak and U(1)B−L symmetry, we can write the mass term as −LM ⊃ νLmDνR + 1 2νc RMRνR + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', (33) where mD = yνvΦ √ 2 and MR = yMvχ √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Now using the fact that Majorana mass terms are symmetric and νc RmT ν νc L = νLmννR, we can write the LM in the following matrix form −LM ⊃ 1 2 � νL (νR)c � � 0 mD mT D MR � � (νL)c νR � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (34) From the above mass matrix, one can easily recover the seesaw formula for light Majorana neutrinos as, Mν ≈ mDM −1 R mT D and the heavy neutrino mass as MN ≈ MR with the assumption mD ≪ MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Fields ( SU(3)C ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1)B−L ) LL (1, 2, −1/2, −1) QL (3, 2, 1/6, 1/3) eR (1, 1, −1, −1) uR (3, 1, 2/3, 1/3) dR (3, 1, −1/3, 1/3) ν1 R (1, 1, 0, 5) ν2,3 R (1, 1, 0, −4) Φ (1, 2, 1/2, 0) ϕ (1, 2, 1/2, −3) σ (1, 1, 0, 3) χd (1, 1, 0, 1/2) TABLE II: Matter content and charge assignment of the chiral B − L model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' For brevity, the generation index is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Chiral B − L Model Another U(1)B−L gauged model will be discussed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' For RHNs, we use a chiral anomaly cancellation solution (5, −4, −4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Apart from the SM particle content and 11 RHNs, in scalar sector we add one more SU(2)L doublet ϕ and a scalar singlet σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' ϕ is with hypercharge +1/2 and U(1)B−L charge −3, whereas scalar σ has U(1)B−L charge +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We also include a scalar DM χd with a charge of U(1)B−L of +1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The advantage is that one does not need to impose any ad hoc Z2 symmetry to stabilise the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Instead, the stability of χd can be guaranteed by this nontrivial B − L charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The most general renormalizable and SU(3)c ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1)B−L gauge invariant scalar sector is given by Ls = (DµΦ)†DµΦ + (Dµϕ)†Dµϕ + (Dµσ)†Dµσ + (Dµχd)†Dµχd − V(Φ, ϕ, σ, χd), (35) where again the covariant derivative is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The scalar potential V(Φ, ϕ, σ, χd) is given by V(Φ, ϕ, σ, χd) = m2 σ(σ∗σ) + 1 2λσ(σ∗σ)2 + m2 Φ(Φ†Φ) + 1 2λΦ(Φ†Φ)2 + m2 ϕ(ϕ†ϕ) + 1 2λϕ(ϕ†ϕ)2 + m2 χd(χ∗ dχd) + 1 2λχd(χ∗ dχd)2 − µ(Φ†ϕ)σ − µ(ϕ†Φ)σ∗ + λΦσ(Φ†Φ)(σσ∗) + λϕσ(ϕ†ϕ)(σσ∗) + λΦϕ1(Φ†Φ)(ϕ†ϕ) + λΦϕ2(Φ†ϕ)(ϕ†Φ) + λΦχd(Φ†Φ)(χ∗ dχd) + λϕχd(ϕ†ϕ)(χ∗ dχd) + λσχd(σσ∗)(χ∗ dχd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (36) Neutral components of Φ and ϕ spontaneously break electroweak symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' A singlet scalar σ, along with ϕ, breaks the U(1)B−L spontaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' First we solve the minimization equations for the mass parameters mΦ, mϕ, mσ in the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We get 2m2 Φ + v2 ΦλΦ − √ 2µ vΦ vϕvσ + v2 χλΦσ + v2 ϕ(λΦϕ1 + λΦϕ2) = 0, (37a) 2m2 σ − √ 2µ vσ vΦvϕ + v2 σλσ + v2 ΦλΦσ + v2 ϕλϕσ = 0, (37b) 2m2 ϕ − √ 2µ vϕ vΦvσ + v2 σλϕσ + v2 ϕλϕ + v2 Φ(λΦϕ1 + λΦϕ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (37c) The fields Φ, ϕ and σ can be written in unitary gauge after symmetry breaking in the following form: Φ = 1 √ 2 � √ 2G+ 1 vΦ + R1 + iI1 � , ϕ = 1 √ 2 � √ 2G+ 2 vϕ + R2 + iI2 � , σ = 1 √ 2(vσ + R3 + iI3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (38) G± 1 and G± 2 will mix and give rises to the Goldstone bosons G± corresponding to the W ± boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' One electrically charged field remains as the physical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The mass matrix of these electrically charged fields in the basis (G+ 1 , G+ 2 ) reads as M2 ± = 1 2 � � � √ 2µvσvϕ vΦ − v2 ϕλΦϕ2 vΦvϕλΦϕ2 − √ 2µvσ vΦvϕλΦϕ2 − √ 2µvσ √ 2µvσvΦ vϕ − v2 ΦλΦϕ2 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (39) 12 Mass eigen states are given as M 2 H± = v2 2vΦvϕ �√ 2µvσ − vΦvϕλΦϕ2 � , (40) where, v = � v2 Φ + v2 ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The two mass eigenstates G±, H± are related with the (G± 1 , G± 2 ) fields through the fol- lowing rotation matrix as � G± H± � = U � G± 1 G± 2 � = � cos α sin α − sin α cos α � � G± 1 G± 2 � , with tan α = vϕ vΦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (41) In pseudo-scalar sector I1, I2 and I3 mix together and gives two Goldstone boson G0 1, G0 2 corresponding to the neutral gauge bosons Z and Z′ and one pseudo scalar field remains as a physical massive field H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The mass matrix in the basis (I1, I2, I3) can be written as M2 I = 1 √ 2 � � � µvϕvσ vΦ − µvσ − µvϕ −µvσ µvΦvσ vϕ µvΦ −µvϕ µvΦ µvΦvϕ vσ � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (42) Mass of the physical eigenstate is given as M 2 H0 = µ √ 2vΦvϕvσ � v2 Φv2 ϕ + v2 σv2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (43) Mass eigenstates G0 1, G0 2, H0 are related with the (I1, I2, I3) fields through the following rotation matrix as � �� G0 1 G0 2 H0 � �� = U � �� I1 I2 I3 � �� = � �� cos α sin α 0 − sin α cos β cos α cos β − sin α − sin α sin β cos α sin β cos β � �� � �� I1 I2 I3 � �� , (44) Where, tan α = vϕ vΦ , tan β = vσv vΦvϕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (45) Three CP-even neutral scalars are mixed together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The mass matrix in the basis (R1, R2, R3) can be expressed as M2 S = 1 2 � ������� 2v2 ΦλΦ + √ 2µ vΦ vϕvσ 2vΦvϕλ12 − √ 2µvσ 2vΦvσλΦσ − √ 2vϕµ 2vΦvϕλ12 − √ 2µvσ 2v2 ϕλϕ + √ 2µ vϕ vΦvσ 2vϕvσλϕσ − √ 2vΦµ 2vΦvσλΦσ − √ 2vϕµ 2vϕvσλϕσ − √ 2vΦµ 2v2 σλσ + √ 2µ vσ vΦvϕ, � ������� , (46) 13 where, λ12 = λΦϕ1 + λΦϕ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The matrix M2 S can be diagonalized by an orthogonal matrix : OT RM 2 ROR = diag(m2 H1, m2 H2, m2 H3) with � � � H1 H2 H3 � � � = OR � � � R1 R2 R3 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (47) We assume the mass eigenstates to be ordered by their masses mH1 ≤ mH2 ≤ mH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' H1 = h is identified with the SM Higgs of 125 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We will use the standard parameterization OR = R23R13R12 where R12 = � � � c12 −s12 0 s12 c12 0 0 0 1 � � � , R13 = � � � c13 0 −s13 0 1 0 s13 0 c13 � � � , R23 = � � � 1 0 0 0 c23 −s23 0 s23 c23 � � � (48) cij = cos θij, sij = sin θij, where the angles θij can be chosen to lie in the range − π 2 ≤ θij ≤ π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Finally the mass of dark matter χd will be given as M 2 DM = 2m2 χd + v2 ΦλΦχd + v2 σλσχd + v2 ϕλϕχd 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (49) Neutrino mass: The Yukawa sector of the model can be written in a gauge-invariant way as −LY = Y ij e L iΦej R + Y ij u Q i ˜Φuj R + Y ij d Q iΦdj R + Y ij ν L i ˜ϕνj R + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (50) We see from the last term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (50) that the three RHNs pair up with the three left-handed L νR ⟨φ⟩ ⟨Φ⟩ ⟨σ⟩ 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 4: Neutrino mass generation in Chiral B − L model through Dirac type II seesaw neutrinos of the SM to form Dirac particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Note the importance of unconventional B − L charges of ϕ and νi R to generate the Dirac neutrino mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' This B − L charge assignment is needed to forbid Majorana mass terms for the νi R while simultaneously enforcing a Yukawa 14 coupling structure in which only ϕ couples to RHNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' After the spontaneous breaking of electroweak and U(1)B−L symmetry, we can write the neutrino mass term as − LM = νLmννR + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (51) where mν = Y ij ν vϕ √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The smallness of the neutrino masses relative to those of the quarks and charged leptons is explained by smallness of the second Higgs doublet vev vϕ ∼ eV for large Yukawa coupling Yν ∼ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In this model, smallness of vϕ arises very naturally and this can be understood from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (43) with the approximation vσ ≫ vΦ,ϕ: vϕ ≈ µvσv2 √ 2M 2 H0vΦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (52) Hence, the vev of the neutral component of the field ϕ is inversely proportional to the mass of the heavy scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' This provided a natural explanation for the low vev and thus low neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Note that in our analysis for simplicity, we assumed one neutrino to be massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Gauge sector: As field ϕ is charged under both SM and U(1)B−L gauge group, even in the absence of kinetic mixing this will introduce mixing between SM neutral boson Z with the new neutral gauge boson Z′ corresponding to U(1)B−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Again to determine the gauge boson mass spectrum we have to expand the following kinetic terms: (DµΦ)†DµΦ + (Dµϕ)†Dµϕ + (Dµσ)†Dµσ, (53) and have to replace the fields Φ, ϕ and σ by the expressions given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The gauge bosons mass matrix in the basis (Bµ, W µ 3 , Xµ) can be written as M2 V = v2 4 � �� g′2 − gg′ − 6u2g′gx −gg′ g2 6u2ggx −6u2g′gx 6u2ggx 36b2g2 x � �� , where u = vϕ v , and b2 = u2 + v2 σ v2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (54) Mass matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (54) can be diagonalized by the following unitary matrix � �� Aµ Zµ Z′µ � �� = � �� cos θw sin θw 0 − cos α′ cos θw cos α′ cos θw − sin α′ − sin α′ sin θw sin α′ cos θw cos α′ � �� � �� Bµ W µ 3 Xµ � �� , (55) where, tan θw = g′ g , and tan 2α′ = C′ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' After rotation we get a massless photon and two heavy bosons: MA = 0, M 2 Z = v2 8 � A′ − √ B′2 + C′2 � and M 2 Z′ = v2 8 � A′ + √ B′2 + C′2 � , (56) where, A′ = 36b2g2 x + (g2 + g′2), B′ = 36b2g2 x − (g2 + g′2) and C′ = 12gxu2� g2 + g′2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 15 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' W MASS AND THE S, T, U PARAMETERS In the SM, the W boson mass can be calculated very precisely in terms of the precisely measured input parameters {GF, αem, MZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The W boson mass is related with these pa- rameters in the following way [24–26]: M 2 W = M 2 Z 2 � 1 + � 1 − 4παem √ 2GFM 2 Z (1 + ∆r) � , (57) where ∆r represents the quantum corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Taking the central values of the input param- eters, MZ = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1876 GeV, α−1 em = 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='036, GF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1663787 × 10−5 GeV−2 and considering the SM value of ∆r ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='038, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (57) gives us the theoretical prediction of W boson mass 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='360 GeV, with the theoretical uncertainty of near about 4 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' This theoretical predic- tion is 7σ away from the recently announced CDF-II results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Note that the new physics contribution to the parameter ∆r can be reparametrised in terms of the self energy correc- tions to the gauge bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Specifically, dominant BSM effects can be written in terms of the three gauge boson self-energy parameters known as the oblique parameters S, T and U provided that the new physics mass scale is greater than the electroweak scale and that it contributes only through virtual loops to the electroweak precision observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The W boson mass in terms of these parameters can be written as [27]: MW = M SM W � 1 − α 4(cos2 θw − sin2 θw) (S − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='55T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='24U) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (58) Recently, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [28] gave the values of these parameters from an analysis of precision elec- troweak data including the CDF-II new result of the W-mass: S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1, T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='12, U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (59) with the correlation ρST = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='90, ρSU = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='59 and ρTU = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (60) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' NEW PHYSICS CONTRIBUTION TO S, T, U Using six dimensional SU(2)L invariant effective operator we can parametrise new physics that only couples to SM vector bosons and Higgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Effects related with dimension 6 operators can be expressed in the following way [20, 29] L = LSM + 2 v2 (cWBOWB + cHOH + cWWOWW + cBBOBB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (61) We presented two generic models with heavy new neutral vector bosons in section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In the first model the doublet scalar is not charged under the new symmetry and hence there is 16 no mass mixing between SM neutral bosons and the new Z′ boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In the second model we introduced a doublet which is charged under new symmetry and hence it can mix Z′ boson with SM neutral bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' This novel physics is characterized by oblique parameters in an effective lagrangian approach, as [30] S = 4 sin2 θw αZ 2M 2 Wg2 x g2g′2M 2 Z′ [Ze − Zφ + ZL] � g2Ze + g′2(Ze + 2ZL) � , (62a) T = 1 αZ 4M 2 Wg2 x g2M 2 Z′ [Ze − Zφ + ZL]2 , (62b) U = 4 sin2 θw αZ 4M 2 Wg2 x g2M 2 Z′ [Ze − Zφ + ZL] [Ze + 2ZL] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (62c) The U(1)B−L charges of the Lepton singlet, Lepton doublet are Ze, ZL respectively and Zφ is the total B − L charge of both the scalar doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' MZ′ is the mass of the new heavy boson, and gx is the new gauge coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Zϕ = 0 Zϕ =-3 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="3 MZ' /gx [TeV] S S = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='16 S = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='04 Zϕ = 0 Zϕ =-3 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="3 MZ' /gx [TeV] T T = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='23 T = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='01 Zϕ = 0 Zϕ =-3 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="4 MZ' /gx [TeV] U U = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='22 U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='04 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 5: The S, T, U parameters versus ratio of the mass of new boson and its gauge coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The red line represents vector B − L model, whereas the blue line represents chiral B − L model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Taking the central values of the input parameters, M CDF W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4335 [5], sin2 θw = 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='23121, αZ = 1/127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='935 [6, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 5, we plotted The S, T, U parameters versus ratio of the mass of new boson and its gauge coupling for two different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The red line depicts the vector B − L model, whereas the blue line depicts the chiral B − L model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The dotted lines correspond to the updated best fit values of the S, T, U parameters after the CDF results [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' S T U 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="0 MZ'[TeV] gx S T U 5 10 15 20 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="0 MZ'[TeV] gx FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 6: The new boson’s gauge coupling versus mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The space permitted by the new S, T, U (3σ) is shown by a different coloured band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The graph on the left is for vector B − L, while the graph on the right is for chiral B − L model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 6 the parameter space that permits us to solve the W anomaly is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The blue line represents the maximum permissible value for T, and the region between the blue line and the MZ′ axis represents the allowable parameter space fulfilled by the best fit T value in 3σ range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The allowed region for S and U is shown by the colours red and cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The allowed parameter space in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 7 to satisfy CDF-II measurements is the overlap zone between the S, T and U bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' As previously stated, the left panel depicts the case with no mass mixing between SM neutral gauge bosons and the new U(1)B−L neutral gauge boson, whereas the right panel shows the case with mass mixing, as ϕ has a charge of −3 under B − L symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Notice that the parameter space is improved when scalar doublet mixes the new heavy boson with SM neutral bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' As a result, we will now concentrate our efforts on the chiral B-L model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Taking the scenario where ϕ has a charge of −3 under B − L symmetry, we showed the points (dark cyan) that satisfy W mass in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 18 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="0 MZ'[TeV] gx 5 10 15 20 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="0 MZ'[TeV] gx FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 7: The green band represents the permitted parameter space that is consistent with CDF II, W mass measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The graph on the left is for Zφ = 0, while the graph on the right is for Zφ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="0 MZ' [TeV] gx FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 8: For chiral B − L model, we showed the space permitted by the new oblique parameters (S, T, U) after CDF-II results (3σ) in green colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Dark Cyan points are consistent with CDF-II, W mass measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' DARK MATTER CONSTRAINTS In this section we collect the results of our analysis of DM phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' As previously stated, in the second model outlined in the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' III, χd is the scalar DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' It carries the 19 U(1)B−L charge 1/2, which forbids any term in the potential that results in the decay of χd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We study the χd relic density and its direct detection prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Specifically, we determine the regions in the parameter space of the model where the DM constraints and S, T, U consistent with CDF-II W mass measurements can be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The SARAH-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='5 [32, 33] package is used to calculate all of the vertices and mass matrices, among other things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' All the expressions are verified analytically and numerical calculation are performed by package SPheno-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The relic abundance, on the other hand, is determined using micrOMEGAS-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='13 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' There are several DM annihilation channels present in this model which is shown in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' They involve annilation to quarks, leptons, neutrinos, gauge bosons (Z, Z′), neutral scalars (Hi, H0) and charged scalar (H±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Altogether, they determine the relic abun- dance of χd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Note that as χd is charged under B − L, the DM χd has both the gauge and scalar interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The gauge interactions allow the annihilation of the dark matter particle into fermions mediated by the gauge boson, χdχ∗ d → Z′∗ → f ¯f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' One should also consider the direct annihilation into two gauge bosons, χdχ∗ d → Z′Z′, when kinematically accessi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Hence in pure gauge interaction case there are very few parameters (MDM, gx, MZ′) which plays the role in determining DM phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Due to the strong experimental constraints on MZ′/gx, the annihilation into Z′Z′ is suppressed, hence only annihilation to fermions turns out to be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Due to the structure of the gauge coupling, the annihila- tion channel χdχ∗ d → Z′∗ → f ¯f is velocity suppressed (∝ v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' This is why the relic density tends to be much higher than the observed value except in a narrow region close to the resonance (MDM ∼ MZ′/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' And even at the resonance, the relic density can be too large to be in agreement with the data, as illustrated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Specifically the requirements to have correct relic is to be near resonance and relatively large coupling gx and low MZ′, which in view of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 2 is ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Besides the gauge interactions, the DM χd, also has scalar interactions, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Scalar Parameter Range mχd [ 0, 104 ] GeV λΦχd [ 10−6, 1 ] λϕχd [ 10−6, 1 ] λσχd [ 10−6, 1 ] µ [ 10−6, 1 ] TABLE III: Ranges of variation of the input parameters used in our numerical scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' interactions between the DM and the SM scalar give rise to the well-known Higgs-portal scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' But there are some differences between this simplistic Higgs-portal scenario and our B − L case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' First of all, the DM field is necessarily complex as it is charged under 20 U(1)B−L – rather than real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In addition to this there will be many additional annihilation channels due to the presence of additional scalars, both neutral and charged, such as H2,3, H0 and H±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In the following, instead of separately studying gauge interaction and scalar interaction, we will focus on the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' To do this we first need to identify the parameters which are relevant for DM analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Even though the model introduces new free parameters, not all of them are important to DM analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' For example, the self quartic couplings and some mixed quartic couplings such as λΦσ, λϕσ, λΦϕ1,2 does not play any role in DM phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Hence, we choose to fix these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The remaining free parameters relevant for DM analysis can be chosen as: mχd, λΦχd, λϕχd, λσχd, gx and MZ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (63) We will look at how the DM phenomenology of this model is affected by the above-mentioned parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' To carry out the numerical scan, we varied these parameters as listed in the Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We varied them on the logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The gauge coupling gx and mass MZ′ are varied according to the allowable parameter space coming from S, T, U restriction consistent with CDF-II W mass measurements (Dark cyan points ), as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 9: Relic density vs the mass of the Dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The blue and grey dots show over and under abundance relic density points, respectively, whilst the magenta points reflect the 3 σ range for cold dark matter obtained from Planck satellite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 9, we show the relic density as a function of the mass of the scalar DM χd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The blue 108 106 104 102 Qh 100 10-2 10-4 10-6E 102 103 10° MpM [GeV]21 and grey points represent the over and under-abundance relic density regions, respectively, whereas the magenta points in the narrow band fall in the 3σ range for cold dark matter derived from the Planck satellite data [4]: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1126 ≤ Ωh2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (64) Various features of the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 9 can be understood from different DM annihilation channels shown in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Annihilation in the low mass region of DM is dominated via exchange of SM Higgs (H1 = h) to SM fermionic final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' As DM approaches half of the Higgs mass ( MDM ≈ mh/2), h becomes on-shell and these annihilation channels become very efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Notice that there is no dip at MDM ≈ MZ/2 because the mixing between Z and Z′ is not strong enough for annihilation through Z exchange to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' For MDM ≥ 80 GeV, annihilation of DM to Z and W final states comes into picture (χdχ∗ d → ZZ, W +W −) and hence we get another dip in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We see a dip in relic density in the DM mass range 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='5 TeV-4 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' This is due to the fact that the combination of gx and MZ′ required to get correct oblique parameters forces the vev of singlet scalar (vσ) to be around 8 TeV-10 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' This high vev pushes the mass of CP-even scalars H2 and H3 to be in the range of 5-8 TeV and we get a dip when DM mass is roughly half of this range due to the the H2,3 mediated s-channel annihilation to SM final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' A sub-dominant role is played by annihilation into HiHj, H0H0 and ZZ, Z′Z′ via the direct 4-point vertices HiHjχdχ∗ d, H0H0χdχ∗ d and ZZχdχ∗ d or ZZ′χdχ∗ d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Also there could be additional contribution from χd exchange in the t-channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' As the annihilation cross section is inversely proportional to the mass of dark matter, at very high value of DM mass the relic density increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Direct Detection: Let us now study the direct detection prospects of our DM candi- dates χd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' A large number of experiments are being conducted to demonstrate the parti- cle nature of dark matter through direct detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Various direct detection experiments, XENON1T [37], LZ[38], XENONnT[39], LUX[40, 41], PandaX-II[42], impose constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' These experiments are designed to measure the tiny recoil in the detector target nuclei pro- duced by the elastic collisions between DM and target nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The effective lagrangian for nucleon-DM interactions is expressed as Leff = aNNNχ2 d, (65) Where aN is the effective nucleon-DM coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The spin independent scattering cross section via the Higgs(H1,2,3) interaction is given by σSI N−χd = µ2M 2 Nf 2 N 4πM 2 DMv2 �λH1χ2 d M 2 H1 (OR)11 + λH2χ2 d M 2 H2 (OR)21 + λH3χ2 d M 2 H3 (OR)31 � , (66) Where (OR)ij is the elements of the mass matrix defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (47), fN is the form factor, which depends on the hadronic matrix elements and µ = MNMDM MN+MDM is the reduced mass for 22 nucleon-DM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The trilinear couplings are given as λHiχ2 d = 2 [vΦλΦχd(OR)i1 + vϕλϕχd(OR)i2 + vσλσχd(OR)i3] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (67) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (66) is an extension of the expression corresponding to the scalar DM case [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The cross section per nucleon for Dark matter-nuclei interaction through Z′ is given as [44] σ0 = 1 π � MDMMn MDM + AMn �2 � g2 x 2M 2 Z′ �2 , (68) Where A is the number of nucleons in the target, we have set it to 131 for Xenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The nucleon mass is Mn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='938919 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 10: WIMP-nucleon spin-independent cross section for the scalar dark matter χd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The colour code has the same meaning as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The solid red line denotes the latest upper bound from the LZ [38] collaboration, the dashed red line corresponds to XENON1T [37] limit and the dashed brown line corresponds to the “neutrino floor” lower limit [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' (10) we imposed the direct detection constraints on our scalar dark matter χd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We performed the numerical scan with the micrOMEGAS-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='13 and varied the parameters as shown in the table (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The colour code has the same meaning as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The LZ and XENON1T experiment imposes the most stringent constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' As a result, we plotted the most recent upper bound from the both LZ and XENON1T collaboration [37], as shown by the solid and dashed red line respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The brown line represents the lower limit, which corresponds to the “neutrino floor” from the coherent elastic neutrino scattering [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 11, we demonstrated the parameter space that is compatible with all of the afore- mentioned requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The green bands show the permitted values for oblique parameters [cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='] 10-44 XENONIT 10-48 10-50 101 102 103 MpM [GeV]23 S, T and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The region with the grey shading is the ATLAS’s most stringent collider con- straint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The magenta points fulfill M CDF W , relic density and limitations from direct detection experiments all at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content="0 MZ'[TeV] gx ATLAS13(2l) FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 11: We showed the space permitted by the new oblique parameters (S, T, U) after CDF-II results in green colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The grey region is ruled out by most stringent collider constraints from ATLAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The magenta points satisfy M CDF W , relic density and limits from direct detection experiments simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' It is worth noting that there is a significant parameter space in the DM high mass region, which is consistent with recent measurements of W boson mass, relic abundance, collider constraints, and the direct detection experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' CONCLUSION The U(1)B−L gauged extension of the SM is very simple in its nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' It’s minimal version just needs three right-handed neutrinos to cancel gauge anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' It naturally explains the small neutrino masses through a seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Despite their simplicity, these type of models can explain the recent CDF-II measurement of the W boson mass, which reveals considerable disagreement with the SM predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The new neutral boson associated with the new U(1)B−L symmetry can provide the loop corrections to gauge boson two-point functions that are compatible with the most recently revised oblique parameter values as a consequence of the CDF-II results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We investigated and find that kinetic mixing alone can not explain the W anomaly at the tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' In addition, we investigated the two models with and without mass mixing between neutral bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We focused our attention on chiral B − L model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' We 24 imposed constraints derived from direct detection of dark matter and relic abundance and demonstrated that the chiral U(1)B−L model can explain the W anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' ACKNOWLEDGMENTS Work of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' has been supported by KIAS Individual Grants (PG086001) at Korea Insti- tute for Advanced Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The work of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' is supported by the Government of India,SERB Startup Grant SRG/2020/002303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' The work of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' is supported by the Prime Minister Research Fellowship (ID: 0401969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Appendix A: Annihilation channels for scalar DM χd In the chiral B − L model the relic abundance of the DM candidate χd is determined by the annihilation diagrams shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='H1/H2/H3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='χd ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='χd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='Hi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='χd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='Z/Z′/Z′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='χ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='χd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='Z/Z/Z′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 12: Feynman diagrams that contributes to the relic density of the scalar dark matter χd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 26 [1] ATLAS Collaboration, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' B 716 (2012) 1–29, arXiv:1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='7214 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [2] CMS Collaboration, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Chatrchyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' B 716 (2012) 30–61, arXiv:1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='7235 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [3] Super-Kamiokande Collaboration, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Fukuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “Evidence for oscillation of atmospheric neutrinos,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 81 (1998) 1562–1567, arXiv:hep-ex/9807003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [4] Planck Collaboration, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Cosmological parameters,” Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 641 (2020) A6, arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='06209 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [Erratum: Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 652, C4 (2021)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [5] CDF Collaboration, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Aaltonen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “High-precision measurement of the W boson mass with the CDF II detector,” Science 376 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 6589, (2022) 170–176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [6] Particle Data Group Collaboration, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Zyla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “Review of Particle Physics,” PTEP 2020 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 8, (2020) 083C01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Schechter and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Valle, “Neutrino Masses in SU(2) x U(1) Theories,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' D 22 (1980) 2227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Schechter and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Valle, “Neutrino Decay and Spontaneous Violation of Lepton Number,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' D 25 (1982) 774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Mohapatra and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Valle, “Neutrino Mass and Baryon Number Nonconservation in Superstring Models,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' D 34 (1986) 1642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Malinsky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Romao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Valle, “Novel supersymmetric SO(10) seesaw mechanism,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 95 (2005) 161801, arXiv:hep-ph/0506296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [11] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Okada and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Okada, “Z′-portal right-handed neutrino dark matter in the minimal U(1)X extended Standard Model,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' D 95 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 3, (2017) 035025, arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='02672 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Das, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Gola, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Mandal, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Sinha, “Two-component scalar and fermionic dark matter candidates in a generic U(1)X model,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' B 829 (2022) 137117, arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='01443 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Das, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Okada, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Raut, “Heavy Majorana neutrino pair productions at the LHC in minimal U(1) extended Standard Model,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' C 78 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 9, (2018) 696, arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='09896 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [14] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Basso, Phenomenology of the minimal B-L extension of the Standard Model at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' PhD thesis, Southampton U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' arXiv:1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4462 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 27 [15] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Emam and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Khalil, “Higgs and Z-prime phenomenology in B-L extension of the standard model at LHC,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' C 52 (2007) 625–633, arXiv:0704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1395 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Basso, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Belyaev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Moretti, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Pruna, “Tree Level Unitarity Bounds for the Minimal B-L Model,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' D 81 (2010) 095018, arXiv:1002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1939 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [17] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rodejohann and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Yaguna, “Scalar dark matter in the B−L model,” JCAP 12 (2015) 032, arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='04036 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Langacker, “The Physics of Heavy Z′ Gauge Bosons,” Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 81 (2009) 1199–1228, arXiv:0801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1345 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Carena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Daleo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Dobrescu, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Tait, “Z′ gauge bosons at the Tevatron,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' D 70 (2004) 093009, arXiv:hep-ph/0408098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [20] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Cacciapaglia, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Csaki, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Marandella, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Strumia, “The Minimal Set of Electroweak Precision Parameters,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' D 74 (2006) 033011, arXiv:hep-ph/0604111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [21] ATLAS Collaboration, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “Search for high-mass dilepton resonances using 139 fb−1 of pp collision data collected at √s =13 TeV with the ATLAS detector,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' B 796 (2019) 68–87, arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='06248 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [22] CMS Collaboration, “Search for a narrow resonance in high-mass dilepton final states in proton-proton collisions using 140 fb−1 of data at √s = 13 TeV,”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Das, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Dev, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Hosotani, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Mandal, “Probing the minimal U(1)X model at future electron-positron colliders via the fermion pair-production channel,” arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='10902 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' L´opez-Val and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Robens, “∆r and the W-boson mass in the singlet extension of the standard model,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' D 90 (2014) 114018, arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1043 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [25] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Hollik, “Electroweak theory,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 53 (2006) 7–43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Amrith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Butterworth, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Deppisch, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Varma, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Yallup, “LHC Constraints on a B − L Gauge Model using Contur,” JHEP 05 (2019) 154, arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='11452 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [27] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Cai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Yu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Zhang, “CEPC Precision of Electroweak Oblique Parameters and Weakly Interacting Dark Matter: the Fermionic Case,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' B 921 (2017) 181–210, arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='02186 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Lu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Wu, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Zhu, “Electroweak Precision Fit and New Physics in light of W Boson Mass,” arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='03796 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [29] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Barbieri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Pomarol, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rattazzi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Strumia, “Electroweak symmetry breaking after LEP-1 and LEP-2,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' B 703 (2004) 127–146, arXiv:hep-ph/0405040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Strumia, “Interpreting electroweak precision data including the W-mass CDF anomaly,” arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='04191 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [31] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Babu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Jana, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “Correlating W-Boson Mass Shift with Muon g − 2 in the 2HDM,” arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='05303 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 28 [32] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Staub, “Exploring new models in all detail with SARAH,” Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 2015 (2015) 840780, arXiv:1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='04200 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Goodsell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Liebler, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Staub, “Generic calculation of two-body partial decay widths at the full one-loop level,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' C 77 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 11, (2017) 758, arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='09237 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [34] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Porod, “SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e- colliders,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 153 (2003) 275–315, arXiv:hep-ph/0301101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [35] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Porod and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Staub, “SPheno 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1: Extensions including flavour, CP-phases and models beyond the MSSM,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 183 (2012) 2458–2469, arXiv:1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='1573 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [36] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' B´elanger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Boudjema, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Goudelis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Pukhov, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Zaldivar, “micrOMEGAs5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='0 : Freeze-in,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 231 (2018) 173–186, arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='03509 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [37] XENON Collaboration, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Aprile et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “Dark Matter Search Results from a One Ton-Year Exposure of XENON1T,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 121 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 11, (2018) 111302, arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='12562 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [38] LUX-ZEPLIN Collaboration, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Aalbers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “First Dark Matter Search Results from the LUX-ZEPLIN (LZ) Experiment,” arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='03764 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [39] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Aprile et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “Search for New Physics in Electronic Recoil Data from XENONnT,” arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='11330 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [40] LUX Collaboration, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Akerib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “First results from the LUX dark matter experiment at the Sanford Underground Research Facility,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 112 (2014) 091303, arXiv:1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='8214 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [41] LUX Collaboration, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Akerib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “Results from a search for dark matter in the complete LUX exposure,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 118 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 2, (2017) 021303, arXiv:1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='07648 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [42] PandaX-II Collaboration, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “Results of dark matter search using the full PandaX-II exposure,” Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' C 44 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 12, (2020) 125001, arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='15469 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Cline, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Kainulainen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Scott, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Weniger, “Update on scalar singlet dark matter,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' D 88 (2013) 055025, arXiv:1306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='4710 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='D 92, 039906 (2015)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [44] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Ma, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Pollard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Srivastava, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Zakeri, “Gauge B − L Model with Residual Z3 Symmetry,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' B 750 (2015) 135–138, arXiv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='03943 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' [45] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Billard, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Strigari, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Figueroa-Feliciano, “Implication of neutrino backgrounds on the reach of next generation dark matter direct detection experiments,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' D 89 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 2, (2014) 023524, arXiv:1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='5458 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' 29 [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=' Billard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content=', “Direct Detection of Dark Matter – APPEC Committee Report,” arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} +page_content='07634 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAzT4oBgHgl3EQfjv3G/content/2301.01522v1.pdf'} diff --git a/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf b/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..984eb3c2c15cfd725ff9987427a88c48ca8e76e3 --- /dev/null +++ b/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b2b39b3a565befc7995fb4b7e833f5090f500ff7189ded8fc6207e8abfbebef7 +size 12899608 diff --git a/K9FJT4oBgHgl3EQfxi3J/content/tmp_files/2301.11635v1.pdf.txt b/K9FJT4oBgHgl3EQfxi3J/content/tmp_files/2301.11635v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7aca7fb3909b2cc94dc438a9dd71a533a717cec4 --- /dev/null +++ b/K9FJT4oBgHgl3EQfxi3J/content/tmp_files/2301.11635v1.pdf.txt @@ -0,0 +1,280 @@ +A tape-peeling model for spatiotemporal pattern formation +by deformed adhesives +Keisuke Taga1 and Yoshihiro Yamazaki1 +1Department of Physics, School of Advanced Science and Engineering, +Waseda University, Tokyo 169-8555, Japan +(*Electronic address: tagaksk@akane.waseda.jp) +(Dated: January 30, 2023) +Abstract +We propose a new model for pattern formation in peeling of an adhesive tape based on the equation of +motion for the displacement of deformed adhesives in the peel front. The spatiotemporal patterns obtained +from the model are consistent with those from previous models and experiments. Moreover, dynamical and +statistical properties of the patterns are investigated. +1 +arXiv:2301.11635v1 [nlin.PS] 27 Jan 2023 + +Introduction: Peeling an adhesive tape is a common daily activity, and if we look closely at +the tape after peeling, we may find interesting problems in nonlinear dynamics and statistical +physics. Previously one of the authors has investigated dynamical and statistical properties of +spatiotemporal patterns composed of two different types of deformed adhesives in peeling [1–4]. +The difference originates from whether a characteristic structure, tunnel structure, exists or not as +shown in Figs.1 (a) and (b). It has been found that the peel speed and stiffness of the system are +the main factors for controlling the formation of the spatiotemporal patterns and the formation of +the tunnel structure. +Figure 1 (c) shows a typical pattern in the case of high stiffness. Peeling proceeds from top +to bottom. Black and white regions show peel states without and with the tunnel structure, re- +spectively. In the previous experimental studies, it has been known that there exists a peel speed +region where a coexistence pattern such as shown in Fig.1(c) is formed. In this speed region, the +following properties of the spatiotemporal patterns have been found [1, 2, 4–6]. (i) As the peel +speed increases, peel state with tunnel structure becomes hard to emerge, and then the ratio of the +white region decreases in the patterns. (ii) Interchange of connectivity between black and white +regions occurs by changing the peel speed. (iii) Furthermore, the ratio of the white region in the +coexistence pattern becomes a monotonically decreasing function of peel speed. [2, 3, 6]. +FIG. 1: Peel states (a) with and (b) without tunnel structures. Peeling proceeds from top to bottom. The +small image at the bottom left in each figure shows the enlarged image of the peel front. (c) A fractal +coexistence pattern. Black and white regions represent the peel states without and with the tunnel structures, +respectively. The actual size of the figure is 25mm×25mm. +To reproduce this pattern formation, several models have been proposed so far [2–6]. Among +these models, we now comment on the model proposed in ref.[2, 5]. In this previous model, the +2 + +1 mm +100um +00state variable for stability of the tunnel structure is introduced, and the peel front is considered to +consist of bistable units described by the state variable. Then, the equation of motion for the peel +front and the time evolution of the state variable for each unit are constructed. The asymmetric +local interaction is introduced between the nearest neighbor units according to the experimental +observation. This asymmetricity between the units causes such an effect that once the tunnel struc- +ture collapses, then the neighboring tunnel structures tend to collapse. The previous model with +the asymmetric interaction succeeds to describe the dynamics of peeling adhesive tape. However, +the relation between the state transition process and the adhesive deformation process is not clear. +If we consider the deformation of adhesives in peeling, we need to construct another model for +adhesive deformation in the peel front. +In this letter, we propose a new model where the asymmetricity of the two states is introduced +not from a spatial interaction but from different local dynamics of the two peel states. Here we +focus on the case of high stiffness and discuss dynamical and statistical properties of the model. +Modelling: It has been experimentally found that the peel front with the tunnel structure has +larger deformation of adhesives than that without it. As in the previous model, the peel front +is divided into discrete units with the size of the tunnel structure. The difference in adhesive +deformation of each unit can be described by its displacement u, or ui(t) at ith unit in the peel +front at time t. Large u and small u correspond to peeling with the tunnel structure and without it, +respectively. Hereafter, the peel states with and without tunnel structure are referred to as the states +A and B, respectively. For the time evolution of u, we consider the Newton’s equation of motion +and assume the phase-space dynamics illustrated in Fig. 2 is realized in the system. For the slow +or fast peeling, the system has only one stable fixed point at large u or small u, which corresponds +to the state A or B as shown in Figs. 2(a) and (b). And at the intermediate peel speed, the tunnel +structure in the state A adjacent to the unit in the state B tends to collapse from the experiment [5], +however, the tunnel structure can regenerate after a meanwhile. This dynamics seems to resemble +a threshold firing dynamics and its dynamics in the phase space can be illustrated as shown in +Fig. 2(c). +Based on the above consideration, the following equation is proposed for each unit: +d2s +dt2 = −dU(u) +du +−bds +dt + +a +1+c(u−d)2 +du +dt , +(1) +where s = u+Vt shows the length of peeled tape, and we assume s = 0 at t = 0. U is a double-well +potential for bistability of the two peel states, b is viscosity of adhesives, and V corresponds to a +3 + +FIG. 2: Illustration of the assumed phase space dynamics. +(a) Small V. The system has only one stable fixed point as state A. (b) Large V. The system has only one +stable fixed point as state B. (c) Intermediate V. The system has two stable fixed points; the states A and B. +If the tunnel structure in the state A collapses with a large perturbation as 1⃝, then the state goes to the state +B as 2⃝. And at last, the state comes back to the state A ( 3⃝). +peel speed. The last term on the right-hand side shows an asymmetric dissipation, which depends +on the existence of the tunnel structure. a, c and d are positive constants. For the dynamics of u +instead of s, we can rewrite eq.(1) as +d2u +dt2 = −dU(u) +du +−b +� +V + du +dt +� ++ +a +1+c(u−d)2 +du +dt . +(2) +On the right-hand side of eq.(2), the terms − +�dU(u) +du ++bV +� +and +� +−b+ +a +1+c(u−d)2 +� du +dt cor- +respond to elastic and viscous forces, respectively. Furthermore, we add the following symmetric +spatial interaction terms in eq.(2): +D1(ui+1 +ui−1 −2ui)+D2 +�dui+1 +dt ++ dui−1 +dt +−2dui +dt +� +, +(3) +where D1 and D2 are positive constants. These two terms also express the effect of the viscoelas- +ticity of adhesives. +Results: Based on eq.(2) with eq.(3), we calculated the following equation, +d2ui +dt2 =−3(ui −1)2(ui −2)− +� +V + dui +dt +� ++ +2 +1+20(ui −1)2 +dui +dt ++(ui+1 +ui−1 −2ui)+0.1 +�dui+1 +dt ++ dui−1 +dt +−2dui +dt +� +, +i = 1,...,N. +(4) +ui = 0 corresponds to the state where no adhesive deformation occurs. And we set ui ≈ 2 and +≈ 1 for the states A and B, respectively. In numerical calculation, in order to take relaxation of +4 + +Cadhesive deformation due to spatial inhomogeneity of adhesives into account, we reset ui(t) = 0 +with a probability p = 0.001 per unit time. As the initial condition, we set ui(0) = 0 and dui +dt = 0. +Periodic boundary condition was adopted. The 4th Runge-Kutta method was used with dt = 0.01 +in time. The discretized system size was N = 1000. +Figure 3 shows the typical spatiotemporal patterns obtained by the numerical calculation. The +black and white regions correspond to the states B and A, respectively. There are five cases with +different values of V. It is found that as V increases the ratio of white regions (the state A) +decreases. And interchange of connectivity between black and white regions is confirmed. These +results are consistent with those in the previous study[2]. +Here the following scaling properties for the coexistent patterns are focused on: (i) the cu- +mulative distribution F(≥ s) ∼ s−ξ where s is the size of the white clusters, (ii) their standard +deviations of the height h(s) ∼ sν∥ and the width w(s) ∼ sν⊥, and (iii) the fractal dimension D of +the spatiotemporal patterns of the white regions. Our numerical results for ξ, D, ν∥, and ν⊥ are +shown in Tbl.I. For comparison, this table also has their values obtained in the previous studies +[5, 6]. It is found that our present result is consistent with the previous results. +FIG. 3: Typical spatiotemporal patterns obtained by eq.(4). Time proceeds from top to bottom (500 ≤ t ≤ +1200). (a) V = 0.29, (b) V = 0.30, (c) V = 0.309, (d) V = 0.32, (e) V = 0.50. +Discussion: Here we comment on difference between the previous model[3] and the present +model. The previous model has an asymmetric spatial interaction which expresses the situation of +the state transition where the tunnel structures (in the state A) adjacent to a part of the peel front +in the state B tend to collapse [5]. In the present model, we assume the local dynamics of the +peel front as shown in Fig. 2 and introduce the local dissipation as a function of u instead of the +previous asymmetric spatial interaction. +Figure 4 shows the nullclines and stable and unstable manifolds of eq.(4) without the spatial +5 + +TABLE I: The estimated values of the exponents. +ξ +D +ν⊥ +ν∥ +eq.(4) at V = 0.309 +0.78 +1.82 +0.63 +0.45 +model A at r = 0.275 [5] +0.85 +1.70 +0.58 +0.41 +model B at r = 0.181 [6] +0.84(±0.04) 1.61(±0.01) 0.58(±0.02) 0.41(±0.03) +model C at r = 0.037 [6] +0.81(±0.05) 1.62(±0.01) 0.58(±0.03) 0.41(±0.034) +peeling at V = 0.48mm [5] +0.85 +1.70 +0.59 +0.45 +FIG. 4: Phase spaces of eq. (4) without spatial interaction. Red and Green lines are the nullcline for u and +du/dt. Orange and Blue lines are stable and unstable manifolds of the saddle point. (a) V = 0. The system +has only one fixed point as the state A. (b) V = 0.5. The system has only one fixed point as the state B. +(c) V = 0.309. The system has two fixed points as the state A and B, and a threshold firing dynamics is +realized. +interaction terms. For the slow or fast peeling, the system has only one stable fixed point at large +u ≈ 2 or small u ≈ 1, which corresponds to the states A or B in Figs.4(a) and (b). And the assumed +dynamics at the intermediate speed is also realized as shown in Fig. 4(c); 1⃝ if the state A crosses +over the stable manifold then 2⃝ the state goes to the state B and 3⃝ the state comes back to the +state A. We consider that such a threshold firing mechanism is essential for the pattern formation +of peeling adhesive tapes, although we have to experimentally verify the validity of each term in +the present phenomenological model in the future. +Regarding the effect of noise, the noise term is necessary for the reproduction of the fractal +spatiotemporal patterns in the previous model[3]. However, the present model does not need any +noise term in the time evolution of u to reproduce the patterns as shown in Fig.3, although we +6 + +BFIG. 5: A spatiotemporal pattern obtained from eq.(2) with eq.(3), dU +du = 0.2(u + 0.5)(u − 1.5)(u − 4), +V = 0.66, b = 1, a = 1, c = 2, d = 0, D1 = 1,D2 = 0.1, and p = 0. 500 ≤ t ≤ 1200. Initial conditions of +both ui(0) and dui +dt (0) are given with a uniform random number between −2 and 2. +added the noise in our numerical calculation for stabilizing the state B. Actually, Fig.5 shows an +example of the spatiotemporal pattern obtained from eq.(4) without noise term; we considered the +randomness in u only for the initial condition. This result suggests that the fractal spatiotemporal +patterns originate not from stochasticity but from chaoticity. +Finally, we note that the fractal patterns as shown in Figs.3(c) and 5 are reproducible by +Bonhoffer-van der Pol type equation and other reaction-diffusion equations [7–10]. The relation- +ship between our present model and these models can be found by transforming our model into +the 2-component activator-inhibitor system as the Liénard system by introducing a new parameter +wi ≡ dui +dt +bui −a′ arctan(c′(ui −d))−D2(ui+1 +ui−1 −2ui): +dui +dt += −wi −bui +a′ arctan(c′(ui −d))+D2(ui+1 +ui−1 −2ui) +(5) +dwi +dt += dU +du +���� +u=ui ++bV −D1(ui+1 +ui−1 −2ui), +(6) +where a′ = a/√c,c′ = √c. If we change the term dU +du +bV in eq.(6) to u, then the reaction terms of +eqs.(5) and (6) correspond to those of the Bonhoffer-van der Pol type equation[7]. The presence +of such correspondence also suggests that the fractal patterns emerge chaotically. +Conclusion: In this letter, we have proposed a new model for pattern formation in peeling +adhesive tapes which focuses on the equation of motion for the displacement of adhesives in the +peel front. The model reproduces the dynamical and statistical properties of the spatiotemporal +patterns which consist of the two different peel states. The previous experimental results also +support our numerical results. Note that the spatiotemporal pattern obtained by the present model +7 + +is seemingly relevant to directed-percolation universality class [11]. Detailed discussions of this +relevancy will be reported in near future. +[1] Y. Yamazaki and A. Toda, Journal of the Physics Society Japan 71 1618 (2002). +[2] Y. Yamazaki and A. Toda, Journal of the Physics Society Japan 73 2342 (2004). +[3] Y. Yamazaki and A. Toda, Physica D 214 120 (2006). +[4] Y. Yamazaki, K. Yamamoto, D. Kadono, and A. Toda, J. Phys. Soc. Jpn. 81 043002 (2012). +[5] Y. Yamazaki, Prog. Theor. Phys. 125 641 (2011). +[6] S. Ohmori and Y. Yamazaki, J. Phys. Soc. Jpn. 88 105001 (2019). +[7] Y. Hayase, J. Phys. Soc. jpn. 66 2584 (1997). +[8] Y. Hayase and T. Ohta, Phys. Rev. Lett. 81 1726 (1998). +[9] H. Chaté, A. Pikovsky, and O. Rudzick 131 17 (1999). +[10] Y. Hayase and T. Ohta 62 5998 (2000). +[11] H. Hinrichsen, Adv. in Phys. 49 815 (2000). +8 + diff --git a/K9FJT4oBgHgl3EQfxi3J/content/tmp_files/load_file.txt b/K9FJT4oBgHgl3EQfxi3J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce4b7dabf1180f5d02443a03719cb61b4b2a8833 --- /dev/null +++ b/K9FJT4oBgHgl3EQfxi3J/content/tmp_files/load_file.txt @@ -0,0 +1,257 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf,len=256 +page_content='A tape-peeling model for spatiotemporal pattern formation by deformed adhesives Keisuke Taga1 and Yoshihiro Yamazaki1 1Department of Physics, School of Advanced Science and Engineering, Waseda University, Tokyo 169-8555, Japan (*Electronic address: tagaksk@akane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='waseda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='jp) (Dated: January 30, 2023) Abstract We propose a new model for pattern formation in peeling of an adhesive tape based on the equation of motion for the displacement of deformed adhesives in the peel front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The spatiotemporal patterns obtained from the model are consistent with those from previous models and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Moreover, dynamical and statistical properties of the patterns are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='11635v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='PS] 27 Jan 2023 Introduction: Peeling an adhesive tape is a common daily activity, and if we look closely at the tape after peeling, we may find interesting problems in nonlinear dynamics and statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Previously one of the authors has investigated dynamical and statistical properties of spatiotemporal patterns composed of two different types of deformed adhesives in peeling [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The difference originates from whether a characteristic structure, tunnel structure, exists or not as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='1 (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' It has been found that the peel speed and stiffness of the system are the main factors for controlling the formation of the spatiotemporal patterns and the formation of the tunnel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Figure 1 (c) shows a typical pattern in the case of high stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Peeling proceeds from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Black and white regions show peel states without and with the tunnel structure, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' In the previous experimental studies, it has been known that there exists a peel speed region where a coexistence pattern such as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='1(c) is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' In this speed region, the following properties of the spatiotemporal patterns have been found [1, 2, 4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (i) As the peel speed increases, peel state with tunnel structure becomes hard to emerge, and then the ratio of the white region decreases in the patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (ii) Interchange of connectivity between black and white regions occurs by changing the peel speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (iii) Furthermore, the ratio of the white region in the coexistence pattern becomes a monotonically decreasing function of peel speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [2, 3, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 1: Peel states (a) with and (b) without tunnel structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Peeling proceeds from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The small image at the bottom left in each figure shows the enlarged image of the peel front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (c) A fractal coexistence pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Black and white regions represent the peel states without and with the tunnel structures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The actual size of the figure is 25mm×25mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' To reproduce this pattern formation, several models have been proposed so far [2–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Among these models, we now comment on the model proposed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [2, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' In this previous model, the 2 1 mm 100um 00state variable for stability of the tunnel structure is introduced, and the peel front is considered to consist of bistable units described by the state variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Then, the equation of motion for the peel front and the time evolution of the state variable for each unit are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The asymmetric local interaction is introduced between the nearest neighbor units according to the experimental observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' This asymmetricity between the units causes such an effect that once the tunnel struc- ture collapses, then the neighboring tunnel structures tend to collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The previous model with the asymmetric interaction succeeds to describe the dynamics of peeling adhesive tape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' However, the relation between the state transition process and the adhesive deformation process is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' If we consider the deformation of adhesives in peeling, we need to construct another model for adhesive deformation in the peel front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' In this letter, we propose a new model where the asymmetricity of the two states is introduced not from a spatial interaction but from different local dynamics of the two peel states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Here we focus on the case of high stiffness and discuss dynamical and statistical properties of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Modelling: It has been experimentally found that the peel front with the tunnel structure has larger deformation of adhesives than that without it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' As in the previous model, the peel front is divided into discrete units with the size of the tunnel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The difference in adhesive deformation of each unit can be described by its displacement u, or ui(t) at ith unit in the peel front at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Large u and small u correspond to peeling with the tunnel structure and without it, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Hereafter, the peel states with and without tunnel structure are referred to as the states A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' For the time evolution of u, we consider the Newton’s equation of motion and assume the phase-space dynamics illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 2 is realized in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' For the slow or fast peeling, the system has only one stable fixed point at large u or small u, which corresponds to the state A or B as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 2(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' And at the intermediate peel speed, the tunnel structure in the state A adjacent to the unit in the state B tends to collapse from the experiment [5], however, the tunnel structure can regenerate after a meanwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' This dynamics seems to resemble a threshold firing dynamics and its dynamics in the phase space can be illustrated as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Based on the above consideration, the following equation is proposed for each unit: d2s dt2 = −dU(u) du −bds dt + a 1+c(u−d)2 du dt , (1) where s = u+Vt shows the length of peeled tape, and we assume s = 0 at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' U is a double-well potential for bistability of the two peel states, b is viscosity of adhesives, and V corresponds to a 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 2: Illustration of the assumed phase space dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (a) Small V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The system has only one stable fixed point as state A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (b) Large V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The system has only one stable fixed point as state B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (c) Intermediate V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The system has two stable fixed points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' the states A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' If the tunnel structure in the state A collapses with a large perturbation as 1⃝, then the state goes to the state B as 2⃝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' And at last, the state comes back to the state A ( 3⃝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' peel speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The last term on the right-hand side shows an asymmetric dissipation, which depends on the existence of the tunnel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' a, c and d are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' For the dynamics of u instead of s, we can rewrite eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (1) as d2u dt2 = −dU(u) du −b � V + du dt � + a 1+c(u−d)2 du dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (2) On the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (2), the terms − �dU(u) du +bV � and � −b+ a 1+c(u−d)2 � du dt cor- respond to elastic and viscous forces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Furthermore, we add the following symmetric spatial interaction terms in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (2): D1(ui+1 +ui−1 −2ui)+D2 �dui+1 dt + dui−1 dt −2dui dt � , (3) where D1 and D2 are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' These two terms also express the effect of the viscoelas- ticity of adhesives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Results: Based on eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (2) with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (3), we calculated the following equation, d2ui dt2 =−3(ui −1)2(ui −2)− � V + dui dt � + 2 1+20(ui −1)2 dui dt +(ui+1 +ui−1 −2ui)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='1 �dui+1 dt + dui−1 dt −2dui dt � , i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=',N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (4) ui = 0 corresponds to the state where no adhesive deformation occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' And we set ui ≈ 2 and ≈ 1 for the states A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' In numerical calculation, in order to take relaxation of 4 Cadhesive deformation due to spatial inhomogeneity of adhesives into account, we reset ui(t) = 0 with a probability p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='001 per unit time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' As the initial condition, we set ui(0) = 0 and dui dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Periodic boundary condition was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The 4th Runge-Kutta method was used with dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='01 in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The discretized system size was N = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Figure 3 shows the typical spatiotemporal patterns obtained by the numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The black and white regions correspond to the states B and A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' There are five cases with different values of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' It is found that as V increases the ratio of white regions (the state A) decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' And interchange of connectivity between black and white regions is confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' These results are consistent with those in the previous study[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Here the following scaling properties for the coexistent patterns are focused on: (i) the cu- mulative distribution F(≥ s) ∼ s−ξ where s is the size of the white clusters, (ii) their standard deviations of the height h(s) ∼ sν∥ and the width w(s) ∼ sν⊥, and (iii) the fractal dimension D of the spatiotemporal patterns of the white regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Our numerical results for ξ, D, ν∥, and ν⊥ are shown in Tbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' For comparison, this table also has their values obtained in the previous studies [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' It is found that our present result is consistent with the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 3: Typical spatiotemporal patterns obtained by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Time proceeds from top to bottom (500 ≤ t ≤ 1200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (a) V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='29, (b) V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='30, (c) V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='309, (d) V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='32, (e) V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Discussion: Here we comment on difference between the previous model[3] and the present model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The previous model has an asymmetric spatial interaction which expresses the situation of the state transition where the tunnel structures (in the state A) adjacent to a part of the peel front in the state B tend to collapse [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' In the present model, we assume the local dynamics of the peel front as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 2 and introduce the local dissipation as a function of u instead of the previous asymmetric spatial interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Figure 4 shows the nullclines and stable and unstable manifolds of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (4) without the spatial 5 TABLE I: The estimated values of the exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' ξ D ν⊥ ν∥ eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (4) at V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='45 model A at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='275 [5] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='41 model B at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='181 [6] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='84(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='04) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='61(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='58(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='02) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='41(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='03) model C at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='037 [6] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='81(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='05) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='62(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='58(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='41(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='034) peeling at V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='48mm [5] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='45 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 4: Phase spaces of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (4) without spatial interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Red and Green lines are the nullcline for u and du/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Orange and Blue lines are stable and unstable manifolds of the saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (a) V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The system has only one fixed point as the state A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (b) V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The system has only one fixed point as the state B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (c) V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The system has two fixed points as the state A and B, and a threshold firing dynamics is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' interaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' For the slow or fast peeling, the system has only one stable fixed point at large u ≈ 2 or small u ≈ 1, which corresponds to the states A or B in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='4(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' And the assumed dynamics at the intermediate speed is also realized as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 4(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 1⃝ if the state A crosses over the stable manifold then 2⃝ the state goes to the state B and 3⃝ the state comes back to the state A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' We consider that such a threshold firing mechanism is essential for the pattern formation of peeling adhesive tapes, although we have to experimentally verify the validity of each term in the present phenomenological model in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Regarding the effect of noise, the noise term is necessary for the reproduction of the fractal spatiotemporal patterns in the previous model[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' However, the present model does not need any noise term in the time evolution of u to reproduce the patterns as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='3, although we 6 BFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 5: A spatiotemporal pattern obtained from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (2) with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (3), dU du = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='2(u + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='5)(u − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='5)(u − 4), V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='66, b = 1, a = 1, c = 2, d = 0, D1 = 1,D2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='1, and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 500 ≤ t ≤ 1200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Initial conditions of both ui(0) and dui dt (0) are given with a uniform random number between −2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' added the noise in our numerical calculation for stabilizing the state B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Actually, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='5 shows an example of the spatiotemporal pattern obtained from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (4) without noise term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' we considered the randomness in u only for the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' This result suggests that the fractal spatiotemporal patterns originate not from stochasticity but from chaoticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Finally, we note that the fractal patterns as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content='3(c) and 5 are reproducible by Bonhoffer-van der Pol type equation and other reaction-diffusion equations [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The relation- ship between our present model and these models can be found by transforming our model into the 2-component activator-inhibitor system as the Liénard system by introducing a new parameter wi ≡ dui dt +bui −a′ arctan(c′(ui −d))−D2(ui+1 +ui−1 −2ui): dui dt = −wi −bui +a′ arctan(c′(ui −d))+D2(ui+1 +ui−1 −2ui) (5) dwi dt = dU du ���� u=ui +bV −D1(ui+1 +ui−1 −2ui), (6) where a′ = a/√c,c′ = √c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' If we change the term dU du +bV in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (6) to u, then the reaction terms of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' (5) and (6) correspond to those of the Bonhoffer-van der Pol type equation[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The presence of such correspondence also suggests that the fractal patterns emerge chaotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Conclusion: In this letter, we have proposed a new model for pattern formation in peeling adhesive tapes which focuses on the equation of motion for the displacement of adhesives in the peel front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The model reproduces the dynamical and statistical properties of the spatiotemporal patterns which consist of the two different peel states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' The previous experimental results also support our numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Note that the spatiotemporal pattern obtained by the present model 7 is seemingly relevant to directed-percolation universality class [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Detailed discussions of this relevancy will be reported in near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Yamazaki and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Toda, Journal of the Physics Society Japan 71 1618 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Yamazaki and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Toda, Journal of the Physics Society Japan 73 2342 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [3] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Yamazaki and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Toda, Physica D 214 120 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Yamazaki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Yamamoto, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Kadono, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Toda, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 81 043002 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Yamazaki, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 125 641 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Ohmori and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Yamazaki, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 88 105001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Hayase, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 66 2584 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Hayase and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Ohta, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 81 1726 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Chaté, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Pikovsky, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Rudzick 131 17 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Hayase and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Ohta 62 5998 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' Hinrichsen, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' in Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 49 815 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} +page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FJT4oBgHgl3EQfxi3J/content/2301.11635v1.pdf'} diff --git a/MdE4T4oBgHgl3EQf8w5p/content/tmp_files/2301.05350v1.pdf.txt b/MdE4T4oBgHgl3EQf8w5p/content/tmp_files/2301.05350v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..14bd17328f81c30665a1eeb1604c1c53a9cadb1f --- /dev/null +++ b/MdE4T4oBgHgl3EQf8w5p/content/tmp_files/2301.05350v1.pdf.txt @@ -0,0 +1,1897 @@ +Sublinear Algorithms for TSP via Path Covers +Soheil Behnezhad +Northeastern University +Mohammad Roghani +Stanford University +Aviad Rubinstein +Stanford University +Amin Saberi +Stanford University +Abstract +a We study sublinear time algorithms for the traveling salesman problem (TSP). First, we +focus on the closely related maximum path cover problem, which asks for a collection of vertex +disjoint paths that include the maximum number of edges. We show that for any fixed ε > 0, +there is an algorithm that (1/2 − ε)-approximates the maximum path cover size of an n-vertex +graph in �O(n) time. This improves upon a (3/8 − ε)-approximate �O(n√n)-time algorithm of +Chen, Kannan, and Khanna [ICALP’20]. +Equipped with our path cover algorithm, we give �O(n) time algorithms that estimate the +cost of graphic TSP and (1, 2)-TSP up to factors of 1.83 and (1.5 + ε), respectively. +Our +algorithm for graphic TSP improves over a 1.92-approximate �O(n) time algorithm due to +[CHK ICALP’20, Behnezhad FOCS’21]. Our algorithm for (1, 2)-TSP improves over a folklore +(1.75 + ε)-approximate �O(n)-time algorithm, as well as a (1.625 + ε)-approximate �O(n√n)-time +algorithm of [CHK ICALP’20]. +Our analysis of the running time uses connections to parallel algorithms and is information- +theoretically optimal up to poly log n factors. Additionally, we show that our approximation +guarantees for path cover and (1, 2)-TSP hit a natural barrier: We show better approximations +require better sublinear time algorithms for the well-studied maximum matching problem. +1 +arXiv:2301.05350v1 [cs.DS] 13 Jan 2023 + +Contents +1 +Introduction +1 +2 +Technical Overview +3 +3 +Preliminaries +4 +4 +New Meta Algorithms for Maximum Path Cover +5 +5 +A Local Query Process for Algorithm 2 and its Complexity +8 +6 +Our Estimator for Maximum Path Cover +15 +7 +Our Estimator for (1,2)-TSP +19 +8 +Our Estimator for Graphic TSP +20 +9 +Further Improvement for Graphic TSP +23 +10 Lower Bound for Approximating Maximum Path Cover +27 +10.1 “Conditional” Hardness for the Approximation Ratio . . . . . . . . . . . . . . . . . . +27 +10.2 Information-Theoretic Lower Bounds on the Running Time . . . . . . . . . . . . . . +29 +A Implementation Details +32 + +1 +Introduction +The traveling salesman problem (TSP) is a central problem in combinatorial optimization. Given +a set V of n vertices and their pairwise distances, it asks for a Hamiltonian cycle of the minimum +cost. In this paper, we study sublinear time algorithms for TSP. The algorithm is given query +access to the distance pairs, and the goal is to estimate the solution cost in time sublinear in the +input size (which is Θ(n2)). +TSP is NP-hard to approximate within a polynomial factor for an arbitrary distance function. +As such, much of the work in the literature has been on more specific distance functions. Some +notable examples include graphic TSP [13, 20, 21, 25, 8] where the distances are the shortest paths +over an arbitrary unweighted undirected graph, (1, 2)-TSP [1, 8, 5, 16, 19] where the distances are 1 +or 2, and more generally metric TSP [15, 12, 10, 26] where the distances satsify triangle inequality. +In 2003, Czumaj and Sohler [11, 12] showed that for any fixed ε > 0, a (1+ε)-approximation of +the cost of metric minimum spanning tree (MST) and thus a (2 + ε)-approximation of the cost of +metric TSP can be found in �O(n) time. Twenty years later, it still remains a major open problem to +either break two-approximation in n2−Ω(1) time or prove a lower bound.1 However, better bounds +are known for both graphic TSP and (1, 2)-TSP. In this paper, we present improved algorithms for +these two well-studied variants of TSP. Our main tool to achieve this is an improved algorithm for +the closely related maximum path cover problem which might be of independent interest. +Maximum Path Cover: +The maximum path cover in a graph is a collection of vertex disjoint +paths with the maximum number of edges in it. The (almost) 1/2-approximate maximum match- +ing size estimator of Behnezhad [2] immediately implies an (almost) 1/4-approximation for the +maximum path cover problem in �O(n) time.2 This can be improved to an (almost) (3/8 = .375)- +approximation using the matching-pair idea of Chen, Kannan, and Khanna [8] in �O(n√n)-time.3 +Our first main contribution is an improvement over both of these results: +Result 1 (Formally as Theorem 6.8). For any ε > 0, there is a randomized algorithm that w.h.p. +(1/2 − ε)-approximates the size of maximum path cover in �O(n · poly(1/ε)) time. +Besides quantitavely improving prior work both in the running time and the approximation +ratio, Result 1 reaches a qualitatively important milestone as well. +First, the running time of +Result 1 is information-theoretically optimal up to poly log n factors (the lower bound holds for +any constant approximation — see Section 10.2). Second, its approximation ratio hits a rather +important barrier. We give a non-trivial reduction that shows a (1/2+Ω(1))-approximation in �O(n) +time for maximum path cover would imply the same bound for maximum matching in bipartite +graphs. Such a result has remained elusive for matching, which is one of the most extensively +studied problems in the literature of sublinear time algorithms. See Section 10. +It is also worth noting that in bounding the running time of our algorithm in Result 1, we use +connections to parallel algorithms. Such a connection was previously only used for matchings [2]. +1See e.g. Open Problem 71 on sublinear.info [14]. +2The application of sublinear time maximum matching algorithms for approximating maximum path cover was +first proposed by Gupta and Onak. See [14]. +3We note that even though a subsequent result of Behnezhad [2] improved the running time for maximal matchings +and graphic TSP from O(n√n) in [8] to �O(n), it is not immediately clear whether the same holds for path cover and +(1, 2)-TSP as they rely on a different notion of a matching pair. +1 + +(1, 2)-TSP: +The (1, 2)-TSP problem has been studied extensively in the classical setting. In his +landmark paper, Karp [16] showed that (1, 2)-TSP is NP-hard. Papadimitriou and Yannakakis +[23] then proved its APX-hardness. Since then there has been a significant amount of work on +(1, 2)-TSP in the classical setting. The current best known inapproximability bound for (1, 2)-TSP +is 535/534 [17]. After a series of works, the best known polynomial time approximation is 8/7 [5] +which can be implemented in O(n3) time [1]. For sublinear time algorithms, an �O(n)-time (almost) +1.75-approximation is folklore [14]. Chen, Kannan, and Khanna [8] improved the approximation to +(almost) 1.625 in �O(n√n) time. +It is not hard to see that up to a small additive error of 1, (1, 2)-TSP is equivalent to finding +a maximum path cover on the weight-1 edges and then connecting their endpoints via weight-2 +edges. A simple calculation shows that any α-approximation for the maximum path cover problem +leads to a (2 − α)-approximation for (1, 2)-TSP. Our path cover algorithm of Result 1 immediately +implies the following result as a corollary: +Result 2. For any ε > 0, there is a randomized algorithm that w.h.p. (1.5 + ε)-approximates +the cost of (1, 2)-TSP in �O(n · poly(1/ε)) time. +Similar to Result 1, the running time of Result 2 is information-theoretically optimal up to +poly log n factors, and its approximation ratio hits a natural barrier due to a connection to sublinear +time matching that we establish in this work. +Graphic TSP: +The graphic TSP problem is equivalent to finding a tour of the minimium size +that visits all the vertices. This is an important instance of TSP that has received a lot of attention +over the years. For polynomial time algorithms, a 1.5-approximation of Christofides [10] (which +also works more generally for metric TSP) had remained the best known until a series of works +over the last decade improved it to (1.5 − ε0) [13], 1.461 [20], 1.444 [21], and finally to 1.4 [25]. For +sublinear time algorithms, Chen, Kannan, and Khanna [8] showed that an (almost) (27/14 ≈ 1.928)- +approximation of graphic TSP can be obtained in �O(n√n) time. The running time was subsequently +improved to �O(n) by Behnezhad [2]. +We first show that plugging Result 1 into the framework of [8] immediately improves their +approximation from 1.928 to (almost) 1.9 while keeping the running time �O(n). We then give a +more fine tuned algorithm that obtains a much improved approximation ratio of (11/6 ≈ 1.833). +Result 3. For any ε > 0, there is a randomized algorithm that w.h.p. (1 + ε)( 11 +6 ≈ 1.833)- +approximates the cost of graphic TSP in �O(n · poly(1/ε)) time. +Further related work: +Finally, we note that in a recent paper, Chen, Khanna, and Tan [9] +show that assuming that the metric has a spanning tree supported on weight 1 edges, one can +obtain a (2 − ε0)-approximation with �O(n√n) queries for some small unspecified constant ε0 > 0. +While this is a more general metric than graphic TSP and (1,2)-TSP that we study in this paper, +we note that the two papers are orthogonal and their results are incomparable. In particular, the +techniques developed in this paper are specifically designed to improve the approximation to much +below 2, whereas [9] focuses on generalizing the distance function while beating 2. +Independent work: +In two concurrent papers, Bhattacharya, Kiss, and Saranurak [6], and +Behnezhad, Roghani, and Rubinstein [3], gave an almost 2/3-approximation for maximum match- +2 + +ing in n2−Ω(1) time. Combining these two algorithms with the framework of Chen, Kannan, and +Khanna [8] implies a 40/21-approximation for graphic TSP in n2−Ω(1) time (see Theorem 1.6 in +[6]). Our algorithm makes an improvement on both the running time and approximation ratio for +the graphic TSP over the recent results. In terms of techniques, our work is entirely disjoint. +2 +Technical Overview +In this section, we give an overview of our algorithms, especially our sublinear time maximum path +cover algorithm of Result 1 which is the key to the other results as well. +Let us start with using matchings to approximate maximum path cover. Consider a graph that +has a Hamiltonian path. Here, the optimal maximum path cover has size n − 1. On the other +hand, any maximum matching can have at most n/2 edges, which is by a factor 2 smaller than +our optimal path cover. On top of this, we only know close to 2/3 approximations for maximum +matching in the sublinear time model [3, 6], thus can only achieve an approximation close to 1/3. +Instead of a single matching, Chen, Kannan, and Khanna [8] showed how to estimate the +number of edges in a maximal matching pair in �O(n√n) time, where a matching pair is simply two +edge disjoint matchings. It is not hard to see that the number of edges in a maximal matching +pair is at least half the number of edges in a maximum path cover. The problem, however, is that +a maximal matching pair is not a collection of paths! In particular, the two matchings can form +cycles of length as small as four. Therefore, one may only be able to use 3/4 fraction of the edges of +a matching pair in a path cover. This is precisely why the algorithm of [8] only obtains a 1 +2 × 3 +4 = 3 +8 +approximation for path cover, and a 2 − 3 +8 = 1.625 approximation for (1, 2)-TSP. +If we could modify the matching pair algorithm of [8], and avoid cycles by manually excluding +edges whose endpoints are the endpoints of a path in the current matching pair, then we could +avoid the 3/4 factor loss discussed above and achieve a 1/2-approximation. Unfortunately, checking +whether the endpoints of an edge are endpoints of a path requires knowledge about whether a series +of other edges belong to the solution, which seems hard to implement in sublinear time. +Instead of checking for cycles manually, we introduce the following Algorithm 1 which avoids +cycles more naturally. While our final algorithm is a modified variant of Algorithm 1 described +below, we start with Algorithm 1 as we believe it provides the right intuition. +Algorithm 1: A new algorithm for path cover. +1 Initialize P ← ∅. +2 Each vertex v has two ports that we denote by v0 and v1. Each of these ports throughout the +algorithm will be either free or occupied. Initially, all ports are free. +3 Iterate over the edges in some ordering π. Upon visiting an edge e = (u, v): +• If v0 and u0 are free, add e to P, mark v0 and u0 as occupied, and skip to the next edge. +• If v1 and u0 are free, add e to P, mark v1 and u0 as occupied, and skip to the next edge. +• If v0 and u1 are free, add e to P, mark v0 and u1 as occupied, and skip to the next edge. +4 Return P. +Two properties of Algorithm 1 are crucial. First, it prioritizes occupying (u0, v0) (compared to +(u1, v0) or (u0, v1)) which in particular implies that any component in P must have a (u0, v0) edge. +Second, it never occupies (u1, v1) with an edge (u, v). While it is easy to see that the output of +Algorithm 1 has maximum degree 2, and is thus a collection of paths or cycles, the two properties +3 + +above actually guarantee that it never includes any cycle. See Figure 1. We provide the formal +proof of this later in Section 4. Additionally, we show that the output of Algorithm 1 must be at +least half the size of a maximum path cover, as we prove next. Hence, if we manage to estimate +the size of the output P of Algorithm 1, then we have proved Result 1. +0 +0 +1 +0 +0 +0 +0 +1 +1 +0 +1 +1 +1 +1 +1 +0 +0 +0 +0 +1 +1 +0 +1 +1 +1 +0 +0 +0 +1 +0 +1 +0 +1 +0 +0 +1 +1 +0 +1 +1 +0 +0 +0 +1 +0 +0 +1 +0 +Invalid: +has no (0, 0) edge +0 +1 +1 +1 +1 +1 +1 +0 +0 +0 +0 +0 +0 +1 +Invalid: +a vertex has two 0’s +Invalid: +(1,1) not allowed +Valid +(not a cycle) +Figure 1: Examples of why the output of Algorithm 1 will not have cycles. +Our final algorithm is slightly different from Algorithm 1 discussed above. In particular, we +slightly relax it—see Algorithm 2—so that it can be solved via a randomized greedy maximal in- +dependent set (RGMIS), for which we have a rich toolkit of sublinear time estimators. Existing +approaches (particularly the algorithm of Yoshida, Yamamoto, and Ito [27] and its two-step im- +plementation by Chen, Kannan, and Khanna [8]) can be employed to estimate the value of this +modified Algorithm 2 in �O(n√n) time. We achieve the improved, and near tight, �O(n) time bound +guarantee of Result 1 by building on the analysis of Behnezhad [2] for maximal independent set +on the line graphs (i.e., maximal matchings). Though we note that several new ideas are needed, +because the MIS graph in our case will not be exactly a line graph. We defer more discussions +about this to Sections 4 and 5. +Implications for TSP: +By having an α-approximate maximum path cover algorithm, we imme- +diately obtain a (2 − α)-approximation for (1, 2)-TSP. Therefore, the algorithm above immediately +proves Result 2 that we can (almost) 1.5-approximate (1, 2)-TSP in �O(n) time. For our Result 3 for +graphic TSP, we first observe that our improved path cover algorithm can be employed to provide +a better lower bound for the optimal TSP solution. This improves the 1.92-approximation of [8] +as black-box to 1.9-approximation (Section 8). However, the final improvement to 1.83 requires +more ideas, in particular, on how to better estimate the number of certain bridges in the graph. +See Section 9 for more details about this. +3 +Preliminaries +Problem Definition: +In the sublinear TSP problem, we have a set V of n vertices and a distance +function d : V × V → R+. The algorithm has query access to this distance function. Namely, for +any pair (u, v) of the vertices of its choice, the algorithm may query the value of d(u, v). The goal +is to design an algorithm that runs in sublinear time in the input size, which is Θ(n2) (all the +distance pairs), and produces an estimate of the size of the optimal TSP solution. Denoting the +optimal TSP value by τ(V ), we say an estimate �τ(V ) provides an α-approximation for α ≥ 1 if +τ(V ) ≤ �τ(V ) ≤ α · τ(V ). +4 + +We focus specifically on graph TSP and (1, 2)-TSP problems. In graphic TSP, the distance function +d is the shortest path metric on an unweighted undirected graph G that is unknown to the algorithm. +Note, however, that the distance queries essentially provide adjacency matrix access to this graph +G. In (1, 2)-TSP, the assumption is that d(u, v) ∈ {1, 2} for every pair u, v. In the case of (1, 2)-TSP +we may use G to refer to the subgraph induced on the pairs with distance 1. +Defining graph G as above, we use n to denote the number of its vertices, m to denote the +number of its edges, ∆ to denote its maximum degree, µ(G) to denote its maximum matching size, +ν(G) to denote its minimum vertex cover size, and ¯d to denote its average degree. +Path Cover Definitions: +Given an unweighted graph G, a path cover in G is a collection of +vertex disjoint paths in G. A maximum path cover is a path cover of G with the maximum number +of edges in it (note that we are not counting the number of paths, but rather the total number +of edges in them). We use ρ(G) to denote the size of the maximum path cover in G. We say an +estimate �ρ(G) for ρ(G) provides an (α, ε)-approximation for α, ε ∈ [0, 1] if +α · ρ(G) − εn ≤ �ρ(G) ≤ ρ(G). +We may also use α-approximation instead of (α, 0)-approximation. +Graph Theory Definitions/Tools: +A bridge (cut edge) in a graph is an edge whose deletion +increases the number of connected components. Similarly, a cut vertex is a vertex whose deletion +(along with its edges) increases the number of connected components. A biconnected graph is a +connected graph with no cut vertex. Also, a biconnected component (block) of a graph is a maximal +biconnected subgraph of the original graph. A non-trivial biconnected component is a block that +is not a bridge. We say a graph is 2-edge-connected if there is no bridge in the graph. A 2-edge- +connected component of a graph is maximal 2-edge-connected subgraph of the original graph. The +bridge-block tree of a graph is a tree obtained by contracting the 2-edge-connected components; +note that the edge set of a bridge-block tree correspond to the bridges in the original graph. +We use the following classic theorem of K¨onig [18] that the size of the minimum vertex cover is +equal to the size of maximum matching in bipartite graphs. Namely: +Proposition 3.1 (K¨onig’s Theorem). In any bipartite graph G, µ(G) = ν(G). +Probabilistic Tools: +In our proofs, we use the following standard concentration inequalities. +Proposition 3.2 (Chernoff Bound). Let X1, X2, . . . , Xn be independent Bernoulli random vari- +ables. Let X = �n +i=1 Xi. For any t > 0, Pr[|X − E[X]| ≥ t] ≤ 2 exp +� +− +t2 +3 E[X] +� +. +Proposition 3.3 (Hoeffding’s Inequality). Let X1, X2, . . . , Xn be independent random variables +such that a ≤ Xi ≤ b. Let ¯X = (�n +i=1 Xi)/n. For any t > 0, Pr[| ¯X −E[X]| ≥ t] ≤ 2 exp +� +− +2nt +(b−a)2 +� +. +4 +New Meta Algorithms for Maximum Path Cover +In this section, we present a new meta algorithm for maximum path cover that obtains a 1/2- +approximation. The algorithm, as we will state it in this section, will not be particularly in the +sublinear time model. We discuss its sublinear time implementation later in Sections 5 and 6. +Our starting point is the Algorithm 1 described in Section 2. Let us first formally prove that it +obtains a 1/2-approximation, and that no component in it is a cycle. +5 + +Claim 4.1. The output of Algorithm 1 is a collection of disjoint paths. +Proof. Since P has maximum degree two, it suffices to show none of its connected components are +cycles. Property (i) above implies that at any point during the algorithm, any degree one vertex +v has its port v0 occupied. Now take an edge e = (u, v) that forms a cycle if added to P. Both u +and v must have degree one and so u0 and v0 are occupied. Since by property (ii) edge e does not +occupy both v1 and u1, the algorithm does not add e to P thus not completing a cycle. +Claim 4.2. Let P ⋆ be any path cover using weight one edges. Then the output of Algorithm 1 has +size at least 1 +2|P ⋆|. +Proof. For any edge e = (u, v) ∈ P ⋆ define φ(e) = 1 +4(degP (u) + degP (v)). We first claim that +for every edge e = (u, v) in G, we have φ(e) ≥ 1/2 (or, equivalently, degP (u) + degP (v) ≥ 2). +This is clear for edges e ∈ P due to the contribution of e itself to its endpoints’ degrees, so fix +e ̸∈ P. Consider the time that we process e = (u, v) in the algorithm and decide not to add it to +P. We claim that out of v0, v1, u0, u1 at least two ports must be occupied. Suppose w.l.o.g. and +for contradiction that only vx is occupied for x ∈ {0, 1}. Then (u, v) can occupy v1−x and ux and +be added to P. This contradicts (u, v) not being added to P and proves our claim that φ(e) ≥ 1/2. +From the discussion above, we get that +� +e∈P ⋆ +φ(e) ≥ +� +e∈P ⋆ +1/2 = |P ⋆|/2. +Moreover, because every vertex has degree at most two in P ⋆, we get +� +e∈P ⋆ +φ(e) = 1 +4 +� +(u,v)∈P ⋆ +degP (u) + degP (v) ≤ 1 +4 · 2 +� +v∈V +degP (v) = |P|. +The two inequalities above combined imply that |P| ≥ |P ⋆|/2. +As discussed, our final algorithm is different from Algorithm 1 discussed above. One problem +with Algorithm 1 is that it cannot be cast as an instance of the randomized greedy maximal +independent set (RGMIS) algorithm for which there is a rich toolkit of sublinear time estimators. +To remedy this, we present a modified variant of Algorithm 1 whose output is (almost) as good, +but in addition can be modeled as an instance of RGMIS. We denote the output of RGMIS on a +graph G with a permutation π on its vertices by RGMIS(G, π). +The algorithm is stated below as Algorithm 2. Similar to the output of Algorithm 1, the output +of Algorithm 2 can be verified to have maximum degree two. Thus, it is a collection of paths and +cycles. But unlike Algorithm 1, the output of Algorithm 2 can have cycles. This happens since, +unlike Algorithm 1, each connected component of the output of Algorithm 2 is not guaranteed to +have an edge (u, v) occupying both u0 and v0. Nonetheless, we are able to show that this bad +event only happens for a small fraction of connected components of the output of Algorithm 2 in +expectation, and so once we remove one edge of each of these cycles, the resulting collection of +6 + +disjoint paths has almost the same size. +Algorithm 2: A modification of Algorithm 1 that uses RGMIS. +1 Parameter: K (think of it as a large constant integer). +2 Let G = (V, E) be the subgraph of weight one edges. We construct a graph H = (VH, EH) +from G on which we run RGMIS. +3 Each vertex in H corresponds to an edge e in G and two ports (as in Algorithm 1) of the +endpoints of e that it occupies. Formally, for any (u, v) ∈ E we have K + 2 vertices in H: +• One vertex that corresponds to occuping u0 and v1. +• One vertex that corresponds to occuping u1 and v0. +• K vertices that each corresponds to occuping u0 and v0. +4 Consider two distinct vertices a and b in H corresponding to edges ea and eb in G: +• If ea = eb then we add an edge between a and b in H. +• If ea and eb share exactly one endpoint v and both a and b occupy the same port of v, we +add an edge between a and b in H. +5 Find a randomized greedy maximal independent set I of H. +6 Let P be the set of edges in G corresponding to the vertices in I. +7 Return P. +Observation 4.3. Let C be a connected component in the output of Algorithm 2. If C is a cycle, +then every edge in C occupies one 0-port and one 1-port (that is, no edge occupies two 0-ports). +Proof. Suppose that C has n′ vertices. Since each vertex in a cycle has degree two, both ports +of each vertex in C must be occupied. +Hence, n′ 0-ports and n′ 1-ports of C are occupied in +total. Given that any edge occupies at least one 0-port by the algorithm, we cannot have an edge +that occupies two 0-ports, or else we should occupy more 0-ports than 1-ports of C, which is a +contradiction. +Next, we show that up to a factor of (1 + 2/k) which is negligible for K in the order 1/ε, the +output of Algorithm 2 is an (almost) 1/2-approximation of the maximum path cover value. +Observation 4.4. Let C be a connected component in the output of Algorithm 2. If C is a path, +then it contains at most one edge that occupies two 0-ports. +Proof. Let C be the path (v1, v2, . . . , vr). Since the degree of any vertex vi for 1 < i < r is two in +the path, both ports of vi must be occupied. For v1 and vr, on the other hand, only one port is +occupied. Hence, the total number of 0-ports that are occupied by C minus the number of 1-ports +occupied by it is at most two. This means that there is at most one edge that occupies two 0-ports +since all other types of edges occupy exactly one 0-port and one 1-port. +Lemma 4.5. let P be the output of Algorithm 2 on graph G. Then +1 +2ρ(G) ≤ E |P| ≤ +� +1 + 2 +K +� +ρ(G), +where the expectation is taken over the randomization of computing RGMIS in Algorithm 2. +7 + +Proof. Let P ∗ be a maximum path cover. For any edge e = (u, v) ∈ P ⋆ define φ(e) = 1 +4(degP (u) + +degP (v)). With the exact same argument as in the proof of Claim 4.2, we get that φ(e) ≥ 1/2, +which implies +� +e∈P ⋆ +φ(e) ≥ +� +e∈P ⋆ +1/2 = ρ(G)/2. +Since the degree of each vertex in P is at most two, we get +� +e∈P ⋆ +φ(e) = 1 +4 +� +(u,v)∈P ⋆ +degP (u) + degP (v) ≤ 1 +4 · 2 +� +v∈V +degP (v) = |P|. +By combining above inequalities we get 1 +2ρ(G) ≤ |P|. Note that we do not need the randomization +for the proof of the lower bound. +By construction of P, every vertex has degree at most two in P. Hence, all connected compo- +nents of P are cycles and paths. We claim that at most +2 +K+2 fraction of connected components are +cycles in expectation. Since the expected number of connected components is at most E |P|, from +this we get that the expected number of cycles is at most 2 E |P|/(K + 2). By removing one edge +from each cycle, we obtain a valid solution for maximum path cover problem. Thus, +E |P| − 2 E |P| +K + 2 = +K +K + 2 E |P| ≤ ρ(G) +⇒ +E |P| ≤ +� +1 + 2 +K +� +· ρ(G). +So it remains to show that at most +2 +K+2 fraction of connected components are cycles in expec- +tation. As we process edges one by one according to the ordering of RGMIS, let A be the set of +edges that none of their incident edges are added to the solution of Algorithm 2. By definition of +A, if one copy of edge (u, v) is in A, then all other copies of (u, v) are also in A. Therefore, at any +point during running RGMIS, if a new component is added to the solution, the edge (u, v) that +gets added to the solution occupies (u0, v0) with probability at least +K +K+2 since K copies out of the +K + 2 copies are for (u0, v0). Let C0 be the number of times that the newly added component is +an edge occupying two 0-ports, and C1 be the number of times that the newly added component +is an edge occupying one 0-port and one 1-port. By the above argument, we have +E[C0] +E[C0] + E[C1] = +K +K + 2. +(1) +Note that after running Algorithm 2, it is possible that the number of connected components is +actually smaller than C0 +C1, since some of the components may merge as the algorithm proceeds. +However, by Observation 4.4, two components that their first edge occupies two 0-ports will not +merge together. Also, by Observation 4.3, none of the cycle components have an edge that occupies +two 0-ports. Therefore, in the end, there exists at most E[C0] + E[C1] connected components and +at least E[C0] of them will not be cycles. This completes the proof. +5 +A Local Query Process for Algorithm 2 and its Complexity +In this section, we define a query process to estimate the size of the output of Algorithm 2. +In graph H of Algorithm 2, each vertex corresponds to an edge in the original graph. More +precisely, we make K + 2 copies of each edge (u, v) such that one of the copies corresponds to an +8 + +edge occupying (u0, v1), one for (u1, v0), and K for (u0, v0). We use G′ = (V, E′) to show the new +graph with these parallel edges. During the course of Algorithm 2, two different edges that share +the same endpoint and port cannot appear in the solution together. We use the following definition +to formalize this notion. +Definition 5.1 (Conflicting Pair of Edges). Two edges e, e′ ∈ E′ that share an endpoint v are +conflicting if both e and e′ correspond to same port vi for i ∈ {0, 1}. We call (e, e′) a conflicting +pair of edges. +In order to estimate the size of the output of Algorithm 2, we define a vertex oracle that given +a vertex v and a permutation π on E′, returns the degree of vertex v in the output of Algorithm 2. +These are akin to the query processes used before in the works of [2, 27], but are specific to our +Algorithm 2. +Algorithm 3: “vertex oracle” VO(u, π) to determine the degree of vertex u in RGMIS(G′, π). +1 Let e1 = (u, v1), . . . , er = (u, vr) be the edges incident to u with π(e1) < . . . < π(er). +2 d ← 0 +3 for i in 1 . . . r do +4 +if EO(ei, vi, π) = True then d ← d + 1; +5 return d +Algorithm 4: “edge oracle” EO(e, u, π) to determine an edge e is in RGMIS(G′, π). Also, u +must be an endpoint of e. +1 if EO(e, u, π) computed before then return the computed result.; +2 Let e1 = (u, v1), . . . , er = (u, vr) be the edges incident to e such that +π(e1) < . . . < π(er) < π(e). Also, (e, ei) is a conflicting pair for all 1 ≤ i ≤ r. +3 for i in 1 . . . r do +4 +if EO(ei, vi, π) = True then return False; +5 return True +Note that in Line 2 of the Algorithm 4 we only recursively call the function on edges that their +label, conflict with edge e since if other edges appear in the RMGIS subgraph, we can still have e +in the RGMIS subgraph. Before analyzing the query complexity of the vertex oracle, we prove the +correctness of the vertex oracle. +Claim 5.2. For any edge e = (u, z) ∈ E′ that is occupying ports ui and zj, if EO(e, u, π) is called +while computing VO(v, π), then EO(e, u, π) = True iff e ∈ RGMIS(G′, π). +Proof. We prove the claim using induction on ranking of edge e. Assume that the claim is true +for all edges with ranking smaller than π(e). If EO(e, u, π) is called by EO(e′ = (w, z), z, π) or +directly by VO(v, π), then by definition of Algorithm 4 and Algorithm 3, all edges e′′ = (w′, z) with +π(e′′) < π(e′) that are occupying zj are queried before e′ which means that none of them return +True. Hence, by induction hypothesis, none of the edges incident to z that are occupying zj with +lower rank are in the RGMIS(G′, π). Moreover, EO(e, u, π) calls all incident edges to u with lower +rank that are occupying ui and return Trueif none of them are in the RGMIS(G′, π) by induction +hypothesis. Therefore, EO(e, u, π) = True iff e ∈ RGMIS(G′, π). +Claim 5.3. Let v ∈ V and d be the output of VO(v, π). Then d is equal to the degree of vertex v +in the subgraph outputted by RGMIS(G′, π). +9 + +Proof. The observation follows by combining the fact that the vertex oracle queries edges in in- +creasing order and Claim 5.2. +Let T(v, π) denote the number of recursive calls to the edge oracle during the execution of +VO(v, π). +Theorem 5.4. For a randomly chosen vertex v and permutation π on E′, we have that +Ev,π[T(v, π)] = O( ¯d · log2 n) +where ¯d is the average degree of the graph G. +Let Q(e, v, π) be the number of EO(e, ·, π) calls during the execution of VO(v, π). Moreover, +let Q(e, π) be the number of EO(e, ·, π) calls starting from any vertex. In other words, we have +that Q(e, π) = � +v∈V Q(e, v, π). +Observation 5.5. For every edge e and permutation π, Q(e, π) ≤ O(n2). +Proof. Let e = {x, y}. For a fixed vertex u, either the vertex oracle VO(u, π) queries the edge +oracle for e directly, or through some incident edge e′. Hence, the edge oracle of e is called through +at most (K + 2)(deg(x) − 1) + (K + 2)(deg(y) − 1) of its incident edges (K + 2 appears since each +edge has K + 2 copies), which implies that Q(e, u, π) ≤ (2K + 4)(n − 1) + 1. Therefore, +Q(e, π) ≤ +� +u∈V +Q(e, u, π) ≤ n ((2K + 4)(n − 1) + 1) ≤ O(n2). +The main contribution of this section is to show that the expected number of EO(e, π) calls +over all permutations π is O(log2 n), which is formalized in the following lemma. +Lemma 5.6. For any edge e ∈ E′, we have Eπ[Q(e, ·, π)] = O(log2 n). +Assuming the correctness of Lemma 5.6, we can complete the proof of Theorem 5.4. +Proof of Theorem 5.4. +Ev,π[T(v, π)] = 1 +n Eπ +� � +v∈V +T(v, π) +� += 1 +n Eπ +� � +v∈V +� +e∈E′ +Q(e, v, π) +� += 1 +n Eπ +� � +e∈E′ +� +v∈V +Q(e, v, π) +� += 1 +n Eπ +� � +e∈E′ +Q(e, π) +� += 1 +n +� +e∈E′ +Eπ[Q(e, π)] = 1 +n +� +e∈E′ +O(log2 n) += 1 +nO(|E′| · log2 n) = O( ¯d · log2 n). +During the recursive calls to the edge oracle that starts from vertex v, the edges in the stack of +recursive calls create a trail. +Observation 5.7. Let S = (e1 = (v, u), e2, . . . , er) be the stack of recursive calls starting from +vertex v. Then (e1, e2, . . . , er) is a trail in G′. +10 + +Proof. Since in Line 2 of Algorithm 4, edge oracle only queries incident edges, (e1, e2, . . . , er) is a +walk. It remains to show that all edges are distinct. Suppose that ei = ej for some i < j which +implies π(ei) = π(ej). Since the edge oracle queries edges in decreasing order, we have π(ej) < π(ei) +which is a contradiction. +We direct the edges of the trail from v to the other endpoint. We call a trail that starts from +v on the graph with edge permutation π, a (v, π)-query-trail. For an edge e = (x, y), let ⃗e denote +the directed edge from x to y and +⃗ +e denote a directed edge from y to x. +Observation 5.8. Let ⃗P = (⃗e1, ⃗e2, . . . , ⃗ek) be a (v, π)-query-trail; then π(e1) > π(e2) > . . . > π(ek). +Proof. During the answering whether an edge is in RGMIS(G′, π), Algorithm 4 recursively calls on +edges with π values lower than the value of the current edge. Therefore, the stack of recursive calls +will be decreasing with respect to π values. +Let Q(⃗e, π) ⊆ Q(e, π) be the set of all query trails that end at ⃗e (with the same direction). +In what follows, we obtain a bound for the query complexity for ⃗e. We use this lemma to prove +Lemma 5.6. +Lemma 5.9. For any edge e, we have Eπ[Q(⃗e, π)] = O(log2 n). +Proof of Lemma 5.6. Since Q(e, π) = Q(⃗e, π) ∪ Q( +⃗ +e, π), by Lemma 5.9 we have +Eπ[Q(e, π)] ≤ Eπ[Q(⃗e, π)] + Eπ[Q( +⃗ +e, π)] = O(log2 n) + O(log2 n) = O(log2 n). +Given a permutation π and a trail ⃗P = (⃗e1, ⃗e2, . . . , ⃗ek), we define φ(π, ⃗P) to be another permu- +tation σ over the edges such that: +(σ(e1), σ(e2), . . . , σ(ek−1), σ(ek)) := (π(e2), π(e3), . . . , π(ek), π(e1)) +π(e′) = σ(e′) +∀e′ /∈ ⃗P +Given an edge ⃗e, by using the above mapping function we can construct a bipartite graph H +with two parts A and B such that each part has |E′|! vertices showing different permutations of +edges. For a permutation π ∈ A and a (v, π)-query-trail ⃗P that ends at ⃗e for some arbitrary vertex +v, we connect π in A to φ(π, ⃗P) in B. Note that by construction of H, deg(πA) = Q(⃗e, πA) for +all πA ∈ A, since we have a unique edge for each query-trail that ends at ⃗e with permutation πA. +Hence, in order to prove Lemma 5.6, it is sufficient to prove that EπA∼A[degH(πA)] = O(log2 n). +Let Q(⃗e, π) be the set of all query-trails for permutation π that ends at ⃗e. Let β = c log2 n for some +large c. We partition permutations into two sets of likely and unlikely permutations called L and +U as follows: +L := +� +π ∈ Π +��� +max +⃗P∈Q(⃗e,π) +|⃗P| ≤ β +� +U := Π \ L. +Likely permutations are those permutations that the longest query-trail ending at ⃗e has length +at most β and unlikely permutations are the remaining permutations. Let AL be the set of ver- +tices corresponding to the likely permutations in A and AU be the set of vertices corresponding +to the unlikely permutations. The intuition behind this partitioning is that the set of unlikely +permutations makes up a tiny fraction of all permutations which is formalized in Lemma 5.10. +11 + +Lemma 5.10. If c is a large enough constant, then we have |AU| ≤ |E′|!/n2. +Before proving Lemma 5.10, we introduce the parallel implementation of the greedy maximal +independent set. +Parallel Randomized Greedy Maximal Independent Set: Let G be a graph and π be a +permutation over its edges. In each iteration, we pick all vertices whose rank is less than all their +neighbors and remove all their neighbors. We denote the number of rounds in this algorithm until +G becomes empty as round complexity and we show it with ρ(G, π). +It is clear that the output of the parallel randomized greedy MIS is the same as RGMIS(G, π). +We have the following known result about the round complexity of parallel randomized greedy MIS. +Lemma 5.11 ([7, Theorem 3.5]). For a uniformly random chosen permutation π over edges of G, +we have ρ(G, π) = O(log2 n), with probability of at least 1 − 1 +n2 . +In order to use the above lemma, we need to show that for an unlikely permutation, the +round complexity is large and therefore, small fraction of permutations are unlikely as a result of +Lemma 5.11. +Claim 5.12. Let ⃗P be query-trail in G′ with permutation π. Then ρ(G′, π) ≥ ⌊| ⃗P| +2 ⌋. +Proof. Let ⃗P = (⃗e1, ⃗e2, . . . , ⃗ek) be a query-trail. By Observation 5.8, we have π(e1) > π(e2) > . . . > +π(ek), where ek is the last edge on the trail. Let ρ(e) show the round in which edge e is deleted by the +parallel algorithm. If we can show that for i < k−1, ρ(ei) > ρ(ei+2), then we have that ρ(e2) ≥ ⌊k +2⌋ +which completes the proof. We prove it using a contradiction. Assume that ρ(ei) ≤ ρ(ei+2) for +some 1 < i < k − 1. Note that ρ(ei+1) ≥ ρ(ei), otherwise, when ei+1 is deleted from the graph, +one of its corresponding ports that is shared with ei and ei+2 was occupied which implies that at +least one of ei and ei+2 should be deleted at the same time. Hence, in round ρ(ei), edge ei+1 is +still present in the graph. Therefore, ei is not a local minimum in round ρ(ei) and is deleted due +to presence of an edge e′ in the solution. Note that e′ ̸= ei+1 since ei+1 is not the minimum edge +because ei+2 is still in the graph. If e′ is only incident to ei, EO(ei−1, ·, π) should call EO(e′, ·, π) +before EO(ei, ·, π) since e′ is the local minimum in round ρ(ei) and therefore π(e′) < π(ei). If +e′ is incident to both ei and ei+1, EO(ei, ·, π) should call EO(e′, ·, π) before EO(ei+1, ·, π) since +e′ is local minimum at round ρ(ei) and therefore π(e′) < π(ei+1). In both cases, the edge oracle +terminates and will not query edge ei+2. Hence, the assumption that ρ(ei) ≤ ρ(ei+2) leads to a +contradiction and the proof is complete. +Now we are ready to prove Lemma 5.10. +Proof of Lemma 5.10. For each unlikely permutation π ∈ U, there exists a query-trail of length +larger than β. By Claim 5.12, we have ρ(G, π) ≥ ⌊ β+1 +2 ⌋. Since β = c log2 n, by choosing c large +enough and Lemma 5.11, we have that |U|/|Π| ≤ 1/n2. Therefore, |U| ≤ |E′|!/n2 which implies +that |AU| ≤ |E′|!/n2 since AU represents vertices that correspond to unlikely permutations. +Next, we show that each vertex πB ∈ B, has at most β neighbors between likely permutations +in part A in bipartite graph H. +Lemma 5.13. Let πY be a vertex in Y . Then πY has most β neighbors in XL. +12 + +Before proving this lemma, we show how we can prove Lemma 5.9 using Lemma 5.10 and +Lemma 5.13. +Proof of Lemma 5.9. Note that by Observation 5.5, degree of each vertex πA ∈ A is at most O(n2). +Combining Lemma 5.10, we have +E(AU, B) ≤ |E′|!/n2 · O(n2) ≤ O +� +|E′|!). +Moreover, by Lemma 5.13, each vertex πB ∈ B has at most O(β) neighbors in AL. Since H is +a bipartite graph, E(AL, B) ≤ O(β) · |AL|. Therefore, sum of degrees of all vertices in A is at most +E(AL, B) + E(AU, B) ≤ O(β) · |AL| + O(|E′|!) ≤ O(β · |E′|!). +For a random vertex in A, the expected degree is O(β·|E′|!) +|E′|! += O(|E′|). Combining with β = +c log2 n and deg(πA) = Q(⃗e, πA) completes the proof. +The rest of this section, we prove Lemma 5.13. Before proving Lemma 5.13, we prove that if +two different query-trails that are mapped to two different permutations of AL to πB ∈ B by φ, +the shorter query-trail must be subgraph of the longer one. +Lemma 5.14. Let π and π′ be two different permutations, and ⃗P and ⃗P ′ be (v, π)- and (v′, π′)- +query-trail, respectively, that both end at edge ⃗e. If φ(π, ⃗P) = φ(π′, ⃗P ′) and |⃗P| ≥ |⃗P ′|, then ⃗P ′ is a +subgraph of ⃗P. +We prove this lemma by series of observations and claims. Let ⃗P = (⃗ek, . . . , ⃗e1) and ⃗P ′ = +(⃗er′, . . . , ⃗e1′) such that e = e1 = e′ +1. If ⃗P ′ is not a subgraph of ⃗P, then it must branch after an edge +⃗eb +′. This means that ⃗ei = ⃗ei′ for i ≤ b and +⃗ +eb+1 ̸= +⃗ +eb+1 +′. Note that +⃗ +eb+1 and +⃗ +eb+1 +′ can be copy of +the same edge. +Observation 5.15. Let π be a random permutation over E′. For a (u, π)-query-trail, if f and f′ +are two consecutive edges in the trail, then (f, f′) is a conflicting pair. +Proof. Since the edge oracle calls EO(f′, ·, π) in EO(f, ·, π), (f, f′) must be a conflicting pair. +Observation 5.16. Let f1, f2, f3 be three different edges incident to some vertex u and let π be a +random permutation over E′. Let ⃗P1 be a (x, π)-query-trail that calls EO(f3, ·, π) in EO(f1, ·, π). +Also, let ⃗P2 be a (y, π′)-query-trail that calls EO(f3, ·, π′) in EO(f2, ·, π′). Then (f1, f2) is a con- +flicting pair. +Proof. By Observation 5.15, (f1, f3) is a conflicting pair. Assume that both f1 and f3 occupied port +ui. Moreover, since (f2, f3) is a conflicting pair, then f2 is also occupying ui. Therefore, (f1, f2) is +a conflicting pair. +Observation 5.17. π(eb) = π′(eb+1). +Proof. Since +⃗ +eb+1 is not in ⃗P ′, we have that φ(π′, ⃗P ′)(eb+1) = π′(eb+1). Also, φ(π, ⃗P)(eb+1) = π(eb) +since φ(π, ⃗P) shifts edges of the trail ⃗P by one. Given that permutation φ(π, ⃗P) is equal to φ(π′, ⃗P ′), +we have π(eb) = π′(eb+1). +Without loss of generality, we can assume that π(eb) ≤ π′(eb) since we did not make any +difference between π and π′ until this point. +13 + +Observation 5.18. π′(eb+1) < π′(eb). +Proof. By combining Observation 5.17, our assumption that π(eb) ≤ π′(eb), and the fact that π′ is +a permutation, we have that π′(eb+1) < π′(eb). +Claim 5.19. If π(f) < π(eb) or π′(f) < π(eb) for some edge ⃗f, then π(f) = π′(f). +Proof. There are five different possible cases for f: +• ⃗f /∈ ⃗P ∪ ⃗P ′: Since φ only changes the edge on the query-trail and φ(π, ⃗P) = φ(π′, ⃗P ′), we +have π(f) = π′(f). +• ⃗f ∈ {⃗e1, . . . ,⃗eb−1}: Since φ(π, ⃗P)(ei+1) = φ(π′, ⃗P ′)(ei+1) for 1 ≤ i < b, we have π(ei) = π′(ei). +Hence, π(f) = π′(f). +• ⃗f = ⃗eb: In this case, condition π(f) < π(eb) does not hold since π(f) = π(eb). +Also, +π′(f) = π′(eb) ≥ π(eb). Therefore, condition π′(f) < π(eb) does not hold. +• ⃗f ∈ {⃗eb+1, . . . ,⃗ek}: By Observation 5.8, we have that π(f) > π(eb). Therefore, condition +π(f) < π(eb) does not hold. +Let ⃗f = ⃗ei for i > b. +Since φ(π, ⃗P) = φ(π′, ⃗P ′), we have +π′(f) = π(ei−1) ≥ π(eb). Therefore, none of the conditions in the claim statement holds. +• ⃗f ∈ { ⃗ +eb+1 +′, . . . , ⃗er′}: By Observation 5.8, we have that π′(f) > π′(eb). Combining by our +assumption that π′(eb) ≥ π(eb), we have π′(f) ≥ π(eb). +Let ⃗f = ⃗ei′ for i > b. +Since +φ(π, ⃗P) = φ(π′, ⃗P ′), we have that π(f) = π′(e′ +i−1) ≥ π′(eb) ≥ π(eb). Therefore, none of the +conditions in the claim statement holds. +The proof is thus complete. +Claim 5.20. eb+1 ∈ RGMIS(G′, π′). +Proof. We prove the claim by contradiction. Assume that eb+1 /∈ RGMIS(G′, π′). Hence, there +exists an edge f which is incident to eb+1 such that π′(f) < π′(eb+1). Thus, EO(eb+1, ·, π′) will +recursively call EO(f, ·, π′). Let f be incident to ei and ei+1 for i ∈ {b, b + 1}. In the query-trail +⃗P, EO(ei+1, ·, π) calls EO(ei, ·, π). Therefore, using the Observation 5.16, we have that (f, ei) is a +conflicting pair. Note that by Observation 5.17, we have π′(f) < π(eb). Hence, π(f) = π′(f) < π(eb) +by Claim 5.19. Since both permutations are identical for ranks lower than π(eb), edge f must appear +in RGMIS(G′, π) and the query-trail ⃗P is not a valid query-trail since EO(ei, ·, π) terminates upon +calling EO(f, ·, π) (see Figure 2). +Proof of Lemma 5.14. We prove that query-trail ⃗P ′ is not a valid (v, π′)-query-trail. Note that +by Observation 5.18, EO(e′ +b+1, ·, π′) calls EO(eb+1, ·, π′) before EO(eb, ·, π′). Thus, by Claim 5.20, +EO(eb+1, ·, π′) will return True and execution of EO(e′ +b+1, ·, π′) terminates at this point. Therefore, +query-trail ⃗P ′ is a subgraph of query-trail ⃗P. +Now we are ready to complete the proof of Lemma 5.13. +Proof of Lemma 5.13. For each edge between πA ∈ AL and πB ∈ B in graph H, we write a label +χ(πA, πB) on the edge which is equal to the length of the query-trail corresponding to this edge in +H. By Lemma 5.14, all the labels for edges of a fixed vertex πB ∈ B that are incident to AL should +be different. Moreover, by the definition of likely permutations, all query-trails of permutation AL +have length less than or equal to β. Thus, each vertex πB ∈ B has at most β neighbors in AL. +14 + +0 +0 +0 +0 +1 +1 1 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +1 +1 1 +1 +0 +0 +0 +0 +1 +0 +Figure 2: Illustration of proof of Claim 5.20. The highlighted blue trails show query-trails ⃗P and +⃗P ′. Query-trail ⃗P is not valid since EO(ei, ·, π) terminates upon calling EO(f, ·, π). +6 +Our Estimator for Maximum Path Cover +In this section, we use the oracle of the previous section to estimate the number of edges in the +output of Algorithm 2. In Section 5, we provide a lower bound on the number of recursive calls +to our local query process. Note that this bound does not necessarily imply the same running +time algorithm. For example, if we generate the whole permutation over all copies of edges before +running the algorithm, it takes Θ(m) which is no longer sublinear. Using by now standard ideas of +the literature, we show in Appendix A how we can implement the query process in almost the same +running time (multiplied by a polylogarithmic factor) which is formalized in the following lemma. +Lemma 6.1. There exists a data structure that given a graph G in the adjacency list format, +(implicitly) fixes a random permutation π over its edges. Then for any vertex v, the data structure +returns the degree of vertex v in the subgraph P produced by Algorithm 2 according to a random +permutation π. Each query v to the data structure is answered in ˜O(T(v, π)) time w.h.p. where +T(v, π) is as defined in Section 5. +Note that in our local query process, we need access to the adjacency list of weight-one edges. +So the challenge that arises here is how to estimate the size of the output of Algorithm 2 in +the adjacency matrix model. We present a reduction from adjacency matrix to adjacency list that +appeared in the literature [2]. In this reduction, each query to the adjacency list can be implemented +with O(1) queries to the adjacency matrix and still we are able to estimate the maximum path +cover with some additive error. +Let γ = 16Kn. We construct a graph ˆG = (V ˆG, E ˆG) for weight-one edges of graph G as follows: +• V ˆG is the union of V1, V2 and U1, U2, . . . , Un such that: +– V1 and V2 are two copies of the vertex set of the original graph G. +15 + +– Ui is a vertex set of size γ for each i ∈ [n]. +• We define the edge set such that degree of each vertex is in {1, n, n + γ}: +– Degree of each vertex v ∈ V1 is n. The i-th neighbor of v is the i-th vertex in V1 if +(v, i) ∈ E, otherwise its i-th neighbor is the i-th vertex in V2 for i ≤ n. Note that graph +(V1, EH ∩ (V1 × V1)) is isomorphic to G. +– Degree of each vertex v ∈ V2 is n + γ. The i-th neighbor of v is the i-th vertex in +V2 if (v, i) ∈ E, otherwise, its i-th neighbor is the i-th vertex in V1 for i ≤ n. For all +n < i ≤ n + γ, the i-th neighbor of v is i-th vertex in Uv. +– Degree of each vertex u ∈ Ui is one for i ∈ [n]. The only neighbor of u is the i-th vertex +of V2. +By the construction of ˆG, the only neighbor of v ∈ �n +i=1 Ui can be determined without any +query to the adjacency matrix. Also, the i-th neighbor of each vertex in V1 ∪ V2 can be determined +with one query. +Observation 6.2. For each vertex v ∈ V ˆG and i ∈ [deg ˆG(v)], the i-th neighbor of vertex v can be +determined using at most one query to the adjacency matrix. +Fix a vertex v ∈ V2. When we run Algorithm 2, intuitively with high probability the first +edge that is incident to v and occupies port v0 is between v and u ∈ Uv. Furthermore, with high +probability the first two edges that are incident to v and occupies port v1 are between v and u ∈ Uv. +A vertex v ∈ V2 is an abnormal vertex if the above properties do not hold for v. Let R ∈ V2 be the +set of abnormal vertices. In the following observation, we show that for each vertex v ∈ V2 \ R, all +incident edges of v in the output of Algorithm 2 are between v and vertices of Uv. +Claim 6.3. Eπ |R| ≤ n/(4K) +Proof. Fix a vertex v ∈ V2. For a random permutation over copies of edges of ˆG, the first incident +edge to v that occupies port v0 is between v and Uv with a probability of at least +(K+1)γ +(n+γ)(K+1) ≥ 1− 1 +8K . +Moreover, the first two edges that occupy v1 are between v and Uv with probability of at least +γ(γ−1) +(n+γ)(n+γ−1) ≥ 1 − +1 +8K . +Since both events are independent, the probability of v not being an +abnormal vertex is at least +� +1 − 1 +8K +�2 +≥ 1 − 1 +4K , +which implies Eπ |R| ≤ n/(4K). +Claim 6.4. For each v ∈ V2 \ R, all incident edges of v in the output of Algorithm 2 are between +v and vertices of Uv. +Proof. By definition of an abnormal vertex, let the first edge in the permutation incident to v be +between v and w ∈ Uv which occupies v0. Since all copies of edges incident to w are between v +and w, this edge will be added to the solution of Algorithm 2. Moreover, we know that the first +two edges that are incident to v and occupy port v1 are between v and Uv. Let these two edges +be (v, u1) and (v, u2) where u1, u2 ∈ Uv. Note that the only way that (v, u1) is not added to the +solution of Algorithm 2 is when u1 = w. In this case, since there is only one copy for each edge +that occupied port v1, then u2 ̸= w. Therefore, Algorithm 2 adds (v, u2) to its output if it has not +added (v, u1). Since both ports of v are occupied in this case, all incident edges of v in the output +of Algorithm 2 are between v and vertices of Uv. +16 + +Observation 6.5. Let P be the output of Algorithm 2 on ˆG. Then +1 +2ρ( ˆG[V1 ∪ R]) ≤ E |P ∩ (V1 ∪ R) × (V1 ∪ R)| ≤ (1 + 2 +K ) · ρ( ˆG[V1 ∪ R]). +Proof. By Claim 6.4, if we run Algorithm 2 on ˆG, for any vertex v ∈ V2 \ R, all incident edges of v +in the output are between v and Uv. Hence, none of the edges between V2 \ R and V1 ∪ R will be +added to the output of Algorithm 2. Since, the permutation over edges of V1 ∪ R is uniformly at +random, by Lemma 4.5, we obtain the claimed bound. +In the above sequence of observations, we show that there are few abnormal vertices in V2, +which implies that most of the incident edges to vertices of V1 in the output of Algorithm 2 are in +ˆG[V1] (only those between V1 and R violate this property). Therefore, a natural way to estimate +the number of edges in the output of Algorithm 2 on G, is to estimate the number of edges in +ˆG[V1] in the output of Algorithm 2 on ˆG. With this intuition in mind, we need to bound the query +complexity of the algorithm for a random vertex in V1. +Claim 6.6. Let v be a random vertex in V1 and π be a random permutation over edges of graph +that is created by copying E ˆG according to Algorithm 2. Then +Ev∼V1,π[T(v, π)] = ˜O(n). +Proof. By Theorem 5.4, we have that +Ev∼V ˆ +G,π[T(v, π)] = O(K · |E ˆG| +|V ˆG| +· log2 |V ˆG|). +Summing over all vertices in V ˆG, we obtain +� +v∈V ˆ +G +Eπ[T(v, π)] = O(K · |E ˆG| · log2 |V ˆG|) = ˜O(n2), +since |V ˆG| = O(n2), K = O(1/ε), and |E ˆG| = O(n2). Therefore, for a random vertex in V1, we get +Ev∼V1,π[T(v, π)] ≤ +� +� � +v∈V ˆ +G +Eπ[T(v, π)] +� +� /|V1| = ˜O(n). +Lemma 6.7. Let ˜ρ be the output of Algorithm 5 on input graph G. With high probability, +�1 +2 − 1 +K +� +· ρ(G) − n +K ≤ ˜ρ ≤ ρ(G), +where K is the parameter which is defined in Algorithm 2. +Proof. Let ˆP be the set of edges outputted by Algorithm 2 on ˆG with both endpoints in V1. By +Lemma 4.5, we have that E | ˆP| ≤ (1 + 2 +K ) · ρ(G). Furthermore, by Claim 6.3 and the fact that the +degree of each vertex in the output of Algorithm 2 is at most two, in the output of Algorithm 2 on +ˆG[V1∪R] we have at most n/(2K) edges with one endpoint in R. Hence, combining with Lemma 4.5 +and Observation 6.5 we get +1 +2ρ(G) − n +2K ≤ E | ˆP| ≤ (1 + 2 +K ) · ρ(G). +(2) +17 + +Algorithm 5: Final algorithm for maximum path cover. +1 Let ˆG = (V ˆG, E ˆG) as described above. +2 r ← 192 · K2 · log n. +3 Sample r vertices u1, u2, . . . , ur uniformly at random from V1 with replacement. +4 Sample r ports p1, p2, . . . , pr uniformly at random from {0, 1}. +5 Run vertex oracle for each ui and let Xi be the indicator if port upi +i is occupied with an edge +in ˆG[V1] in output of Algorithm 2. +6 Let X = � +i∈[r] Xi and f = X/r. +7 Let ˜ρ = +K +2(K+2) · (f · n − +n +4K ). +8 return ˜ρ +Since each edge in the output of Algorithm 2 counted twice in Algorithm 5, we have +E[Xi] = Pr[Xi = 1] = 2 E | ˆP| +n +, +and, +E[X] = 2r E | ˆP| +n +. +(3) +Since X is sum of r independent random variables, by Chernoff bound (Proposition 3.2) we get +Pr[|X − E[X]| ≤ +� +6 E[X] log n] ≤ 2 exp +� +−6 E[X] log n +3 E[X] +� += 2 +n2 . +Combining fn = Xn/r and the above bound, with probability of at least 1 − 2/n2 we have +fn ∈ n(E[X] ± +� +6 E[X] log n) +r += n E[X] +r +± +� +6n2 E[X] log n +r2 += 2 E | ˆP| ± +� +12n E | ˆP| log n +r +(By (3)) += 2 E | ˆP| ± +� +n E | ˆP| +16K2 +(Since r = 192 · K2 · log n) +∈ 2 E | ˆP| ± n +4K +(Since E | ˆP| ≤ n). +Since, ˜ρ = +K +2(K+2) · (f · n − +n +4K ), hence +K +K + 2 +� +E | ˆP| − n +2K +� +≤ ˜ρ ≤ +K +K + 2 · E | ˆP|. +Combining with (2), implies the claimed bound. +Theorem 6.8. Given an adjacency matrix access for input graph G, there exists a randomized +algorithm that w.h.p. runs in �O(n) time and produces an estimate ˜ρ, such that +�1 +2 − ε +� +· ρ(G) − εn ≤ ˜ρ ≤ ρ(G). +18 + +Proof. Let K = +1 +ε and ˜ρ be the output of Algorithm 5 on G. +In Algorithm 5, by combining +Lemma 6.1 and Claim 6.6, the running time for each sample is ˜O(n). Since the number of samples +is r = 192K2 log n, and K is a constant, the total running time of the algorithm is ˜O(n). Moreover, +by Lemma 6.7 we get the approximation ratio in the statement. +7 +Our Estimator for (1,2)-TSP +In this section, we use the algorithm for estimating the size of maximum path cover as a black +box to estimate the size of (1,2)-TSP. First, note that if there is no Hamiltonian cycle with weight +one edges of the graph, then the set of weight-one edges of the graph (1,2)-TSP is a solution for +maximum path cover of graph G. Also, in the case that there exists a Hamiltonian cycle, then the +size of maximum path cover is n − 1. Moreover, if P ∗ is the maximum path cover of a graph G, +then it is possible to create a TSP by connecting these paths using edges with weight two. This +intuition helps to formalize the following observation. +Observation 7.1. Let τ(V ) be the cost of (1,2)-TSP of graph G = (V, E). Then, we have +2n − ρ(G) − 1 ≤ τ(V ) ≤ 2n − ρ(G). +Now we are ready to present the final algorithm for estimating (1,2)-TSP. +Algorithm 6: Final algorithm for (1,2)-TSP. +1 Construct ˆG = (V ˆG, E ˆG) implicitly as desribed in Section 6. +2 Let ˜ρ be the output of Algorithm 5 on ˆG. +3 ˜τ = 2n − ˜ρ +4 return ˜τ +Lemma 7.2. Let ˜τ be the output of Algorithm 6 and τ(V ) be the cost of (1,2)-TSP of graph +G = (V, E). With high probability, +τ(V ) ≤ ˜τ ≤ (3 +2 + 4 +K ) · τ(V ), +where K is the parameter which is defined in Algorithm 2. +Proof. By Observation 7.1, we have 2n − ρ(G) − 1 ≤ τ(V ) ≤ 2n − ρ(G). Algorithm 6 outputs +˜τ = 2n − ˜ρ as the estimate, where ˜ρ is the output of Algorithm 5. Hence, by Lemma 6.7, we have +2n − ˜ρ ≥ 2n − ρ(G). Also, by Lemma 6.7, we have +2n − ˜ρ ≤ 2n − (1 +2 − 1 +K ) · ρ(G) + n +K +≤ 3n − 3ρ(G) +2 ++ 4n +K − 2ρ(G) +K +− 1 +(Since ρ(G) < n) +≤ (3 +2 + 4 +K )(2n − ρ(G) − 1) +(Since K ≪ n) +≤ (3 +2 + 4 +K ) · τ(V ) +(Since τ(V ) = 2n − ρ(G)), +which completes the proof. +19 + +Theorem 7.3. Let τ(V ) be the cost of (1,2)-TSP of graph G = (V, E). For any ε > 0, there exists +an algorithm that estimate the cost of (1,2)-TSP, ˜τ, such that +τ(V ) ≤ ˜τ ≤ (3 +2 + ε) · τ(V ), +w.h.p in ˜O(n) running time. +Proof. We choose K = 4 +ε. By Lemma 7.2, if ˜τ is the output of Algorithm 6, we get +τ(V ) ≤ ˜τ ≤ (3 +2 + ε) · τ(V ). +Also, since the running time of Algorithm 6 is the same as the running time of Algorithm 5, by +Theorem 6.8, the total running time is ˜O(n), which completes the proof. +8 +Our Estimator for Graphic TSP +In this section, we use our algorithm for estimating the size of maximum path cover to estimate +the size of graphic TSP. In a recent work, Chen et al. [8] showed that it is possible to obtain a +(27/14)-approximate algorithm for graphic TSP by estimating the matching size and the number +of biconnected components in the graph. Since the size of graphic TSP is at most 2n (the cost of +MST is n − 1), they proved that if a graph has large matching and a few biconnected components, +the cost of graphic TSP is significantly lower than 2n. Since estimating the number of biconnected +components is not an easy task in sublinear time, they use a proxy quantity that can be estimated +in sublinear time. +We show that if we use our estimator for maximum path cover as a black-box instead of matching +estimator in algorithm of [8], we can improve the approximation ratio to 19/10. Moreover, we +show that we can estimate the number of bridges in ˜O(n). We exploit this estimation for further +improvement to get a 11/6-approximate algorithm for graphic TSP. +Chen et al. [8] introduced the following definition of bad vertex as a proxy for estimating the +number of biconnected components. +Definition 8.1 (Bad Vertex). We say a vertex v ∈ V is a bad vertex, if one of the following holds: +• degree of v is 1, +• v is a cut vertex with degree 2. +In the following series of lemmas, we bound the cost of graphic TSP based on the size of +maximum path cover and number of bad vertices. Almost all the steps of this part are similar to +the algorithm for graphic TSP of [8] — except the path cover subroutine that we use instead of +maximal matching subroutine. We restate some of the useful lemmas to achieve the approximation +bound that the black-box algorithm can get, and in the next subsection we improve this bound. +First, we prove that if the size of the maximum path cover is small, the cost of graphic TSP is +bounded away from n. +Claim 8.2. If the size of maximum path cover of graph G is at most ρ, then the cost of graphic +TSP is at least 2n − ρ. +20 + +Proof. Let (v0, v2, . . . , vn = v0) be the optimal graphic TSP of graph G. Note that the subgraph +induced by weight-one edges of this cycle is a solution for path cover. Hence, at most ρ edges in +cycle (v0, v2, . . . , vn = v0) have weight one. All the remaining edges have a weight of at least two +which implies the claimed bound. +Furthermore, the following lemma from [8], provides a lower bound for a graphic TSP of graph +in terms of number of bad vertices. +Lemma 8.3 ([8, Lemma 2.8]). If the number of bad vertices of graph G is at least β, then the cost +of graphic TSP is at least n + β − 2. +Chen et al. [8] showed that in a biconnected graph, if there exists a matching of large size, the +cost of graphic TSP is significantly smaller than 2n. +Lemma 8.4 ([8, Lemma 2.11]). Let G be a graph and M′ be a matching that none of its edges is +bridge. Then the cost of graphic TSP is at most 2n − 2 +3|M′|. +We now upper bound the cost of graphic TSP in terms of size of maximum path cover and +number of bad vertices. +Lemma 8.5. If the size of maximum path cover of graph G is ρ(G) and it has β bad vertices, then +the cost of graphic TSP is at most 2n − 1 +5(ρ(G) − 2β). +Proof. Let l be the number of non-trivial biconnected components and M′ be a maximum matching +in graph G that none of its edges is bridge. Also, let B be the number of bridges in G. By the proof +of Lemma 2.9 of [8], the cost of graphic TSP is at most min{2n − 2 +3|M′|, 2n − l}. Note that there +are at least ρ(G) − B edges of the maximum path cover that are not bridge. Since all non-bridge +edges of the maximum path cover are still union of several disjoint paths, there exists a matching +with size of at least half of the edges of these paths. Hence, there exist a matching of size at least +1 +2(ρ(G) − B). On the other hand, in the proof of the same lemma, they showed that l ≥ B +2 − β +which implies that the cost of graphic TSP is at most +min +� +2n − 2 +3|M′|, 2n − l +� +≤ min +� +2n − 1 +3(ρ(G) − B), 2n − B +2 + β +� +. +There are two possible cases: +• If B ≤ 2 +5ρ(G) + 6 +5β, then we have +2n − 1 +3(ρ(G) − B) ≤ 2n − 1 +3(ρ(G) − 2 +5ρ(G) − 6 +5β) = 2n − 1 +5(ρ(G) − 2β). +• If B > 2 +5ρ(G) + 6 +5β, then we have +2n − B +2 + β ≤ 2n − 1 +5ρ(G) − 3 +5β + β = 2n − 1 +5(ρ(G) − 2β). +Therefore, the cost of graphic TSP is at most +min +� +2n − 1 +3(ρ(G) − B), 2n − B +2 + β +� +≤ 2n − 1 +5(ρ(G) − 2β). +21 + +Now we are ready to introduce the first algorithm for estimating the cost of graphic TSP, which +uses our maximum path cover subroutine instead of the matching subroutine as a black-box. In +Algorithm 7, we first estimate the size of the maximum path cover and the number of bad vertices +of the graph and report the graphic TSP cost in terms of the two estimations. The subroutine used +for counting number of bad vertices is similar to the one in section 2.2 of [8]. +Lemma 8.6 ([8]). Let β be the number of bad vertices. For any constant ε > 0, there exists an +algorithm that w.h.p estimates the number of bad vertices ˜β, such that β ≤ ˜β ≤ β + εn, in ˜O(n) +running time. +Algorithm 7: First algorithm for graphic TSP. +1 Construct ˆG = (V ˆG, E ˆG) implicitly as desribed in Section 6. +2 Let ˜ρ be the output of Algorithm 5 on ˆG. +3 Let ˜β be the estimate of number of bad vertices. +4 ˜τ = 2n − 1 +5(˜ρ − 2˜β) +5 return ˜T +Lemma 8.7. Let ˜τ be the output of Algorithm 7 and τ(V ) be the cost of graphic TSP of graph +G = (V, E). With high probability, +τ(V ) ≤ ˜τ ≤ (19 +10 + 1 +K ) · τ(V ), +where K is the parameter which is defined in Algorithm 2. +Proof. Let β be the number of bad vertices. By Lemma 6.7 and Lemma 8.6 ˜ρ ≤ ρ(G) and β ≤ ˜β. +Hence, we have τ(V ) ≤ ˜τ. +By Lemma 6.7 and Lemma 8.6, we can estimate ρ(G) and β such that +� 1 +2 − 1 +K +� +ρ(G) − n +K ≤ ˜ρ +and ˜β ≤ β + n +K , if we choose ε = 1 +K . Thus, we have +˜τ ≤ 2n − 1 +5 +� +(1 +2 − 1 +K ) · ρ(G) − n +K − 2(β + n +K ) +� +≤ 2n − 1 +5(ρ(G) +2 +− 2β) + 4n +5K . +(Since ρ(G) ≤ n). +On the other hand, assume that the approximation ratio that the algorithm obtains is α + 1/K for +some α ≤ 2. Thus, we get +(α + 1 +K ) · τ(V ) ≥ α · τ(V ) + n +K +(Since τ(V ) ≥ n) +≥ α · max{2n − ρ(G), n + β − 2} + n +K +(By Claim 8.2 and Lemma 8.3) +≥ α · max{2n − ρ(G), n + β} + n +K − 4. +So in order to show that ˜τ ≤ (α + 1 +K ) · τ(V ), it is sufficient to show that +2n − 1 +5(ρ(G) +2 +− 2β) + 4n +5K ≤ α · max{2n − ρ(G), n + β} + n +K − 4. +22 + +If n is large enough, then we have 4n +5K ≤ n +K − 4, which implies that we need to prove +2n − 1 +5(ρ(G) +2 +− 2β) ≤ α · max{2n − ρ(G), n + β}. +Now, let ρ(G) = xn and β = yn for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. To obtain α, it suffices to solve the +following program +maximize +α +subject to +2− 1 +5 ( x +2 −2y) +max{2−x,1+y} ≤ α, +0 ≤ x ≤ 1, +0 ≤ y ≤ 1. +This is a constant size program that can be easily solved; the solution is 19/10.4 This completes +the proof. +Theorem 8.8. Let τ(V ) be the cost of graphic TSP of graph G = (V, E). For any ε > 0, there +exists an algorithm that estimate the cost of graphic TSP, ˜τ, such that +τ(V ) ≤ ˜τ ≤ (19 +10 + ε) · τ(V ), +w.h.p in ˜O(n) running time. +Proof. Let ˜τ be the output of Algorithm 7. If we choose K = 1 +ε, then by Lemma 8.7, we have +τ(V ) ≤ ˜τ ≤ (19 +10 + ε) · τ(V ). +Also, by Theorem 6.8 and Lemma 8.6, estimating ˜ρ and ˜β can be done in ˜O(n). +9 +Further Improvement for Graphic TSP +In this section, we design an algorithm to estimate the number of bridges in given graph G. +Equipped with this tool, we are able to estimate the number of non-bridge edges in the path +cover which helps to improve the approximation ratio. Before describing the techniques for esti- +mating the number of bridges, we prove the following lemma that provides a lower bound on the +cost of graphic TSP based on the number of bridges in the graph. +Claim 9.1. If the number of bridges in the graph G is at least B, then the cost of the graphic TSP +is at least n + B. +Proof. Since the metric in the graphic TSP is corresponding to the shortest path distances in graph +G, then a TSP tour is corresponding to a closed walk that contains all vertices. Thus, each bridge +should be crossed at least two times in this walk in order for the walk to be closed and cover all +vertices. Therefore, the cost of graphic TSP is at least n + B. +4See e.g. this WolframAlpha link. +23 + +In the following series of lemmas, first, we prove that there are a few bridges that both of their +endpoints have a high degree and then we show an efficient way to estimate the number of bridges +that have at least one endpoint with a low degree. Combining the above arguments is the main +idea to estimate the number of bridges. +Lemma 9.2. For any integer c ≥ 2, there exists at most 2n +c bridges that both of their endpoints +have a degree larger than c. +Proof. Let B be the set of bridges that both of their endpoints have a degree larger than c. We +construct a tree, TB, with edge set equal to B such that each vertex of TB corresponds to a component +of vertices that are compressed to a single vertex. We construct TB iteratively. In the beginning, we +consider the bridge-block tree of the original graph. In each step, if there exists a bridge e = (u, v) +(note that u and v are vertices of the tree and corresponding to a set of vertices of the original +graph) such that at least one of its endpoints has a degree less than or equal to c, we merge u with +v and add all edges of u to v. We continue this process until there is no bridge with an endpoint +of degree less than or equal to c. +Now, we prove that |B| ≤ 2n +c . Let xv denote the number of vertices in the original graph that +are compressed to vertex v ∈ TB. We remove vertices of TB one by one until there is no vertex in +the tree. At each step, we remove a leaf v ∈ TB and at the end when only one vertex is remaining, +we remove that vertex. Let yv be the number of incident edges to v in TB that are removed before +removing v. At the time that we are removing leaf v, we have xv +yv +1 ≥ c+1, since the endpoint +of the leaf that is the component of v has at most xv incident edges in the same component in the +original graph, yv incident edges to the other components that are removed before, and there is +only one remaining incident edge to other components (the other endpoint of the leaf). Thus, +� +v∈TB +xv ≥ +� +v∈TB +(c − yv) = (|B| + 1)c − +� +v∈TB +yv. +(4) +Since vertices of each component are disjoint, we have � +v∈TB xv = n. +Moreover, we have +� +v∈TB yv = |B| since each edge of B counted when one of its endpoints is deleted from the tree. +Combining above bounds and inequality (4), we have +n = +� +v∈TB +xv ≥ (|B| + 1)c − |B| +Therefore, +|B| ≤ n − c +c − 1 ≤ 2n +c , +where the last inequality holds for sufficiently large n. +Lemma 9.3. Let c ≥ 2 be a constant and u is a vertex such that deg(u) ≤ c. Then we can test if +each of incident edges of u is a bridge in O(n) total running time. +Proof. We can query all neighbors of u in O(n). Assume that {v1, v2, . . . , vr} are neighbors of u for +r ≤ c. Now we divide the vertices of the graph except u into r sets V1, V2, . . . , Vr. For each vertex +w ̸= u, we query the distance of w to all {v1, v2, . . . , vr}. Let vi be the closest one to w (if there is +a tie, choose the one with the lowest index). Then we put w in Vi. Note that since c is a constant +and r ≤ c, this step can be done in O(n). +Now we claim that (u, vj) is a bridge iff the following conditions hold: +24 + +• For each w ∈ Vj and i ̸= j, d(w, vi) − d(w, vj) = 2. +• For each w ∈ Vi such that i ̸= j, d(w, vj) − d(w, vi) = 2. +Suppose that e = (u, vj) is a bridge. Since removing e creates two connected components Cu +and Cvj, all vertices in Cvj (resp. Cu) have a closer distance to vj (resp. u). In other words, all +shortest paths between w ∈ Vj to vi for i ̸= j, cross edges (vj, u) and (u, vi). In addition, all the +shortest paths between w ∈ Vi and vj for i ̸= j, cross edges (vj, u) and (u, vi). Therefore, both +conditions hold. +Now suppose that e = (u, vj) is not a bridge. In this case, there must be an edge between Vj +and at least one of Vi as otherwise, Vj will be disconnected from the rest of the graph by removing +e. Without loss of generality, assume that this edge is (w, w′) such that w ∈ Vj, w′ ∈ Vi, and i ̸= j. +Also, w.l.o.g., we assume d(w, vj) ≤ d(w′, vi). Since there is an edge between w and w′, we have +d(w′, vj) ≤ 1 + d(w, vj) ≤ 1 + d(w′, vi), which contradicts the conditions. +To test whether the conditions hold, we need to query the distance of each vertex to all +{v0, v1, . . . , vr} which can be done in O(n) in total since r is a constant. +Lemma 9.4. Let B be the number of bridges in graph G. For any ε > 0, there exists an algorithm +that outputs an estimate ˜B in ˜O(n) such that B ≤ ˜B ≤ B + εn. +Proof. By Lemma 9.2, there are at most εn +2 bridges with both endpoints have degree larger than 4 +ε. +Let ˆB be the number of bridges that at least one of their endpoint has degree of at most 4 +ε. Thus, +B − εn +2 ≤ ˆB ≤ B. +(5) +We sample r = 256 · ε−4 · log n vertices uniformly at random with replacement. Let u be the +i-th sampled vertex. If the degree of the vertex is larger than 4 +ε, we let Xi = 0. Otherwise, let +{v1, v2, . . . , vk} be the neighbors of u where k ≤ 4 +ε. By Lemma 9.3, we can test if each of the +incident edges of u is bridge in O(n) total running time. For each edge (u, vj) if deg(u) < deg(vj) +or deg(u) = deg(vj) and index of u is smaller than vj, we test if the edge is bridge or not. Let Xi +show the number of successful tests for incident edges of u. Note that in the above algorithm, each +bridge with low-degree endpoints only counted once. +Let ¯X = (�r +i Xi)/r and n ¯X + 3εn +4 +be our final estimate of the number of bridges. Hence, +E[ ¯X] = ˆB/n. Since ¯X is the average of r independent random variables such that 0 ≤ Xi ≤ 4/ε, +by Hoeffding’s inequality (Proposition 3.3) we obtain +Pr +��� ¯X − E[ ¯X] +�� ≥ ε +4 +� +≤ 2 exp +� +− rε4 +128 +� += 2 +n2 , +where the last inequality follows from r = 256 · ε−4 · log n. Therefore, with probability of 1 − 2 +n2 , +n ¯X ∈ n E[ ¯X] ± nε +4 += ˆB ± nε +4 +(Since E[ ¯X] = ˆB/n). +Combining above range and inequality (5), we get +B ≤ n ¯X + 3εn +4 +≤ B + εn. +Since the number of sampled vertices is r = 256 · ε−4 · log n, the total running time is ˜O(n). +25 + +Now we are ready to introduce the improved algorithm for graphic TSP. +Algorithm 8: Second algorithm for graphic TSP. +1 Construct ˆG = (V ˆG, E ˆG) implicitly as described in Section 6. +2 Let ˜ρ be the output of Algorithm 5 on ˆG. +3 Let ˜B be the estimate of number of bridges. +4 ˜τ = 2n − 1 +3(˜ρ − ˜B) +5 return ˜T +Lemma 9.5. Let ˜τ be the output of Algorithm 8 and τ(V ) be the cost of graphic TSP of graph +G = (V, E). With high probability, +τ(V ) ≤ ˜τ ≤ (11 +6 + 1 +K ) · τ(V ), +where K is the parameter which is defined in Algorithm 2. +Proof. Let ρ(G) be the size of maximum path cover and B be the number of bridges in the graph. +There are at least ρ(G)−B edges of maximum path cover that are not bridge. These edges construct +disjoint paths which implies there exists a matching of size 1 +2(ρ(G) − B) that none of its edges is +bridge. Hence, by Lemma 8.4, the cost of graphic TSP is at most 2n − 1 +3(ρ(G) − B). Therefore, +since ˜ρ ≤ ρ(G) and B ≤ ˜B, we get τ(V ) ≤ ˜τ. +By Lemma 6.7 and Lemma 9.4, we have +� 1 +2 − 1 +K +� +· ρ(G) − n +K ≤ ˜ρ and ˜B ≤ B + n +K which implies +˜τ ≤ 2n − 1 +3 +� +(1 +2 − 1 +K ) · ρ(G) − 2(B + n +K ) +� +≤ 2n − 1 +3(ρ(G) +2 +− 2B) + n +K +(Since ρ(G) ≤ n). +Also, assume that the approximation ratio that the algorithm obtains is α + 1/K for some α ≤ 2. +Thus, +(α + 1 +K ) · τ(V ) ≥ α · τ(V ) + n +K +(Since τ(V ) ≥ n) +≥ α · max{2n − ρ(G), n + B} + n +K +(By Claim 8.2 and Claim 9.1). +Therefore, to show (α + 1 +K ) · τ(V ) ≥ ˜τ, it is sufficient to show +2n − 1 +3(ρ(G) +2 +− 2B) ≤ α · max{2n − ρ(G), n + B}. +Now, let ρ(G) = xn and B = yn for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. To obtain α, we write the +following maximization problem, +maximize +α +subject to +2− 1 +3 ( x +2 −2y) +max{2−x,1+y} ≤ α, +0 ≤ x ≤ 1, +0 ≤ y ≤ 1. +The solution to this problem is 11/6.5 This completes the proof. +5See e.g. this WolframAlpha link. +26 + +Theorem 9.6. Let τ(V ) be the cost of graphic TSP of graph G = (V, E). For any ε > 0, there +exists an algorithm that estimate the cost of graphic TSP, ˜τ, such that +τ(V ) ≤ ˜τ ≤ (11 +6 + ε) · τ(V ), +w.h.p in ˜O(n) running time. +Proof. Let ˜τ be the output of Algorithm 8. If we choose K = 1 +ε, then by Lemma 9.5, we have +τ(V ) ≤ ˜τ ≤ (11 +6 + ε) · τ(V ). +Also, by Theorem 6.8 and Lemma 9.4, estimating ˜ρ and ˜B can be done in ˜O(n). +10 +Lower Bound for Approximating Maximum Path Cover +10.1 +“Conditional” Hardness for the Approximation Ratio +In this section, we prove that if there exists a constant α > 0 and an algorithm that returns a +( 1 +2 + α)-approximate estimate for the size of maximum path cover in ˜O(n) time in a bipartite +graph, then there is a (1 +2 + α)-approximate algorithm for estimating the maximum matching size +in ˜O(n) time. This remains an important open problem in the study of sublinear time maximum +matching algorithms. See in particular [4]. This implies that short of a major result in the study +maximum matchings in the sublinear time model, which have received significant attention in the +literature (see [27, 2, 4, 3, 6] and references therein), our path cover algorithm has an optimal +approximation ratio. +Let G = (V, U, E) be a bipartite graph. We construct a graph G′ = (V ′, U′, E′) such that +a better than 1 +2-approximate estimate of maximum path cover on G′ leads to a better than 1 +2- +approximate estimate of maximum matching in G. Let r be a large constant. We create r copies of +G, showing the i-th copy with Gi = (Vi, Ui, E). Also, we create another r − 1 copies H1, . . . , Hr−1 +of G with Hi = (Vi, Ui+1, E). Now we let the G′ = (�r +i=1 Gi) ∪ (�r−1 +i=1 Hi). Now we claim that the +size of maximum path cover of the graph G′ is roughly 2r · µ(G) which can be used as an estimator +for the maximum matching of G. +Before proving the main result of this section, we characterize some properties of the constructed +graph G′. +Claim 10.1. µ(G′) = r · µ(G). +Proof. First, since all graphs {Gi}r +i=1 are the same as G and are vertex-disjoint, if we consider the +maximum matching of G in each of the r graphs, we will have a matching of size r · µ(G). Thus, +µ(G′) ≥ r · µ(G). +Let ˆV ∪ ˆU be the minimum vertex cover of G such that ˆV ∈ V and ˆU ∈ U. By K¨onig’s Theorem +(Proposition 3.1), we have | ˆV ∪ ˆU| = µ(G). Now we show there exists a vertex cover of size r ·µ(G) +for graph G′. Let ˆVi ∈ Vi (resp. ˆUi ∈ Ui) be the copy of vertices V (resp. U) in graph Gi. We +claim (�r +i=1 ˆVi ∪ ˆUi) is a vertex cover for G′. If an edge is in Gi, then at least one of its endpoints is +in ˆVi ∪ ˆUi since ˆVi ∪ ˆUi is a vertex cover of Gi. Moreover, by the construction, ˆVi ∪ ˆUi+1 is a vertex +cover of Hi. Hence, each edge of Hi is also covered by the vertex cover. Therefore, since there +exists a vertex cover of size |(�r +i=1 ˆVi ∪ ˆUi)| = r · | ˆV ∪ ˆU| = r · µ(G), then we have µ(G′) ≤ r · µ(G) +which completes the proof. +27 + +... +Figure 3: Illustration of graph G′ = (V ′, U′, E′). Each Gi is shown by a rectangle and each Hi is +shown by a parallelogram. Top and bottom horizontal lines illustrate Vi and Ui. Blue highlighted +parts represent the vertex cover of the graph. +Observation 10.2. It holds (2r − 1) · µ(G) ≤ ρ(G′) ≤ 2r · µ(G). +Proof. Since the union of maximum matching of all graphs {Gi}r +i=1 and {Hi}r−1 +i=1 creates a path +cover, we get (2r − 1) · µ(G) ≤ ρ(G′). Futhermore, if there exists a path cover of size larger than +2r ·µ(G), then the maximum matching of these paths will be larger than r ·µ(G) which contradicts +Claim 10.1. Thus, ρ(G′) ≤ 2r · µ(G). +Now we are ready to show the reduction. +Lemma 10.3. For any constant α > 0, if there exists an algorithm that can estimate the maximum +path cover within a ( 1 +2 +α)-factor in O(T(n)) time, then the same algorithm can be used to estimate +the maximum matching of bipartite graph G within a (1 − ε) · ( 1 +2 + α)-factor in O(T(n/ε)) time. +Proof. We construct graph G′ as described at the beginning of the section with r = +1 +2ε. +By +Observation 10.2, (1 +ε − 1) · µ(G) ≤ ρ(G′) ≤ 1 +ε · µ(G). Let ˜ρ be the estimate of the algorithm for the +maximum path cover of G′. Hence, we have +(1 +2 + α)(1 +ε − 1) · µ(G) ≤ ˜ρ ≤ 1 +ε · µ(G). +Now let ˜µ = ε · ˜ρ be the estimate for the maximum matching of G. Hence, +(1 − ε)(1 +2 + α) · µ(G) ≤ ˜µ ≤ µ(G). +Since the number of vertices and number of edges of G′ is r = +1 +2ε times more than G, then the +running time will be O(T(n/ε)). +A reduction to matchings can also be proved for (1, 2)-TSP, albeit with an extra promise for +the matching instance that the matching is either perfect or half-perfect. This problem, formalized +below, also remains open in the study matchings. We show that a better than 1.5-approximation +for (1, 2)-TSP in �O(n) time would resolve this question. +Problem 10.4. Suppose that we are given a bipartite graph G = (L, R, E) with |L| = |R| = n +and are promised that either µ(G) = n or µ(G) = ( 1 +2 + ε)n/2 for any desirably small constant +ε > 0. Provided adjacency matrix access to the graph, does there exist an �O(n) time algorithm that +distinguishes the two? +28 + +Corollary 10.5. If there is an algorithm that estimates the size of (1, 2)-TSP within a ( 3 +2 − ε0)- +factor for some fixed constant ε0 ∈ (0, 1 +4] in �O(n), then Problem 10.4 can indeed be solved in �O(n) +time. +Proof. Let G1 and G2 be two graphs with n vertices such that µ(G1) = n and µ(G2) = ( 1 +2 + ε0 +16). +We construct graph G′ +1 = (V ′ +1, E′ +1) and G′ +2 = (V ′ +2, E′ +2) as described at the beginning of the section +with r = +1 +ε0 . By Observation 10.2, we have ρ(G′ +1) ≥ ( 2 +ε0 − 1)n and ρ(G′ +2) ≤ ( 1 +ε0 + 1 +8)n. Thus, by +Observation 7.1, we get +τ(V ′ +1) ≤ 4 +ε0 +n − ( 2 +ε0 +− 1)n = ( 2 +ε0 ++ 1)n, +τ(V ′ +2) ≥ 4 +ε0 +n − ( 1 +ε0 ++ 1 +8)n − 1 ≥ ( 3 +ε0 +− 1 +4)n, +for sufficiently large n, which implies +τ(V ′ +2) +τ(V1) = 3 − ε0/4 +2 + ε0 +≥ 3 +2 − ε0. +Therefore, the algorithm for (1,2)-TSP can distinguish between G′ +1 and G′ +2 which implies Prob- +lem 10.4 can be solved in �O(n) time for ε = ε0/16. +10.2 +Information-Theoretic Lower Bounds on the Running Time +Since any constant approximation algorithm for estimating maximum path cover can be used to +estimate the size of matching within a constant factor, then all of the lower bounds for O(1)- +approximating maximum matching in sublinear time also hold for (1)-approximating maximum +path cover in sublinear time. We restate some of these lower bounds along with a short proof (see +[2] for a detailed discussion). +Lemma 10.6. Any algorithm that estimates maximum path cover within a constant multiplicative +factor requires Ω(n) queries in the adjacency list model. +Proof. Consider two graphs that the first one does not have any edge and the second one has only +a single edge. In order to give any multiplicative approximation for maximum path cover, the +algorithm needs to find the edge which requires Ω(n) queries in the adjacency list model. +Lemma 10.7. Any algorithm that estimates maximum path cover within a constant multiplicative +factor require Ω(n2) queries in the adjacency matrix model. +Proof. Consider the same construction as Lemma 10.6. To give any multiplicative approximation +for maximum path cover, the algorithm needs to find the edge which requires Ω(n2) queries in the +adjacency matrix model. +Lemma 10.8. Any algorithm that estimates maximum path cover within a multiplicative-additive +requires Ω(n) queries in the adjacency matrix model. +Proof. Consider a graph with no edge and a graph with one Hamiltonian cycle and no other edges. +In order for the algorithm to distinguish between these two graphs, it must find at least one edge +of the second graph which requires Ω(n) queries in the adjacency matrix model. +29 + +There is also a lower bound for multiplicative-additive estimation of matching in adjacency list +model [24] that also holds for maximum path cover. +Lemma 10.9. Any algorithm that estimates maximum path cover within a constant multiplicative- +additive factor requires Ω( ¯d) queries in the adjacency list model. +Acknowledgements. +Mohammad Roghani and Amin Saberi were supported by NSF award +1812919 and ONR award 141912550. Soheil Behnezhad and Aviad Rubinstein were supported by +NSF CCF-1954927, and a David and Lucile Packard Fellowship. Soheil Behnezhad was additionally +supported by NSF Awards 1942123, 1812919 and by Moses Charikar’s Simons Investigator Award. +References +[1] Anna Adamaszek, Matthias Mnich, and Katarzyna Paluch. New approximation algorithms +for (1, 2)-tsp. In 45th International Colloquium on Automata, Languages, and Programming, +ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 9:1–9:14. +Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2018. +[2] Soheil Behnezhad. Time-Optimal Sublinear Algorithms for Matching and Vertex Cover. In +62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, +CO, USA, February 7-10, 2022, pages 873–884. IEEE, 2021. +[3] Soheil Behnezhad, Mohammad Roghani, and Aviad Rubinstein. Sublinear time algorithms +and complexity of approximate maximum matching. CoRR, abs/2211.15843, 2022. +[4] Soheil Behnezhad, Mohammad Roghani, Aviad Rubinstein, and Amin Saberi. Beating Greedy +Matching in Sublinear Time. In Proceedings of the 2023 ACM-SIAM Symposium on Discrete +Algorithms, SODA 2023, to appear. SIAM, 2023. +[5] Piotr Berman and Marek Karpinski. 8/7-approximation algorithm for (1, 2)-tsp. In Proceed- +ings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, +Miami, Florida, USA, January 22-26, 2006, pages 641–648. ACM Press, 2006. +[6] Sayan Bhattacharya, Peter Kiss, and Thatchaphol Saranurak. +Sublinear algorithms for +(1.5+ε)-approximate matching. CoRR, abs/2212.00189, 2022. +[7] Guy E. Blelloch, Jeremy T. Fineman, and Julian Shun. Greedy Sequential Maximal Inde- +pendent Set and Matching are Parallel on Average. In 24th ACM Symposium on Parallelism +in Algorithms and Architectures, SPAA ’12, Pittsburgh, PA, USA, June 25-27, 2012, pages +308–317. ACM, 2012. +[8] Yu Chen, Sampath Kannan, and Sanjeev Khanna. Sublinear Algorithms and Lower Bounds for +Metric TSP Cost Estimation. In 47th International Colloquium on Automata, Languages, and +Programming (ICALP 2020), volume 168 of Leibniz International Proceedings in Informatics +(LIPIcs), pages 30:1–30:19, Dagstuhl, Germany, 2020. Schloss Dagstuhl–Leibniz-Zentrum f¨ur +Informatik. ISBN 978-3-95977-138-2. +[9] Yu Chen, Sanjeev Khanna, and Zihan Tan. +Sublinear Algorithms and Lower Bounds for +Estimating MST and TSP Cost in General Metrics. CoRR, abs/2203.14798, 2022. +30 + +[10] Nicos Christofides. Worst-case analysis of a new heuristic for the travelling salesman problem. +Technical report, Carnegie-Mellon Univ Pittsburgh Pa Management Sciences Research Group, +1976. +[11] Artur Czumaj and Christian Sohler. +Estimating the weight of metric minimum spanning +trees in sublinear-time. In Proceedings of the 36th Annual ACM Symposium on Theory of +Computing, Chicago, IL, USA, June 13-16, 2004, pages 175–183, 2004. +[12] Artur Czumaj and Christian Sohler. Estimating the weight of metric minimum spanning trees +in sublinear time. SIAM J. Comput., 39(3):904–922, 2009. +[13] Shayan Oveis Gharan, Amin Saberi, and Mohit Singh. A randomized rounding approach to the +traveling salesman problem. In IEEE 52nd Annual Symposium on Foundations of Computer +Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 550–559. IEEE +Computer Society, 2011. +[14] Anupam Gupta and Krzysztof Onak. +Sublinear.info Open Problem 71: Metric TSP Cost +Approximation. +2016. +Available at https://sublinear.info/index.php?title=Open_ +Problems:71. +[15] Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. A (slightly) improved approximation +algorithm for metric TSP. In STOC ’21: 53rd Annual ACM SIGACT Symposium on Theory +of Computing, Virtual Event, Italy, June 21-25, 2021, pages 32–45. ACM, 2021. +[16] Richard M Karp. Reducibility among combinatorial problems. In Complexity of computer +computations, pages 85–103. Springer, 1972. +[17] Marek Karpinski and Richard Schmied. On approximation lower bounds for TSP with bounded +metrics. Electron. Colloquium Comput. Complex., page 8, 2012. +[18] D. K¨onig. +¨Uber graphen und ihre anwendung auf determinantentheorie und mengenlehre. +Mathematische Annalen, 77:453–465, 1916. URL http://eudml.org/doc/158740. +[19] Matthias Mnich and Tobias M¨omke. +Improved integrality gap upper bounds for traveling +salesperson problems with distances one and two. Eur. J. Oper. Res., 266(2):436–457, 2018. +[20] Tobias M¨omke and Ola Svensson. Approximating graphic TSP by matchings. In IEEE 52nd +Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, +USA, October 22-25, 2011, pages 560–569. IEEE Computer Society, 2011. +[21] Marcin Mucha. 13/9-approximation for graphic TSP. In 29th International Symposium on +Theoretical Aspects of Computer Science, STACS 2012, February 29th - March 3rd, 2012, +Paris, France, volume 14 of LIPIcs, pages 30–41. Schloss Dagstuhl - Leibniz-Zentrum f¨ur +Informatik, 2012. +[22] Krzysztof Onak, Dana Ron, Michal Rosen, and Ronitt Rubinfeld. A near-optimal sublinear- +time algorithm for approximating the minimum vertex cover size. In Proceedings of the Twenty- +Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, +January 17-19, 2012, pages 1123–1131. SIAM, 2012. +[23] Christos H. Papadimitriou and Mihalis Yannakakis. +The traveling salesman problem with +distances one and two. Math. Oper. Res., 18(1):1–11, 1993. +31 + +[24] Michal Parnas and Dana Ron. Approximating the minimum vertex cover in sublinear time +and a connection to distributed algorithms. Theoretical Computer Science, 381(1):183–196, +2007. ISSN 0304-3975. +[25] Andr´as Seb¨o and Jens Vygen. Shorter tours by nicer ears: 7/5-approximation for the graph-tsp, +3/2 for the path version, and 4/3 for two-edge-connected subgraphs. Comb., 34(5):597–629, +2014. +[26] Anatoliy I Serdyukov. O nekotorykh ekstremal’nykh obkhodakh v grafakh. Upravlyayemyye +sistemy, 17:76–79, 1978. +[27] Yuichi Yoshida, Masaki Yamamoto, and Hiro Ito. An improved constant-time approximation +algorithm for maximum matchings. In Proceedings of the 41st Annual ACM Symposium on +Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 225– +234. ACM, 2009. +A +Implementation Details +In this section, we discuss why Lemma 6.1, restated below, holds. +Lemma 6.1. There exists a data structure that given a graph G in the adjacency list format, +(implicitly) fixes a random permutation π over its edges. Then for any vertex v, the data structure +returns the degree of vertex v in the subgraph P produced by Algorithm 2 according to a random +permutation π. Each query v to the data structure is answered in ˜O(T(v, π)) time w.h.p. where +T(v, π) is as defined in Section 5. +The proof of Lemma 6.1 uses standard ideas from the literature [22, 2]. The only modification, +essentially, is to show that these algorithms also work for multi-graphs. +Let us focus on the +specific algorithm proposed in [2, Appendix A]. Given the adjacency list of a graph G = (V, E), it +defines gives a procedure LOWEST(v, i) that first draws a random rank E → [0, 1] on each edge +(implicitly), then for any input vertex v and an integer i ≤ degG(v), returns a vertex u such that +(v, u) is the i-th lowest rank edge incident to v. It is proved in [2] that if the procedure is called for a +fix vertex v and all indices i with 1 ≤ i ≤ r, then the total running time is ˜O(r). The only difference +between the implementation of our algorithm and the one in [2] is that we have multiple copies of a +single edge in the original graph. First, we observe that the procedure LOWEST(v, i), in addition +to returning the neighbor u, can also return the rank of the edge (v, u). (This is explicitly computed +by LOWEST(v, i) in [2].) Now let G′ be the multigraph with K copies of each edge of G. Instead +of a multigraph, we can assume that we have K copies of same graph G called G1, G2, . . . , GK. +Also, for each i, let LOWESTGi be the LOWEST procedure corresponding to graph Gi. For each +vertex v, we use a balanced binary search tree (BST) that stores the ranks of the lowest incident +edge to v in each graph. So at any point during the course of the algorithm, there are at most K +different values in the BST of vertex v. Now for the next LOWEST query to the multigraph graph +G′ for vertex v, we can return the minimum edge in the BST of vertex v. Since K is a constant +and the any query to a BST is answered in O(log n) time, the total running time will be the same +as [2, Appendix A] within a O(log n)-factor. +32 + diff --git a/MdE4T4oBgHgl3EQf8w5p/content/tmp_files/load_file.txt b/MdE4T4oBgHgl3EQf8w5p/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce7ae90e0fbfad3165a6bdb53990a96b57b02758 --- /dev/null +++ b/MdE4T4oBgHgl3EQf8w5p/content/tmp_files/load_file.txt @@ -0,0 +1,1400 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf,len=1399 +page_content='Sublinear Algorithms for TSP via Path Covers Soheil Behnezhad Northeastern University Mohammad Roghani Stanford University Aviad Rubinstein Stanford University Amin Saberi Stanford University Abstract a We study sublinear time algorithms for the traveling salesman problem (TSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' First, we focus on the closely related maximum path cover problem, which asks for a collection of vertex disjoint paths that include the maximum number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We show that for any fixed ε > 0, there is an algorithm that (1/2 − ε)-approximates the maximum path cover size of an n-vertex graph in �O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This improves upon a (3/8 − ε)-approximate �O(n√n)-time algorithm of Chen, Kannan, and Khanna [ICALP’20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Equipped with our path cover algorithm, we give �O(n) time algorithms that estimate the cost of graphic TSP and (1, 2)-TSP up to factors of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='83 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5 + ε), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Our algorithm for graphic TSP improves over a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='92-approximate �O(n) time algorithm due to [CHK ICALP’20, Behnezhad FOCS’21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Our algorithm for (1, 2)-TSP improves over a folklore (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='75 + ε)-approximate �O(n)-time algorithm, as well as a (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='625 + ε)-approximate �O(n√n)-time algorithm of [CHK ICALP’20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Our analysis of the running time uses connections to parallel algorithms and is information- theoretically optimal up to poly log n factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Additionally, we show that our approximation guarantees for path cover and (1, 2)-TSP hit a natural barrier: We show better approximations require better sublinear time algorithms for the well-studied maximum matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='05350v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='DS] 13 Jan 2023 Contents 1 Introduction 1 2 Technical Overview 3 3 Preliminaries 4 4 New Meta Algorithms for Maximum Path Cover 5 5 A Local Query Process for Algorithm 2 and its Complexity 8 6 Our Estimator for Maximum Path Cover 15 7 Our Estimator for (1,2)-TSP 19 8 Our Estimator for Graphic TSP 20 9 Further Improvement for Graphic TSP 23 10 Lower Bound for Approximating Maximum Path Cover 27 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1 “Conditional” Hardness for the Approximation Ratio .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 27 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2 Information-Theoretic Lower Bounds on the Running Time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 29 A Implementation Details 32 1 Introduction The traveling salesman problem (TSP) is a central problem in combinatorial optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Given a set V of n vertices and their pairwise distances, it asks for a Hamiltonian cycle of the minimum cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In this paper, we study sublinear time algorithms for TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The algorithm is given query access to the distance pairs, and the goal is to estimate the solution cost in time sublinear in the input size (which is Θ(n2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' TSP is NP-hard to approximate within a polynomial factor for an arbitrary distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' As such, much of the work in the literature has been on more specific distance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Some notable examples include graphic TSP [13, 20, 21, 25, 8] where the distances are the shortest paths over an arbitrary unweighted undirected graph, (1, 2)-TSP [1, 8, 5, 16, 19] where the distances are 1 or 2, and more generally metric TSP [15, 12, 10, 26] where the distances satsify triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In 2003, Czumaj and Sohler [11, 12] showed that for any fixed ε > 0, a (1+ε)-approximation of the cost of metric minimum spanning tree (MST) and thus a (2 + ε)-approximation of the cost of metric TSP can be found in �O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Twenty years later, it still remains a major open problem to either break two-approximation in n2−Ω(1) time or prove a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1 However, better bounds are known for both graphic TSP and (1, 2)-TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In this paper, we present improved algorithms for these two well-studied variants of TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Our main tool to achieve this is an improved algorithm for the closely related maximum path cover problem which might be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Maximum Path Cover: The maximum path cover in a graph is a collection of vertex disjoint paths with the maximum number of edges in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The (almost) 1/2-approximate maximum match- ing size estimator of Behnezhad [2] immediately implies an (almost) 1/4-approximation for the maximum path cover problem in �O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2 This can be improved to an (almost) (3/8 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='375)- approximation using the matching-pair idea of Chen, Kannan, and Khanna [8] in �O(n√n)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3 Our first main contribution is an improvement over both of these results: Result 1 (Formally as Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any ε > 0, there is a randomized algorithm that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' (1/2 − ε)-approximates the size of maximum path cover in �O(n · poly(1/ε)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Besides quantitavely improving prior work both in the running time and the approximation ratio, Result 1 reaches a qualitatively important milestone as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' First, the running time of Result 1 is information-theoretically optimal up to poly log n factors (the lower bound holds for any constant approximation — see Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Second, its approximation ratio hits a rather important barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We give a non-trivial reduction that shows a (1/2+Ω(1))-approximation in �O(n) time for maximum path cover would imply the same bound for maximum matching in bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Such a result has remained elusive for matching, which is one of the most extensively studied problems in the literature of sublinear time algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' See Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' It is also worth noting that in bounding the running time of our algorithm in Result 1, we use connections to parallel algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Such a connection was previously only used for matchings [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 1See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Open Problem 71 on sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='info [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 2The application of sublinear time maximum matching algorithms for approximating maximum path cover was first proposed by Gupta and Onak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' See [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 3We note that even though a subsequent result of Behnezhad [2] improved the running time for maximal matchings and graphic TSP from O(n√n) in [8] to �O(n), it is not immediately clear whether the same holds for path cover and (1, 2)-TSP as they rely on a different notion of a matching pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 1 (1, 2)-TSP: The (1, 2)-TSP problem has been studied extensively in the classical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In his landmark paper, Karp [16] showed that (1, 2)-TSP is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Papadimitriou and Yannakakis [23] then proved its APX-hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since then there has been a significant amount of work on (1, 2)-TSP in the classical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The current best known inapproximability bound for (1, 2)-TSP is 535/534 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' After a series of works, the best known polynomial time approximation is 8/7 [5] which can be implemented in O(n3) time [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For sublinear time algorithms, an �O(n)-time (almost) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='75-approximation is folklore [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Chen, Kannan, and Khanna [8] improved the approximation to (almost) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='625 in �O(n√n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' It is not hard to see that up to a small additive error of 1, (1, 2)-TSP is equivalent to finding a maximum path cover on the weight-1 edges and then connecting their endpoints via weight-2 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' A simple calculation shows that any α-approximation for the maximum path cover problem leads to a (2 − α)-approximation for (1, 2)-TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Our path cover algorithm of Result 1 immediately implies the following result as a corollary: Result 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any ε > 0, there is a randomized algorithm that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5 + ε)-approximates the cost of (1, 2)-TSP in �O(n · poly(1/ε)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Similar to Result 1, the running time of Result 2 is information-theoretically optimal up to poly log n factors, and its approximation ratio hits a natural barrier due to a connection to sublinear time matching that we establish in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Graphic TSP: The graphic TSP problem is equivalent to finding a tour of the minimium size that visits all the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This is an important instance of TSP that has received a lot of attention over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For polynomial time algorithms, a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5-approximation of Christofides [10] (which also works more generally for metric TSP) had remained the best known until a series of works over the last decade improved it to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5 − ε0) [13], 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='461 [20], 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='444 [21], and finally to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For sublinear time algorithms, Chen, Kannan, and Khanna [8] showed that an (almost) (27/14 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='928)- approximation of graphic TSP can be obtained in �O(n√n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The running time was subsequently improved to �O(n) by Behnezhad [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We first show that plugging Result 1 into the framework of [8] immediately improves their approximation from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='928 to (almost) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='9 while keeping the running time �O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We then give a more fine tuned algorithm that obtains a much improved approximation ratio of (11/6 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='833).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Result 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any ε > 0, there is a randomized algorithm that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' (1 + ε)( 11 6 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='833)- approximates the cost of graphic TSP in �O(n · poly(1/ε)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Further related work: Finally, we note that in a recent paper, Chen, Khanna, and Tan [9] show that assuming that the metric has a spanning tree supported on weight 1 edges, one can obtain a (2 − ε0)-approximation with �O(n√n) queries for some small unspecified constant ε0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' While this is a more general metric than graphic TSP and (1,2)-TSP that we study in this paper, we note that the two papers are orthogonal and their results are incomparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In particular, the techniques developed in this paper are specifically designed to improve the approximation to much below 2, whereas [9] focuses on generalizing the distance function while beating 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Independent work: In two concurrent papers, Bhattacharya, Kiss, and Saranurak [6], and Behnezhad, Roghani, and Rubinstein [3], gave an almost 2/3-approximation for maximum match- 2 ing in n2−Ω(1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Combining these two algorithms with the framework of Chen, Kannan, and Khanna [8] implies a 40/21-approximation for graphic TSP in n2−Ω(1) time (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6 in [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Our algorithm makes an improvement on both the running time and approximation ratio for the graphic TSP over the recent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In terms of techniques, our work is entirely disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 2 Technical Overview In this section, we give an overview of our algorithms, especially our sublinear time maximum path cover algorithm of Result 1 which is the key to the other results as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let us start with using matchings to approximate maximum path cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Consider a graph that has a Hamiltonian path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Here, the optimal maximum path cover has size n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' On the other hand, any maximum matching can have at most n/2 edges, which is by a factor 2 smaller than our optimal path cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' On top of this, we only know close to 2/3 approximations for maximum matching in the sublinear time model [3, 6], thus can only achieve an approximation close to 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Instead of a single matching, Chen, Kannan, and Khanna [8] showed how to estimate the number of edges in a maximal matching pair in �O(n√n) time, where a matching pair is simply two edge disjoint matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' It is not hard to see that the number of edges in a maximal matching pair is at least half the number of edges in a maximum path cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The problem, however, is that a maximal matching pair is not a collection of paths!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In particular, the two matchings can form cycles of length as small as four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, one may only be able to use 3/4 fraction of the edges of a matching pair in a path cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This is precisely why the algorithm of [8] only obtains a 1 2 × 3 4 = 3 8 approximation for path cover, and a 2 − 3 8 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='625 approximation for (1, 2)-TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If we could modify the matching pair algorithm of [8], and avoid cycles by manually excluding edges whose endpoints are the endpoints of a path in the current matching pair, then we could avoid the 3/4 factor loss discussed above and achieve a 1/2-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Unfortunately, checking whether the endpoints of an edge are endpoints of a path requires knowledge about whether a series of other edges belong to the solution, which seems hard to implement in sublinear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Instead of checking for cycles manually, we introduce the following Algorithm 1 which avoids cycles more naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' While our final algorithm is a modified variant of Algorithm 1 described below, we start with Algorithm 1 as we believe it provides the right intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Algorithm 1: A new algorithm for path cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 1 Initialize P ← ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 2 Each vertex v has two ports that we denote by v0 and v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Each of these ports throughout the algorithm will be either free or occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Initially, all ports are free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 3 Iterate over the edges in some ordering π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Upon visiting an edge e = (u, v): If v0 and u0 are free, add e to P, mark v0 and u0 as occupied, and skip to the next edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If v1 and u0 are free, add e to P, mark v1 and u0 as occupied, and skip to the next edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If v0 and u1 are free, add e to P, mark v0 and u1 as occupied, and skip to the next edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 4 Return P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Two properties of Algorithm 1 are crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' First, it prioritizes occupying (u0, v0) (compared to (u1, v0) or (u0, v1)) which in particular implies that any component in P must have a (u0, v0) edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Second, it never occupies (u1, v1) with an edge (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' While it is easy to see that the output of Algorithm 1 has maximum degree 2, and is thus a collection of paths or cycles, the two properties 3 above actually guarantee that it never includes any cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We provide the formal proof of this later in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Additionally, we show that the output of Algorithm 1 must be at least half the size of a maximum path cover, as we prove next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, if we manage to estimate the size of the output P of Algorithm 1, then we have proved Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 0 0 1 0 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 0 1 0 0 1 1 0 1 1 0 0 0 1 0 0 1 0 Invalid: has no (0, 0) edge 0 1 1 1 1 1 1 0 0 0 0 0 0 1 Invalid: a vertex has two 0’s Invalid: (1,1) not allowed Valid (not a cycle) Figure 1: Examples of why the output of Algorithm 1 will not have cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Our final algorithm is slightly different from Algorithm 1 discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In particular, we slightly relax it—see Algorithm 2—so that it can be solved via a randomized greedy maximal in- dependent set (RGMIS), for which we have a rich toolkit of sublinear time estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Existing approaches (particularly the algorithm of Yoshida, Yamamoto, and Ito [27] and its two-step im- plementation by Chen, Kannan, and Khanna [8]) can be employed to estimate the value of this modified Algorithm 2 in �O(n√n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We achieve the improved, and near tight, �O(n) time bound guarantee of Result 1 by building on the analysis of Behnezhad [2] for maximal independent set on the line graphs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=', maximal matchings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Though we note that several new ideas are needed, because the MIS graph in our case will not be exactly a line graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We defer more discussions about this to Sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Implications for TSP: By having an α-approximate maximum path cover algorithm, we imme- diately obtain a (2 − α)-approximation for (1, 2)-TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, the algorithm above immediately proves Result 2 that we can (almost) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5-approximate (1, 2)-TSP in �O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For our Result 3 for graphic TSP, we first observe that our improved path cover algorithm can be employed to provide a better lower bound for the optimal TSP solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This improves the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='92-approximation of [8] as black-box to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='9-approximation (Section 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' However, the final improvement to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='83 requires more ideas, in particular, on how to better estimate the number of certain bridges in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' See Section 9 for more details about this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 3 Preliminaries Problem Definition: In the sublinear TSP problem, we have a set V of n vertices and a distance function d : V × V → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The algorithm has query access to this distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Namely, for any pair (u, v) of the vertices of its choice, the algorithm may query the value of d(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The goal is to design an algorithm that runs in sublinear time in the input size, which is Θ(n2) (all the distance pairs), and produces an estimate of the size of the optimal TSP solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Denoting the optimal TSP value by τ(V ), we say an estimate �τ(V ) provides an α-approximation for α ≥ 1 if τ(V ) ≤ �τ(V ) ≤ α · τ(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 4 We focus specifically on graph TSP and (1, 2)-TSP problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In graphic TSP, the distance function d is the shortest path metric on an unweighted undirected graph G that is unknown to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note, however, that the distance queries essentially provide adjacency matrix access to this graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In (1, 2)-TSP, the assumption is that d(u, v) ∈ {1, 2} for every pair u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In the case of (1, 2)-TSP we may use G to refer to the subgraph induced on the pairs with distance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Defining graph G as above, we use n to denote the number of its vertices, m to denote the number of its edges, ∆ to denote its maximum degree, µ(G) to denote its maximum matching size, ν(G) to denote its minimum vertex cover size, and ¯d to denote its average degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Path Cover Definitions: Given an unweighted graph G, a path cover in G is a collection of vertex disjoint paths in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' A maximum path cover is a path cover of G with the maximum number of edges in it (note that we are not counting the number of paths, but rather the total number of edges in them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We use ρ(G) to denote the size of the maximum path cover in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We say an estimate �ρ(G) for ρ(G) provides an (α, ε)-approximation for α, ε ∈ [0, 1] if α · ρ(G) − εn ≤ �ρ(G) ≤ ρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We may also use α-approximation instead of (α, 0)-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Graph Theory Definitions/Tools: A bridge (cut edge) in a graph is an edge whose deletion increases the number of connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Similarly, a cut vertex is a vertex whose deletion (along with its edges) increases the number of connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' A biconnected graph is a connected graph with no cut vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, a biconnected component (block) of a graph is a maximal biconnected subgraph of the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' A non-trivial biconnected component is a block that is not a bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We say a graph is 2-edge-connected if there is no bridge in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' A 2-edge- connected component of a graph is maximal 2-edge-connected subgraph of the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The bridge-block tree of a graph is a tree obtained by contracting the 2-edge-connected components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' note that the edge set of a bridge-block tree correspond to the bridges in the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We use the following classic theorem of K¨onig [18] that the size of the minimum vertex cover is equal to the size of maximum matching in bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Namely: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1 (K¨onig’s Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In any bipartite graph G, µ(G) = ν(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Probabilistic Tools: In our proofs, we use the following standard concentration inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2 (Chernoff Bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , Xn be independent Bernoulli random vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let X = �n i=1 Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any t > 0, Pr[|X − E[X]| ≥ t] ≤ 2 exp � − t2 3 E[X] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3 (Hoeffding’s Inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , Xn be independent random variables such that a ≤ Xi ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ¯X = (�n i=1 Xi)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any t > 0, Pr[| ¯X −E[X]| ≥ t] ≤ 2 exp � − 2nt (b−a)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 4 New Meta Algorithms for Maximum Path Cover In this section, we present a new meta algorithm for maximum path cover that obtains a 1/2- approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The algorithm, as we will state it in this section, will not be particularly in the sublinear time model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We discuss its sublinear time implementation later in Sections 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Our starting point is the Algorithm 1 described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let us first formally prove that it obtains a 1/2-approximation, and that no component in it is a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 5 Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The output of Algorithm 1 is a collection of disjoint paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since P has maximum degree two, it suffices to show none of its connected components are cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Property (i) above implies that at any point during the algorithm, any degree one vertex v has its port v0 occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now take an edge e = (u, v) that forms a cycle if added to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Both u and v must have degree one and so u0 and v0 are occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since by property (ii) edge e does not occupy both v1 and u1, the algorithm does not add e to P thus not completing a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let P ⋆ be any path cover using weight one edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then the output of Algorithm 1 has size at least 1 2|P ⋆|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any edge e = (u, v) ∈ P ⋆ define φ(e) = 1 4(degP (u) + degP (v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We first claim that for every edge e = (u, v) in G, we have φ(e) ≥ 1/2 (or, equivalently, degP (u) + degP (v) ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This is clear for edges e ∈ P due to the contribution of e itself to its endpoints’ degrees, so fix e ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Consider the time that we process e = (u, v) in the algorithm and decide not to add it to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We claim that out of v0, v1, u0, u1 at least two ports must be occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Suppose w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' and for contradiction that only vx is occupied for x ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then (u, v) can occupy v1−x and ux and be added to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This contradicts (u, v) not being added to P and proves our claim that φ(e) ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' From the discussion above, we get that � e∈P ⋆ φ(e) ≥ � e∈P ⋆ 1/2 = |P ⋆|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, because every vertex has degree at most two in P ⋆, we get � e∈P ⋆ φ(e) = 1 4 � (u,v)∈P ⋆ degP (u) + degP (v) ≤ 1 4 · 2 � v∈V degP (v) = |P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The two inequalities above combined imply that |P| ≥ |P ⋆|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' As discussed, our final algorithm is different from Algorithm 1 discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' One problem with Algorithm 1 is that it cannot be cast as an instance of the randomized greedy maximal independent set (RGMIS) algorithm for which there is a rich toolkit of sublinear time estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' To remedy this, we present a modified variant of Algorithm 1 whose output is (almost) as good, but in addition can be modeled as an instance of RGMIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We denote the output of RGMIS on a graph G with a permutation π on its vertices by RGMIS(G, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The algorithm is stated below as Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Similar to the output of Algorithm 1, the output of Algorithm 2 can be verified to have maximum degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, it is a collection of paths and cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' But unlike Algorithm 1, the output of Algorithm 2 can have cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This happens since, unlike Algorithm 1, each connected component of the output of Algorithm 2 is not guaranteed to have an edge (u, v) occupying both u0 and v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Nonetheless, we are able to show that this bad event only happens for a small fraction of connected components of the output of Algorithm 2 in expectation, and so once we remove one edge of each of these cycles, the resulting collection of 6 disjoint paths has almost the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Algorithm 2: A modification of Algorithm 1 that uses RGMIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 1 Parameter: K (think of it as a large constant integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 2 Let G = (V, E) be the subgraph of weight one edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We construct a graph H = (VH, EH) from G on which we run RGMIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 3 Each vertex in H corresponds to an edge e in G and two ports (as in Algorithm 1) of the endpoints of e that it occupies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Formally, for any (u, v) ∈ E we have K + 2 vertices in H: One vertex that corresponds to occuping u0 and v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' One vertex that corresponds to occuping u1 and v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' K vertices that each corresponds to occuping u0 and v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 4 Consider two distinct vertices a and b in H corresponding to edges ea and eb in G: If ea = eb then we add an edge between a and b in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If ea and eb share exactly one endpoint v and both a and b occupy the same port of v, we add an edge between a and b in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 5 Find a randomized greedy maximal independent set I of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 6 Let P be the set of edges in G corresponding to the vertices in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 7 Return P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let C be a connected component in the output of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If C is a cycle, then every edge in C occupies one 0-port and one 1-port (that is, no edge occupies two 0-ports).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Suppose that C has n′ vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since each vertex in a cycle has degree two, both ports of each vertex in C must be occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, n′ 0-ports and n′ 1-ports of C are occupied in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Given that any edge occupies at least one 0-port by the algorithm, we cannot have an edge that occupies two 0-ports, or else we should occupy more 0-ports than 1-ports of C, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Next, we show that up to a factor of (1 + 2/k) which is negligible for K in the order 1/ε, the output of Algorithm 2 is an (almost) 1/2-approximation of the maximum path cover value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let C be a connected component in the output of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If C is a path, then it contains at most one edge that occupies two 0-ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let C be the path (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , vr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since the degree of any vertex vi for 1 < i < r is two in the path, both ports of vi must be occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For v1 and vr, on the other hand, only one port is occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, the total number of 0-ports that are occupied by C minus the number of 1-ports occupied by it is at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This means that there is at most one edge that occupies two 0-ports since all other types of edges occupy exactly one 0-port and one 1-port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' let P be the output of Algorithm 2 on graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then 1 2ρ(G) ≤ E |P| ≤ � 1 + 2 K � ρ(G), where the expectation is taken over the randomization of computing RGMIS in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let P ∗ be a maximum path cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any edge e = (u, v) ∈ P ⋆ define φ(e) = 1 4(degP (u) + degP (v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' With the exact same argument as in the proof of Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2, we get that φ(e) ≥ 1/2, which implies � e∈P ⋆ φ(e) ≥ � e∈P ⋆ 1/2 = ρ(G)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since the degree of each vertex in P is at most two, we get � e∈P ⋆ φ(e) = 1 4 � (u,v)∈P ⋆ degP (u) + degP (v) ≤ 1 4 · 2 � v∈V degP (v) = |P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By combining above inequalities we get 1 2ρ(G) ≤ |P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that we do not need the randomization for the proof of the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By construction of P, every vertex has degree at most two in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, all connected compo- nents of P are cycles and paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We claim that at most 2 K+2 fraction of connected components are cycles in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since the expected number of connected components is at most E |P|, from this we get that the expected number of cycles is at most 2 E |P|/(K + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By removing one edge from each cycle, we obtain a valid solution for maximum path cover problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, E |P| − 2 E |P| K + 2 = K K + 2 E |P| ≤ ρ(G) ⇒ E |P| ≤ � 1 + 2 K � ρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' So it remains to show that at most 2 K+2 fraction of connected components are cycles in expec- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' As we process edges one by one according to the ordering of RGMIS, let A be the set of edges that none of their incident edges are added to the solution of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By definition of A, if one copy of edge (u, v) is in A, then all other copies of (u, v) are also in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, at any point during running RGMIS, if a new component is added to the solution, the edge (u, v) that gets added to the solution occupies (u0, v0) with probability at least K K+2 since K copies out of the K + 2 copies are for (u0, v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let C0 be the number of times that the newly added component is an edge occupying two 0-ports, and C1 be the number of times that the newly added component is an edge occupying one 0-port and one 1-port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By the above argument, we have E[C0] E[C0] + E[C1] = K K + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' (1) Note that after running Algorithm 2, it is possible that the number of connected components is actually smaller than C0 +C1, since some of the components may merge as the algorithm proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' However, by Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4, two components that their first edge occupies two 0-ports will not merge together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, by Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3, none of the cycle components have an edge that occupies two 0-ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, in the end, there exists at most E[C0] + E[C1] connected components and at least E[C0] of them will not be cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 5 A Local Query Process for Algorithm 2 and its Complexity In this section, we define a query process to estimate the size of the output of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In graph H of Algorithm 2, each vertex corresponds to an edge in the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' More precisely, we make K + 2 copies of each edge (u, v) such that one of the copies corresponds to an 8 edge occupying (u0, v1), one for (u1, v0), and K for (u0, v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We use G′ = (V, E′) to show the new graph with these parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' During the course of Algorithm 2, two different edges that share the same endpoint and port cannot appear in the solution together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We use the following definition to formalize this notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1 (Conflicting Pair of Edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Two edges e, e′ ∈ E′ that share an endpoint v are conflicting if both e and e′ correspond to same port vi for i ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We call (e, e′) a conflicting pair of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In order to estimate the size of the output of Algorithm 2, we define a vertex oracle that given a vertex v and a permutation π on E′, returns the degree of vertex v in the output of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' These are akin to the query processes used before in the works of [2, 27], but are specific to our Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Algorithm 3: “vertex oracle” VO(u, π) to determine the degree of vertex u in RGMIS(G′, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 1 Let e1 = (u, v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , er = (u, vr) be the edges incident to u with π(e1) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' < π(er).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 2 d ← 0 3 for i in 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' r do 4 if EO(ei, vi, π) = True then d ← d + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 5 return d Algorithm 4: “edge oracle” EO(e, u, π) to determine an edge e is in RGMIS(G′, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, u must be an endpoint of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 1 if EO(e, u, π) computed before then return the computed result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 2 Let e1 = (u, v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , er = (u, vr) be the edges incident to e such that π(e1) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' < π(er) < π(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, (e, ei) is a conflicting pair for all 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 3 for i in 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' r do 4 if EO(ei, vi, π) = True then return False;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 5 return True Note that in Line 2 of the Algorithm 4 we only recursively call the function on edges that their label, conflict with edge e since if other edges appear in the RMGIS subgraph, we can still have e in the RGMIS subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Before analyzing the query complexity of the vertex oracle, we prove the correctness of the vertex oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any edge e = (u, z) ∈ E′ that is occupying ports ui and zj, if EO(e, u, π) is called while computing VO(v, π), then EO(e, u, π) = True iff e ∈ RGMIS(G′, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We prove the claim using induction on ranking of edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Assume that the claim is true for all edges with ranking smaller than π(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If EO(e, u, π) is called by EO(e′ = (w, z), z, π) or directly by VO(v, π), then by definition of Algorithm 4 and Algorithm 3, all edges e′′ = (w′, z) with π(e′′) < π(e′) that are occupying zj are queried before e′ which means that none of them return True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, by induction hypothesis, none of the edges incident to z that are occupying zj with lower rank are in the RGMIS(G′, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, EO(e, u, π) calls all incident edges to u with lower rank that are occupying ui and return Trueif none of them are in the RGMIS(G′, π) by induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, EO(e, u, π) = True iff e ∈ RGMIS(G′, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let v ∈ V and d be the output of VO(v, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then d is equal to the degree of vertex v in the subgraph outputted by RGMIS(G′, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The observation follows by combining the fact that the vertex oracle queries edges in in- creasing order and Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let T(v, π) denote the number of recursive calls to the edge oracle during the execution of VO(v, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For a randomly chosen vertex v and permutation π on E′, we have that Ev,π[T(v, π)] = O( ¯d · log2 n) where ¯d is the average degree of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let Q(e, v, π) be the number of EO(e, ·, π) calls during the execution of VO(v, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, let Q(e, π) be the number of EO(e, ·, π) calls starting from any vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In other words, we have that Q(e, π) = � v∈V Q(e, v, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For every edge e and permutation π, Q(e, π) ≤ O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let e = {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For a fixed vertex u, either the vertex oracle VO(u, π) queries the edge oracle for e directly, or through some incident edge e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, the edge oracle of e is called through at most (K + 2)(deg(x) − 1) + (K + 2)(deg(y) − 1) of its incident edges (K + 2 appears since each edge has K + 2 copies), which implies that Q(e, u, π) ≤ (2K + 4)(n − 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, Q(e, π) ≤ � u∈V Q(e, u, π) ≤ n ((2K + 4)(n − 1) + 1) ≤ O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The main contribution of this section is to show that the expected number of EO(e, π) calls over all permutations π is O(log2 n), which is formalized in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any edge e ∈ E′, we have Eπ[Q(e, ·, π)] = O(log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Assuming the correctness of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6, we can complete the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Ev,π[T(v, π)] = 1 n Eπ � � v∈V T(v, π) � = 1 n Eπ � � v∈V � e∈E′ Q(e, v, π) � = 1 n Eπ � � e∈E′ � v∈V Q(e, v, π) � = 1 n Eπ � � e∈E′ Q(e, π) � = 1 n � e∈E′ Eπ[Q(e, π)] = 1 n � e∈E′ O(log2 n) = 1 nO(|E′| · log2 n) = O( ¯d · log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' During the recursive calls to the edge oracle that starts from vertex v, the edges in the stack of recursive calls create a trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let S = (e1 = (v, u), e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , er) be the stack of recursive calls starting from vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then (e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , er) is a trail in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since in Line 2 of Algorithm 4, edge oracle only queries incident edges, (e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , er) is a walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' It remains to show that all edges are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Suppose that ei = ej for some i < j which implies π(ei) = π(ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since the edge oracle queries edges in decreasing order, we have π(ej) < π(ei) which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We direct the edges of the trail from v to the other endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We call a trail that starts from v on the graph with edge permutation π, a (v, π)-query-trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For an edge e = (x, y), let ⃗e denote the directed edge from x to y and ⃗ e denote a directed edge from y to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ⃗P = (⃗e1, ⃗e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , ⃗ek) be a (v, π)-query-trail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' then π(e1) > π(e2) > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' > π(ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' During the answering whether an edge is in RGMIS(G′, π), Algorithm 4 recursively calls on edges with π values lower than the value of the current edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, the stack of recursive calls will be decreasing with respect to π values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let Q(⃗e, π) ⊆ Q(e, π) be the set of all query trails that end at ⃗e (with the same direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In what follows, we obtain a bound for the query complexity for ⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We use this lemma to prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any edge e, we have Eπ[Q(⃗e, π)] = O(log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since Q(e, π) = Q(⃗e, π) ∪ Q( ⃗ e, π), by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='9 we have Eπ[Q(e, π)] ≤ Eπ[Q(⃗e, π)] + Eπ[Q( ⃗ e, π)] = O(log2 n) + O(log2 n) = O(log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Given a permutation π and a trail ⃗P = (⃗e1, ⃗e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , ⃗ek), we define φ(π, ⃗P) to be another permu- tation σ over the edges such that: (σ(e1), σ(e2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , σ(ek−1), σ(ek)) := (π(e2), π(e3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , π(ek), π(e1)) π(e′) = σ(e′) ∀e′ /∈ ⃗P Given an edge ⃗e, by using the above mapping function we can construct a bipartite graph H with two parts A and B such that each part has |E′|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' vertices showing different permutations of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For a permutation π ∈ A and a (v, π)-query-trail ⃗P that ends at ⃗e for some arbitrary vertex v, we connect π in A to φ(π, ⃗P) in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that by construction of H, deg(πA) = Q(⃗e, πA) for all πA ∈ A, since we have a unique edge for each query-trail that ends at ⃗e with permutation πA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, in order to prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6, it is sufficient to prove that EπA∼A[degH(πA)] = O(log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let Q(⃗e, π) be the set of all query-trails for permutation π that ends at ⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let β = c log2 n for some large c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We partition permutations into two sets of likely and unlikely permutations called L and U as follows: L := � π ∈ Π ��� max ⃗P∈Q(⃗e,π) |⃗P| ≤ β � U := Π \\ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Likely permutations are those permutations that the longest query-trail ending at ⃗e has length at most β and unlikely permutations are the remaining permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let AL be the set of ver- tices corresponding to the likely permutations in A and AU be the set of vertices corresponding to the unlikely permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The intuition behind this partitioning is that the set of unlikely permutations makes up a tiny fraction of all permutations which is formalized in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 11 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If c is a large enough constant, then we have |AU| ≤ |E′|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='/n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Before proving Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='10, we introduce the parallel implementation of the greedy maximal independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Parallel Randomized Greedy Maximal Independent Set: Let G be a graph and π be a permutation over its edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In each iteration, we pick all vertices whose rank is less than all their neighbors and remove all their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We denote the number of rounds in this algorithm until G becomes empty as round complexity and we show it with ρ(G, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' It is clear that the output of the parallel randomized greedy MIS is the same as RGMIS(G, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We have the following known result about the round complexity of parallel randomized greedy MIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='11 ([7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For a uniformly random chosen permutation π over edges of G, we have ρ(G, π) = O(log2 n), with probability of at least 1 − 1 n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In order to use the above lemma, we need to show that for an unlikely permutation, the round complexity is large and therefore, small fraction of permutations are unlikely as a result of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ⃗P be query-trail in G′ with permutation π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then ρ(G′, π) ≥ ⌊| ⃗P| 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ⃗P = (⃗e1, ⃗e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , ⃗ek) be a query-trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='8, we have π(e1) > π(e2) > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' > π(ek), where ek is the last edge on the trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ρ(e) show the round in which edge e is deleted by the parallel algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If we can show that for i < k−1, ρ(ei) > ρ(ei+2), then we have that ρ(e2) ≥ ⌊k 2⌋ which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We prove it using a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Assume that ρ(ei) ≤ ρ(ei+2) for some 1 < i < k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that ρ(ei+1) ≥ ρ(ei), otherwise, when ei+1 is deleted from the graph, one of its corresponding ports that is shared with ei and ei+2 was occupied which implies that at least one of ei and ei+2 should be deleted at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, in round ρ(ei), edge ei+1 is still present in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, ei is not a local minimum in round ρ(ei) and is deleted due to presence of an edge e′ in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that e′ ̸= ei+1 since ei+1 is not the minimum edge because ei+2 is still in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If e′ is only incident to ei, EO(ei−1, ·, π) should call EO(e′, ·, π) before EO(ei, ·, π) since e′ is the local minimum in round ρ(ei) and therefore π(e′) < π(ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If e′ is incident to both ei and ei+1, EO(ei, ·, π) should call EO(e′, ·, π) before EO(ei+1, ·, π) since e′ is local minimum at round ρ(ei) and therefore π(e′) < π(ei+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In both cases, the edge oracle terminates and will not query edge ei+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, the assumption that ρ(ei) ≤ ρ(ei+2) leads to a contradiction and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now we are ready to prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For each unlikely permutation π ∈ U, there exists a query-trail of length larger than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='12, we have ρ(G, π) ≥ ⌊ β+1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since β = c log2 n, by choosing c large enough and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='11, we have that |U|/|Π| ≤ 1/n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, |U| ≤ |E′|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='/n2 which implies that |AU| ≤ |E′|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='/n2 since AU represents vertices that correspond to unlikely permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Next, we show that each vertex πB ∈ B, has at most β neighbors between likely permutations in part A in bipartite graph H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let πY be a vertex in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then πY has most β neighbors in XL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 12 Before proving this lemma, we show how we can prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='9 using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='10 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that by Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5, degree of each vertex πA ∈ A is at most O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Combining Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='10, we have E(AU, B) ≤ |E′|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='/n2 · O(n2) ≤ O � |E′|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='13, each vertex πB ∈ B has at most O(β) neighbors in AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since H is a bipartite graph, E(AL, B) ≤ O(β) · |AL|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, sum of degrees of all vertices in A is at most E(AL, B) + E(AU, B) ≤ O(β) · |AL| + O(|E′|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=') ≤ O(β · |E′|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For a random vertex in A, the expected degree is O(β·|E′|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=') |E′|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' = O(|E′|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Combining with β = c log2 n and deg(πA) = Q(⃗e, πA) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The rest of this section, we prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Before proving Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='13, we prove that if two different query-trails that are mapped to two different permutations of AL to πB ∈ B by φ, the shorter query-trail must be subgraph of the longer one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let π and π′ be two different permutations, and ⃗P and ⃗P ′ be (v, π)- and (v′, π′)- query-trail, respectively, that both end at edge ⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If φ(π, ⃗P) = φ(π′, ⃗P ′) and |⃗P| ≥ |⃗P ′|, then ⃗P ′ is a subgraph of ⃗P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We prove this lemma by series of observations and claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ⃗P = (⃗ek, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , ⃗e1) and ⃗P ′ = (⃗er′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , ⃗e1′) such that e = e1 = e′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If ⃗P ′ is not a subgraph of ⃗P, then it must branch after an edge ⃗eb ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This means that ⃗ei = ⃗ei′ for i ≤ b and ⃗ eb+1 ̸= ⃗ eb+1 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that ⃗ eb+1 and ⃗ eb+1 ′ can be copy of the same edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let π be a random permutation over E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For a (u, π)-query-trail, if f and f′ are two consecutive edges in the trail, then (f, f′) is a conflicting pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since the edge oracle calls EO(f′, ·, π) in EO(f, ·, π), (f, f′) must be a conflicting pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let f1, f2, f3 be three different edges incident to some vertex u and let π be a random permutation over E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ⃗P1 be a (x, π)-query-trail that calls EO(f3, ·, π) in EO(f1, ·, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, let ⃗P2 be a (y, π′)-query-trail that calls EO(f3, ·, π′) in EO(f2, ·, π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then (f1, f2) is a con- flicting pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='15, (f1, f3) is a conflicting pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Assume that both f1 and f3 occupied port ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, since (f2, f3) is a conflicting pair, then f2 is also occupying ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, (f1, f2) is a conflicting pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' π(eb) = π′(eb+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since ⃗ eb+1 is not in ⃗P ′, we have that φ(π′, ⃗P ′)(eb+1) = π′(eb+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, φ(π, ⃗P)(eb+1) = π(eb) since φ(π, ⃗P) shifts edges of the trail ⃗P by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Given that permutation φ(π, ⃗P) is equal to φ(π′, ⃗P ′), we have π(eb) = π′(eb+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Without loss of generality, we can assume that π(eb) ≤ π′(eb) since we did not make any difference between π and π′ until this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 13 Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' π′(eb+1) < π′(eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By combining Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='17, our assumption that π(eb) ≤ π′(eb), and the fact that π′ is a permutation, we have that π′(eb+1) < π′(eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If π(f) < π(eb) or π′(f) < π(eb) for some edge ⃗f, then π(f) = π′(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' There are five different possible cases for f: ⃗f /∈ ⃗P ∪ ⃗P ′: Since φ only changes the edge on the query-trail and φ(π, ⃗P) = φ(π′, ⃗P ′), we have π(f) = π′(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ⃗f ∈ {⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ,⃗eb−1}: Since φ(π, ⃗P)(ei+1) = φ(π′, ⃗P ′)(ei+1) for 1 ≤ i < b, we have π(ei) = π′(ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, π(f) = π′(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ⃗f = ⃗eb: In this case, condition π(f) < π(eb) does not hold since π(f) = π(eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, π′(f) = π′(eb) ≥ π(eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, condition π′(f) < π(eb) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ⃗f ∈ {⃗eb+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ,⃗ek}: By Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='8, we have that π(f) > π(eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, condition π(f) < π(eb) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ⃗f = ⃗ei for i > b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since φ(π, ⃗P) = φ(π′, ⃗P ′), we have π′(f) = π(ei−1) ≥ π(eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, none of the conditions in the claim statement holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ⃗f ∈ { ⃗ eb+1 ′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , ⃗er′}: By Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='8, we have that π′(f) > π′(eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Combining by our assumption that π′(eb) ≥ π(eb), we have π′(f) ≥ π(eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ⃗f = ⃗ei′ for i > b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since φ(π, ⃗P) = φ(π′, ⃗P ′), we have that π(f) = π′(e′ i−1) ≥ π′(eb) ≥ π(eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, none of the conditions in the claim statement holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The proof is thus complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' eb+1 ∈ RGMIS(G′, π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We prove the claim by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Assume that eb+1 /∈ RGMIS(G′, π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, there exists an edge f which is incident to eb+1 such that π′(f) < π′(eb+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, EO(eb+1, ·, π′) will recursively call EO(f, ·, π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let f be incident to ei and ei+1 for i ∈ {b, b + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In the query-trail ⃗P, EO(ei+1, ·, π) calls EO(ei, ·, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, using the Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='16, we have that (f, ei) is a conflicting pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that by Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='17, we have π′(f) < π(eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, π(f) = π′(f) < π(eb) by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since both permutations are identical for ranks lower than π(eb), edge f must appear in RGMIS(G′, π) and the query-trail ⃗P is not a valid query-trail since EO(ei, ·, π) terminates upon calling EO(f, ·, π) (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We prove that query-trail ⃗P ′ is not a valid (v, π′)-query-trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that by Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='18, EO(e′ b+1, ·, π′) calls EO(eb+1, ·, π′) before EO(eb, ·, π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='20, EO(eb+1, ·, π′) will return True and execution of EO(e′ b+1, ·, π′) terminates at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, query-trail ⃗P ′ is a subgraph of query-trail ⃗P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now we are ready to complete the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For each edge between πA ∈ AL and πB ∈ B in graph H, we write a label χ(πA, πB) on the edge which is equal to the length of the query-trail corresponding to this edge in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='14, all the labels for edges of a fixed vertex πB ∈ B that are incident to AL should be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, by the definition of likely permutations, all query-trails of permutation AL have length less than or equal to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, each vertex πB ∈ B has at most β neighbors in AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 14 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 Figure 2: Illustration of proof of Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The highlighted blue trails show query-trails ⃗P and ⃗P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Query-trail ⃗P is not valid since EO(ei, ·, π) terminates upon calling EO(f, ·, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 6 Our Estimator for Maximum Path Cover In this section, we use the oracle of the previous section to estimate the number of edges in the output of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In Section 5, we provide a lower bound on the number of recursive calls to our local query process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that this bound does not necessarily imply the same running time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For example, if we generate the whole permutation over all copies of edges before running the algorithm, it takes Θ(m) which is no longer sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Using by now standard ideas of the literature, we show in Appendix A how we can implement the query process in almost the same running time (multiplied by a polylogarithmic factor) which is formalized in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' There exists a data structure that given a graph G in the adjacency list format, (implicitly) fixes a random permutation π over its edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then for any vertex v, the data structure returns the degree of vertex v in the subgraph P produced by Algorithm 2 according to a random permutation π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Each query v to the data structure is answered in ˜O(T(v, π)) time w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' where T(v, π) is as defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that in our local query process, we need access to the adjacency list of weight-one edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' So the challenge that arises here is how to estimate the size of the output of Algorithm 2 in the adjacency matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We present a reduction from adjacency matrix to adjacency list that appeared in the literature [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In this reduction, each query to the adjacency list can be implemented with O(1) queries to the adjacency matrix and still we are able to estimate the maximum path cover with some additive error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let γ = 16Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We construct a graph ˆG = (V ˆG, E ˆG) for weight-one edges of graph G as follows: V ˆG is the union of V1, V2 and U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , Un such that: – V1 and V2 are two copies of the vertex set of the original graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 15 – Ui is a vertex set of size γ for each i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We define the edge set such that degree of each vertex is in {1, n, n + γ}: – Degree of each vertex v ∈ V1 is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The i-th neighbor of v is the i-th vertex in V1 if (v, i) ∈ E, otherwise its i-th neighbor is the i-th vertex in V2 for i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that graph (V1, EH ∩ (V1 × V1)) is isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' – Degree of each vertex v ∈ V2 is n + γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The i-th neighbor of v is the i-th vertex in V2 if (v, i) ∈ E, otherwise, its i-th neighbor is the i-th vertex in V1 for i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For all n < i ≤ n + γ, the i-th neighbor of v is i-th vertex in Uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' – Degree of each vertex u ∈ Ui is one for i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The only neighbor of u is the i-th vertex of V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By the construction of ˆG, the only neighbor of v ∈ �n i=1 Ui can be determined without any query to the adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, the i-th neighbor of each vertex in V1 ∪ V2 can be determined with one query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Observation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For each vertex v ∈ V ˆG and i ∈ [deg ˆG(v)], the i-th neighbor of vertex v can be determined using at most one query to the adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Fix a vertex v ∈ V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' When we run Algorithm 2, intuitively with high probability the first edge that is incident to v and occupies port v0 is between v and u ∈ Uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Furthermore, with high probability the first two edges that are incident to v and occupies port v1 are between v and u ∈ Uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' A vertex v ∈ V2 is an abnormal vertex if the above properties do not hold for v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let R ∈ V2 be the set of abnormal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In the following observation, we show that for each vertex v ∈ V2 \\ R, all incident edges of v in the output of Algorithm 2 are between v and vertices of Uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Eπ |R| ≤ n/(4K) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Fix a vertex v ∈ V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For a random permutation over copies of edges of ˆG, the first incident edge to v that occupies port v0 is between v and Uv with a probability of at least (K+1)γ (n+γ)(K+1) ≥ 1− 1 8K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, the first two edges that occupy v1 are between v and Uv with probability of at least γ(γ−1) (n+γ)(n+γ−1) ≥ 1 − 1 8K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since both events are independent, the probability of v not being an abnormal vertex is at least � 1 − 1 8K �2 ≥ 1 − 1 4K , which implies Eπ |R| ≤ n/(4K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For each v ∈ V2 \\ R, all incident edges of v in the output of Algorithm 2 are between v and vertices of Uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By definition of an abnormal vertex, let the first edge in the permutation incident to v be between v and w ∈ Uv which occupies v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since all copies of edges incident to w are between v and w, this edge will be added to the solution of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, we know that the first two edges that are incident to v and occupy port v1 are between v and Uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let these two edges be (v, u1) and (v, u2) where u1, u2 ∈ Uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that the only way that (v, u1) is not added to the solution of Algorithm 2 is when u1 = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In this case, since there is only one copy for each edge that occupied port v1, then u2 ̸= w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, Algorithm 2 adds (v, u2) to its output if it has not added (v, u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since both ports of v are occupied in this case, all incident edges of v in the output of Algorithm 2 are between v and vertices of Uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 16 Observation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let P be the output of Algorithm 2 on ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then 1 2ρ( ˆG[V1 ∪ R]) ≤ E |P ∩ (V1 ∪ R) × (V1 ∪ R)| ≤ (1 + 2 K ) · ρ( ˆG[V1 ∪ R]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4, if we run Algorithm 2 on ˆG, for any vertex v ∈ V2 \\ R, all incident edges of v in the output are between v and Uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, none of the edges between V2 \\ R and V1 ∪ R will be added to the output of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since, the permutation over edges of V1 ∪ R is uniformly at random, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5, we obtain the claimed bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In the above sequence of observations, we show that there are few abnormal vertices in V2, which implies that most of the incident edges to vertices of V1 in the output of Algorithm 2 are in ˆG[V1] (only those between V1 and R violate this property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, a natural way to estimate the number of edges in the output of Algorithm 2 on G, is to estimate the number of edges in ˆG[V1] in the output of Algorithm 2 on ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' With this intuition in mind, we need to bound the query complexity of the algorithm for a random vertex in V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let v be a random vertex in V1 and π be a random permutation over edges of graph that is created by copying E ˆG according to Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then Ev∼V1,π[T(v, π)] = ˜O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4, we have that Ev∼V ˆ G,π[T(v, π)] = O(K · |E ˆG| |V ˆG| log2 |V ˆG|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Summing over all vertices in V ˆG, we obtain � v∈V ˆ G Eπ[T(v, π)] = O(K · |E ˆG| · log2 |V ˆG|) = ˜O(n2), since |V ˆG| = O(n2), K = O(1/ε), and |E ˆG| = O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, for a random vertex in V1, we get Ev∼V1,π[T(v, π)] ≤ � � � v∈V ˆ G Eπ[T(v, π)] � � /|V1| = ˜O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ˜ρ be the output of Algorithm 5 on input graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' With high probability, �1 2 − 1 K � ρ(G) − n K ≤ ˜ρ ≤ ρ(G), where K is the parameter which is defined in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ˆP be the set of edges outputted by Algorithm 2 on ˆG with both endpoints in V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5, we have that E | ˆP| ≤ (1 + 2 K ) · ρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Furthermore, by Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3 and the fact that the degree of each vertex in the output of Algorithm 2 is at most two, in the output of Algorithm 2 on ˆG[V1∪R] we have at most n/(2K) edges with one endpoint in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, combining with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5 and Observation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5 we get 1 2ρ(G) − n 2K ≤ E | ˆP| ≤ (1 + 2 K ) · ρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' (2) 17 Algorithm 5: Final algorithm for maximum path cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 1 Let ˆG = (V ˆG, E ˆG) as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 2 r ← 192 · K2 · log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 3 Sample r vertices u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , ur uniformly at random from V1 with replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 4 Sample r ports p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , pr uniformly at random from {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 5 Run vertex oracle for each ui and let Xi be the indicator if port upi i is occupied with an edge in ˆG[V1] in output of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 6 Let X = � i∈[r] Xi and f = X/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 7 Let ˜ρ = K 2(K+2) · (f · n − n 4K ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 8 return ˜ρ Since each edge in the output of Algorithm 2 counted twice in Algorithm 5, we have E[Xi] = Pr[Xi = 1] = 2 E | ˆP| n , and, E[X] = 2r E | ˆP| n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' (3) Since X is sum of r independent random variables, by Chernoff bound (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2) we get Pr[|X − E[X]| ≤ � 6 E[X] log n] ≤ 2 exp � −6 E[X] log n 3 E[X] � = 2 n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Combining fn = Xn/r and the above bound, with probability of at least 1 − 2/n2 we have fn ∈ n(E[X] ± � 6 E[X] log n) r = n E[X] r ± � 6n2 E[X] log n r2 = 2 E | ˆP| ± � 12n E | ˆP| log n r (By (3)) = 2 E | ˆP| ± � n E | ˆP| 16K2 (Since r = 192 · K2 · log n) ∈ 2 E | ˆP| ± n 4K (Since E | ˆP| ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since, ˜ρ = K 2(K+2) · (f · n − n 4K ), hence K K + 2 � E | ˆP| − n 2K � ≤ ˜ρ ≤ K K + 2 · E | ˆP|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Combining with (2), implies the claimed bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Given an adjacency matrix access for input graph G, there exists a randomized algorithm that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' runs in �O(n) time and produces an estimate ˜ρ, such that �1 2 − ε � ρ(G) − εn ≤ ˜ρ ≤ ρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 18 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let K = 1 ε and ˜ρ be the output of Algorithm 5 on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In Algorithm 5, by combining Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1 and Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6, the running time for each sample is ˜O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since the number of samples is r = 192K2 log n, and K is a constant, the total running time of the algorithm is ˜O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='7 we get the approximation ratio in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 7 Our Estimator for (1,2)-TSP In this section, we use the algorithm for estimating the size of maximum path cover as a black box to estimate the size of (1,2)-TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' First, note that if there is no Hamiltonian cycle with weight one edges of the graph, then the set of weight-one edges of the graph (1,2)-TSP is a solution for maximum path cover of graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, in the case that there exists a Hamiltonian cycle, then the size of maximum path cover is n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, if P ∗ is the maximum path cover of a graph G, then it is possible to create a TSP by connecting these paths using edges with weight two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This intuition helps to formalize the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Observation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let τ(V ) be the cost of (1,2)-TSP of graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then, we have 2n − ρ(G) − 1 ≤ τ(V ) ≤ 2n − ρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now we are ready to present the final algorithm for estimating (1,2)-TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Algorithm 6: Final algorithm for (1,2)-TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 1 Construct ˆG = (V ˆG, E ˆG) implicitly as desribed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 2 Let ˜ρ be the output of Algorithm 5 on ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 3 ˜τ = 2n − ˜ρ 4 return ˜τ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ˜τ be the output of Algorithm 6 and τ(V ) be the cost of (1,2)-TSP of graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' With high probability, τ(V ) ≤ ˜τ ≤ (3 2 + 4 K ) · τ(V ), where K is the parameter which is defined in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Observation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1, we have 2n − ρ(G) − 1 ≤ τ(V ) ≤ 2n − ρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Algorithm 6 outputs ˜τ = 2n − ˜ρ as the estimate, where ˜ρ is the output of Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='7, we have 2n − ˜ρ ≥ 2n − ρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='7, we have 2n − ˜ρ ≤ 2n − (1 2 − 1 K ) · ρ(G) + n K ≤ 3n − 3ρ(G) 2 + 4n K − 2ρ(G) K − 1 (Since ρ(G) < n) ≤ (3 2 + 4 K )(2n − ρ(G) − 1) (Since K ≪ n) ≤ (3 2 + 4 K ) · τ(V ) (Since τ(V ) = 2n − ρ(G)), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 19 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let τ(V ) be the cost of (1,2)-TSP of graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any ε > 0, there exists an algorithm that estimate the cost of (1,2)-TSP, ˜τ, such that τ(V ) ≤ ˜τ ≤ (3 2 + ε) · τ(V ), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='p in ˜O(n) running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We choose K = 4 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2, if ˜τ is the output of Algorithm 6, we get τ(V ) ≤ ˜τ ≤ (3 2 + ε) · τ(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, since the running time of Algorithm 6 is the same as the running time of Algorithm 5, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='8, the total running time is ˜O(n), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 8 Our Estimator for Graphic TSP In this section, we use our algorithm for estimating the size of maximum path cover to estimate the size of graphic TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In a recent work, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [8] showed that it is possible to obtain a (27/14)-approximate algorithm for graphic TSP by estimating the matching size and the number of biconnected components in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since the size of graphic TSP is at most 2n (the cost of MST is n − 1), they proved that if a graph has large matching and a few biconnected components, the cost of graphic TSP is significantly lower than 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since estimating the number of biconnected components is not an easy task in sublinear time, they use a proxy quantity that can be estimated in sublinear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We show that if we use our estimator for maximum path cover as a black-box instead of matching estimator in algorithm of [8], we can improve the approximation ratio to 19/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, we show that we can estimate the number of bridges in ˜O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We exploit this estimation for further improvement to get a 11/6-approximate algorithm for graphic TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [8] introduced the following definition of bad vertex as a proxy for estimating the number of biconnected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1 (Bad Vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We say a vertex v ∈ V is a bad vertex, if one of the following holds: degree of v is 1, v is a cut vertex with degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In the following series of lemmas, we bound the cost of graphic TSP based on the size of maximum path cover and number of bad vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Almost all the steps of this part are similar to the algorithm for graphic TSP of [8] — except the path cover subroutine that we use instead of maximal matching subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We restate some of the useful lemmas to achieve the approximation bound that the black-box algorithm can get, and in the next subsection we improve this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' First, we prove that if the size of the maximum path cover is small, the cost of graphic TSP is bounded away from n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If the size of maximum path cover of graph G is at most ρ, then the cost of graphic TSP is at least 2n − ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 20 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let (v0, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , vn = v0) be the optimal graphic TSP of graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that the subgraph induced by weight-one edges of this cycle is a solution for path cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, at most ρ edges in cycle (v0, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , vn = v0) have weight one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' All the remaining edges have a weight of at least two which implies the claimed bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Furthermore, the following lemma from [8], provides a lower bound for a graphic TSP of graph in terms of number of bad vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3 ([8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If the number of bad vertices of graph G is at least β, then the cost of graphic TSP is at least n + β − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [8] showed that in a biconnected graph, if there exists a matching of large size, the cost of graphic TSP is significantly smaller than 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4 ([8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let G be a graph and M′ be a matching that none of its edges is bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then the cost of graphic TSP is at most 2n − 2 3|M′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We now upper bound the cost of graphic TSP in terms of size of maximum path cover and number of bad vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If the size of maximum path cover of graph G is ρ(G) and it has β bad vertices, then the cost of graphic TSP is at most 2n − 1 5(ρ(G) − 2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let l be the number of non-trivial biconnected components and M′ be a maximum matching in graph G that none of its edges is bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, let B be the number of bridges in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='9 of [8], the cost of graphic TSP is at most min{2n − 2 3|M′|, 2n − l}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that there are at least ρ(G) − B edges of the maximum path cover that are not bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since all non-bridge edges of the maximum path cover are still union of several disjoint paths, there exists a matching with size of at least half of the edges of these paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, there exist a matching of size at least 1 2(ρ(G) − B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' On the other hand, in the proof of the same lemma, they showed that l ≥ B 2 − β which implies that the cost of graphic TSP is at most min � 2n − 2 3|M′|, 2n − l � ≤ min � 2n − 1 3(ρ(G) − B), 2n − B 2 + β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' There are two possible cases: If B ≤ 2 5ρ(G) + 6 5β, then we have 2n − 1 3(ρ(G) − B) ≤ 2n − 1 3(ρ(G) − 2 5ρ(G) − 6 5β) = 2n − 1 5(ρ(G) − 2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If B > 2 5ρ(G) + 6 5β, then we have 2n − B 2 + β ≤ 2n − 1 5ρ(G) − 3 5β + β = 2n − 1 5(ρ(G) − 2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, the cost of graphic TSP is at most min � 2n − 1 3(ρ(G) − B), 2n − B 2 + β � ≤ 2n − 1 5(ρ(G) − 2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 21 Now we are ready to introduce the first algorithm for estimating the cost of graphic TSP, which uses our maximum path cover subroutine instead of the matching subroutine as a black-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In Algorithm 7, we first estimate the size of the maximum path cover and the number of bad vertices of the graph and report the graphic TSP cost in terms of the two estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The subroutine used for counting number of bad vertices is similar to the one in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2 of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6 ([8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let β be the number of bad vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any constant ε > 0, there exists an algorithm that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='p estimates the number of bad vertices ˜β, such that β ≤ ˜β ≤ β + εn, in ˜O(n) running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Algorithm 7: First algorithm for graphic TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 1 Construct ˆG = (V ˆG, E ˆG) implicitly as desribed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 2 Let ˜ρ be the output of Algorithm 5 on ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 3 Let ˜β be the estimate of number of bad vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 4 ˜τ = 2n − 1 5(˜ρ − 2˜β) 5 return ˜T Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ˜τ be the output of Algorithm 7 and τ(V ) be the cost of graphic TSP of graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' With high probability, τ(V ) ≤ ˜τ ≤ (19 10 + 1 K ) · τ(V ), where K is the parameter which is defined in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let β be the number of bad vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='7 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6 ˜ρ ≤ ρ(G) and β ≤ ˜β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, we have τ(V ) ≤ ˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='7 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6, we can estimate ρ(G) and β such that � 1 2 − 1 K � ρ(G) − n K ≤ ˜ρ and ˜β ≤ β + n K , if we choose ε = 1 K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, we have ˜τ ≤ 2n − 1 5 � (1 2 − 1 K ) · ρ(G) − n K − 2(β + n K ) � ≤ 2n − 1 5(ρ(G) 2 − 2β) + 4n 5K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' (Since ρ(G) ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' On the other hand, assume that the approximation ratio that the algorithm obtains is α + 1/K for some α ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, we get (α + 1 K ) · τ(V ) ≥ α · τ(V ) + n K (Since τ(V ) ≥ n) ≥ α · max{2n − ρ(G), n + β − 2} + n K (By Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3) ≥ α · max{2n − ρ(G), n + β} + n K − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' So in order to show that ˜τ ≤ (α + 1 K ) · τ(V ), it is sufficient to show that 2n − 1 5(ρ(G) 2 − 2β) + 4n 5K ≤ α · max{2n − ρ(G), n + β} + n K − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 22 If n is large enough, then we have 4n 5K ≤ n K − 4, which implies that we need to prove 2n − 1 5(ρ(G) 2 − 2β) ≤ α · max{2n − ρ(G), n + β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now, let ρ(G) = xn and β = yn for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' To obtain α, it suffices to solve the following program maximize α subject to 2− 1 5 ( x 2 −2y) max{2−x,1+y} ≤ α, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This is a constant size program that can be easily solved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' the solution is 19/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4 This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let τ(V ) be the cost of graphic TSP of graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any ε > 0, there exists an algorithm that estimate the cost of graphic TSP, ˜τ, such that τ(V ) ≤ ˜τ ≤ (19 10 + ε) · τ(V ), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='p in ˜O(n) running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ˜τ be the output of Algorithm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If we choose K = 1 ε, then by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='7, we have τ(V ) ≤ ˜τ ≤ (19 10 + ε) · τ(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='8 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6, estimating ˜ρ and ˜β can be done in ˜O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 9 Further Improvement for Graphic TSP In this section, we design an algorithm to estimate the number of bridges in given graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Equipped with this tool, we are able to estimate the number of non-bridge edges in the path cover which helps to improve the approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Before describing the techniques for esti- mating the number of bridges, we prove the following lemma that provides a lower bound on the cost of graphic TSP based on the number of bridges in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If the number of bridges in the graph G is at least B, then the cost of the graphic TSP is at least n + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since the metric in the graphic TSP is corresponding to the shortest path distances in graph G, then a TSP tour is corresponding to a closed walk that contains all vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, each bridge should be crossed at least two times in this walk in order for the walk to be closed and cover all vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, the cost of graphic TSP is at least n + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 4See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' this WolframAlpha link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 23 In the following series of lemmas, first, we prove that there are a few bridges that both of their endpoints have a high degree and then we show an efficient way to estimate the number of bridges that have at least one endpoint with a low degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Combining the above arguments is the main idea to estimate the number of bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any integer c ≥ 2, there exists at most 2n c bridges that both of their endpoints have a degree larger than c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let B be the set of bridges that both of their endpoints have a degree larger than c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We construct a tree, TB, with edge set equal to B such that each vertex of TB corresponds to a component of vertices that are compressed to a single vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We construct TB iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In the beginning, we consider the bridge-block tree of the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In each step, if there exists a bridge e = (u, v) (note that u and v are vertices of the tree and corresponding to a set of vertices of the original graph) such that at least one of its endpoints has a degree less than or equal to c, we merge u with v and add all edges of u to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We continue this process until there is no bridge with an endpoint of degree less than or equal to c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now, we prove that |B| ≤ 2n c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let xv denote the number of vertices in the original graph that are compressed to vertex v ∈ TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We remove vertices of TB one by one until there is no vertex in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' At each step, we remove a leaf v ∈ TB and at the end when only one vertex is remaining, we remove that vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let yv be the number of incident edges to v in TB that are removed before removing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' At the time that we are removing leaf v, we have xv +yv +1 ≥ c+1, since the endpoint of the leaf that is the component of v has at most xv incident edges in the same component in the original graph, yv incident edges to the other components that are removed before, and there is only one remaining incident edge to other components (the other endpoint of the leaf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, � v∈TB xv ≥ � v∈TB (c − yv) = (|B| + 1)c − � v∈TB yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' (4) Since vertices of each component are disjoint, we have � v∈TB xv = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, we have � v∈TB yv = |B| since each edge of B counted when one of its endpoints is deleted from the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Combining above bounds and inequality (4), we have n = � v∈TB xv ≥ (|B| + 1)c − |B| Therefore, |B| ≤ n − c c − 1 ≤ 2n c , where the last inequality holds for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let c ≥ 2 be a constant and u is a vertex such that deg(u) ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then we can test if each of incident edges of u is a bridge in O(n) total running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We can query all neighbors of u in O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Assume that {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , vr} are neighbors of u for r ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now we divide the vertices of the graph except u into r sets V1, V2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , Vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For each vertex w ̸= u, we query the distance of w to all {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , vr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let vi be the closest one to w (if there is a tie, choose the one with the lowest index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then we put w in Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that since c is a constant and r ≤ c, this step can be done in O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now we claim that (u, vj) is a bridge iff the following conditions hold: 24 For each w ∈ Vj and i ̸= j, d(w, vi) − d(w, vj) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For each w ∈ Vi such that i ̸= j, d(w, vj) − d(w, vi) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Suppose that e = (u, vj) is a bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since removing e creates two connected components Cu and Cvj, all vertices in Cvj (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Cu) have a closer distance to vj (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In other words, all shortest paths between w ∈ Vj to vi for i ̸= j, cross edges (vj, u) and (u, vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In addition, all the shortest paths between w ∈ Vi and vj for i ̸= j, cross edges (vj, u) and (u, vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, both conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now suppose that e = (u, vj) is not a bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In this case, there must be an edge between Vj and at least one of Vi as otherwise, Vj will be disconnected from the rest of the graph by removing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Without loss of generality, assume that this edge is (w, w′) such that w ∈ Vj, w′ ∈ Vi, and i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=', we assume d(w, vj) ≤ d(w′, vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since there is an edge between w and w′, we have d(w′, vj) ≤ 1 + d(w, vj) ≤ 1 + d(w′, vi), which contradicts the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' To test whether the conditions hold, we need to query the distance of each vertex to all {v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , vr} which can be done in O(n) in total since r is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let B be the number of bridges in graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any ε > 0, there exists an algorithm that outputs an estimate ˜B in ˜O(n) such that B ≤ ˜B ≤ B + εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2, there are at most εn 2 bridges with both endpoints have degree larger than 4 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ˆB be the number of bridges that at least one of their endpoint has degree of at most 4 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, B − εn 2 ≤ ˆB ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' (5) We sample r = 256 · ε−4 · log n vertices uniformly at random with replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let u be the i-th sampled vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If the degree of the vertex is larger than 4 ε, we let Xi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Otherwise, let {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , vk} be the neighbors of u where k ≤ 4 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3, we can test if each of the incident edges of u is bridge in O(n) total running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For each edge (u, vj) if deg(u) < deg(vj) or deg(u) = deg(vj) and index of u is smaller than vj, we test if the edge is bridge or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let Xi show the number of successful tests for incident edges of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Note that in the above algorithm, each bridge with low-degree endpoints only counted once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ¯X = (�r i Xi)/r and n ¯X + 3εn 4 be our final estimate of the number of bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, E[ ¯X] = ˆB/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since ¯X is the average of r independent random variables such that 0 ≤ Xi ≤ 4/ε, by Hoeffding’s inequality (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3) we obtain Pr ��� ¯X − E[ ¯X] �� ≥ ε 4 � ≤ 2 exp � − rε4 128 � = 2 n2 , where the last inequality follows from r = 256 · ε−4 · log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, with probability of 1 − 2 n2 , n ¯X ∈ n E[ ¯X] ± nε 4 = ˆB ± nε 4 (Since E[ ¯X] = ˆB/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Combining above range and inequality (5), we get B ≤ n ¯X + 3εn 4 ≤ B + εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since the number of sampled vertices is r = 256 · ε−4 · log n, the total running time is ˜O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 25 Now we are ready to introduce the improved algorithm for graphic TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Algorithm 8: Second algorithm for graphic TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 1 Construct ˆG = (V ˆG, E ˆG) implicitly as described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 2 Let ˜ρ be the output of Algorithm 5 on ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 3 Let ˜B be the estimate of number of bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 4 ˜τ = 2n − 1 3(˜ρ − ˜B) 5 return ˜T Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ˜τ be the output of Algorithm 8 and τ(V ) be the cost of graphic TSP of graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' With high probability, τ(V ) ≤ ˜τ ≤ (11 6 + 1 K ) · τ(V ), where K is the parameter which is defined in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ρ(G) be the size of maximum path cover and B be the number of bridges in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' There are at least ρ(G)−B edges of maximum path cover that are not bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' These edges construct disjoint paths which implies there exists a matching of size 1 2(ρ(G) − B) that none of its edges is bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4, the cost of graphic TSP is at most 2n − 1 3(ρ(G) − B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, since ˜ρ ≤ ρ(G) and B ≤ ˜B, we get τ(V ) ≤ ˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='7 and Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4, we have � 1 2 − 1 K � ρ(G) − n K ≤ ˜ρ and ˜B ≤ B + n K which implies ˜τ ≤ 2n − 1 3 � (1 2 − 1 K ) · ρ(G) − 2(B + n K ) � ≤ 2n − 1 3(ρ(G) 2 − 2B) + n K (Since ρ(G) ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, assume that the approximation ratio that the algorithm obtains is α + 1/K for some α ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, (α + 1 K ) · τ(V ) ≥ α · τ(V ) + n K (Since τ(V ) ≥ n) ≥ α · max{2n − ρ(G), n + B} + n K (By Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2 and Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, to show (α + 1 K ) · τ(V ) ≥ ˜τ, it is sufficient to show 2n − 1 3(ρ(G) 2 − 2B) ≤ α · max{2n − ρ(G), n + B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now, let ρ(G) = xn and B = yn for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' To obtain α, we write the following maximization problem, maximize α subject to 2− 1 3 ( x 2 −2y) max{2−x,1+y} ≤ α, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The solution to this problem is 11/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5 This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 5See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' this WolframAlpha link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 26 Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let τ(V ) be the cost of graphic TSP of graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any ε > 0, there exists an algorithm that estimate the cost of graphic TSP, ˜τ, such that τ(V ) ≤ ˜τ ≤ (11 6 + ε) · τ(V ), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='p in ˜O(n) running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ˜τ be the output of Algorithm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If we choose K = 1 ε, then by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5, we have τ(V ) ≤ ˜τ ≤ (11 6 + ε) · τ(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='8 and Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4, estimating ˜ρ and ˜B can be done in ˜O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 10 Lower Bound for Approximating Maximum Path Cover 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1 “Conditional” Hardness for the Approximation Ratio In this section, we prove that if there exists a constant α > 0 and an algorithm that returns a ( 1 2 + α)-approximate estimate for the size of maximum path cover in ˜O(n) time in a bipartite graph, then there is a (1 2 + α)-approximate algorithm for estimating the maximum matching size in ˜O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This remains an important open problem in the study of sublinear time maximum matching algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' See in particular [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This implies that short of a major result in the study maximum matchings in the sublinear time model, which have received significant attention in the literature (see [27, 2, 4, 3, 6] and references therein), our path cover algorithm has an optimal approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let G = (V, U, E) be a bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We construct a graph G′ = (V ′, U′, E′) such that a better than 1 2-approximate estimate of maximum path cover on G′ leads to a better than 1 2- approximate estimate of maximum matching in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let r be a large constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We create r copies of G, showing the i-th copy with Gi = (Vi, Ui, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, we create another r − 1 copies H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , Hr−1 of G with Hi = (Vi, Ui+1, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now we let the G′ = (�r i=1 Gi) ∪ (�r−1 i=1 Hi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now we claim that the size of maximum path cover of the graph G′ is roughly 2r · µ(G) which can be used as an estimator for the maximum matching of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Before proving the main result of this section, we characterize some properties of the constructed graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' µ(G′) = r · µ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' First, since all graphs {Gi}r i=1 are the same as G and are vertex-disjoint, if we consider the maximum matching of G in each of the r graphs, we will have a matching of size r · µ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, µ(G′) ≥ r · µ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ˆV ∪ ˆU be the minimum vertex cover of G such that ˆV ∈ V and ˆU ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By K¨onig’s Theorem (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1), we have | ˆV ∪ ˆU| = µ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now we show there exists a vertex cover of size r ·µ(G) for graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ˆVi ∈ Vi (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ˆUi ∈ Ui) be the copy of vertices V (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' U) in graph Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We claim (�r i=1 ˆVi ∪ ˆUi) is a vertex cover for G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If an edge is in Gi, then at least one of its endpoints is in ˆVi ∪ ˆUi since ˆVi ∪ ˆUi is a vertex cover of Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Moreover, by the construction, ˆVi ∪ ˆUi+1 is a vertex cover of Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, each edge of Hi is also covered by the vertex cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, since there exists a vertex cover of size |(�r i=1 ˆVi ∪ ˆUi)| = r · | ˆV ∪ ˆU| = r · µ(G), then we have µ(G′) ≤ r · µ(G) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 27 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Figure 3: Illustration of graph G′ = (V ′, U′, E′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Each Gi is shown by a rectangle and each Hi is shown by a parallelogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Top and bottom horizontal lines illustrate Vi and Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Blue highlighted parts represent the vertex cover of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Observation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' It holds (2r − 1) · µ(G) ≤ ρ(G′) ≤ 2r · µ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since the union of maximum matching of all graphs {Gi}r i=1 and {Hi}r−1 i=1 creates a path cover, we get (2r − 1) · µ(G) ≤ ρ(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Futhermore, if there exists a path cover of size larger than 2r ·µ(G), then the maximum matching of these paths will be larger than r ·µ(G) which contradicts Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, ρ(G′) ≤ 2r · µ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now we are ready to show the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For any constant α > 0, if there exists an algorithm that can estimate the maximum path cover within a ( 1 2 +α)-factor in O(T(n)) time, then the same algorithm can be used to estimate the maximum matching of bipartite graph G within a (1 − ε) · ( 1 2 + α)-factor in O(T(n/ε)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We construct graph G′ as described at the beginning of the section with r = 1 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Observation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2, (1 ε − 1) · µ(G) ≤ ρ(G′) ≤ 1 ε · µ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let ˜ρ be the estimate of the algorithm for the maximum path cover of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, we have (1 2 + α)(1 ε − 1) · µ(G) ≤ ˜ρ ≤ 1 ε · µ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now let ˜µ = ε · ˜ρ be the estimate for the maximum matching of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Hence, (1 − ε)(1 2 + α) · µ(G) ≤ ˜µ ≤ µ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since the number of vertices and number of edges of G′ is r = 1 2ε times more than G, then the running time will be O(T(n/ε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' A reduction to matchings can also be proved for (1, 2)-TSP, albeit with an extra promise for the matching instance that the matching is either perfect or half-perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' This problem, formalized below, also remains open in the study matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We show that a better than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5-approximation for (1, 2)-TSP in �O(n) time would resolve this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Problem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Suppose that we are given a bipartite graph G = (L, R, E) with |L| = |R| = n and are promised that either µ(G) = n or µ(G) = ( 1 2 + ε)n/2 for any desirably small constant ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Provided adjacency matrix access to the graph, does there exist an �O(n) time algorithm that distinguishes the two?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 28 Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' If there is an algorithm that estimates the size of (1, 2)-TSP within a ( 3 2 − ε0)- factor for some fixed constant ε0 ∈ (0, 1 4] in �O(n), then Problem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4 can indeed be solved in �O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let G1 and G2 be two graphs with n vertices such that µ(G1) = n and µ(G2) = ( 1 2 + ε0 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We construct graph G′ 1 = (V ′ 1, E′ 1) and G′ 2 = (V ′ 2, E′ 2) as described at the beginning of the section with r = 1 ε0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' By Observation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2, we have ρ(G′ 1) ≥ ( 2 ε0 − 1)n and ρ(G′ 2) ≤ ( 1 ε0 + 1 8)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Thus, by Observation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1, we get τ(V ′ 1) ≤ 4 ε0 n − ( 2 ε0 − 1)n = ( 2 ε0 + 1)n, τ(V ′ 2) ≥ 4 ε0 n − ( 1 ε0 + 1 8)n − 1 ≥ ( 3 ε0 − 1 4)n, for sufficiently large n, which implies τ(V ′ 2) τ(V1) = 3 − ε0/4 2 + ε0 ≥ 3 2 − ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Therefore, the algorithm for (1,2)-TSP can distinguish between G′ 1 and G′ 2 which implies Prob- lem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='4 can be solved in �O(n) time for ε = ε0/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='2 Information-Theoretic Lower Bounds on the Running Time Since any constant approximation algorithm for estimating maximum path cover can be used to estimate the size of matching within a constant factor, then all of the lower bounds for O(1)- approximating maximum matching in sublinear time also hold for (1)-approximating maximum path cover in sublinear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' We restate some of these lower bounds along with a short proof (see [2] for a detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Any algorithm that estimates maximum path cover within a constant multiplicative factor requires Ω(n) queries in the adjacency list model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Consider two graphs that the first one does not have any edge and the second one has only a single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In order to give any multiplicative approximation for maximum path cover, the algorithm needs to find the edge which requires Ω(n) queries in the adjacency list model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Any algorithm that estimates maximum path cover within a constant multiplicative factor require Ω(n2) queries in the adjacency matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Consider the same construction as Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' To give any multiplicative approximation for maximum path cover, the algorithm needs to find the edge which requires Ω(n2) queries in the adjacency matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Any algorithm that estimates maximum path cover within a multiplicative-additive requires Ω(n) queries in the adjacency matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Consider a graph with no edge and a graph with one Hamiltonian cycle and no other edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In order for the algorithm to distinguish between these two graphs, it must find at least one edge of the second graph which requires Ω(n) queries in the adjacency matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 29 There is also a lower bound for multiplicative-additive estimation of matching in adjacency list model [24] that also holds for maximum path cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Any algorithm that estimates maximum path cover within a constant multiplicative- additive factor requires Ω( ¯d) queries in the adjacency list model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Mohammad Roghani and Amin Saberi were supported by NSF award 1812919 and ONR award 141912550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Soheil Behnezhad and Aviad Rubinstein were supported by NSF CCF-1954927, and a David and Lucile Packard Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Soheil Behnezhad was additionally supported by NSF Awards 1942123, 1812919 and by Moses Charikar’s Simons Investigator Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' References [1] Anna Adamaszek, Matthias Mnich, and Katarzyna Paluch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' New approximation algorithms for (1, 2)-tsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 9:1–9:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [2] Soheil Behnezhad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Time-Optimal Sublinear Algorithms for Matching and Vertex Cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 873–884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [3] Soheil Behnezhad, Mohammad Roghani, and Aviad Rubinstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Sublinear time algorithms and complexity of approximate maximum matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' CoRR, abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='15843, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [4] Soheil Behnezhad, Mohammad Roghani, Aviad Rubinstein, and Amin Saberi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Beating Greedy Matching in Sublinear Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' SIAM, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [5] Piotr Berman and Marek Karpinski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 8/7-approximation algorithm for (1, 2)-tsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In Proceed- ings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22-26, 2006, pages 641–648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ACM Press, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [6] Sayan Bhattacharya, Peter Kiss, and Thatchaphol Saranurak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Sublinear algorithms for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='5+ε)-approximate matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' CoRR, abs/2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='00189, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [7] Guy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Blelloch, Jeremy T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Fineman, and Julian Shun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Greedy Sequential Maximal Inde- pendent Set and Matching are Parallel on Average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In 24th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA ’12, Pittsburgh, PA, USA, June 25-27, 2012, pages 308–317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ACM, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [8] Yu Chen, Sampath Kannan, and Sanjeev Khanna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Sublinear Algorithms and Lower Bounds for Metric TSP Cost Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), volume 168 of Leibniz International Proceedings in Informatics (LIPIcs), pages 30:1–30:19, Dagstuhl, Germany, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Schloss Dagstuhl–Leibniz-Zentrum f¨ur Informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ISBN 978-3-95977-138-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [9] Yu Chen, Sanjeev Khanna, and Zihan Tan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Sublinear Algorithms and Lower Bounds for Estimating MST and TSP Cost in General Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' CoRR, abs/2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='14798, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 30 [10] Nicos Christofides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Worst-case analysis of a new heuristic for the travelling salesman problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Technical report, Carnegie-Mellon Univ Pittsburgh Pa Management Sciences Research Group, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [11] Artur Czumaj and Christian Sohler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Estimating the weight of metric minimum spanning trees in sublinear-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 175–183, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [12] Artur Czumaj and Christian Sohler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Estimating the weight of metric minimum spanning trees in sublinear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=', 39(3):904–922, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [13] Shayan Oveis Gharan, Amin Saberi, and Mohit Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' A randomized rounding approach to the traveling salesman problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 550–559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' IEEE Computer Society, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [14] Anupam Gupta and Krzysztof Onak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='info Open Problem 71: Metric TSP Cost Approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Available at https://sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='info/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='title=Open_ Problems:71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [15] Anna R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Karlin, Nathan Klein, and Shayan Oveis Gharan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' A (slightly) improved approximation algorithm for metric TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In STOC ’21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 32–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ACM, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [16] Richard M Karp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Reducibility among combinatorial problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In Complexity of computer computations, pages 85–103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Springer, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [17] Marek Karpinski and Richard Schmied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' On approximation lower bounds for TSP with bounded metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Colloquium Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=', page 8, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' K¨onig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ¨Uber graphen und ihre anwendung auf determinantentheorie und mengenlehre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Mathematische Annalen, 77:453–465, 1916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' URL http://eudml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='org/doc/158740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [19] Matthias Mnich and Tobias M¨omke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Improved integrality gap upper bounds for traveling salesperson problems with distances one and two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=', 266(2):436–457, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [20] Tobias M¨omke and Ola Svensson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Approximating graphic TSP by matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 560–569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' IEEE Computer Society, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [21] Marcin Mucha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 13/9-approximation for graphic TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, February 29th - March 3rd, 2012, Paris, France, volume 14 of LIPIcs, pages 30–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [22] Krzysztof Onak, Dana Ron, Michal Rosen, and Ronitt Rubinfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' A near-optimal sublinear- time algorithm for approximating the minimum vertex cover size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In Proceedings of the Twenty- Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1123–1131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' SIAM, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [23] Christos H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Papadimitriou and Mihalis Yannakakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The traveling salesman problem with distances one and two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=', 18(1):1–11, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 31 [24] Michal Parnas and Dana Ron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Theoretical Computer Science, 381(1):183–196, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ISSN 0304-3975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [25] Andr´as Seb¨o and Jens Vygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Shorter tours by nicer ears: 7/5-approximation for the graph-tsp, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=', 34(5):597–629, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [26] Anatoliy I Serdyukov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' O nekotorykh ekstremal’nykh obkhodakh v grafakh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Upravlyayemyye sistemy, 17:76–79, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' [27] Yuichi Yoshida, Masaki Yamamoto, and Hiro Ito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' An improved constant-time approximation algorithm for maximum matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 225– 234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' ACM, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' A Implementation Details In this section, we discuss why Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1, restated below, holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' There exists a data structure that given a graph G in the adjacency list format, (implicitly) fixes a random permutation π over its edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Then for any vertex v, the data structure returns the degree of vertex v in the subgraph P produced by Algorithm 2 according to a random permutation π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Each query v to the data structure is answered in ˜O(T(v, π)) time w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' where T(v, π) is as defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content='1 uses standard ideas from the literature [22, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The only modification, essentially, is to show that these algorithms also work for multi-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Let us focus on the specific algorithm proposed in [2, Appendix A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Given the adjacency list of a graph G = (V, E), it defines gives a procedure LOWEST(v, i) that first draws a random rank E → [0, 1] on each edge (implicitly), then for any input vertex v and an integer i ≤ degG(v), returns a vertex u such that (v, u) is the i-th lowest rank edge incident to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' It is proved in [2] that if the procedure is called for a fix vertex v and all indices i with 1 ≤ i ≤ r, then the total running time is ˜O(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' The only difference between the implementation of our algorithm and the one in [2] is that we have multiple copies of a single edge in the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' First, we observe that the procedure LOWEST(v, i), in addition to returning the neighbor u, can also return the rank of the edge (v, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' (This is explicitly computed by LOWEST(v, i) in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=') Now let G′ be the multigraph with K copies of each edge of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Instead of a multigraph, we can assume that we have K copies of same graph G called G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' , GK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Also, for each i, let LOWESTGi be the LOWEST procedure corresponding to graph Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' For each vertex v, we use a balanced binary search tree (BST) that stores the ranks of the lowest incident edge to v in each graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' So at any point during the course of the algorithm, there are at most K different values in the BST of vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Now for the next LOWEST query to the multigraph graph G′ for vertex v, we can return the minimum edge in the BST of vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' Since K is a constant and the any query to a BST is answered in O(log n) time, the total running time will be the same as [2, Appendix A] within a O(log n)-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} +page_content=' 32' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQf8w5p/content/2301.05350v1.pdf'} diff --git a/N9E3T4oBgHgl3EQfZQq9/vector_store/index.faiss b/N9E3T4oBgHgl3EQfZQq9/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e1f63c7cd09e5018a5479bb55175c7154d5cb635 --- /dev/null +++ b/N9E3T4oBgHgl3EQfZQq9/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:00f5a2ea98c1c0266792bc24193ed9fe73b4d572343a343b4cb7a47f90e3eb9a +size 1310765 diff --git a/P9AzT4oBgHgl3EQfIvtp/content/tmp_files/2301.01067v1.pdf.txt b/P9AzT4oBgHgl3EQfIvtp/content/tmp_files/2301.01067v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd1368850a51824992e25551b9343a5b65652c80 --- /dev/null +++ b/P9AzT4oBgHgl3EQfIvtp/content/tmp_files/2301.01067v1.pdf.txt @@ -0,0 +1,1578 @@ +Towards Knowledge-Intensive Text-to-SQL Semantic Parsing with +Formulaic Knowledge +Longxu Dou1∗, Yan Gao2, Xuqi Liu1, Mingyang Pan1, Dingzirui Wang1, +Wanxiang Che1, Min-Yen Kan3, Dechen Zhan1, Jian-Guang Lou2 +1 Harbin Institute of Technology 2 Microsoft Research Asia +3 National University of Singapore +{lxdou, xqliu, mypan, dzrwang, car}@ir.hit.edu.cn, dechen@hit.edu.cn, +{yan.gao, jlou}@microsoft.com, kanmy@comp.nus.edu.sg +Abstract +In this paper, +we study the problem of +knowledge-intensive text-to-SQL, in which +domain knowledge is necessary to parse ex- +pert questions into SQL queries over domain- +specific tables. +We formalize this sce- +nario by building a new Chinese benchmark +KNOWSQL +consisting of domain-specific +questions covering various domains. We then +address this problem by presenting formulaic +knowledge, rather than by annotating addi- +tional data examples. +More concretely, we +construct a formulaic knowledge bank as a do- +main knowledge base and propose a frame- +work (REGROUP) to leverage this formulaic +knowledge during parsing. Experiments using +REGROUP demonstrate a significant 28.2% +improvement overall on KNOWSQL. +1 +Introduction +Text-to-SQL translates user queries into executable +SQL, greatly facilitating interactions between users +and relational databases. Along with the release +of large-scale benchmarks (Zhong et al., 2017; Yu +et al., 2018, 2019a,b) and developments in model +design (Wang et al., 2020a; Cao et al., 2021), text- +to-SQL works are now achieving promising results +in both research and practical applications (Zeng +et al., 2020). +However, in the professional application of text- +to-SQL, such as in the data analysis of financial +reports, models require external knowledge to map +the expert query with the domain-specific database. +Take the financial query for example: What’s the +EBIT1 of Walmart?, where the underlying database +has component columns that can be used to calcu- +late the EBIT. We treat this problem as knowledge- +intensive text-to-SQL, where domain knowledge +is highly necessary to parse expert questions over +∗Contribution during the internship at Microsoft Research +Asia. +1EBIT is Earnings Before Interest and Tax, and is calcu- +lated as Revenue – Cost of Goods Sold – Operating Expenses. +What‘s the balance of trade of BRIC countries? +SELECT Ship_Out – Ship_In, Nation +FROM Reports WHERE +Nation in (‘Brazil’, ‘Russia’, ‘India’, +‘China’) +Ground +Parse +Retrieval +Balance of Trade = Exports - Imports +BRIC countries: Country in {Brazil, Russia, +India, China} +Trade Surplus : Export > Import +Formulaic Knowledge Bank +Nation +GDP +Ship In +Ship Out +A +14.72 +1550 +2650 +B +5.04 +581 +623 +Figure 1: Harnessing REGROUP with formulaic knowl- +edge for knowledge-intensive text-to-SQL with three +steps: +(1) Retrieval the formulaic knowledge; (2) +Ground the concept of formulaic knowledge; (3) Parse +the question. +domain-specific tables. This problem prevents text- +to-SQL techniques from being fielded in novel, +professional applications to assist the experts in +processing data. +Traditional approaches would address this prob- +lem by annotating specific question/SQL pairs on +a target domain (Wang et al., 2015; Herzig and +Berant, 2019). Then such mappings are induced +during the training process. This approach does +work but has the drawback that any induced infor- +mation is both fragile and expertise-heavy: such +knowledge does not port across domains and re- +quires expert knowledge to craft. +We propose to solve this problem by modeling +how a non-expert person might tackle this prob- +lem. When meeting unseen examples (as in the +EBIT case above), they may first search for the re- +lated mathematical formulas from public resources, +then ground the concepts referenced in the formu- +las with schema elements presented in their partic- +ular databases. This process leverages common, +encoded formulaic knowledge that are already de- +scribed in publicly-available resources such as tu- +arXiv:2301.01067v1 [cs.CL] 3 Jan 2023 + +torials, textbooks, encyclopedias, and references. +Inspired by this, we propose to address the +knowledge-intensive text-to-SQL through formu- +laic knowledge which provides the evidence of +mapping from domain-specific phrases presented +in questions to actual SQL operations over schema +elements. More concretely, we define a taxonomy +of three types of formulaic knowledge: calcula- +tion, union, and condition, each corresponding to +a particular snippet of SQL. Then we propose RE- +GROUP, a text-to-SQL framework (Fig. 1), con- +sisting of three stages: (1) Retrieve the formu- +laic knowledge from formulaic knowledge bank +as an external knowledge source; (2) Ground the +concept of formulaic knowledge to the schema +elements; (3) Parse the results with the ques- +tion, schema, and grounded formulaic knowledge. +The external formulaic knowledge bank imbues +REGROUP with formulaic knowledge, making it +knowledgeable. REGROUP is also extensible be- +cause updating the formulaic knowledge bank does +not require retraining any modules. +Moreover, we construct a Chinese benchmark +KNOWSQL, to examine the effectiveness of RE- +GROUP framework. +It advances the existing +knowledge-intensive text-to-SQL beyond the previ- +ous work (Wang et al., 2020b; Zhao et al., 2022) by +considering more SQL operations and challenging +domains. Experimental results demonstrate the RE- +GROUP with formulaic knowledge would improve +the performance by 23.4% overall. Furthermore, +we classify error cases into three classes, which are +resolvable by advancing the corresponding mod- +ule of REGROUP. Finally, we discuss the potential +future work such as expanding the scope of knowl- +edge and advancing REGROUP model design. +Our contributions are summarised as follows: +• To the best of our knowledge, we are the first +to explore knowledge-intensive text-to-SQL +and propose a challenging Chinese benchmark +KNOWSQL, which requires domain-specific +knowledge. +• We propose a novel framework REGROUP +to address knowledge-intensive text-to-SQL +by retrieving and grounding formulaic knowl- +edge, which is knowledge-extensible. +• Experimental results demonstrate the effec- +tiveness of REGROUP with formulaic knowl- +edge which achieves 28.2% overall improve- +ment on KNOWSQL. +2 +Knowledge-Intensive Text-to-SQL +2.1 +Problem Analysis +After studying the real cases in professional data +analysis, we roughly categorize the required knowl- +edge for knowledge-intensive text-to-SQL into +three classes : (1) linguistic knowledge enables the +model to adapt to linguistic diversity; (2) domain +knowledge allows the model to perceive domain- +specific sayings and concepts; (3) mathematical +knowledge yields the specific SQL operations (e.g., +Density phrase to division operation). These three +sets of knowledge jointly provide the evidence of +mapping from domain-specific phrases of questions +to actual SQL operations over schema elements. +However, most text-to-SQL researches focus on +general scenario (Yu et al., 2018; Zhong et al., +2017), where linguistic knowledge is mainly re- +quired. Recently, Wang et al. (2020b) and Zhao +et al. (2022) promote text-to-SQL to more chal- +lenging scenarios via involving the calculation +questions. In this paper, we further explore the +knowledge-intensive text-to-SQL by considering +more operations (e.g., calculation, union, and condi- +tion) with more challenging domains which require +all these three classes of knowledge. +2.2 +Challenge +Despite that pre-trained language models contain +linguistic knowledge, they lack domain knowledge +and mathematical knowledge. +Therefore, the +model would meet two problems: +(1) don’t +know which operations to use: +if an opera- +tion (e.g., +density = total number / space) has +never occurred in training data, +the model +rarely employ this unseen operation during +the inference; (2) don’t know how to adapt +operations: the model would fail to generalize +the operation across domains. +For instance, +the model cannot generalize the calculation of +Population Density (number of people / land area) +to Car Density (number of cars / parking lot area). +Accordingly, we consider that the vanilla pre- +trained language model is (1) narrow since it +only supports the limited operation and (2) in- +efficient since it can’t generalize the operation +across domains. However, it’s time-consuming and +expertise-heavy to directly increase the amount of +annotated data examples. In contrast, we address +this challenge from the view of formulaic knowl- +edge in Sec 3, which is more knowledge-extensible. + +2.3 +KNOWSQL Benchmark +#DB +#Question +#Formulaic +Train +160 +23, 157 +328 +Dev +40 +2, 731 +122 +Finance +217 +1, 392 +219 +Estate +35 +749 +79 +Transportation +36 +439 +82 +Table 1: The dataset statistic of KNOWSQL. +To uncover the knowledge-intensive text-to- +SQL problem and advance the research, we con- +struct a challenging Chinese text-to-SQL bench- +mark named KNOWSQL. Roughly, it consists of +two parts: training/dev sets built on the existing +DuSQL (Wang et al., 2020b) dataset and a newly +constructed test set on three professional domains +with discovered knowledge in DuSQL. +2.3.1 +Building Training/Dev Set on DuSQL +We build the training/dev set of KNOWSQL based +on the existing DuSQL, a Chinese multi-table +text-to-SQL benchmark. We categorize its 200 +databases into 16 domains like sports, energy, +health care, foods, etc. Given the high quality +of DuSQL schema and broad domain coverage, +it’s a satisfactory start-point to build a challeng- +ing knowledge-intensive text-to-SQL benchmark. +However, the domain-specific question is not well +included in DuSQL, where most of the questions +could be answered easily without relying on exter- +nal knowledge and only considers one SQL oper- +ation (i.e., calculation). Given that, we extend the +original DuSQL by adding more domain-specific +questions and involving more operations in both the +train set and the dev set. Eventually, KNOWSQL +expands the size of DuSQL train set from 22,521 +to 23,157 and the dev set from 2,482 to 2,731. +2.3.2 +Building Test Set from Scratch +To simulate the professional data analysis scenario, +we create a challenging test set covering three do- +mains (finance, estate, and transportation). These +three domains have high data analysis requirements +in real life. Different from the train/dev sets, we +construct the test set from the scratch by: (1) col- +lecting the domain-specific tables, and (2) annotat- +ing the domain-specific questions and correspond- +ing SQL queries. +Table Collection. +For collecting table schema, +we collect the tables from the following source: +(1) the public annual reports of the company (2) +the industry reports (3) academic papers (4) the +statistical reports released by the government. To +ensure the table quality, we conduct several pre- +processing procedures. Firstly, we convert matrix +tables (present in annual reports) into relational +tables to make the question SQL-answerable. Next, +to ensure the table data quality, we conduct data +cleaning (e.g., filtering out the irrelevant columns +to simplify the table structure, and normalizing the +headers to reduce the noise). Finally, to avoid data +privacy issues, we conduct value anonymization +(e.g., removing direct identifiers and anonymizing +geo-related data). +Question Annotation. +It’s challenging for anno- +tators to propose the domain-specific questions +without background knowledge 2. +Thus, we +train the annotators first about the domain-specific +knowledge via (1) collecting the jargon (i.e., ab- +breviation, terminology) from the domain-specific +open resources, which are widely adopted by do- +main experts (e.g., EBIT for finance) but unusual +for a layperson; (2) to mimic the domain expert by +asking questions using the jargon with the above +materials. +After that, the annotators would annotate the +questions and SQL with the following criteria: (1) +be faithful to the given table (i.e., don’t exceed the +scope of table columns and table content); (2) not +be directly answerable by the single element of the +table but could be answered by the operation over +existing columns; (3) limited to first-order opera- +tion (i.e., excludes multi-hop questions like ‘What +is the gross profit?’, where the table only contains +‘Sales’, ‘Average Price’ and ‘Cost of Goods Sold’ +so that model needs to compute the ‘revenue’ first). +2.3.3 +Dataset Quality and Data Statistic +To guarantee the data quality, we conduct a multi- +rounds check. Finally, the inter-agreement of anno- +tators reaches 94.7% 3. During each round, we ask +each annotator to review others’ annotations based +on the criteria (stated above), then ask them to fur- +ther improve annotations that do not meet the cri- +teria. As shown in Tab.1, the test set contains 288 +databases and 2,580 questions. Notably, all these +challenging data examples in the test set could be +covered by 380 formulaic knowledge, which will +be discussed in Sec. 3. +2See Sec. 8 for annotator payment and profile. +3The inter-annotator agreement is calculated as the per- +centage of overlapping votes about whether it’s a correct and +domain-specific question. + +Textual Knowledge +Formulaic Knowledge +Figure 2: Two types of knowledge in expressing the +calculation of EBIT: textual knowledge and formulaic +knowledge. +3 +Approach: Formulaic Knowledge +3.1 +Motivation +When meeting unseen examples, the human may +first search the related mathematical knowledge +or domain knowledge from textbooks or encyclo- +pedias. As shown in Fig. 2, the information of +calculation of EBIT is returned in both textual and +formulaic format. Intuitively, the formulaic for- +mat is preferred because it’s (1) more concise and +precise: for instance, a adds b times c is more am- +biguous than a+b*c or (a+b)*c; (2) easy to obtain: +most description of calculation is stored in a formu- +laic format in the textbook, tutorials, and academic +paper; (3) SQL parser friendly: the formulaic for- +mat is closed to the snippet of SQL then easily for +the parser to generate4. +3.2 +Formulaic Knowledge for Text-to-SQL +Following this idea, we focus on three categories +of operations (Fig. 3): calculation, union, and +condition. Besides the popular calculation knowl- +edge, we also consider the taxonomy information +as union knowledge and the judgment standard as +condition knowledge. The design insight here is +that the left part is the name of the knowledge item, +and the right part expresses its semantic meaning +represented by operations over concepts. Note that +4In Sec. 6, experimental results prove that formulaic format +receives better performance than textual format. +all operations are consistent with SQL grammar, +making it closer to SQL query. Besides the entity, +the left part of formulaic knowledge might also be +the SQL function (e.g., NOW()) or constant (e.g., +threshold of Real Estate Bubble). +3.3 +Formulaic Knowledge Bank +We further build a formulaic knowledge bank with +1,954 formulaic knowledge items, which supports +19 domains involved in KNOWSQL. Importantly, +the bank covers all these examples of KNOWSQL +as shown in Tab. 2. +Note that this bank is a +domain-related resource, not one tied to the spe- +cific database. Thus, this bank is more general +and could be utilized in other applications natural +language applications (e.g., question answering) 5. +Criteria +The design of the formulaic knowledge +follows three criteria: (1) Only the first-order (flat) +formulaic knowledge is considered (i.e., the con- +cept in the formulaic item should be align-able to +the schema elements rather than another formulaic +item) ; (2) The stored formulaic knowledge should +be both faithful (i.e., acknowledged by the expert) +and standardized (i.e., shared at the domain level); +(3) The formulaic knowledge should be domain- +level (i.e., not tied to the specific schema elements). +Collection +We collect the formulaic knowledge +from the following public resource: (1) Baidu +Wenku, the platform where the domain experts +usually share the domain knowledge of various +domain6; (2) CNKI, China’s largest academic web- +site7; (3) the data analysis websites of a specific +domain, like ESPN for sports and Yahoo for fi- +nance. We also collect some knowledge from the +English resource and let annotators translate this +domain knowledge into Chinese. +Abstraction +To make the formulaic knowledge +more generic, we propose to accumulate the for- +mulaic knowledge at the domain level instead of +database-specific. Specifically, we abstract the con- +cept of formulaic knowledge before storing them in +the knowledge bank, which indicates the operation +over concept rather than specific schema. For ex- +ample, we would extract the formulaic knowledge +from ‘People Density in China 2020 = total num- +ber of Chinese in 2020 / Chinese Land Area’ to +‘People Density = total number of People / Area’. +5See fine-grained statistic of bank in Appendix A.1. +6https://wenku.baidu.com/ +7https://oversea.cnki.net/index/ + +how to compute EBIT +X +Q +Q All + Images + Videos +国 News + Shopping +: More +Tools +About196,000results(0.44seconds) +EBIT is calculated by subtracting a company's cost of goods sold (COGs) and its +operating expenses from its revenue. EBlT can also be calculated as operating revenue +and non-operating income, less operating expenses. +https://www.investopedia.com>...>FinancialStatements +: +Earnings Before Interest and Taxes (EBIT) - Investopedia +? +Aboutfeaturedsnippets· +DFeedback +People also ask : +What is the formula to calculate EBIT? +How to Calculate EBIT +1. EBIT = Net Income + Interest + Taxes. +2. EBIT = Revenue - COGS - Operating Expenses. +3. EBIT = Gross Profit - Operating Expenses. +Feb25,2022 +https://www.masterclass.com>articles>ebit-explained +Earnings Before Interest and Taxes: How to Calculate EBlT +Search for: What is the formula to calculate EBIT?Operation +Calculation +Union +Condition +Formulaic +Knowledge +Trade Balance = Exports – Imports +BRIC Countries : +Country in {Brazil, Russia, India, China} +Trade Surplus : Export > Import +Abstract +Phrase = Schema1 op Schema2 +Phrase : Schema in Set +Phrase : Schema1 op Schema2 +Example +What‘s the balance of trade of China? +SELECT Exports -Imports FROM Reports +WHERE Country=China +Show me the sum of GDP of BRIC countries? +SELECT sum(GDP) FROM Reports WHERE +Country in (Brazil, Russia, India, China) +GROUP By Name +Which country has a trade +surplus problem? +SELECT Country FROM Reports +WHERE Export > Import +Figure 3: We consider three types of formulaic knowledge to address knowledge-intensive text-to-SQL. +#Formulaic +#Calculation +#Union +#Condition +Formulaic Knowledge Bank +1, 954 +1, 102 +346 +506 +KNOWSQL involved +891 +656 +52 +183 +Table 2: The dataset statistic of formulaic knowledge bank and its overlap with KNOWSQL. +Consequently, only ONE formulaic knowledge is +required to address MANY schema elements to cal- +culate the density of animals/cars/shops. +Mapping within KNOWSQL +We further exam- +ine the overlap between formulaic knowledge bank +and KNOWSQL benchmark. As stated in Sec 2.3.3, +all questions from KNOWSQL are covered by for- +mulaic knowledge banks. Specifically, there are +1,954 knowledge items in the bank, and 891 items +are used for answering the KNOWSQL questions +as shown in Table 2. Especially, there are extra +1,063 knowledge items beyond KNOWSQL which +could support future work in applying formulaic +knowledge. +4 +REGROUP Framework +To address the knowledge-intensive text-to-SQL +problem, we propose a novel framework named +REGROUP, consisting of three stages: (1) Retrieve +the formulaic knowledge from the formulaic knowl- +edge bank as an external knowledge source; (2) +Ground the concept of formulaic knowledge to +the schema elements (e.g., Exports to Ship_Out); +(3) Parse the results with the question, schema, +and grounded formulaic knowledge. As shown in +Fig. 4, REGROUP consists of three models: re- +triever, grounding model, and parser. We will give +a brief introduction of each model in the follow- +ing8. +8More details of the model implementation could be found +in Appendix B. +4.1 +Retriever Model +The goal of the retriever is to extract the rele- +vant formulaic knowledge items from the formu- +laic knowledge bank (Fig. 4). The challenge is +the fine-grained modeling of the formulaic knowl- +edge to disambiguate the ones with the same intent +but differing in operation over concepts, such as +calculating EBIT in different ways. We directly +utilize the off-the-shelf Dense Passage Retriever +(DPR) (Karpukhin et al., 2020) which was origi- +nally designed for open-domain QA. It employs a +bi-encoder architecture to learn the dense represen- +tation of sentences and passages, then it computes +the dot-product between the representations as the +similarity score. +To adapt the DPR in the formulaic knowledge re- +trieval task, we treat the formula knowledge bank as +the passage candidate and concatenate the question +with flattened schema (separated by special token +‘|’) to enrich the semantics of the question. Then +we follow the standard DPR training procedure to +optimize the bi-encoder. Specifically, during the +training process, we derive the positive knowledge +items from KNOWSQL annotation and sample five +negative examples from the formulaic knowledge +bank. During the inference process, we first cache +the embedding of formulaic knowledge items, then +leverage the FAISS algorithm (Johnson et al., 2017) +to rank each formulaic knowledge item. +4.2 +Grounding Model +Given the retrieved knowledge items, the goal of +the grounding model is to edit the formulaic knowl- + +Parser +Which company from BRIC countries +has the largest profit ? +SELECT Name, Revenue - Cost +FROM Company WHERE Nation +in (‘Brazil’, ‘Russia’, ‘India’, ‘China') +ORDER BY Revenue - Cost +DESC LIMIT 1 +Name +Nation +Continents +Revenue +Cost +A +China +Asia +124.5 +10.9 +B +America +North America +223.1 +48.1 +Retriever +BRIC Countries: Country in +{Brazil, Russia, India, China} +Profit = Revenue – Cost of Revenue +– Selling and Maintenance Expense ++ Other Income - Tax +Grounding Model +BRIC Countries: Nation in +{Brazil, Russia, India, China} +Profit = Revenue – Cost +A, 113.6 +• +Age = now() - Date of Birth +• +Density = Number / Area +• +Speed = Distance / Time +• +BRIC Countries: Country in +{Brazil, Russia, India, China} +• +EBIT = Net Income + +Interest + Taxes +• +EBIT = Revenue – Cost of +Goods Solds – Operation +Expense +• +Profit = Revenue – Cost of +Revenue – Selling and +Maintenance Expense + +Other Income - Tax +Formulaic Knowledge Bank +Figure 4: Pipeline of REGROUP: (1) Retrieve the formulaic knowledge item from the bank; (2) Ground the +concept of formulaic knowledge into schema elements; (3) Parse the question with grounded formulaic knowledge +into SQL. +edge items w.r.t specific schema through (1) remov- +ing the irrelevant concept and (2) instantiating the +concept with the schema elements. The main chal- +lenge is the expensive annotations of grounding +(i.e. supervision). Therefore, the weakly super- +vised grounding approaches would be more suit- +able. Specifically, we leverage the Erasing-then- +Awakening (ETA) model proposed by (Liu et al., +2021), which was originally designed for ground- +ing the entity from the knowledge base to the entity +mentioned in the question. The output of ETA is a +confidence matrix, indicating the possible ground- +ing relations between entity mentions and entities. +To adapt the ETA in the formulaic knowledge +grounding task, we treat each knowledge item as +the ‘question’ and attempt to figure out which spe- +cific schema elements are grounded in the knowl- +edge item. Specifically, it’s determined by a hyper- +parameter H to indicate the threshold of confi- +dence (whether it’s grounded and which one it’s +grounded). As shown in Fig. 4, we filter the concept +(cross outed parts) under the confidence threshold +H and replace the concept with aligned elements +(green parts). +4.3 +Parser Model +The goal of the parser is to predict the executable +SQL according to question and database schema. +The main challenge is how to model the database +structure to infer the implicit schema mentioned, +and how to make use of the grounded knowl- +edge (i.e., knowledge-fusion) to leverage grounded +knowledge. We are inspired by the recent progress +in adopting the large pre-trained language model in +semantic parsing problems. For instance, (Scholak +et al., 2021; Shin et al., 2021; Dou et al., 2022; +Xie et al., 2022) achieves excellent performance +on several semantic parsing tasks under the sim- +ple pretrained language model framework, such as +BART (Lewis et al., 2020) and T5 (Raffel et al., +2020). +Given that, we propose to adopt UniSAr (Dou +et al., 2022) as the base parser in this work. It im- +proves the vanilla BART with three non-invasive +extensions and achieves SOTA or competitive per- +formance on seven text-to-SQL benchmarks. Con- +cretely, the input of the model is the concatena- +tion of the question, serialized schema, and re- +trieved formulaic knowledge. We propose that the +parser should correctly adopt the grounded formu- +laic knowledge during SQL generation. +5 +Experimental Results and Analysis +To evaluate our approach: REGROUP with exter- +nal formulaic knowledge bank, we conduct several +experiments on KNOWSQL benchmark. We report +both the overall results of the pipeline and the fine- +grained results of each module. We also conduct +error analysis and categorize the bad cases into +three main classes. Note that we report the average +experimental results of each setting during three +runs9. +9Code and data are available at link. + +Model +Dev +Finance +Estate +Transportation +Average +Vanilla +69.3 +8.7 +5.7 +6.9 +22.7 +REGROUP (w/o Grounding) +71.7 +38.1 +25.1 +32.7 +41.9 +REGROUP +74.6 +43.7 +46.1 +39.1 +50.9 +REGROUP (Oracle) +78.4 +71.4 +84.8 +64.7 +74.8 +Table 3: Overall results on different KNOWSQL splits. Oracle refers to the use of the oracle formulaic knowledge. +The evaluation metric is SQL exact set match. Average indicates the micro-average score of the first four columns. +Data +Model +R@1 +R@3 +R@10 +Dev +BM-25 +67.9 +89.1 +96.5 +REGROUP +73.0 +89.8 +96.5 +Finance +BM-25 +39.4 +66.5 +85.9 +REGROUP +46.0 +68.1 +86.1 +Table 4: Results of REGROUP retriever compared with +BM-25 on KNOWSQL dev and finance splits. The eval- +uation metric is the Recall. +Data +Model +Precision +Recall +F1 +Dev +FuzzyMatch +69.3 +62.5 +65.7 +REGROUP +71.3 +70.4 +70.8 +Finance +FuzzyMatch +35.3 +31.5 +33.2 +REGROUP +42.9 +44.7 +43.8 +Table 5: Results of REGROUP grounding model com- +pared with fuzzy string match on KNOWSQL dev and +finance splits. +5.1 +Experimental Setup +The retriever returns the top-3 retrieved formu- +laic knowledge items from the bank. The ground- +ing model further aligns the concept in formulaic +knowledge into schema elements and the decision +threshold H is 0.6 which is decided empirically. +The parser receives the grounded knowledge, table +schema, and user query as the input and then out- +puts the SQL. For the parsing baseline, we adopt +UniSAr (Dou et al., 2022) as the vanilla parser10. +5.2 +Overall Results +As shown in Tab. 3, we could observe that: (1) +REGROUP exceeds the vanilla model by 28.2%, +which indicates the effectiveness of using formu- +laic knowledge; (2) grounding the formulaic knowl- +edge improves the REGROUP by 9.0%; (3) the or- +acle formulaic knowledge (retrieve correctly and +grounding correctly) reaches the upper bound of +REGROUP 74.8%, which implies the potential im- +provement room for KNOWSQL. +10More implementation details could be found in Ap- +pendix B +5.3 +Fine-grained Results +We compare the retriever and grounding model +with other baselines, on both the dev set and the +test set of KNOWSQL in the finance domain, to +examine the performance in general and domain- +specific scenarios. +Retriever +We compare the retriever of RE- +GROUP (bi-encoder) with BM-25 (Robertson and +Zaragoza, 2009). The evaluation metric is the re- +call score over retrieved results. We observe that +the finance domain is more challenging than the +general domain (dev split) since it contains many +homogeneous formulaic knowledge items that ex- +press the same intention in the left part but with +different computation ways in the right part. For +example, there are two ways to compute the ‘EBIT’ +in Fig. 4. +Grounding +We compare the grounding model +of REGROUP with the fuzzy string match-based +method 11. Following the previous work (Lei et al., +2020; Liu et al., 2021), we report the micro-average +precision, recall, and F1-score. We could observe +that: (1) the model-based grounding improves the +performance by 5.1% and 10.6% respectively; (2) +the domain-specific data like Finance poses more +challenging cases than the general domain, where +finance is behind the dev by about 27.0%. +5.4 +Error Analysis +We sample 300 cases from the dev split and 100 +cases from finance/estate/transportation in the test +split respectively (600 in total) for error analysis. +Vanilla Model Error +We first compare the cor- +rect case of REGROUP while predicted incorrectly +by the vanilla model. As the example in the first +part of Fig. 5, the vanilla model is unable to pre- +dict the unseen operation during training. In con- +trast, the grounded formulaic knowledge enables +11It enumerates all n-gram (n ≤ 5) of the concepts in for- +mulaic knowledge, and links each of them to schema element +by fuzzy string matching. + +Vanilla Model Error +Formulaic Knowledge +Question: 东三省每省的一胎出生率是多少? +(What is the first birth rate in each of the three northeastern provinces in China?) +Schema : 省份 ｜ 婴儿出生率 | 二胎出生率 | 人口 +(Province | Birth Rate | Second Birth Rate | Population) +Vanilla: SELECT 婴儿出生率 FROM 各省人口出生及死亡率 WHERE 省份 = "辽宁" +ReGrouP: SELECT 婴儿出生率 - 二胎出生率 FROM 各省人口出生及死亡率 WHERE 省 +份 IN ("辽宁" , "吉林" , "黑龙江" ) +Grounded Formulaic Knowledge: +东三省 : { 辽宁 , 吉林 , 黑龙江 } +(Three Northeastern Provinces: { Liaoning , Jilin , Heilongjiang }) +一胎出生率 = 婴儿出生率 - 二胎出生率 +(First birth rate = Birth rate - Second Birth Rate) +Retriever Error (43%) +Retrieval Knowledge +Question:息税前利润是多少? +(Please return the Earnings Before Interest and Taxes ) +Schema: 收入 | 净收入 | 销售费用 |营业费用 | 销售额 +(Revenue| Net Income | Cost of Goods Sold Expenses | Operating Expenses | Sales) +Gold SQL: SELECT 收入 - 销售费用 - 营业费用 FROM 报表 +Pred SQL: SELECT 净收入 + 销售额 FROM 报表 +Oracle Formulaic Knowledge: +息税前利润 = 收入 - 销售成本 - 营业费用 +(Earnings Before Interest and Taxes = Revenue – Cost of Goods Sold – +Operating Expenses ) +Retrieved Formulaic Knowledge: +息税前利润 = 净收入 + 利息 + 税 +(Earnings Before Interest and Taxes = Net Income + Interest + Taxes ) +Grounding Error (41%) +Grounded Knowledge +Question: A公司的流动资产是多少? +(What is company A's current assets?) +Schema:现金 | 应收款项 | 可销售证券|商品成本| 运营费用 +(Cash | Trade Receivables | Marketable Securities | Cost of Goods | Operating Expenses) +Gold SQL: SELECT 应收款项 + 可销售证券 +现金 FROM 报表 +Pred SQL: SELECT 应收款项 + 现金 FROM 报表 +Undergrounded Formulaic Knowledge: +流动资产 = 短期资本 + 应收帐款 + 股票 + 存款余额 +(Current Assets = Short Term Capital + Debtors + Stock + Cash and bank) +Correct Grounded Formulaic Knowledge: +流动资产 = 应收款项 + 可销售证券 + 现金 +(Current Assets = Trade Receivables + Marketable Securities + Cash) +Prediced Grounded Formulaic Knowledge: +流动资产 = 应收款项 + 现金 +(Current Assets = Trade Receivables + Cash) +Parser Error (12%) +Leveraging Knowledge +Question: 哪个城市的房地产市场发展合理? +(Which city's real estate market is developing reasonably?) +Schema: 城市 | 吸纳率 | 空置率 +(City | Commercial Housing Absorption Rate | Commercial Housing Vacancy Rate) +Gold SQL: SELECT 城市 FROM 报表 where 空置率 > 15% and 空置率 < 30% +Pred SQL: SELECT 城市 FROM 报表 where 空置率 > 15% +Grounded Formulaic Knowledge: +房地产市场良性发展 : 空置率 > 15% AND 空置率 < 30% +(Good development of real estate market: Commercial Housing Vacancy +Rate > 15% AND Commercial Housing Vacancy Rate < 30%) +Figure 5: Case studies of REGROUP. We first compare it with the vanilla parsing model. Then we classify the bad +cases of REGROUP into three categories: (1) Retriever Error: not getting the knowledge from bank; (2) Grounding +Error: not learning the knowledge by alignment; (3) Parser Error: not using the grounded knowledge in generation. +REGROUP to predict the operation over schema +elements correctly. +Then we categorize the error of REGROUP into +three main classes and list their percentage in Fig. 5. +Finally, we discuss the potential future work in +improving each part of REGROUP. An advantage +of REGROUP is the decoupled framework could +track each type of bad case individually, avoiding +the catastrophic forgetting problem. +Retrieval Error +About 43% errors are attributed +to the retriever where the model doesn’t get the +correct knowledge from bank since it can’t distin- +guish the semantic difference between the closed +formulaic knowledge items. Future work should +improve its distinguishing ability by fine-grained +modeling, like attention mechanism Huang et al. +(2019). +Grounding Error +About 41% errors are caused +by incorrect grounded knowledge where the model +doesn’t learn the knowledge by alignment since +it can’t correctly align the concept to schema ele- +ments. Future work should focus on how to derive +the grounding information under weak supervision +or even without supervision. It would greatly allevi- +ate the severe annotation effort in specific domains. +Parsing Error +There are still 8% error cases +caused by parsing, where the formulaic knowledge +is correctly retrieved and grounded but the parser +still doesn’t use the grounded knowledge in gen- +eration well. Future work should improve it by +explicitly modeling the copy process of knowledge +from the input to the SQL snippet position, such as +implementing the additional gate mechanism. +Other Error +The remaining 8% errors are about +the SQL generation, such as the GROUP-BY or +nested SQL. Since it’s not our main focus, we ig- +nore these cases in Fig. 5 for brevity. +6 +Discussion +Is formulaic knowledge better than textual +knowledge for text-to-SQL? +In Sec.3.1, we ar- +gue that formulaic knowledge is preferred over tex- +tual knowledge intuitively. Empirically, we con- +duct the experiments by the following steps: (1) +transforming the formulaic knowledge to textual + +knowledge through annotators; (2) training the re- +triever and parser with textual knowledge under the +same experiment setting as formulaic knowledge. +Experimental results reveal that textual knowledge +receives an overall performance degradation of +13.6% compared with Table 3. We conclude that +REGROUP prefers formulaic knowledge since it’s +more close to the SQL snippets or schema repre- +sentation. Moreover, formulaic knowledge is both +precise and concise. In contrast, textual knowledge +is redundant and much more diverse in expressing +the equivalent meaning. +What’s the cost of collecting formulaic knowl- +edge? +During the collection process of formulaic +knowledge bank (19 domains), we found most do- +mains have the public knowledge resource. More- +over, the effort spent on collection formulaic knowl- +edge is also acceptable compared with annotating +data examples. For example, we spent 4 hours in +collecting 219 formulaic knowledge in the finance +domain, which is far more effective than annotat- +ing the equivalent data examples. Eventually, for- +mulaic knowledge improves the performance by +35.0% without retraining the model as shown in +Table 3 (from 8.7% to 43.7%). +How to expand the scope of formulaic knowl- +edge further? +In this paper, we mainly focus +on domain knowledge and mathematical knowl- +edge and transfer them into formulaic knowledge +format for model learning. Other types of knowl- +edge would improve the knowledge-intensive text- +to-SQL further, such as the commonsense knowl- +edge (e.g., water freezing point: temperature=0) +or personalized information (e.g., favourite food: +Tiramisu). Thus, we could package these types of +knowledge into a formulaic format in future work. +7 +Related Work +7.1 +Domain Generalization of Text-to-SQL +To be applicable in real scenarios, a text-to-SQL +model should generalize to new domains with- +out relying on expensive domain-specific labeled +data. Previous work has shown that current text- +to-SQL usually fails on domain generalization sce- +narios (Finegan-Dollak et al., 2018). Recent ap- +proaches track this problem including data synthe- +sis (Yin et al., 2021), meta-learning (Wang et al., +2021) and encoder pretraining (Yin et al., 2020; +Herzig et al., 2020). Most recently, Zhao et al. +(2022) proposed to adopt schema expansion and +scheme pruning to preprocess the table schemas. +We highlight that compared with the schema- +expansion approach, the advantage of our ap- +proach (REGROUP with formulaic knowledge) is +the broad knowledge scope: we not only consider +the calculation knowledge but also union knowl- +edge and condition knowledge. Moreover, our ap- +proach is extensible with an external and maintain- +able formulaic knowledge bank. +7.2 +Retrieval Enhanced Semantic Parsing +There has been a recent trend toward leveraging +retrieval-enhanced methods in various NLP tasks +such as machine translation (Cai et al., 2019) and +question answering (Karpukhin et al., 2020). Sim- +ilar with REGROUP, previous work (Gupta et al., +2022; Pasupat et al., 2021) leverage a retrieval step +to provides examples as the context of input for +seq2seq model learning. +However, our approach differs in two ways: (1) +our retrieval object is grounded formulaic knowl- +edge which contains more condensed information +than data example; (2) prior work directly leverage +the retrieved results. We leverage the grounding +model to edit the retrieved formulaic knowledge to +make it more relevant to the question and schema. +8 +Conclusion and Future Work +This paper explores formulaic knowledge to ad- +dress the knowledge-intensive text-to-SQL prob- +lem, which would advance the professional appli- +cation of text-to-SQL such as data analysis for do- +main experts. First, we analyze the challenge of +knowledge-intensive text-to-SQL and construct a +new challenging benchmark KNOWSQL. Then we +propose to address this problem from the view of +formulaic knowledge. Concretely, we propose a +simple framework REGROUP to leverage an exter- +nal formulaic knowledge bank. Experimental re- +sults reveal that REGROUP with formulaic knowl- +edge achieves the 28.2% improvements overall. +We further discuss three directions in improv- +ing the REGROUP via analyzing different types +of bad cases: (1) iterative filling in the blank of +formulaic knowledge bank; (2) mitigating the gap +between formulaic knowledge and specific schema +via improving the grounding model; (3) driving the +parser to fully make use of more complicated (e.g., +commonsense) formulaic knowledge. + +Ethical Considerations +This work presents KNOWSQL, a free and open +dataset for the research community to study +the knowledge-intensive text-to-SQL problem. +Data in KNOWSQL are constructed based on +DuSQL (Wang et al., 2020b) , a free and open +cross-database Chinese text-to-SQL dataset. We +also collect formulaic and table data from CNKI12 +and Baidu Wenku13, which are also free and open +for academic usage. The content of the table is +anonymized to address the privacy issue. To anno- +tate the KNOWSQL, we recruit 3 Chinese college +students (1 female and 2 males). Each student is +paid 4 yuan ($0.6 USD) for annotating the (SQL, +question) pairs and 2 yuan ($0.3USD) for collect- +ing the formulaic knowledge items. This compen- +sation is determined according to the prior simi- +lar dataset construction (Guo et al., 2021). Since +all question sequences are collected against open- +access databases or public tables, there is no pri- +vacy issue. +Limitations +(1) KNOWSQL is built based on DuSQL, a Chinese +large-scale text-to-SQL dataset. Thus the language +coverage of this paper is limited to Chinese. We +leave the extension to other languages for future +work. (2) For the scope of formulaic knowledge, +we mainly address three types of knowledge to +associate with each SQL phrase: calculation, union, +and condition. Some types of knowledge are under- +explored such as commonsense knowledge. (3) For +the model design of REGROUP, we build it from +improving many existing works. Despite achieving +promising evaluation results, the case studies reveal +that many challenging remains during the retrieval, +grounding, or parsing. +Acknowledgement +We thank all anonymous reviewers for their con- +structive comments. Wanxiang Che was supported +via the grant 2020AAA0106501 and NSFC grants +62236004 and 61976072. Dechen Zhan is the cor- +responding author. +References +Deng Cai, Yan Wang, Wei Bi, Zhaopeng Tu, Xi- +aojiang Liu, and Shuming Shi. 2019. +Retrieval- +12https://oversea.cnki.net/index/ +13https://wenku.baidu.com/ +guided dialogue response generation via a matching- +to-generation framework. +In Proceedings of the +2019 Conference on Empirical Methods in Natu- +ral Language Processing and the 9th International +Joint Conference on Natural Language Processing +(EMNLP-IJCNLP), pages 1866–1875. Association +for Computational Linguistics. +Ruisheng Cao, Lu Chen, Zhi Chen, Yanbin Zhao, +Su Zhu, and Kai Yu. 2021. LGESQL: Line graph en- +hanced text-to-SQL model with mixed local and non- +local relations. In Proceedings of the 59th Annual +Meeting of the Association for Computational Lin- +guistics and the 11th International Joint Conference +on Natural Language Processing (Volume 1: Long +Papers), pages 2541–2555. Association for Compu- +tational Linguistics. +Yiming Cui, Wanxiang Che, Ting Liu, Bing Qin, and +Ziqing Yang. 2021. Pre-training with whole word +masking for chinese bert. +IEEE/ACM Transac- +tions on Audio, Speech, and Language Processing, +29:3504–3514. +Longxu Dou, Yan Gao, Mingyang Pan, Dingzirui +Wang, Wanxiang Che, Dechen Zhan, and Jian- +Guang Lou. 2022. +Unisar: +A unified structure- +aware autoregressive language model for text-to-sql. +arXiv. +Catherine Finegan-Dollak, Jonathan K. Kummerfeld, +Li Zhang, Karthik Ramanathan, Sesh Sadasivam, +Rui Zhang, and Dragomir Radev. 2018. +Improv- +ing text-to-SQL evaluation methodology. +In Pro- +ceedings of the 56th Annual Meeting of the Associa- +tion for Computational Linguistics (Volume 1: Long +Papers), pages 351–360. Association for Computa- +tional Linguistics. +Jiaqi Guo, Ziliang Si, Yu Wang, Qian Liu, Ming Fan, +Jian-Guang Lou, Zijiang Yang, and Ting Liu. 2021. +Chase: A large-scale and pragmatic Chinese dataset +for cross-database context-dependent text-to-SQL. +In Proceedings of the 59th Annual Meeting of the +Association for Computational Linguistics and the +11th International Joint Conference on Natural Lan- +guage Processing (Volume 1: Long Papers), pages +2316–2331, Online. Association for Computational +Linguistics. +Vivek Gupta, Akshat Shrivastava, Adithya Sagar, +Armen Aghajanyan, and Denis Savenkov. 2022. +RetroNLU: Retrieval augmented task-oriented se- +mantic parsing. In Proceedings of the 4th Workshop +on NLP for Conversational AI, pages 184–196. As- +sociation for Computational Linguistics. +Jonathan Herzig and Jonathan Berant. 2019. +Don’t +paraphrase, detect! rapid and effective data collec- +tion for semantic parsing. +In Proceedings of the +2019 Conference on Empirical Methods in Natu- +ral Language Processing and the 9th International +Joint Conference on Natural Language Processing +(EMNLP-IJCNLP), pages 3810–3820, Hong Kong, +China. Association for Computational Linguistics. + +Jonathan Herzig, Pawel Krzysztof Nowak, Thomas +Müller, Francesco Piccinno, and Julian Eisenschlos. +2020. TaPas: Weakly supervised table parsing via +pre-training. +In Proceedings of the 58th Annual +Meeting of the Association for Computational Lin- +guistics, pages 4320–4333. Association for Compu- +tational Linguistics. +Hsin-Yuan Huang, Eunsol Choi, and Wen-tau Yih. +2019. +Flowqa: Grasping flow in history for con- +versational machine comprehension. +In 7th Inter- +national Conference on Learning Representations, +ICLR. +Jeff Johnson, Matthijs Douze, and Hervé Jégou. 2017. +Billion-scale similarity search with gpus. arXiv. +Vladimir Karpukhin, Barlas Oguz, Sewon Min, Patrick +Lewis, Ledell Wu, Sergey Edunov, Danqi Chen, and +Wen-tau Yih. 2020. +Dense passage retrieval for +open-domain question answering. In Proceedings of +the 2020 Conference on Empirical Methods in Nat- +ural Language Processing (EMNLP), pages 6769– +6781. +Wenqiang Lei, Weixin Wang, Zhixin Ma, Tian Gan, +Wei Lu, Min-Yen Kan, and Tat-Seng Chua. 2020. +Re-examining the role of schema linking in text-to- +SQL. +In Proceedings of the 2020 Conference on +Empirical Methods in Natural Language Process- +ing (EMNLP), pages 6943–6954, Online. Associa- +tion for Computational Linguistics. +Mike +Lewis, +Yinhan +Liu, +Naman +Goyal, +Mar- +jan Ghazvininejad, Abdelrahman Mohamed, Omer +Levy, Veselin Stoyanov, and Luke Zettlemoyer. +2020. BART: Denoising sequence-to-sequence pre- +training for natural language generation, translation, +and comprehension. In Proceedings of the 58th An- +nual Meeting of the Association for Computational +Linguistics, pages 7871–7880, Online. Association +for Computational Linguistics. +Qian Liu, Dejian Yang, Jiahui Zhang, Jiaqi Guo, Bin +Zhou, and Jian-Guang Lou. 2021. +Awakening la- +tent grounding from pretrained language models for +semantic parsing. +In Findings of the Association +for Computational Linguistics: ACL-IJCNLP 2021, +pages 1174–1189. +Myle Ott, Sergey Edunov, Alexei Baevski, Angela +Fan, Sam Gross, Nathan Ng, David Grangier, and +Michael Auli. 2019. +fairseq: A fast, extensible +toolkit for sequence modeling. +In Proceedings of +the 2019 Conference of the North American Chap- +ter of the Association for Computational Linguistics +(Demonstrations), pages 48–53, Minneapolis, Min- +nesota. Association for Computational Linguistics. +Panupong Pasupat, Yuan Zhang, and Kelvin Guu. 2021. +Controllable semantic parsing via retrieval augmen- +tation. In Proceedings of the 2021 Conference on +Empirical Methods in Natural Language Processing, +pages 7683–7698. +Adam Paszke, Sam Gross, Francisco Massa, Adam +Lerer, James Bradbury, Gregory Chanan, Trevor +Killeen, Zeming Lin, Natalia Gimelshein, Luca +Antiga, Alban Desmaison, Andreas Kopf, Edward +Yang, Zachary DeVito, Martin Raison, Alykhan Te- +jani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, +Junjie Bai, and Soumith Chintala. 2019. +Pytorch: +An imperative style, high-performance deep learn- +ing library. In Advances in Neural Information Pro- +cessing Systems 32, pages 8024–8035. +Peng Qi, Yuhao Zhang, Yuhui Zhang, Jason Bolton, +and Christopher D. Manning. 2020. +Stanza: +A +python natural language processing toolkit for many +human languages. In Proceedings of the 58th An- +nual Meeting of the Association for Computational +Linguistics: +System Demonstrations, pages 101– +108. +Colin Raffel, Noam Shazeer, Adam Roberts, Kather- +ine Lee, Sharan Narang, Michael Matena, Yanqi +Zhou, Wei Li, and Peter J. Liu. 2020. +Exploring +the limits of transfer learning with a unified text-to- +text transformer. Journal of Machine Learning Re- +search, 21(140):1–67. +Stephen Robertson and Hugo Zaragoza. 2009. +The +Probabilistic Relevance Framework: BM25 and Be- +yond. Foundations and Trends in Information Re- +trieval. +Torsten Scholak, Nathan Schucher, and Dzmitry Bah- +danau. 2021. PICARD: Parsing incrementally for +constrained auto-regressive decoding from language +models. In Proceedings of the 2021 Conference on +Empirical Methods in Natural Language Processing, +pages 9895–9901. +Richard Shin, Christopher H. Lin, Sam Thomson, +Charles Chen, Subhro Roy, Emmanouil Antonios +Platanios, Adam Pauls, Dan Klein, Jason Eisner, and +Benjamin Van Durme. 2021. Constrained language +models yield few-shot semantic parsers. +CoRR, +abs/2104.08768. +Bailin Wang, Mirella Lapata, and Ivan Titov. 2021. +Meta-learning for domain generalization in semantic +parsing. In Proceedings of the 2021 Conference of +the North American Chapter of the Association for +Computational Linguistics: Human Language Tech- +nologies, pages 366–379. Association for Computa- +tional Linguistics. +Bailin Wang, Richard Shin, Xiaodong Liu, Oleksandr +Polozov, and Matthew Richardson. 2020a. +RAT- +SQL: Relation-aware schema encoding and linking +for text-to-SQL parsers. In Proceedings of the 58th +Annual Meeting of the Association for Computa- +tional Linguistics, Online. Association for Compu- +tational Linguistics. +Lijie Wang, Ao Zhang, Kun Wu, Ke Sun, Zhenghua +Li, Hua Wu, Min Zhang, and Haifeng Wang. 2020b. +DuSQL: A large-scale and pragmatic Chinese text- +to-SQL dataset. +In Proceedings of the 2020 Con- +ference on Empirical Methods in Natural Language + +Processing (EMNLP), pages 6923–6935, Online. As- +sociation for Computational Linguistics. +Yushi Wang, Jonathan Berant, and Percy Liang. 2015. +Building a semantic parser overnight. In Proceed- +ings of the 53rd Annual Meeting of the Association +for Computational Linguistics and the 7th Interna- +tional Joint Conference on Natural Language Pro- +cessing (Volume 1: Long Papers), pages 1332–1342, +Beijing, China. Association for Computational Lin- +guistics. +Thomas Wolf, Lysandre Debut, Victor Sanh, Julien +Chaumond, Clement Delangue, Anthony Moi, Pier- +ric Cistac, Tim Rault, Remi Louf, Morgan Funtow- +icz, Joe Davison, Sam Shleifer, Patrick von Platen, +Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, +Teven Le Scao, Sylvain Gugger, Mariama Drame, +Quentin Lhoest, and Alexander Rush. 2020. Trans- +formers: State-of-the-art natural language process- +ing. In Proceedings of the 2020 Conference on Em- +pirical Methods in Natural Language Processing: +System Demonstrations, pages 38–45. +Tianbao Xie, Chen Henry Wu, Peng Shi, Ruiqi Zhong, +Torsten Scholak, Michihiro Yasunaga, Chien-Sheng +Wu, Ming Zhong, Pengcheng Yin, Sida I. Wang, Vic- +tor Zhong, Bailin Wang, Chengzu Li, Connor Boyle, +Ansong Ni, Ziyu Yao, Dragomir Radev, Caiming +Xiong, Lingpeng Kong, Rui Zhang, Noah A. Smith, +Luke Zettlemoyer, and Tao Yu. 2022. Unifiedskg: +Unifying and multi-tasking structured knowledge +grounding with text-to-text language models. arXiv +preprint arXiv:2201.05966. +Pengcheng Yin, Graham Neubig, Wen-tau Yih, and Se- +bastian Riedel. 2020. TaBERT: Pretraining for joint +understanding of textual and tabular data. In Pro- +ceedings of the 58th Annual Meeting of the Asso- +ciation for Computational Linguistics, pages 8413– +8426. Association for Computational Linguistics. +Pengcheng Yin, John Wieting, Avirup Sil, and Graham +Neubig. 2021. +On the ingredients of an effective +zero-shot semantic parser. arXiv. +Tao Yu, Rui Zhang, Heyang Er, Suyi Li, Eric Xue, +Bo Pang, Xi Victoria Lin, Yi Chern Tan, Tianze +Shi, Zihan Li, Youxuan Jiang, Michihiro Yasunaga, +Sungrok Shim, Tao Chen, Alexander Fabbri, Zifan +Li, Luyao Chen, Yuwen Zhang, Shreya Dixit, Vin- +cent Zhang, Caiming Xiong, Richard Socher, Walter +Lasecki, and Dragomir Radev. 2019a. CoSQL: A +conversational text-to-SQL challenge towards cross- +domain natural language interfaces to databases. In +Proceedings of the 2019 Conference on Empirical +Methods in Natural Language Processing and the +9th International Joint Conference on Natural Lan- +guage Processing (EMNLP-IJCNLP), pages 1962– +1979, Hong Kong, China. Association for Computa- +tional Linguistics. +Tao Yu, Rui Zhang, Kai Yang, Michihiro Yasunaga, +Dongxu Wang, Zifan Li, James Ma, Irene Li, +Qingning Yao, +Shanelle Roman, +Zilin Zhang, +and Dragomir Radev. 2018. +Spider: +A large- +scale human-labeled dataset for complex and cross- +domain semantic parsing and text-to-SQL task. In +Proceedings of the 2018 Conference on Empirical +Methods in Natural Language Processing, pages +3911–3921. +Tao Yu, Rui Zhang, Michihiro Yasunaga, Yi Chern +Tan, Xi Victoria Lin, Suyi Li, Heyang Er, Irene +Li, Bo Pang, Tao Chen, Emily Ji, Shreya Dixit, +David Proctor, Sungrok Shim, Jonathan Kraft, Vin- +cent Zhang, Caiming Xiong, Richard Socher, and +Dragomir Radev. 2019b. SParC: Cross-domain se- +mantic parsing in context. +In Proceedings of the +57th Annual Meeting of the Association for Com- +putational Linguistics, pages 4511–4523, Florence, +Italy. Association for Computational Linguistics. +Jichuan Zeng, Xi Victoria Lin, Steven C.H. Hoi, +Richard Socher, Caiming Xiong, Michael Lyu, and +Irwin King. 2020. Photon: A robust cross-domain +text-to-SQL system. In Proceedings of the 58th An- +nual Meeting of the Association for Computational +Linguistics: System Demonstrations, pages 204–214. +Association for Computational Linguistics. +Chen Zhao, Yu Su, Adam Pauls, and Emmanouil An- +tonios Platanios. 2022. Bridging the generalization +gap in text-to-SQL parsing with schema expansion. +In Proceedings of the 60th Annual Meeting of the +Association for Computational Linguistics (Volume +1: Long Papers), pages 5568–5578. Association for +Computational Linguistics. +Victor Zhong, Caiming Xiong, and Richard Socher. +2017. +Seq2sql: +Generating structured queries +from natural language using reinforcement learning. +CoRR, abs/1709.00103. + +A +Details of Formulaic Knowledge Bank +A.1 +Knowledge Source +We construct the formulaic knowledge bank across +19 domains following KNOWSQL and 1 misc do- +main. The misc domain stores the infrequent or +general knowledge items in KNOWSQL, such as +the calculation of density, and speed. +In the fol- +lowing, we will briefly analyze the collected bank. +A.2 +Statistic Across Domain +Different domains have different amounts of pub- +licly available data online. As shown in Fig. 6, not +unsurprisingly, finance and estate share the most +plentiful publicly available resource. +A.3 +Distribution within Domain +We also observe the different distribution of knowl- +edge across domains. If the domain focus on cal- +culation (e.g., finance report and fund), we assume +the data analysis tends to be more objective, which +is easier for model learning. If the domain focus +on condition (e.g., estate and awards), we assume +the data analysis tends to be more subjective since +it’s more challenging in learning semantics. +B +Implementation Details of REGROUP +Retriever +We implement the retriever based +on the code of Karpukhin et al. (2020)14. +We +adopt the Chinese BERT-wwm-ext (Cui et al., +2021) as pretrained encoder15. It would return +the top-3 retrieved formulaic knowledge. Future +work could improve the negative sampling by in- +batch sampling or BM25-based sampling following +Karpukhin et al. (2020). +Grounding Model +The code of ETA16 is not re- +leased at the time of submission of this paper. We +re-implement the ETA model based on the paper +using pytorch (Paszke et al., 2019). We evaluate +our implemented model with the original model on +SPIDER-L (Lei et al., 2020) to examine whether the +re-implemented model works. Our model achieves +82.1% column F1 score where Liu et al. (2021) +reported 82.5%. The experiments on KNOWSQL +also employ the Chinese BERT. +14Code of DPR Retrieval Model +15Chinese-BERT-wwm Model +16Code of ETA Grounding Model +Parser +We build the paper based on the code +of Dou et al. (2022)17 We choose the mBART- +CC2518 as the base model to fine-tune. Following +the vanilla model, we build the input of parser as +follows: [schema] | [grounded formulaic knowl- +edge] | [question], where ‘|’ is the delimiter across +different parts. +Resource and Tools +For tokenization, we em- +ploy Stanza (Qi et al., 2020) considering its excel- +lent performance. For the retriever and grounding +model, we import the BERT model with Trans- +former library (Wolf et al., 2020). For parser mode, +we preprocess the data and fine-tune the mBART +with fairseq framework (Ott et al., 2019) +Device and Training Time +We conduct all these +experiments on one NVIDIA TESLA V100-32GB +GPU. The training of the retriever, grounding +model, and parser takes about 4 hours, 3 hours, and +8 hours respectively. The minimum device require- +ment is NVIDIA TESLA P100-16G to fine-tune +mBART. +Hyper-parameters +All the hyper-parameters are +kept the same as cited paper of each model. The +only difference is the batch size of the retriever +and grounding model, we turn it into the maximum +number to fit in the NVIDIA TESLA V100-32G +GPU. +17Code of UniSAr Parser +18mBART Model + +0 +50 +100 +150 +200 +250 +300 +350 +Finance Report +Estate +Transportation +Energe +City +Basketball +Soccer +Swim +Olympics +Fund +Loan +Weather +Awards +Greening +Construction +Health +Foods +Purcahse +Consumption +misc +Statistic of Different Operation +calculation +union +condition +Figure 6: Statistic of three operations in different domains. +0% +10% +20% +30% +40% +50% +60% +70% +80% +90% +100% +Finance Report +Estate +Transportation +Energe +City +Basketball +Soccer +Swim +Olympics +Fund +Loan +Weather +Awards +Greening +Construction +Health +Foods +Purcahse +Consumption +misc +Distribution of Different Operation +calculation +union +condition +Figure 7: Distribution of three operations in different domains. + diff --git a/P9AzT4oBgHgl3EQfIvtp/content/tmp_files/load_file.txt b/P9AzT4oBgHgl3EQfIvtp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4b420ccd21e239e833cffe431d79b4e85a717e0 --- /dev/null +++ b/P9AzT4oBgHgl3EQfIvtp/content/tmp_files/load_file.txt @@ -0,0 +1,776 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf,len=775 +page_content='Towards Knowledge-Intensive Text-to-SQL Semantic Parsing with Formulaic Knowledge Longxu Dou1∗, Yan Gao2, Xuqi Liu1, Mingyang Pan1, Dingzirui Wang1, Wanxiang Che1, Min-Yen Kan3, Dechen Zhan1, Jian-Guang Lou2 1 Harbin Institute of Technology 2 Microsoft Research Asia 3 National University of Singapore {lxdou, xqliu, mypan, dzrwang, car}@ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='cn, dechen@hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='cn, {yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='gao, jlou}@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='com, kanmy@comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='sg Abstract In this paper, we study the problem of knowledge-intensive text-to-SQL, in which domain knowledge is necessary to parse ex- pert questions into SQL queries over domain- specific tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We formalize this sce- nario by building a new Chinese benchmark KNOWSQL consisting of domain-specific questions covering various domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We then address this problem by presenting formulaic knowledge, rather than by annotating addi- tional data examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' More concretely, we construct a formulaic knowledge bank as a do- main knowledge base and propose a frame- work (REGROUP) to leverage this formulaic knowledge during parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Experiments using REGROUP demonstrate a significant 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='2% improvement overall on KNOWSQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 1 Introduction Text-to-SQL translates user queries into executable SQL, greatly facilitating interactions between users and relational databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Along with the release of large-scale benchmarks (Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2018, 2019a,b) and developments in model design (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2021), text- to-SQL works are now achieving promising results in both research and practical applications (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' However, in the professional application of text- to-SQL, such as in the data analysis of financial reports, models require external knowledge to map the expert query with the domain-specific database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Take the financial query for example: What’s the EBIT1 of Walmart?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', where the underlying database has component columns that can be used to calcu- late the EBIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We treat this problem as knowledge- intensive text-to-SQL, where domain knowledge is highly necessary to parse expert questions over ∗Contribution during the internship at Microsoft Research Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 1EBIT is Earnings Before Interest and Tax, and is calcu- lated as Revenue – Cost of Goods Sold – Operating Expenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' What‘s the balance of trade of BRIC countries?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' SELECT Ship_Out – Ship_In, Nation FROM Reports WHERE Nation in (‘Brazil’, ‘Russia’, ‘India’, ‘China’) Ground Parse Retrieval Balance of Trade = Exports - Imports BRIC countries: Country in {Brazil, Russia, India, China} Trade Surplus : Export > Import Formulaic Knowledge Bank Nation GDP Ship In Ship Out A 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='72 1550 2650 B 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='04 581 623 Figure 1: Harnessing REGROUP with formulaic knowl- edge for knowledge-intensive text-to-SQL with three steps: (1) Retrieval the formulaic knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) Ground the concept of formulaic knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) Parse the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' domain-specific tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' This problem prevents text- to-SQL techniques from being fielded in novel, professional applications to assist the experts in processing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Traditional approaches would address this prob- lem by annotating specific question/SQL pairs on a target domain (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Herzig and Berant, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Then such mappings are induced during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' This approach does work but has the drawback that any induced infor- mation is both fragile and expertise-heavy: such knowledge does not port across domains and re- quires expert knowledge to craft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We propose to solve this problem by modeling how a non-expert person might tackle this prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' When meeting unseen examples (as in the EBIT case above), they may first search for the re- lated mathematical formulas from public resources, then ground the concepts referenced in the formu- las with schema elements presented in their partic- ular databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' This process leverages common, encoded formulaic knowledge that are already de- scribed in publicly-available resources such as tu- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='01067v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='CL] 3 Jan 2023 torials, textbooks, encyclopedias, and references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Inspired by this, we propose to address the knowledge-intensive text-to-SQL through formu- laic knowledge which provides the evidence of mapping from domain-specific phrases presented in questions to actual SQL operations over schema elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' More concretely, we define a taxonomy of three types of formulaic knowledge: calcula- tion, union, and condition, each corresponding to a particular snippet of SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Then we propose RE- GROUP, a text-to-SQL framework (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 1), con- sisting of three stages: (1) Retrieve the formu- laic knowledge from formulaic knowledge bank as an external knowledge source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) Ground the concept of formulaic knowledge to the schema elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) Parse the results with the ques- tion, schema, and grounded formulaic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The external formulaic knowledge bank imbues REGROUP with formulaic knowledge, making it knowledgeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' REGROUP is also extensible be- cause updating the formulaic knowledge bank does not require retraining any modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Moreover, we construct a Chinese benchmark KNOWSQL, to examine the effectiveness of RE- GROUP framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' It advances the existing knowledge-intensive text-to-SQL beyond the previ- ous work (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2022) by considering more SQL operations and challenging domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Experimental results demonstrate the RE- GROUP with formulaic knowledge would improve the performance by 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='4% overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Furthermore, we classify error cases into three classes, which are resolvable by advancing the corresponding mod- ule of REGROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Finally, we discuss the potential future work such as expanding the scope of knowl- edge and advancing REGROUP model design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Our contributions are summarised as follows: To the best of our knowledge, we are the first to explore knowledge-intensive text-to-SQL and propose a challenging Chinese benchmark KNOWSQL, which requires domain-specific knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We propose a novel framework REGROUP to address knowledge-intensive text-to-SQL by retrieving and grounding formulaic knowl- edge, which is knowledge-extensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Experimental results demonstrate the effec- tiveness of REGROUP with formulaic knowl- edge which achieves 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='2% overall improve- ment on KNOWSQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2 Knowledge-Intensive Text-to-SQL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 Problem Analysis After studying the real cases in professional data analysis, we roughly categorize the required knowl- edge for knowledge-intensive text-to-SQL into three classes : (1) linguistic knowledge enables the model to adapt to linguistic diversity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) domain knowledge allows the model to perceive domain- specific sayings and concepts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) mathematical knowledge yields the specific SQL operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', Density phrase to division operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' These three sets of knowledge jointly provide the evidence of mapping from domain-specific phrases of questions to actual SQL operations over schema elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' However, most text-to-SQL researches focus on general scenario (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2017), where linguistic knowledge is mainly re- quired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Recently, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2020b) and Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2022) promote text-to-SQL to more chal- lenging scenarios via involving the calculation questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In this paper, we further explore the knowledge-intensive text-to-SQL by considering more operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', calculation, union, and condi- tion) with more challenging domains which require all these three classes of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='2 Challenge Despite that pre-trained language models contain linguistic knowledge, they lack domain knowledge and mathematical knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Therefore, the model would meet two problems: (1) don’t know which operations to use: if an opera- tion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', density = total number / space) has never occurred in training data, the model rarely employ this unseen operation during the inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) don’t know how to adapt operations: the model would fail to generalize the operation across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' For instance, the model cannot generalize the calculation of Population Density (number of people / land area) to Car Density (number of cars / parking lot area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Accordingly, we consider that the vanilla pre- trained language model is (1) narrow since it only supports the limited operation and (2) in- efficient since it can’t generalize the operation across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' However, it’s time-consuming and expertise-heavy to directly increase the amount of annotated data examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In contrast, we address this challenge from the view of formulaic knowl- edge in Sec 3, which is more knowledge-extensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3 KNOWSQL Benchmark #DB #Question #Formulaic Train 160 23, 157 328 Dev 40 2, 731 122 Finance 217 1, 392 219 Estate 35 749 79 Transportation 36 439 82 Table 1: The dataset statistic of KNOWSQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' To uncover the knowledge-intensive text-to- SQL problem and advance the research, we con- struct a challenging Chinese text-to-SQL bench- mark named KNOWSQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Roughly, it consists of two parts: training/dev sets built on the existing DuSQL (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020b) dataset and a newly constructed test set on three professional domains with discovered knowledge in DuSQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 Building Training/Dev Set on DuSQL We build the training/dev set of KNOWSQL based on the existing DuSQL, a Chinese multi-table text-to-SQL benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We categorize its 200 databases into 16 domains like sports, energy, health care, foods, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Given the high quality of DuSQL schema and broad domain coverage, it’s a satisfactory start-point to build a challeng- ing knowledge-intensive text-to-SQL benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' However, the domain-specific question is not well included in DuSQL, where most of the questions could be answered easily without relying on exter- nal knowledge and only considers one SQL oper- ation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', calculation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Given that, we extend the original DuSQL by adding more domain-specific questions and involving more operations in both the train set and the dev set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Eventually, KNOWSQL expands the size of DuSQL train set from 22,521 to 23,157 and the dev set from 2,482 to 2,731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='2 Building Test Set from Scratch To simulate the professional data analysis scenario, we create a challenging test set covering three do- mains (finance, estate, and transportation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' These three domains have high data analysis requirements in real life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Different from the train/dev sets, we construct the test set from the scratch by: (1) col- lecting the domain-specific tables, and (2) annotat- ing the domain-specific questions and correspond- ing SQL queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Table Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' For collecting table schema, we collect the tables from the following source: (1) the public annual reports of the company (2) the industry reports (3) academic papers (4) the statistical reports released by the government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' To ensure the table quality, we conduct several pre- processing procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Firstly, we convert matrix tables (present in annual reports) into relational tables to make the question SQL-answerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Next, to ensure the table data quality, we conduct data cleaning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', filtering out the irrelevant columns to simplify the table structure, and normalizing the headers to reduce the noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Finally, to avoid data privacy issues, we conduct value anonymization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', removing direct identifiers and anonymizing geo-related data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Question Annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' It’s challenging for anno- tators to propose the domain-specific questions without background knowledge 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Thus, we train the annotators first about the domain-specific knowledge via (1) collecting the jargon (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', ab- breviation, terminology) from the domain-specific open resources, which are widely adopted by do- main experts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', EBIT for finance) but unusual for a layperson;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) to mimic the domain expert by asking questions using the jargon with the above materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' After that, the annotators would annotate the questions and SQL with the following criteria: (1) be faithful to the given table (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', don’t exceed the scope of table columns and table content);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) not be directly answerable by the single element of the table but could be answered by the operation over existing columns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) limited to first-order opera- tion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', excludes multi-hop questions like ‘What is the gross profit?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', where the table only contains ‘Sales’, ‘Average Price’ and ‘Cost of Goods Sold’ so that model needs to compute the ‘revenue’ first).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3 Dataset Quality and Data Statistic To guarantee the data quality, we conduct a multi- rounds check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Finally, the inter-agreement of anno- tators reaches 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='7% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' During each round, we ask each annotator to review others’ annotations based on the criteria (stated above), then ask them to fur- ther improve annotations that do not meet the cri- teria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1, the test set contains 288 databases and 2,580 questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Notably, all these challenging data examples in the test set could be covered by 380 formulaic knowledge, which will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 8 for annotator payment and profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 3The inter-annotator agreement is calculated as the per- centage of overlapping votes about whether it’s a correct and domain-specific question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Textual Knowledge Formulaic Knowledge Figure 2: Two types of knowledge in expressing the calculation of EBIT: textual knowledge and formulaic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 3 Approach: Formulaic Knowledge 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 Motivation When meeting unseen examples, the human may first search the related mathematical knowledge or domain knowledge from textbooks or encyclo- pedias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2, the information of calculation of EBIT is returned in both textual and formulaic format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Intuitively, the formulaic for- mat is preferred because it’s (1) more concise and precise: for instance, a adds b times c is more am- biguous than a+b*c or (a+b)*c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) easy to obtain: most description of calculation is stored in a formu- laic format in the textbook, tutorials, and academic paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) SQL parser friendly: the formulaic for- mat is closed to the snippet of SQL then easily for the parser to generate4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='2 Formulaic Knowledge for Text-to-SQL Following this idea, we focus on three categories of operations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 3): calculation, union, and condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Besides the popular calculation knowl- edge, we also consider the taxonomy information as union knowledge and the judgment standard as condition knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The design insight here is that the left part is the name of the knowledge item, and the right part expresses its semantic meaning represented by operations over concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Note that 4In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 6, experimental results prove that formulaic format receives better performance than textual format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' all operations are consistent with SQL grammar, making it closer to SQL query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Besides the entity, the left part of formulaic knowledge might also be the SQL function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', NOW()) or constant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', threshold of Real Estate Bubble).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3 Formulaic Knowledge Bank We further build a formulaic knowledge bank with 1,954 formulaic knowledge items, which supports 19 domains involved in KNOWSQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Importantly, the bank covers all these examples of KNOWSQL as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Note that this bank is a domain-related resource, not one tied to the spe- cific database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Thus, this bank is more general and could be utilized in other applications natural language applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', question answering) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Criteria The design of the formulaic knowledge follows three criteria: (1) Only the first-order (flat) formulaic knowledge is considered (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', the con- cept in the formulaic item should be align-able to the schema elements rather than another formulaic item) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) The stored formulaic knowledge should be both faithful (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', acknowledged by the expert) and standardized (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', shared at the domain level);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) The formulaic knowledge should be domain- level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', not tied to the specific schema elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Collection We collect the formulaic knowledge from the following public resource: (1) Baidu Wenku, the platform where the domain experts usually share the domain knowledge of various domain6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) CNKI, China’s largest academic web- site7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) the data analysis websites of a specific domain, like ESPN for sports and Yahoo for fi- nance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We also collect some knowledge from the English resource and let annotators translate this domain knowledge into Chinese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Abstraction To make the formulaic knowledge more generic, we propose to accumulate the for- mulaic knowledge at the domain level instead of database-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Specifically, we abstract the con- cept of formulaic knowledge before storing them in the knowledge bank, which indicates the operation over concept rather than specific schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' For ex- ample, we would extract the formulaic knowledge from ‘People Density in China 2020 = total num- ber of Chinese in 2020 / Chinese Land Area’ to ‘People Density = total number of People / Area’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 5See fine-grained statistic of bank in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 6https://wenku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='baidu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='com/ 7https://oversea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='cnki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='net/index/ how to compute EBIT X Q Q All Images Videos 国 News Shopping : More Tools About196,000results(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content="44seconds) EBIT is calculated by subtracting a company's cost of goods sold (COGs) and its operating expenses from its revenue." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' EBlT can also be calculated as operating revenue and non-operating income, less operating expenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='investopedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='com>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='>FinancialStatements : Earnings Before Interest and Taxes (EBIT) - Investopedia ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Aboutfeaturedsnippets· DFeedback People also ask : What is the formula to calculate EBIT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' How to Calculate EBIT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' EBIT = Net Income + Interest + Taxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' EBIT = Revenue - COGS - Operating Expenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' EBIT = Gross Profit - Operating Expenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Feb25,2022 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='masterclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='com>articles>ebit-explained Earnings Before Interest and Taxes: How to Calculate EBlT Search for: What is the formula to calculate EBIT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Operation Calculation Union Condition Formulaic Knowledge Trade Balance = Exports – Imports BRIC Countries : Country in {Brazil, Russia, India, China} Trade Surplus : Export > Import Abstract Phrase = Schema1 op Schema2 Phrase : Schema in Set Phrase : Schema1 op Schema2 Example What‘s the balance of trade of China?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' SELECT Exports -Imports FROM Reports WHERE Country=China Show me the sum of GDP of BRIC countries?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' SELECT sum(GDP) FROM Reports WHERE Country in (Brazil, Russia, India, China) GROUP By Name Which country has a trade surplus problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' SELECT Country FROM Reports WHERE Export > Import Figure 3: We consider three types of formulaic knowledge to address knowledge-intensive text-to-SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' #Formulaic #Calculation #Union #Condition Formulaic Knowledge Bank 1, 954 1, 102 346 506 KNOWSQL involved 891 656 52 183 Table 2: The dataset statistic of formulaic knowledge bank and its overlap with KNOWSQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Consequently, only ONE formulaic knowledge is required to address MANY schema elements to cal- culate the density of animals/cars/shops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Mapping within KNOWSQL We further exam- ine the overlap between formulaic knowledge bank and KNOWSQL benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' As stated in Sec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3, all questions from KNOWSQL are covered by for- mulaic knowledge banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Specifically, there are 1,954 knowledge items in the bank, and 891 items are used for answering the KNOWSQL questions as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Especially, there are extra 1,063 knowledge items beyond KNOWSQL which could support future work in applying formulaic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 4 REGROUP Framework To address the knowledge-intensive text-to-SQL problem, we propose a novel framework named REGROUP, consisting of three stages: (1) Retrieve the formulaic knowledge from the formulaic knowl- edge bank as an external knowledge source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) Ground the concept of formulaic knowledge to the schema elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', Exports to Ship_Out);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) Parse the results with the question, schema, and grounded formulaic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 4, REGROUP consists of three models: re- triever, grounding model, and parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We will give a brief introduction of each model in the follow- ing8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 8More details of the model implementation could be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 Retriever Model The goal of the retriever is to extract the rele- vant formulaic knowledge items from the formu- laic knowledge bank (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The challenge is the fine-grained modeling of the formulaic knowl- edge to disambiguate the ones with the same intent but differing in operation over concepts, such as calculating EBIT in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We directly utilize the off-the-shelf Dense Passage Retriever (DPR) (Karpukhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020) which was origi- nally designed for open-domain QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' It employs a bi-encoder architecture to learn the dense represen- tation of sentences and passages, then it computes the dot-product between the representations as the similarity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' To adapt the DPR in the formulaic knowledge re- trieval task, we treat the formula knowledge bank as the passage candidate and concatenate the question with flattened schema (separated by special token ‘|’) to enrich the semantics of the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Then we follow the standard DPR training procedure to optimize the bi-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Specifically, during the training process, we derive the positive knowledge items from KNOWSQL annotation and sample five negative examples from the formulaic knowledge bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' During the inference process, we first cache the embedding of formulaic knowledge items, then leverage the FAISS algorithm (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2017) to rank each formulaic knowledge item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='2 Grounding Model Given the retrieved knowledge items, the goal of the grounding model is to edit the formulaic knowl- Parser Which company from BRIC countries has the largest profit ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=" SELECT Name, Revenue - Cost FROM Company WHERE Nation in (‘Brazil’, ‘Russia’, ‘India’, ‘China') ORDER BY Revenue - Cost DESC LIMIT 1 Name Nation Continents Revenue Cost A China Asia 124." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='9 B America North America 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 Retriever BRIC Countries: Country in {Brazil, Russia, India, China} Profit = Revenue – Cost of Revenue – Selling and Maintenance Expense + Other Income - Tax Grounding Model BRIC Countries: Nation in {Brazil, Russia, India, China} Profit = Revenue – Cost A, 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='6 Age = now() - Date of Birth Density = Number / Area Speed = Distance / Time BRIC Countries: Country in {Brazil, Russia, India, China} EBIT = Net Income + Interest + Taxes EBIT = Revenue – Cost of Goods Solds – Operation Expense Profit = Revenue – Cost of Revenue – Selling and Maintenance Expense + Other Income - Tax Formulaic Knowledge Bank Figure 4: Pipeline of REGROUP: (1) Retrieve the formulaic knowledge item from the bank;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) Ground the concept of formulaic knowledge into schema elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) Parse the question with grounded formulaic knowledge into SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' edge items w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='t specific schema through (1) remov- ing the irrelevant concept and (2) instantiating the concept with the schema elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The main chal- lenge is the expensive annotations of grounding (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' supervision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Therefore, the weakly super- vised grounding approaches would be more suit- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Specifically, we leverage the Erasing-then- Awakening (ETA) model proposed by (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2021), which was originally designed for ground- ing the entity from the knowledge base to the entity mentioned in the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The output of ETA is a confidence matrix, indicating the possible ground- ing relations between entity mentions and entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' To adapt the ETA in the formulaic knowledge grounding task, we treat each knowledge item as the ‘question’ and attempt to figure out which spe- cific schema elements are grounded in the knowl- edge item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Specifically, it’s determined by a hyper- parameter H to indicate the threshold of confi- dence (whether it’s grounded and which one it’s grounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 4, we filter the concept (cross outed parts) under the confidence threshold H and replace the concept with aligned elements (green parts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3 Parser Model The goal of the parser is to predict the executable SQL according to question and database schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The main challenge is how to model the database structure to infer the implicit schema mentioned, and how to make use of the grounded knowl- edge (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', knowledge-fusion) to leverage grounded knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We are inspired by the recent progress in adopting the large pre-trained language model in semantic parsing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' For instance, (Scholak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2022) achieves excellent performance on several semantic parsing tasks under the sim- ple pretrained language model framework, such as BART (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020) and T5 (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Given that, we propose to adopt UniSAr (Dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2022) as the base parser in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' It im- proves the vanilla BART with three non-invasive extensions and achieves SOTA or competitive per- formance on seven text-to-SQL benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Con- cretely, the input of the model is the concatena- tion of the question, serialized schema, and re- trieved formulaic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We propose that the parser should correctly adopt the grounded formu- laic knowledge during SQL generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 5 Experimental Results and Analysis To evaluate our approach: REGROUP with exter- nal formulaic knowledge bank, we conduct several experiments on KNOWSQL benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We report both the overall results of the pipeline and the fine- grained results of each module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We also conduct error analysis and categorize the bad cases into three main classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Note that we report the average experimental results of each setting during three runs9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 9Code and data are available at link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Model Dev Finance Estate Transportation Average Vanilla 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='9 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='7 REGROUP (w/o Grounding) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='7 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='9 REGROUP 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='6 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='7 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='9 REGROUP (Oracle) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='8 Table 3: Overall results on different KNOWSQL splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Oracle refers to the use of the oracle formulaic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The evaluation metric is SQL exact set match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Average indicates the micro-average score of the first four columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Data Model R@1 R@3 R@10 Dev BM-25 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='9 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='5 REGROUP 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='8 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='5 Finance BM-25 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='9 REGROUP 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 Table 4: Results of REGROUP retriever compared with BM-25 on KNOWSQL dev and finance splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The eval- uation metric is the Recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Data Model Precision Recall F1 Dev FuzzyMatch 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='7 REGROUP 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='4 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='8 Finance FuzzyMatch 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='5 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='2 REGROUP 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='9 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='7 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='8 Table 5: Results of REGROUP grounding model com- pared with fuzzy string match on KNOWSQL dev and finance splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 Experimental Setup The retriever returns the top-3 retrieved formu- laic knowledge items from the bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The ground- ing model further aligns the concept in formulaic knowledge into schema elements and the decision threshold H is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='6 which is decided empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The parser receives the grounded knowledge, table schema, and user query as the input and then out- puts the SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' For the parsing baseline, we adopt UniSAr (Dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2022) as the vanilla parser10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='2 Overall Results As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 3, we could observe that: (1) REGROUP exceeds the vanilla model by 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='2%, which indicates the effectiveness of using formu- laic knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) grounding the formulaic knowl- edge improves the REGROUP by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='0%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) the or- acle formulaic knowledge (retrieve correctly and grounding correctly) reaches the upper bound of REGROUP 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='8%, which implies the potential im- provement room for KNOWSQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 10More implementation details could be found in Ap- pendix B 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3 Fine-grained Results We compare the retriever and grounding model with other baselines, on both the dev set and the test set of KNOWSQL in the finance domain, to examine the performance in general and domain- specific scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Retriever We compare the retriever of RE- GROUP (bi-encoder) with BM-25 (Robertson and Zaragoza, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The evaluation metric is the re- call score over retrieved results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We observe that the finance domain is more challenging than the general domain (dev split) since it contains many homogeneous formulaic knowledge items that ex- press the same intention in the left part but with different computation ways in the right part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' For example, there are two ways to compute the ‘EBIT’ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Grounding We compare the grounding model of REGROUP with the fuzzy string match-based method 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Following the previous work (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2021), we report the micro-average precision, recall, and F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We could observe that: (1) the model-based grounding improves the performance by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1% and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='6% respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) the domain-specific data like Finance poses more challenging cases than the general domain, where finance is behind the dev by about 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='4 Error Analysis We sample 300 cases from the dev split and 100 cases from finance/estate/transportation in the test split respectively (600 in total) for error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Vanilla Model Error We first compare the cor- rect case of REGROUP while predicted incorrectly by the vanilla model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' As the example in the first part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 5, the vanilla model is unable to pre- dict the unseen operation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In con- trast, the grounded formulaic knowledge enables 11It enumerates all n-gram (n ≤ 5) of the concepts in for- mulaic knowledge, and links each of them to schema element by fuzzy string matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Vanilla Model Error Formulaic Knowledge Question: 东三省每省的一胎出生率是多少?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (What is the first birth rate in each of the three northeastern provinces in China?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=') Schema : 省份 ｜ 婴儿出生率 | 二胎出生率 | 人口 (Province | Birth Rate | Second Birth Rate | Population) Vanilla: SELECT 婴儿出生率 FROM 各省人口出生及死亡率 WHERE 省份 = "辽宁" ReGrouP: SELECT 婴儿出生率 - 二胎出生率 FROM 各省人口出生及死亡率 WHERE 省 份 IN ("辽宁" , "吉林" , "黑龙江" ) Grounded Formulaic Knowledge: 东三省 : { 辽宁 , 吉林 , 黑龙江 } (Three Northeastern Provinces: { Liaoning , Jilin , Heilongjiang }) 一胎出生率 = 婴儿出生率 - 二胎出生率 (First birth rate = Birth rate - Second Birth Rate) Retriever Error (43%) Retrieval Knowledge Question:息税前利润是多少?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='(Please return the Earnings Before Interest and Taxes ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Schema: 收入 | 净收入 | 销售费用 |营业费用 | 销售额 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='(Revenue| Net Income | Cost of Goods Sold Expenses | Operating Expenses | Sales) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Gold SQL: SELECT 收入 - 销售费用 - 营业费用 FROM 报表 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Pred SQL: SELECT 净收入 + 销售额 FROM 报表 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Oracle Formulaic Knowledge: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='息税前利润 = 收入 - 销售成本 - 营业费用 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='(Earnings Before Interest and Taxes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='= Revenue – Cost of Goods Sold – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Operating Expenses ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Retrieved Formulaic Knowledge: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='息税前利润 = 净收入 + 利息 + 税 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='(Earnings Before Interest and Taxes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='= Net Income + Interest + Taxes ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Grounding Error (41%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Grounded Knowledge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Question: A公司的流动资产是多少?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=" (What is company A's current assets?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Schema:现金 | 应收款项 | 可销售证券|商品成本| 运营费用 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='(Cash | Trade Receivables | Marketable Securities | Cost of Goods | Operating Expenses) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Gold SQL: SELECT 应收款项 + 可销售证券 +现金 FROM 报表 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Pred SQL: SELECT 应收款项 + 现金 FROM 报表 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Undergrounded Formulaic Knowledge: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='流动资产 = 短期资本 + 应收帐款 + 股票 + 存款余额 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='(Current Assets = Short Term Capital + Debtors + Stock + Cash and bank) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Correct Grounded Formulaic Knowledge: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='流动资产 = 应收款项 + 可销售证券 + 现金 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='(Current Assets = Trade Receivables + Marketable Securities + Cash) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Prediced Grounded Formulaic Knowledge: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='流动资产 = 应收款项 + 现金 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='(Current Assets = Trade Receivables + Cash) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Parser Error (12%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Leveraging Knowledge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='Question: 哪个城市的房地产市场发展合理?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=" (Which city's real estate market is developing reasonably?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=') Schema: 城市 | 吸纳率 | 空置率 (City | Commercial Housing Absorption Rate | Commercial Housing Vacancy Rate) Gold SQL: SELECT 城市 FROM 报表 where 空置率 > 15% and 空置率 < 30% Pred SQL: SELECT 城市 FROM 报表 where 空置率 > 15% Grounded Formulaic Knowledge: 房地产市场良性发展 : 空置率 > 15% AND 空置率 < 30% (Good development of real estate market: Commercial Housing Vacancy Rate > 15% AND Commercial Housing Vacancy Rate < 30%) Figure 5: Case studies of REGROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We first compare it with the vanilla parsing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Then we classify the bad cases of REGROUP into three categories: (1) Retriever Error: not getting the knowledge from bank;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) Grounding Error: not learning the knowledge by alignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) Parser Error: not using the grounded knowledge in generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' REGROUP to predict the operation over schema elements correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Then we categorize the error of REGROUP into three main classes and list their percentage in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Finally, we discuss the potential future work in improving each part of REGROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' An advantage of REGROUP is the decoupled framework could track each type of bad case individually, avoiding the catastrophic forgetting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Retrieval Error About 43% errors are attributed to the retriever where the model doesn’t get the correct knowledge from bank since it can’t distin- guish the semantic difference between the closed formulaic knowledge items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Future work should improve its distinguishing ability by fine-grained modeling, like attention mechanism Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Grounding Error About 41% errors are caused by incorrect grounded knowledge where the model doesn’t learn the knowledge by alignment since it can’t correctly align the concept to schema ele- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Future work should focus on how to derive the grounding information under weak supervision or even without supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' It would greatly allevi- ate the severe annotation effort in specific domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Parsing Error There are still 8% error cases caused by parsing, where the formulaic knowledge is correctly retrieved and grounded but the parser still doesn’t use the grounded knowledge in gen- eration well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Future work should improve it by explicitly modeling the copy process of knowledge from the input to the SQL snippet position, such as implementing the additional gate mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Other Error The remaining 8% errors are about the SQL generation, such as the GROUP-BY or nested SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Since it’s not our main focus, we ig- nore these cases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 5 for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 6 Discussion Is formulaic knowledge better than textual knowledge for text-to-SQL?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1, we ar- gue that formulaic knowledge is preferred over tex- tual knowledge intuitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Empirically, we con- duct the experiments by the following steps: (1) transforming the formulaic knowledge to textual knowledge through annotators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) training the re- triever and parser with textual knowledge under the same experiment setting as formulaic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Experimental results reveal that textual knowledge receives an overall performance degradation of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='6% compared with Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We conclude that REGROUP prefers formulaic knowledge since it’s more close to the SQL snippets or schema repre- sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Moreover, formulaic knowledge is both precise and concise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In contrast, textual knowledge is redundant and much more diverse in expressing the equivalent meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' What’s the cost of collecting formulaic knowl- edge?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' During the collection process of formulaic knowledge bank (19 domains), we found most do- mains have the public knowledge resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' More- over, the effort spent on collection formulaic knowl- edge is also acceptable compared with annotating data examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' For example, we spent 4 hours in collecting 219 formulaic knowledge in the finance domain, which is far more effective than annotat- ing the equivalent data examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Eventually, for- mulaic knowledge improves the performance by 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='0% without retraining the model as shown in Table 3 (from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='7% to 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='7%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' How to expand the scope of formulaic knowl- edge further?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In this paper, we mainly focus on domain knowledge and mathematical knowl- edge and transfer them into formulaic knowledge format for model learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Other types of knowl- edge would improve the knowledge-intensive text- to-SQL further, such as the commonsense knowl- edge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', water freezing point: temperature=0) or personalized information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', favourite food: Tiramisu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Thus, we could package these types of knowledge into a formulaic format in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 7 Related Work 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 Domain Generalization of Text-to-SQL To be applicable in real scenarios, a text-to-SQL model should generalize to new domains with- out relying on expensive domain-specific labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Previous work has shown that current text- to-SQL usually fails on domain generalization sce- narios (Finegan-Dollak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Recent ap- proaches track this problem including data synthe- sis (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2021), meta-learning (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2021) and encoder pretraining (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Herzig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Most recently, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2022) proposed to adopt schema expansion and scheme pruning to preprocess the table schemas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We highlight that compared with the schema- expansion approach, the advantage of our ap- proach (REGROUP with formulaic knowledge) is the broad knowledge scope: we not only consider the calculation knowledge but also union knowl- edge and condition knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Moreover, our ap- proach is extensible with an external and maintain- able formulaic knowledge bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='2 Retrieval Enhanced Semantic Parsing There has been a recent trend toward leveraging retrieval-enhanced methods in various NLP tasks such as machine translation (Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2019) and question answering (Karpukhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Sim- ilar with REGROUP, previous work (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Pasupat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2021) leverage a retrieval step to provides examples as the context of input for seq2seq model learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' However, our approach differs in two ways: (1) our retrieval object is grounded formulaic knowl- edge which contains more condensed information than data example;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) prior work directly leverage the retrieved results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We leverage the grounding model to edit the retrieved formulaic knowledge to make it more relevant to the question and schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 8 Conclusion and Future Work This paper explores formulaic knowledge to ad- dress the knowledge-intensive text-to-SQL prob- lem, which would advance the professional appli- cation of text-to-SQL such as data analysis for do- main experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' First, we analyze the challenge of knowledge-intensive text-to-SQL and construct a new challenging benchmark KNOWSQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Then we propose to address this problem from the view of formulaic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Concretely, we propose a simple framework REGROUP to leverage an exter- nal formulaic knowledge bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Experimental re- sults reveal that REGROUP with formulaic knowl- edge achieves the 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='2% improvements overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We further discuss three directions in improv- ing the REGROUP via analyzing different types of bad cases: (1) iterative filling in the blank of formulaic knowledge bank;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) mitigating the gap between formulaic knowledge and specific schema via improving the grounding model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) driving the parser to fully make use of more complicated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', commonsense) formulaic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Ethical Considerations This work presents KNOWSQL, a free and open dataset for the research community to study the knowledge-intensive text-to-SQL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Data in KNOWSQL are constructed based on DuSQL (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020b) , a free and open cross-database Chinese text-to-SQL dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We also collect formulaic and table data from CNKI12 and Baidu Wenku13, which are also free and open for academic usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The content of the table is anonymized to address the privacy issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' To anno- tate the KNOWSQL, we recruit 3 Chinese college students (1 female and 2 males).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Each student is paid 4 yuan ($0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='6 USD) for annotating the (SQL, question) pairs and 2 yuan ($0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3USD) for collect- ing the formulaic knowledge items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' This compen- sation is determined according to the prior simi- lar dataset construction (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Since all question sequences are collected against open- access databases or public tables, there is no pri- vacy issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Limitations (1) KNOWSQL is built based on DuSQL, a Chinese large-scale text-to-SQL dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Thus the language coverage of this paper is limited to Chinese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We leave the extension to other languages for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2) For the scope of formulaic knowledge, we mainly address three types of knowledge to associate with each SQL phrase: calculation, union, and condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Some types of knowledge are under- explored such as commonsense knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (3) For the model design of REGROUP, we build it from improving many existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Despite achieving promising evaluation results, the case studies reveal that many challenging remains during the retrieval, grounding, or parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Acknowledgement We thank all anonymous reviewers for their con- structive comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Wanxiang Che was supported via the grant 2020AAA0106501 and NSFC grants 62236004 and 61976072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Dechen Zhan is the cor- responding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' References Deng Cai, Yan Wang, Wei Bi, Zhaopeng Tu, Xi- aojiang Liu, and Shuming Shi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Retrieval- 12https://oversea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='cnki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='net/index/ 13https://wenku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='baidu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='com/ guided dialogue response generation via a matching- to-generation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 2019 Conference on Empirical Methods in Natu- ral Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pages 1866–1875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Ruisheng Cao, Lu Chen, Zhi Chen, Yanbin Zhao, Su Zhu, and Kai Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' LGESQL: Line graph en- hanced text-to-SQL model with mixed local and non- local relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 59th Annual Meeting of the Association for Computational Lin- guistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 2541–2555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Compu- tational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Yiming Cui, Wanxiang Che, Ting Liu, Bing Qin, and Ziqing Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Pre-training with whole word masking for chinese bert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' IEEE/ACM Transac- tions on Audio, Speech, and Language Processing, 29:3504–3514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Longxu Dou, Yan Gao, Mingyang Pan, Dingzirui Wang, Wanxiang Che, Dechen Zhan, and Jian- Guang Lou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Unisar: A unified structure- aware autoregressive language model for text-to-sql.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Catherine Finegan-Dollak, Jonathan K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Kummerfeld, Li Zhang, Karthik Ramanathan, Sesh Sadasivam, Rui Zhang, and Dragomir Radev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Improv- ing text-to-SQL evaluation methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Pro- ceedings of the 56th Annual Meeting of the Associa- tion for Computational Linguistics (Volume 1: Long Papers), pages 351–360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Jiaqi Guo, Ziliang Si, Yu Wang, Qian Liu, Ming Fan, Jian-Guang Lou, Zijiang Yang, and Ting Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Chase: A large-scale and pragmatic Chinese dataset for cross-database context-dependent text-to-SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Lan- guage Processing (Volume 1: Long Papers), pages 2316–2331, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Vivek Gupta, Akshat Shrivastava, Adithya Sagar, Armen Aghajanyan, and Denis Savenkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' RetroNLU: Retrieval augmented task-oriented se- mantic parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 4th Workshop on NLP for Conversational AI, pages 184–196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' As- sociation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Jonathan Herzig and Jonathan Berant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Don’t paraphrase, detect!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' rapid and effective data collec- tion for semantic parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 2019 Conference on Empirical Methods in Natu- ral Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pages 3810–3820, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Jonathan Herzig, Pawel Krzysztof Nowak, Thomas Müller, Francesco Piccinno, and Julian Eisenschlos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' TaPas: Weakly supervised table parsing via pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 58th Annual Meeting of the Association for Computational Lin- guistics, pages 4320–4333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Compu- tational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Hsin-Yuan Huang, Eunsol Choi, and Wen-tau Yih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Flowqa: Grasping flow in history for con- versational machine comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In 7th Inter- national Conference on Learning Representations, ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Jeff Johnson, Matthijs Douze, and Hervé Jégou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Billion-scale similarity search with gpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Vladimir Karpukhin, Barlas Oguz, Sewon Min, Patrick Lewis, Ledell Wu, Sergey Edunov, Danqi Chen, and Wen-tau Yih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Dense passage retrieval for open-domain question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Nat- ural Language Processing (EMNLP), pages 6769– 6781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Wenqiang Lei, Weixin Wang, Zhixin Ma, Tian Gan, Wei Lu, Min-Yen Kan, and Tat-Seng Chua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Re-examining the role of schema linking in text-to- SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Process- ing (EMNLP), pages 6943–6954, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Associa- tion for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Mike Lewis, Yinhan Liu, Naman Goyal, Mar- jan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Veselin Stoyanov, and Luke Zettlemoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' BART: Denoising sequence-to-sequence pre- training for natural language generation, translation, and comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 58th An- nual Meeting of the Association for Computational Linguistics, pages 7871–7880, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Qian Liu, Dejian Yang, Jiahui Zhang, Jiaqi Guo, Bin Zhou, and Jian-Guang Lou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Awakening la- tent grounding from pretrained language models for semantic parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021, pages 1174–1189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Myle Ott, Sergey Edunov, Alexei Baevski, Angela Fan, Sam Gross, Nathan Ng, David Grangier, and Michael Auli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' fairseq: A fast, extensible toolkit for sequence modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chap- ter of the Association for Computational Linguistics (Demonstrations), pages 48–53, Minneapolis, Min- nesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Panupong Pasupat, Yuan Zhang, and Kelvin Guu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Controllable semantic parsing via retrieval augmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 7683–7698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Te- jani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Pytorch: An imperative style, high-performance deep learn- ing library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Advances in Neural Information Pro- cessing Systems 32, pages 8024–8035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Peng Qi, Yuhao Zhang, Yuhui Zhang, Jason Bolton, and Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Stanza: A python natural language processing toolkit for many human languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 58th An- nual Meeting of the Association for Computational Linguistics: System Demonstrations, pages 101– 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Colin Raffel, Noam Shazeer, Adam Roberts, Kather- ine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Exploring the limits of transfer learning with a unified text-to- text transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Journal of Machine Learning Re- search, 21(140):1–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Stephen Robertson and Hugo Zaragoza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The Probabilistic Relevance Framework: BM25 and Be- yond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Foundations and Trends in Information Re- trieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Torsten Scholak, Nathan Schucher, and Dzmitry Bah- danau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' PICARD: Parsing incrementally for constrained auto-regressive decoding from language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 9895–9901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Richard Shin, Christopher H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Lin, Sam Thomson, Charles Chen, Subhro Roy, Emmanouil Antonios Platanios, Adam Pauls, Dan Klein, Jason Eisner, and Benjamin Van Durme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Constrained language models yield few-shot semantic parsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' CoRR, abs/2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='08768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Bailin Wang, Mirella Lapata, and Ivan Titov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Meta-learning for domain generalization in semantic parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Tech- nologies, pages 366–379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Bailin Wang, Richard Shin, Xiaodong Liu, Oleksandr Polozov, and Matthew Richardson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2020a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' RAT- SQL: Relation-aware schema encoding and linking for text-to-SQL parsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 58th Annual Meeting of the Association for Computa- tional Linguistics, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Compu- tational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Lijie Wang, Ao Zhang, Kun Wu, Ke Sun, Zhenghua Li, Hua Wu, Min Zhang, and Haifeng Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2020b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' DuSQL: A large-scale and pragmatic Chinese text- to-SQL dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 2020 Con- ference on Empirical Methods in Natural Language Processing (EMNLP), pages 6923–6935, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' As- sociation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Yushi Wang, Jonathan Berant, and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Building a semantic parser overnight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceed- ings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th Interna- tional Joint Conference on Natural Language Pro- cessing (Volume 1: Long Papers), pages 1332–1342, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computational Lin- guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pier- ric Cistac, Tim Rault, Remi Louf, Morgan Funtow- icz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Trans- formers: State-of-the-art natural language process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Em- pirical Methods in Natural Language Processing: System Demonstrations, pages 38–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Tianbao Xie, Chen Henry Wu, Peng Shi, Ruiqi Zhong, Torsten Scholak, Michihiro Yasunaga, Chien-Sheng Wu, Ming Zhong, Pengcheng Yin, Sida I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Wang, Vic- tor Zhong, Bailin Wang, Chengzu Li, Connor Boyle, Ansong Ni, Ziyu Yao, Dragomir Radev, Caiming Xiong, Lingpeng Kong, Rui Zhang, Noah A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Smith, Luke Zettlemoyer, and Tao Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Unifiedskg: Unifying and multi-tasking structured knowledge grounding with text-to-text language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='05966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Pengcheng Yin, Graham Neubig, Wen-tau Yih, and Se- bastian Riedel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' TaBERT: Pretraining for joint understanding of textual and tabular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Pro- ceedings of the 58th Annual Meeting of the Asso- ciation for Computational Linguistics, pages 8413– 8426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Pengcheng Yin, John Wieting, Avirup Sil, and Graham Neubig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' On the ingredients of an effective zero-shot semantic parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Tao Yu, Rui Zhang, Heyang Er, Suyi Li, Eric Xue, Bo Pang, Xi Victoria Lin, Yi Chern Tan, Tianze Shi, Zihan Li, Youxuan Jiang, Michihiro Yasunaga, Sungrok Shim, Tao Chen, Alexander Fabbri, Zifan Li, Luyao Chen, Yuwen Zhang, Shreya Dixit, Vin- cent Zhang, Caiming Xiong, Richard Socher, Walter Lasecki, and Dragomir Radev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' CoSQL: A conversational text-to-SQL challenge towards cross- domain natural language interfaces to databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Lan- guage Processing (EMNLP-IJCNLP), pages 1962– 1979, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Tao Yu, Rui Zhang, Kai Yang, Michihiro Yasunaga, Dongxu Wang, Zifan Li, James Ma, Irene Li, Qingning Yao, Shanelle Roman, Zilin Zhang, and Dragomir Radev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Spider: A large- scale human-labeled dataset for complex and cross- domain semantic parsing and text-to-SQL task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pages 3911–3921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Tao Yu, Rui Zhang, Michihiro Yasunaga, Yi Chern Tan, Xi Victoria Lin, Suyi Li, Heyang Er, Irene Li, Bo Pang, Tao Chen, Emily Ji, Shreya Dixit, David Proctor, Sungrok Shim, Jonathan Kraft, Vin- cent Zhang, Caiming Xiong, Richard Socher, and Dragomir Radev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' SParC: Cross-domain se- mantic parsing in context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meeting of the Association for Com- putational Linguistics, pages 4511–4523, Florence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Jichuan Zeng, Xi Victoria Lin, Steven C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Hoi, Richard Socher, Caiming Xiong, Michael Lyu, and Irwin King.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Photon: A robust cross-domain text-to-SQL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 58th An- nual Meeting of the Association for Computational Linguistics: System Demonstrations, pages 204–214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Chen Zhao, Yu Su, Adam Pauls, and Emmanouil An- tonios Platanios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Bridging the generalization gap in text-to-SQL parsing with schema expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 5568–5578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Victor Zhong, Caiming Xiong, and Richard Socher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Seq2sql: Generating structured queries from natural language using reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' CoRR, abs/1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='00103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' A Details of Formulaic Knowledge Bank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1 Knowledge Source We construct the formulaic knowledge bank across 19 domains following KNOWSQL and 1 misc do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The misc domain stores the infrequent or general knowledge items in KNOWSQL, such as the calculation of density, and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' In the fol- lowing, we will briefly analyze the collected bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='2 Statistic Across Domain Different domains have different amounts of pub- licly available data online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 6, not unsurprisingly, finance and estate share the most plentiful publicly available resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='3 Distribution within Domain We also observe the different distribution of knowl- edge across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' If the domain focus on cal- culation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', finance report and fund), we assume the data analysis tends to be more objective, which is easier for model learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' If the domain focus on condition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', estate and awards), we assume the data analysis tends to be more subjective since it’s more challenging in learning semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' B Implementation Details of REGROUP Retriever We implement the retriever based on the code of Karpukhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2020)14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We adopt the Chinese BERT-wwm-ext (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2021) as pretrained encoder15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' It would return the top-3 retrieved formulaic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Future work could improve the negative sampling by in- batch sampling or BM25-based sampling following Karpukhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Grounding Model The code of ETA16 is not re- leased at the time of submission of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We re-implement the ETA model based on the paper using pytorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' We evaluate our implemented model with the original model on SPIDER-L (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020) to examine whether the re-implemented model works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Our model achieves 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='1% column F1 score where Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2021) reported 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The experiments on KNOWSQL also employ the Chinese BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 14Code of DPR Retrieval Model 15Chinese-BERT-wwm Model 16Code of ETA Grounding Model Parser We build the paper based on the code of Dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' (2022)17 We choose the mBART- CC2518 as the base model to fine-tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Following the vanilla model, we build the input of parser as follows: [schema] | [grounded formulaic knowl- edge] | [question], where ‘|’ is the delimiter across different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Resource and Tools For tokenization, we em- ploy Stanza (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020) considering its excel- lent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' For the retriever and grounding model, we import the BERT model with Trans- former library (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' For parser mode, we preprocess the data and fine-tune the mBART with fairseq framework (Ott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=', 2019) Device and Training Time We conduct all these experiments on one NVIDIA TESLA V100-32GB GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The training of the retriever, grounding model, and parser takes about 4 hours, 3 hours, and 8 hours respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The minimum device require- ment is NVIDIA TESLA P100-16G to fine-tune mBART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' Hyper-parameters All the hyper-parameters are kept the same as cited paper of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' The only difference is the batch size of the retriever and grounding model, we turn it into the maximum number to fit in the NVIDIA TESLA V100-32G GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 17Code of UniSAr Parser 18mBART Model 0 50 100 150 200 250 300 350 Finance Report Estate Transportation Energe City Basketball Soccer Swim Olympics Fund Loan Weather Awards Greening Construction Health Foods Purcahse Consumption misc Statistic of Different Operation calculation union condition Figure 6: Statistic of three operations in different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} +page_content=' 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Finance Report Estate Transportation Energe City Basketball Soccer Swim Olympics Fund Loan Weather Awards Greening Construction Health Foods Purcahse Consumption misc Distribution of Different Operation calculation union condition Figure 7: Distribution of three operations in different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfIvtp/content/2301.01067v1.pdf'} diff --git a/P9FIT4oBgHgl3EQffStV/content/tmp_files/2301.11278v1.pdf.txt b/P9FIT4oBgHgl3EQffStV/content/tmp_files/2301.11278v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6cde00f6b98672bf179832f3b9ae4d91ff0422f8 --- /dev/null +++ b/P9FIT4oBgHgl3EQffStV/content/tmp_files/2301.11278v1.pdf.txt @@ -0,0 +1,559 @@ +3LPo1A-04 + +Template version 8.0d, 22 August 2017. IEEE will put copyright information in this area +See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. +1 +Commissioning, Performance, and Effect of the +Quench Current-boosting Device on a Dedicated Su- +perconducting Magnet +S. Stoynev, M. Baldini, S. Feher + + + +Abstract—Superconducting magnet training is one of the accel- +erator related issues attracting attention due to significant opera- +tional costs and time budget associated to it. It is especially worri- +some that magnets based on the “next-generation” Nb3Sn technol- +ogy are affected by long training. While various efforts are under- +way to better understand and resolve the problem a parallel path +could also be investigated, a path bypassing the issue. Following the +concept of fast induced over-current during magnet powering, +FNAL has developed an upgradable capacitor-based device to dis- +charge through a superconducting magnet at quench detection or +operator chosen time. The 0.4 F/1 kV device has been tested on a 1- +m-long dipole-coil in a “mirror” magnet configuration and conclu- +sive results on magnet training elimination have been observed. In +this paper we discuss the main characteristics of the device, com- +pare simulated response and actual performance, elaborate on test +drivers and outcomes. Next steps and perspectives for future use +are debated. + +Index Terms—Accelerator magnets, pulsed power supplies, su- +perconducting magnets. +I. INTRODUCTION +UILDING +state-of-the-art +superconducting +accelerator +magnets is a delicate process and, among other things, it +involves a careful “pre-stress” setting aiming to minimize +conductor degradation and optimize performance. This step +could be considered “pre-conditioning” [1, Chapter 1.2.5] of +the magnet. However, one can expand the meaning of “pre- +conditioning” to include any process that would affect magnet +performance positively. Here we choose to separate action +taken before magnet powering and during magnet powering - +“pre-conditioning” and “operational conditioning”, respective- +ly. “Pre-conditioning” was considered and experimented with +in past [2], [3]. It is known to the authors that “operational +conditioning” was conducted in the past too, but we could find +no clear reference. That for instance includes manipulating +current ramping (levels, rates) to avoid lower current quenches +though it is purely an investigative technique. + +The work described in the present paper builds up on [2] +where capacitors were discharged through a magnet as a “pre- +conditioning” step. We do this as “operational conditioning”. +In our case, a capacitor is discharged through a magnet being + +This work was supported by Fermi Research Alliance, LLC, under Contract +DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, +Office of High Energy Physics.(Corresponding author: Stoyan Stoynev.) +The authors are with Fermilab National Accelerator Laboratory, Batavia, IL +60510 USA (e-mail: stoyan@fnal.gov). +Digital Object Identifier will be inserted here upon acceptance. +powered at user defined time, for instance at quench detection. +This boosts the current through the magnet to levels depending +on circuit parameters, including magnet parameters. Such a +boost could effectively increase the magnet quench current as- +sociated to a given quench although a “quenchless” mode will +be debated later too. Given that magnet training is understood +as the steady increase of quench current after consecutive +spontaneous quenches, changing quench current level is an +important lever to affect training. Authors are aware of possi- +ble phase delays between operating current and local force [4] +in pulsed mode although expectations pointed to time delays +of low tens of milli-seconds at most. Our own measurements +of magnetic field in magnet bore and magnet current during +the sharp decrease of current during system trips and quenches +(with immediate magnet protection) did not indicate any phase +delays between magnet current and bore field beyond our +resolution of a couple of milli-seconds. If present, significant +local phase delays, originating from decaying eddy currents, +could suppress the Lorentz force peak experienced by the su- +perconductor/coil and diminish the effect of fast current boost. +The work to bring the boost ideas to fruition was supported +by LDRD funds at FNAL and the resulting device [5] is in ef- +fect a pulsed power supply integrated into the main power +supply CPS3 [6] with ability to be controlled independently. +We call it Quench Current-boosting Device, or QCD. QCD, +has similarities to CLIQ [7] but is a very different device. +Apart from being used for different purposes, there are two +other major differences: a) the QCD boost current is the same +through the whole magnet, i.e. there is no difference in magnet +Lorentz force distribution with respect to “regular” ramp-up; +b) QCD has no current/voltage oscillation features. +This paper describes the first application of QCD on a dedi- +cated superconducting magnet, points to relevance of simula- +tions before and during testing, reviews choices made during +testing and results obtained; those are followed by a discussion +on future use of QCD and the technique itself. +II. QCD COMMISSIONING AND MAGNET TESTING +A. QCD preparations and simulations +QCD [5] started working as a unit towards the end of 2021 +and was gradually integrated to CPS3. After all major compo- +nents and sub-circuits were verified and tested, the device +went through various full circuit examinations, including +powering. Before using it on a superconducting magnet, a +B + + + +2 +conventional accelerator magnet (fabricated for use in the +FNAL accelerator complex) was utilized as a load to demon- +strate operation. Engineers from Accelerator Division of +FNAL, who developed the device engineering concepts and +worked through the process all the way to commissioning, al- +so helped with circuit simulations. LTSpice software [8] was +employed allowing to explore sensitivity and responses to var- +ious parameters in the circuit, including magnet inductance +and ability for time dependent resistance modeling of the load. +The simulation was initially successfully verified with the +conventional magnet with known inductance and resistance +where discharged currents were limited to several kA. +B. Training of Superconducting Magnets and QCD Baseline +Superconducting magnets still train [9], [10], [11] and this +remains a major issue to resolve [12]. Since QCD is supposed +to affect the training curve, a solid baseline for comparison is +desired. A performance summary of magnet series tested at +FNAL showed that “11 T” (dipoles) and “LARP” (quadru- +poles) short models provided good reproducible training +trends [10]. It was also concluded there that, to a good degree, +coils inside magnets train independently, and coils in mirror +magnets [13] train similarly to coils in “complete” (di- +pole/quadrupole) magnets. Thus, a mirror magnet is well suit- +ed for QCD testing. We chose to start with the “11 T” series as +the training pattern baseline for QCD testing. +There are several features in magnet, or rather coil, training +important in the current context, general discussion is found in +[10]. When quench current is away from conductor limit coil +training is largely independent on liquid helium temperature, +i.e. quench current would be the same at 4.2 K and at 1.9 K. +The behavior is drastically different close to the conductor +limit - transitioning from higher to lower temperature after +training, would initiate an additional (faster) training se- +quence. Damaged coils could exhibit variety of dependencies +and features, depending on the nature of the damage, and cur- +rent may be limited below conductor limit. However, all coils +in the “11 T” baseline behaved “normally” in that respect with +no abnormal dependencies observed. +C. Superconducting Magnet Testing with and without QCD +A “mirror” magnet [13] from the “11 T” series [1, Chapter +8] was assembled specifically for QCD testing. It employed a +coil which was fabricated as the last coil (#12) of the “11 T” +program at FNAL many years ago and was never used. It was +the third “11 T” mirror magnet assembled with similar param- +eters. The first low-voltage QCD discharges at up to few kA +trip-current through this magnet were conducted on March 1st, +2022, as part of magnet check out. The first discharge at spon- +taneous quench occurred on March 2nd. All initial magnet +training was at 4.5 K following the established baseline. +To compare to the baseline as directly as possible, QCD +was discharged at quench detection time while ramping condi- +tions (temperature, ramp rate) were kept nominal with respect +to baseline. Simulations showed that the current boost needed +15-20 ms to reach its peak and that delaying magnet protection +by 50 ms is safe for the magnet for quench currents below 12 +kA. We did not have complete multi-physics simulation to +know the expected effect of quench-back which was inevitable +at such large current differential dI/dt. Thus, we did not know +the expected resistance growth in the magnet, we conserva- +tively ignored it while making protection assessments. +Before testing at high magnet current, we had to make ma- +jor decisions based on partial or no information. Among those, +we did not know the importance of the “over-current” (levels +above the “quench current”) shape or duration on perfor- +mance/training. We hypothesized there may be some “effec- +tive” current, below the peak boost current, which represents +the integral boost effect and is more relevant for training than +the peak current; the only available reference [2] considered +pulse time duration to be of importance. At this time, we had +one magnet and one shot (test sequence) to investigate. Our +main handle was settable QCD voltage, up to 1 kV, affecting +the boost current. There was the possibility that even with high +peak boost current we could be too low in “effective” current +to observe any effect from QCD. On the other hand, the “ef- +fective” current may be close or equal to the peak current +which may be high long enough to damage the magnet and +halt any further QCD testing. There was also the remote pos- +sibility that the fast discharge process at high magnet current +and QCD voltage may affect the magnet integrity negatively. +In our steps we tried to navigate through those risks. +The first spontaneous quench with immediate QCD applica- +tion did occur at expected current level (9 kA, [10]) and we +chose QCD voltage of 800 V providing a substantial boost. +Retrospectively, we found the resistance growth in the magnet +to approximately follow linear trends: 0-35 m from 5 to 22 +ms after quench detection and 35-50 m from 22 to 42 ms af- +ter detection; this dependence was embedded in the LTspice +simulation along with negligibly small quench spot resistance. +Fig. 1 then compares the updated simulation, with the ob- +served real magnet current shape. The good description of cur- +rent development gave us confidence to proceed with a higher +boost current in the next quench allowing for longer “over- +current” time. No abnormal behavior in monitored signals +from the magnet was observed. +The second spontaneous quench occurred at current level +well above the first one but the third went down, well below + +Fig. 1. Magnet current development in the first ramp, driven by QCD. There is +50 ms delay of dump resistor firing and 30 ms delay of protection heater firing +but the latter has its own response time to affect the conductor. The simulation +is not perfect: it assumes constant magnet inductance of 1.4 mH (measured +2.0/1.75 mH at ~ 4/7 kA), magnet resistance development is approximate. + +Magnet current after quench detection +14 +Magnet Current (kA) +12 +10 +: data +6 +: simulation +4 +2 +0 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +time (s) + +3 +the expected level from the baseline. The QCD voltage was at +the maximum 1 kV; no abnormal data signals were observed. + After the third quench the QCD voltage was dropped to +500 V, able to boost the current to ~11 kA if no further mag- +net training. Several more quenches confirmed the magnet is +at a current plateau, within a wide range. At this point our +baseline approach was failing. We continued to follow our test +plan and lowered the temperature to 1.9 K for further testing, +initially keeping QCD voltage of 500 V. Then we moved on to +perform several thermal cycles (TC). +Figure 2. shows the complete quench history of the mirror +magnet at nominal ramp rate (20 A/s) and temperatures +(4.5/1.9 K). The quenches at 1.9 K in the TC1 were all in a +narrow current plateau at the same fraction of Short Sample +Limit (SSL) as the 4.5 K level, namely ~70 %. We stopped us- +ing QCD in the last two 1.9 K quenches, the quench current +levels remained the same. Consequent 4.5 K quenches re- +turned to the current level observed earlier at 4.5 K. Quench +current dependence on ramp rate was determined and was +consistent with earlier 11 T magnet coils [1, Chapter 8], in- +cluding mirror magnet coils. Conclusions at this point were +that the coil reached conductor limit, albeit very low one, +without training between 4.5 K and 1.9 K unlike other 11 T +coils or any other accelerator magnet training ever observed. +TC1 and all following thermal cycles ended at room tempera- +ture with the magnet remaining in the test facility cryostat. +TC2 training started at 1.9 K without any use of QCD. The +magnet forgot its training practically entirely, which is unusual +for Nb3Sn accelerator magnets, and needed 4 training quench- +es to reach the fraction of SSL observed in TC1. Quenches at +4.5 K re-confirmed conductor limitation as in TC1. QCD dis- +charges were re-introduced for TC3 with capacitor voltage of +800 V and 500 V in the first and second quenches (1.9 K), re- +spectively. The first quench of TC3 was at the same current +level as in TC2. The second quench along with few more +quenches were at conductor limit clearly indicating the effect +of QCD on the training curve; later quenches at 4.5 K con- +firmed conductor limitation. The picture from TC1 to TC3 is +quite unambiguous. In TC4 we did not use QCD and wanted +to demonstrate again magnet training, but this time training +was not fully forgotten by the magnet. Still, the first two +quenches were identified as “training” based on quench loca- +tion in the first layer. All limiting quenches in all TCs and at +both temperatures were identified in the outer coil layer, in- +cluding the last ones at 4.5 K in TC4. +III. QCD DISCUSSIONS +A. Discussion on Magnet and QCD Performance +The mirror magnet coil tested clearly underperformed com- +pared to other “11 T” coils (as presented in [10]). A question +arises if this has to do with QCD in any way. +The QCD capacitor discharge in the second quench (highest +boost current reached) drove the current rise at dI/dt ~ 1 MA/s +in the first two ms and ~0.5 MA/s in the next 2 ms, easing +substantially after that. The average increase to peak was 0.3 +MA/s. In comparison, CLIQ discharges, which have some- +what similar dI/dt characteristics in the first 10-15 ms have an +average increase to peak of 0.15 MA/s (dependencies exist, +data from non-“11 T” magnets). Moreover, regular quench +protection itself in small magnets drives dI/dt as ~ 0.5 MA/s in +the first 5 ms. All this is to say QCD pushes to higher differen- +tial current increases but those are still of the same order as +known applications. Analysis of energy flow and energy den- +sity in the magnet, including the QCD energy introduced to +the system, shows that the magnet bulk temperature never ex- +ceeded 150 K and the hot-spot temperature was below 210 K +after current dump. We used the cable enthalpy estimates from +([14], Fig 13) and quench integral calculations from [15]. +The coil used in the present test featured all improvements +made during the “11 T” program. However, it was fabricated +by a partially different (new) team at the time. The team as- +sembling the magnet was also different than earlier magnets. +Fig. 3 shows a prominent non-planarity feature of the coil and +uncharacteristic cracks observed in the non-lead end, outer +layer of the coil. This area is consistent with all limiting +quench locations as observed by quench antenna [16]. +Quenching at this area yielded a characteristic QA signal de- +velopment pattern, up to quench detection, in several channels. +The same pattern was identified immediately after the very +first quench in the inner layer too, pointing to pre-existing +conditions for the under-performance. We hypothesize that the +coil shimming corrections with Kapton layers, based on aver- +age deviations, could not fix in full the abnormal divergence +from flatness observed on the non-lead end and this caused +tension and over-stress on the coil non-lead end outer layer. +QCD developments aimed to investigate timing characteris- +tics of accelerator magnet training. The typical magnet ramp +8 +10 +12 +14 +16 +0 +10 +20 +30 +40 +Magnet current (kA) +Training quench # +MBHSM03 4.5 K +MBHSM03 1.9 K +Boosted Current +TC1 TC2 +TC3 +TC4 +No training + +Fig. 2. Spontaneous quenches at nominal ramp rate – magnet current at +quench detection vs quench number; boosted current at its peak is shown as +well; the two currents differ only if QCD is applied. All four thermal cycles +(TC) are included in the plot. Lack of training quenches is indicated. + +Fig. 3. Unusual non-conformities on the coil. Left: significant non-planarity at +the coil mid-plane (non-lead end), since coil fabrication; Right: cracks (point- +ed to by orange arrows) on the coil non-lead end, outer layer, post-testing. + + + +4 +rate range is 1-300 A/s (nominally ~ 10 A/s), and data suggest +magnets train regardless of ramp rate. Average current gains +after spontaneous training quenches vary but they are of the +order of 100 A. Thus, training mechanisms act, nominally, +within 10 s. With QCD one can test characteristic times up to +tens of ms with low limit driven by practical limitations. QCD +does not change current distribution across the magnet, and +truly emulates known training conditions but at higher ramp +rate. QCD results so far show coil training can be affected at +~30 ms timescale. If it holds that coils train independently [10] +then we can conclude that CLIQ [7] too affects training but +training quenches in magnets shall occur predominantly in +coils with no over-current (due to CLIQ). +Another effect of QCD in support of training reduction can +be seen through the Keiser effect [17], [18] which is well es- +tablished in magnets – they become “quieter” at “known” cur- +rent levels. Fig. 4 presents acoustic (mechanical) activity rec- +orded during current ramps in TC1 and TC3 (with QCD) and +TC2 (no QCD). After QCD is applied, the magnet is quieter in +ramps above previously reached current levels. +B. Discussion on Future of QCD +There are more tests planned to perform with QCD on small +magnet models: we still do not know what the relevance of +over-current length or shape on training is; nor we know how +different magnet designs may behave. However, another im- +portant question is on applications to large accelerator mag- +nets. The main hurdle is their larger inductance which limits +the current boost level along with the induced by QCD large +normal zone. Although the existing QCD can be upgraded in +terms of capacitance (C), voltage (U) increase would be more +relevant as an upgrade (energy ~ CU2). Fig. 5 shows simula- +tions of QCD with nominal and upgraded parameters for a +magnet of 35 mH (a typical HL-LHC Nb3Sn quadrupole). +There is a non-negligible positive effect that can be expected +from a higher voltage QCD likely at the expense of magnet in- +sulation requirements. Novel designs with decreased induct- +ance, like bi-filar windings [19], or multi-magnet circuit de- +signs could benefit more significantly by QCD as it is. +Although QCD is in effect also a protection device its use +does require additional delay for other protection mechanisms. +Depending on the concrete case this may not be a real problem +– heater protection, for instance, has internal delays. More im- +portantly, QCD does not need a quench to operate. In this +mode, there is no hot-spot, per se, and there is no quench de- +tection delay. Operating QCD with series of step-like high- +current trips to eventually reach a quench plateau in a magnet +will be tested in following magnet experiments. Ultimately, +protection issues do not appear to be prohibiting for using +QCD though deeper case-by-case analysis is needed. +IV. CONCLUSIONS +A new device (QCD) aiming to affect superconducting +magnet training, has been commissioned, and tested. Results +support the notion that QCD-like discharges could eliminate +training in superconducting magnets. Possible negative effects +of QCD on magnets were investigated but no clear evidence or +clues were found. More experiments are needed to determine +the limits of capacitor discharges to affect training. Larger ac- +celerator magnets can also benefit from QCD, but they proba- +bly need to have better insulation scheme allowing for higher +QCD voltage. Novel magnet designs with lower inductance +would be more susceptible to QCD. +ACKNOWLEDGMENTS +We thank Howie Pfeffer for actively supporting the phase +of QCD commissioning, Matt Kufer for invaluable electrical +engineering and help with LTSpice simulations, Chris Jensen +for leading the overall efforts; MSD and T&I colleagues who +contributed to QCD commissioning, data taking and engaged +in analysis discussions; LBNL for help with instrumentation. + +Fig. 5. Simulations with QCD on a 35 mH load at 14 275 kA “first” quench +current (both targeting a HL-LHC quadrupole magnet). “R” is the resistance +dependence described earlier; actual resistance growth is not known. Com- +pared to the discussed mirror magnet, HL-LHC magnets have ~15 times the +resistance at 300 K, and about twice as large RRR. With 2 kV discharge the +current is boosted over 500-600 A with 30-40 ms over-current duration. + +Fig. 4. Acoustics data from the three consecutive thermal cycles. The plots +are “stitched” together (note dotted lines) from different ramps to quench +(“Qi”) at the previously highest quench current for that TC. The highest +reached currents on all figures are at approximately the same level (quench +plateau). Once QCD is applied (TC1 and TC3) the magnet becomes quieter. +The lower right figure shows data from the whole ramp to quench after the +magnet reached quench current plateau in TC2. The instrumental noise level +was not well controlled over time for both current and acoustics. + +TC3,Q1TC3,Q2 +.Magnetcurrent +.Magnet current +Acousticsignal +1@1.9K +Q2@4.5K +.Acousticsignal +Q1@4.5K +Arbitrary units +Arbitrary units +TC1, +TC1, +TC1, +Time (arbitrary units) +Time (arbitrary units) +.Magnet current +.Magnet current +Acousticsignal +Q3 +Q4 +Q5 +Acousticsignal +Q1 +Arbitraryunits +TC2, +Arbitrary units +TC2,Q6 +TC2, +Time (arbitrary units) +Time (arbitrary units)Magnet current after quench detection +15 +14 +.1kV,~0Ohm +:1kV,5xR +13 +: 2 kV,5 x R +simulation, 35 mH +•2 kV,10x R +12 +0 +10 +20 +30 +40 +50 +time (ms) + +5 +REFERENCES +[1] “Nb3Sn Accelerator Magnets”, Schoerling, D., Zlobin, A.V., Eds.; +Springer: Berlin/Heidelberg, Germany, 2019. +[2] Krivykh, A. & Anashkin, O. & Keilin, V., “Elimination of training and +degradation of superconducting magnets by electrodynamic treatment”, +Soviet Physics Doklady (1985). (Available in English on researchgate, +record # 241206217) +[3] M. Tigner, “Magnet R&D for SSC”, Proceedings of Workshop on Su- +perconducting Magnets and Cryogenics, p. 6, BNL, May 1986. +[4] Emmanuele Ravaioli, private communications. +[5] C. C. Jensen et al., “Pulsed Power Supply for Magnet Quench Training”, +United States. https://doi.org/10.2172/1875872 +[6] R. Carcagno et al., “New 30 kA Power System at Fermilab and Its Use +for Measuring the Effects of Ripple Current on the Performance of +Superconducting High Field Magnets," in IEEE Trans. Appl. Supercond, +vol. +15, +no. +2, +pp. +1520-1523, +June +2005, +doi: +10.1109/TASC.2005.849153. +[7] E. Ravaioli, “CLIQ A new quench protection technology for supercon- +ducting magnets”, PhD Thesis, 2015, 10.3990/1.9789036539081. +[8] LTspice +by +Analog +Devices, +https://www.analog.com/en/design- +center/design-tools-and-calculators/ltspice-simulator.html (accessed 25 +July 2022). +[9] E. Todesco et al., “Training Behavior of the Main Dipoles in the Large +Hadron Collider,” IEEE Trans. Appl. Supercond. , vol. 27, no. 4, +4702807, June 2017 +[10] S. Stoynev et al., “Analysis of Nb3Sn accelerator magnet training,” +IEEE Trans. Appl. Supercond., vol. 29, no. 5, Aug. 2019, Art. no. +4001206, doi:10.1109/TASC.2019.289555 +[11] P. Ferracin et al., "The HL-LHC Low-β Quadrupole Magnet MQXF: +From Short Models to Long Prototypes," IEEE Trans. Appl. Supercond., +vol. 29, no. 5, pp. 1-9, Aug. 2019, Art no. 4001309, doi: +10.1109/TASC.2019.2895908. +[12] S. Gourlay et al., The U.S. Magnet Development Program Plan:. Law- +rence Berkeley National Laboratory. LBNL Report #: LBNL- +100604(2016), Retrieved from https://escholarship.org/uc/item/5178744r +[13] N. Andreev et al., "Magnetic Mirror Structure for Testing Shell-Type +Quadrupole Coils," IEEE Trans. Appl. Supercond., vol. 20, no. 3, pp. +288-291, June 2010, doi: 10.1109/TASC.2009.2039704. +[14] S. I. Bermudez et al., “Analytical method for the prediction of quench +initiation and development in accelerator magnets,” Cryogenics, Volume +95, +October +2018, +Pages +102-109, +https://doi.org/10.1016/j.cryogenics.2018.09.004. +[15] A. Zlobin et al., “Quench Protection Analysis of a Single-Aperture 11T +Nb3Sn Demonstrator Dipole for LHC Upgrades,” Conf..Proc..C +1205201 (2012), 3599-3601. +[16] S. Stoynev and J. DiMarco, “Assessment and Performance of Flexible +Quench Antenna Array Diagnostics for Superconducting Magnets”, this +conference (2LOr2B). +[17] H. M.Tensi, “The Kaiser effect and its scientific background,” in Proc. +26th 335 Eur. Conf. Acoust. Emission Testing, Berlin, Germany, 2004, +pp. 31–42. 336 +[18] J. Kaiser, “A study of acoustic phenomena in tensile test,” Ph.D. +dissertation, Tech. Univ., Munich, 1950. +[19] Steve Krave, private communications. + + diff --git a/P9FIT4oBgHgl3EQffStV/content/tmp_files/load_file.txt b/P9FIT4oBgHgl3EQffStV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8d1a4997eceecf21cd46b1c031e09ce6feddb26c --- /dev/null +++ b/P9FIT4oBgHgl3EQffStV/content/tmp_files/load_file.txt @@ -0,0 +1,389 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf,len=388 +page_content='3LPo1A-04 Template version 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='0d, 22 August 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' IEEE will put copyright information in this area See http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='org/publications_standards/publications/rights/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='html for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 1 Commissioning, Performance, and Effect of the Quench Current-boosting Device on a Dedicated Su- perconducting Magnet S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Stoynev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Baldini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Feher Abstract—Superconducting magnet training is one of the accel- erator related issues attracting attention due to significant opera- tional costs and time budget associated to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' It is especially worri- some that magnets based on the “next-generation” Nb3Sn technol- ogy are affected by long training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' While various efforts are under- way to better understand and resolve the problem a parallel path could also be investigated, a path bypassing the issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Following the concept of fast induced over-current during magnet powering, FNAL has developed an upgradable capacitor-based device to dis- charge through a superconducting magnet at quench detection or operator chosen time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='4 F/1 kV device has been tested on a 1- m-long dipole-coil in a “mirror” magnet configuration and conclu- sive results on magnet training elimination have been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' In this paper we discuss the main characteristics of the device, com- pare simulated response and actual performance, elaborate on test drivers and outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Next steps and perspectives for future use are debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Index Terms—Accelerator magnets, pulsed power supplies, su- perconducting magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' INTRODUCTION UILDING state-of-the-art superconducting accelerator magnets is a delicate process and, among other things, it involves a careful “pre-stress” setting aiming to minimize conductor degradation and optimize performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' This step could be considered “pre-conditioning” [1, Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5] of the magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' However, one can expand the meaning of “pre- conditioning” to include any process that would affect magnet performance positively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Here we choose to separate action taken before magnet powering and during magnet powering - “pre-conditioning” and “operational conditioning”, respective- ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' “Pre-conditioning” was considered and experimented with in past [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' It is known to the authors that “operational conditioning” was conducted in the past too, but we could find no clear reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' That for instance includes manipulating current ramping (levels, rates) to avoid lower current quenches though it is purely an investigative technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The work described in the present paper builds up on [2] where capacitors were discharged through a magnet as a “pre- conditioning” step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' We do this as “operational conditioning”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' In our case, a capacitor is discharged through a magnet being This work was supported by Fermi Research Alliance, LLC, under Contract DE-AC02-07CH11359 with the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of High Energy Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' (Corresponding author: Stoyan Stoynev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=') The authors are with Fermilab National Accelerator Laboratory, Batavia, IL 60510 USA (e-mail: stoyan@fnal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='gov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Digital Object Identifier will be inserted here upon acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' powered at user defined time, for instance at quench detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' This boosts the current through the magnet to levels depending on circuit parameters, including magnet parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Such a boost could effectively increase the magnet quench current as- sociated to a given quench although a “quenchless” mode will be debated later too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Given that magnet training is understood as the steady increase of quench current after consecutive spontaneous quenches, changing quench current level is an important lever to affect training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Authors are aware of possi- ble phase delays between operating current and local force [4] in pulsed mode although expectations pointed to time delays of low tens of milli-seconds at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Our own measurements of magnetic field in magnet bore and magnet current during the sharp decrease of current during system trips and quenches (with immediate magnet protection) did not indicate any phase delays between magnet current and bore field beyond our resolution of a couple of milli-seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' If present, significant local phase delays, originating from decaying eddy currents, could suppress the Lorentz force peak experienced by the su- perconductor/coil and diminish the effect of fast current boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The work to bring the boost ideas to fruition was supported by LDRD funds at FNAL and the resulting device [5] is in ef- fect a pulsed power supply integrated into the main power supply CPS3 [6] with ability to be controlled independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' We call it Quench Current-boosting Device, or QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' QCD, has similarities to CLIQ [7] but is a very different device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Apart from being used for different purposes, there are two other major differences: a) the QCD boost current is the same through the whole magnet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' there is no difference in magnet Lorentz force distribution with respect to “regular” ramp-up;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' b) QCD has no current/voltage oscillation features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' This paper describes the first application of QCD on a dedi- cated superconducting magnet, points to relevance of simula- tions before and during testing, reviews choices made during testing and results obtained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' those are followed by a discussion on future use of QCD and the technique itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' QCD COMMISSIONING AND MAGNET TESTING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' QCD preparations and simulations QCD [5] started working as a unit towards the end of 2021 and was gradually integrated to CPS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' After all major compo- nents and sub-circuits were verified and tested, the device went through various full circuit examinations, including powering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Before using it on a superconducting magnet, a B 2 conventional accelerator magnet (fabricated for use in the FNAL accelerator complex) was utilized as a load to demon- strate operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Engineers from Accelerator Division of FNAL, who developed the device engineering concepts and worked through the process all the way to commissioning, al- so helped with circuit simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' LTSpice software [8] was employed allowing to explore sensitivity and responses to var- ious parameters in the circuit, including magnet inductance and ability for time dependent resistance modeling of the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The simulation was initially successfully verified with the conventional magnet with known inductance and resistance where discharged currents were limited to several kA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Training of Superconducting Magnets and QCD Baseline Superconducting magnets still train [9], [10], [11] and this remains a major issue to resolve [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Since QCD is supposed to affect the training curve, a solid baseline for comparison is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' A performance summary of magnet series tested at FNAL showed that “11 T” (dipoles) and “LARP” (quadru- poles) short models provided good reproducible training trends [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' It was also concluded there that, to a good degree, coils inside magnets train independently, and coils in mirror magnets [13] train similarly to coils in “complete” (di- pole/quadrupole) magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Thus, a mirror magnet is well suit- ed for QCD testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' We chose to start with the “11 T” series as the training pattern baseline for QCD testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' There are several features in magnet, or rather coil, training important in the current context, general discussion is found in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' When quench current is away from conductor limit coil training is largely independent on liquid helium temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' quench current would be the same at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='2 K and at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='9 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The behavior is drastically different close to the conductor limit - transitioning from higher to lower temperature after training, would initiate an additional (faster) training se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Damaged coils could exhibit variety of dependencies and features, depending on the nature of the damage, and cur- rent may be limited below conductor limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' However, all coils in the “11 T” baseline behaved “normally” in that respect with no abnormal dependencies observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Superconducting Magnet Testing with and without QCD A “mirror” magnet [13] from the “11 T” series [1, Chapter 8] was assembled specifically for QCD testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' It employed a coil which was fabricated as the last coil (#12) of the “11 T” program at FNAL many years ago and was never used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' It was the third “11 T” mirror magnet assembled with similar param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The first low-voltage QCD discharges at up to few kA trip-current through this magnet were conducted on March 1st, 2022, as part of magnet check out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The first discharge at spon- taneous quench occurred on March 2nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' All initial magnet training was at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5 K following the established baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' To compare to the baseline as directly as possible, QCD was discharged at quench detection time while ramping condi- tions (temperature, ramp rate) were kept nominal with respect to baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Simulations showed that the current boost needed 15-20 ms to reach its peak and that delaying magnet protection by 50 ms is safe for the magnet for quench currents below 12 kA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' We did not have complete multi-physics simulation to know the expected effect of quench-back which was inevitable at such large current differential dI/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Thus, we did not know the expected resistance growth in the magnet, we conserva- tively ignored it while making protection assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Before testing at high magnet current, we had to make ma- jor decisions based on partial or no information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Among those, we did not know the importance of the “over-current” (levels above the “quench current”) shape or duration on perfor- mance/training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' We hypothesized there may be some “effec- tive” current, below the peak boost current, which represents the integral boost effect and is more relevant for training than the peak current;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' the only available reference [2] considered pulse time duration to be of importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' At this time, we had one magnet and one shot (test sequence) to investigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Our main handle was settable QCD voltage, up to 1 kV, affecting the boost current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' There was the possibility that even with high peak boost current we could be too low in “effective” current to observe any effect from QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' On the other hand, the “ef- fective” current may be close or equal to the peak current which may be high long enough to damage the magnet and halt any further QCD testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' There was also the remote pos- sibility that the fast discharge process at high magnet current and QCD voltage may affect the magnet integrity negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' In our steps we tried to navigate through those risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The first spontaneous quench with immediate QCD applica- tion did occur at expected current level (9 kA, [10]) and we chose QCD voltage of 800 V providing a substantial boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Retrospectively, we found the resistance growth in the magnet to approximately follow linear trends: 0-35 m\uf057 from 5 to 22 ms after quench detection and 35-50 m\uf057 from 22 to 42 ms af- ter detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' this dependence was embedded in the LTspice simulation along with negligibly small quench spot resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 1 then compares the updated simulation, with the ob- served real magnet current shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The good description of cur- rent development gave us confidence to proceed with a higher boost current in the next quench allowing for longer “over- current” time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' No abnormal behavior in monitored signals from the magnet was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The second spontaneous quench occurred at current level well above the first one but the third went down, well below Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Magnet current development in the first ramp, driven by QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' There is 50 ms delay of dump resistor firing and 30 ms delay of protection heater firing but the latter has its own response time to affect the conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The simulation is not perfect: it assumes constant magnet inductance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='4 mH (measured 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='0/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='75 mH at ~ 4/7 kA), magnet resistance development is approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Magnet current after quench detection 14 Magnet Current (kA) 12 10 : data 6 : simulation 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='06 time (s) 3 the expected level from the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The QCD voltage was at the maximum 1 kV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' no abnormal data signals were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' After the third quench the QCD voltage was dropped to 500 V, able to boost the current to ~11 kA if no further mag- net training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Several more quenches confirmed the magnet is at a current plateau, within a wide range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' At this point our baseline approach was failing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' We continued to follow our test plan and lowered the temperature to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='9 K for further testing, initially keeping QCD voltage of 500 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Then we moved on to perform several thermal cycles (TC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' shows the complete quench history of the mirror magnet at nominal ramp rate (20 A/s) and temperatures (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='9 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The quenches at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='9 K in the TC1 were all in a narrow current plateau at the same fraction of Short Sample Limit (SSL) as the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5 K level, namely ~70 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' We stopped us- ing QCD in the last two 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='9 K quenches, the quench current levels remained the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Consequent 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5 K quenches re- turned to the current level observed earlier at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Quench current dependence on ramp rate was determined and was consistent with earlier 11 T magnet coils [1, Chapter 8], in- cluding mirror magnet coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Conclusions at this point were that the coil reached conductor limit, albeit very low one, without training between 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5 K and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='9 K unlike other 11 T coils or any other accelerator magnet training ever observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' TC1 and all following thermal cycles ended at room tempera- ture with the magnet remaining in the test facility cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' TC2 training started at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='9 K without any use of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The magnet forgot its training practically entirely, which is unusual for Nb3Sn accelerator magnets, and needed 4 training quench- es to reach the fraction of SSL observed in TC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Quenches at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5 K re-confirmed conductor limitation as in TC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' QCD dis- charges were re-introduced for TC3 with capacitor voltage of 800 V and 500 V in the first and second quenches (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='9 K), re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The first quench of TC3 was at the same current level as in TC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The second quench along with few more quenches were at conductor limit clearly indicating the effect of QCD on the training curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' later quenches at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5 K con- firmed conductor limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The picture from TC1 to TC3 is quite unambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' In TC4 we did not use QCD and wanted to demonstrate again magnet training, but this time training was not fully forgotten by the magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Still, the first two quenches were identified as “training” based on quench loca- tion in the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' All limiting quenches in all TCs and at both temperatures were identified in the outer coil layer, in- cluding the last ones at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5 K in TC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' QCD DISCUSSIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Discussion on Magnet and QCD Performance The mirror magnet coil tested clearly underperformed com- pared to other “11 T” coils (as presented in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' A question arises if this has to do with QCD in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The QCD capacitor discharge in the second quench (highest boost current reached) drove the current rise at dI/dt ~ 1 MA/s in the first two ms and ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5 MA/s in the next 2 ms, easing substantially after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The average increase to peak was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='3 MA/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' In comparison, CLIQ discharges, which have some- what similar dI/dt characteristics in the first 10-15 ms have an average increase to peak of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='15 MA/s (dependencies exist, data from non-“11 T” magnets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Moreover, regular quench protection itself in small magnets drives dI/dt as ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5 MA/s in the first 5 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' All this is to say QCD pushes to higher differen- tial current increases but those are still of the same order as known applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Analysis of energy flow and energy den- sity in the magnet, including the QCD energy introduced to the system, shows that the magnet bulk temperature never ex- ceeded 150 K and the hot-spot temperature was below 210 K after current dump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' We used the cable enthalpy estimates from ([14], Fig 13) and quench integral calculations from [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The coil used in the present test featured all improvements made during the “11 T” program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' However, it was fabricated by a partially different (new) team at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The team as- sembling the magnet was also different than earlier magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 3 shows a prominent non-planarity feature of the coil and uncharacteristic cracks observed in the non-lead end, outer layer of the coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' This area is consistent with all limiting quench locations as observed by quench antenna [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Quenching at this area yielded a characteristic QA signal de- velopment pattern, up to quench detection, in several channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The same pattern was identified immediately after the very first quench in the inner layer too, pointing to pre-existing conditions for the under-performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' We hypothesize that the coil shimming corrections with Kapton layers, based on aver- age deviations, could not fix in full the abnormal divergence from flatness observed on the non-lead end and this caused tension and over-stress on the coil non-lead end outer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' QCD developments aimed to investigate timing characteris- tics of accelerator magnet training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The typical magnet ramp 8 10 12 14 16 0 10 20 30 40 Magnet current (kA) Training quench # MBHSM03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5 K MBHSM03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='9 K Boosted Current TC1 TC2 TC3 TC4 No training Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Spontaneous quenches at nominal ramp rate – magnet current at quench detection vs quench number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' boosted current at its peak is shown as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' the two currents differ only if QCD is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' All four thermal cycles (TC) are included in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Lack of training quenches is indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Unusual non-conformities on the coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Left: significant non-planarity at the coil mid-plane (non-lead end), since coil fabrication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Right: cracks (point- ed to by orange arrows) on the coil non-lead end, outer layer, post-testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 4 rate range is 1-300 A/s (nominally ~ 10 A/s), and data suggest magnets train regardless of ramp rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Average current gains after spontaneous training quenches vary but they are of the order of 100 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Thus, training mechanisms act, nominally, within 10 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' With QCD one can test characteristic times up to tens of ms with low limit driven by practical limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' QCD does not change current distribution across the magnet, and truly emulates known training conditions but at higher ramp rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' QCD results so far show coil training can be affected at ~30 ms timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' If it holds that coils train independently [10] then we can conclude that CLIQ [7] too affects training but training quenches in magnets shall occur predominantly in coils with no over-current (due to CLIQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Another effect of QCD in support of training reduction can be seen through the Keiser effect [17], [18] which is well es- tablished in magnets – they become “quieter” at “known” cur- rent levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 4 presents acoustic (mechanical) activity rec- orded during current ramps in TC1 and TC3 (with QCD) and TC2 (no QCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' After QCD is applied, the magnet is quieter in ramps above previously reached current levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Discussion on Future of QCD There are more tests planned to perform with QCD on small magnet models: we still do not know what the relevance of over-current length or shape on training is;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' nor we know how different magnet designs may behave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' However, another im- portant question is on applications to large accelerator mag- nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The main hurdle is their larger inductance which limits the current boost level along with the induced by QCD large normal zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Although the existing QCD can be upgraded in terms of capacitance (C), voltage (U) increase would be more relevant as an upgrade (energy ~ CU2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 5 shows simula- tions of QCD with nominal and upgraded parameters for a magnet of 35 mH (a typical HL-LHC Nb3Sn quadrupole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' There is a non-negligible positive effect that can be expected from a higher voltage QCD likely at the expense of magnet in- sulation requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Novel designs with decreased induct- ance, like bi-filar windings [19], or multi-magnet circuit de- signs could benefit more significantly by QCD as it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Although QCD is in effect also a protection device its use does require additional delay for other protection mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Depending on the concrete case this may not be a real problem – heater protection, for instance, has internal delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' More im- portantly, QCD does not need a quench to operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' In this mode, there is no hot-spot, per se, and there is no quench de- tection delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Operating QCD with series of step-like high- current trips to eventually reach a quench plateau in a magnet will be tested in following magnet experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Ultimately, protection issues do not appear to be prohibiting for using QCD though deeper case-by-case analysis is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' CONCLUSIONS A new device (QCD) aiming to affect superconducting magnet training, has been commissioned, and tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Results support the notion that QCD-like discharges could eliminate training in superconducting magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Possible negative effects of QCD on magnets were investigated but no clear evidence or clues were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' More experiments are needed to determine the limits of capacitor discharges to affect training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Larger ac- celerator magnets can also benefit from QCD, but they proba- bly need to have better insulation scheme allowing for higher QCD voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Novel magnet designs with lower inductance would be more susceptible to QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Howie Pfeffer for actively supporting the phase of QCD commissioning, Matt Kufer for invaluable electrical engineering and help with LTSpice simulations, Chris Jensen for leading the overall efforts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' MSD and T&I colleagues who contributed to QCD commissioning, data taking and engaged in analysis discussions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' LBNL for help with instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Simulations with QCD on a 35 mH load at 14 275 kA “first” quench current (both targeting a HL-LHC quadrupole magnet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' “R” is the resistance dependence described earlier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' actual resistance growth is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Com- pared to the discussed mirror magnet, HL-LHC magnets have ~15 times the resistance at 300 K, and about twice as large RRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' With 2 kV discharge the current is boosted over 500-600 A with 30-40 ms over-current duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Acoustics data from the three consecutive thermal cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The plots are “stitched” together (note dotted lines) from different ramps to quench (“Qi”) at the previously highest quench current for that TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The highest reached currents on all figures are at approximately the same level (quench plateau).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Once QCD is applied (TC1 and TC3) the magnet becomes quieter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The lower right figure shows data from the whole ramp to quench after the magnet reached quench current plateau in TC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' The instrumental noise level was not well controlled over time for both current and acoustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' TC3,Q1TC3,Q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='Magnetcurrent .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='Magnet current Acousticsignal 1@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='9K Q2@4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='Acousticsignal Q1@4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='5K Arbitrary units Arbitrary units TC1, TC1, TC1, Time (arbitrary units) Time (arbitrary units) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='Magnet current .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='Magnet current Acousticsignal Q3 Q4 Q5 Acousticsignal Q1 Arbitraryunits TC2, Arbitrary units TC2,Q6 TC2, Time (arbitrary units) Time (arbitrary units)Magnet current after quench detection 15 14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='1kV,~0Ohm :1kV,5xR 13 : 2 kV,5 x R simulation, 35 mH 2 kV,10x R 12 0 10 20 30 40 50 time (ms) 5 REFERENCES [1] “Nb3Sn Accelerator Magnets”, Schoerling, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', Zlobin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Springer: Berlin/Heidelberg, Germany, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' [2] Krivykh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' & Anashkin, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' & Keilin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', “Elimination of training and degradation of superconducting magnets by electrodynamic treatment”, Soviet Physics Doklady (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' (Available in English on researchgate, record # 241206217) [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Tigner, “Magnet R&D for SSC”, Proceedings of Workshop on Su- perconducting Magnets and Cryogenics, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 6, BNL, May 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' [4] Emmanuele Ravaioli, private communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', “Pulsed Power Supply for Magnet Quench Training”, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='2172/1875872 [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Carcagno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', “New 30 kA Power System at Fermilab and Its Use for Measuring the Effects of Ripple Current on the Performance of Superconducting High Field Magnets," in IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Supercond, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 1520-1523, June 2005, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='1109/TASC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='849153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Ravaioli, “CLIQ A new quench protection technology for supercon- ducting magnets”, PhD Thesis, 2015, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='3990/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='9789036539081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' [8] LTspice by Analog Devices, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='com/en/design- center/design-tools-and-calculators/ltspice-simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='html (accessed 25 July 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' [9] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Todesco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', “Training Behavior of the Main Dipoles in the Large Hadron Collider,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Supercond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' , vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 4, 4702807, June 2017 [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Stoynev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', “Analysis of Nb3Sn accelerator magnet training,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Supercond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 5, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 2019, Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 4001206, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='1109/TASC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='289555 [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Ferracin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', "The HL-LHC Low-β Quadrupole Magnet MQXF: From Short Models to Long Prototypes," IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Supercond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 1-9, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 2019, Art no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 4001309, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='1109/TASC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='2895908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Gourlay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', The U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Magnet Development Program Plan:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Law- rence Berkeley National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' LBNL Report #: LBNL- 100604(2016), Retrieved from https://escholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='org/uc/item/5178744r [13] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', "Magnetic Mirror Structure for Testing Shell-Type Quadrupole Coils," IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Supercond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 288-291, June 2010, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='1109/TASC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='2039704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Bermudez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', “Analytical method for the prediction of quench initiation and development in accelerator magnets,” Cryogenics, Volume 95, October 2018, Pages 102-109, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='cryogenics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Zlobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', “Quench Protection Analysis of a Single-Aperture 11T Nb3Sn Demonstrator Dipole for LHC Upgrades,” Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='.Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='.C 1205201 (2012), 3599-3601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Stoynev and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' DiMarco, “Assessment and Performance of Flexible Quench Antenna Array Diagnostics for Superconducting Magnets”, this conference (2LOr2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' [17] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='Tensi, “The Kaiser effect and its scientific background,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 26th 335 Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Acoust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Emission Testing, Berlin, Germany, 2004, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 31–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' 336 [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Kaiser, “A study of acoustic phenomena in tensile test,” Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' dissertation, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=', Munich, 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} +page_content=' [19] Steve Krave, private communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FIT4oBgHgl3EQffStV/content/2301.11278v1.pdf'} diff --git a/PtAyT4oBgHgl3EQf7fqt/content/2301.00840v1.pdf b/PtAyT4oBgHgl3EQf7fqt/content/2301.00840v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..238c712955873c669d14f76c01ce142b129484ea --- /dev/null +++ b/PtAyT4oBgHgl3EQf7fqt/content/2301.00840v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6ef0011a5b1e17aa9a860a8bdd1192097460d2e1c4866f3fb34e5a0b1a9c5c41 +size 13711716 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We focus +on experimental settings in which the treatment assignment mechanism is known to researchers and +develop a new randomization test utilizing a hierarchical relationship between different exposures. +Compared with existing approaches, our approach is essentially applicable to any null exposure +specifications and produces powerful test statistics without a priori knowledge of the true interference +structure. As empirical illustrations, we revisit two existing social network experiments: one on +farmers’ insurance adoption and the other on anti-conflict education programs. +Keywords: causal inference, exposure mapping, network interference, spillover effects, specification +tests, treatment effects. +∗School of Political Science and Economics, Waseda University, 1-6-1 Nishi-waseda, Shinjuku-ku, Tokyo 169-8050, +Japan. Email: thoshino@waseda.jp. +†Graduate School of Economics, Kyoto University, Yoshida Honmachi, Sakyo, Kyoto 606-8501, Japan. +Email: +yanagi@econ.kyoto-u.ac.jp +1 +arXiv:2301.05580v1 [stat.ME] 13 Jan 2023 + +1 +Introduction +Causal inference in the presence of cross-unit treatment interference has gained increasing attention in +the literature. Previous studies have highlighted the importance of accounting for potential treatment +spillovers through empirical applications in many fields, including economics, education, epidemiology, +and political science. Because individuals generally have different interaction networks, it is typically +impossible to identify any meaningful causal parameters without some simplifying assumptions on the +interference structure (Imbens and Rubin, 2015). A popular approach in the literature is to introduce +some exposure mapping that summarizes the impacts from other individuals’ treatments into lower- +dimensional sufficient statistics (Aronow et al., 2021). For example, Hong and Raudenbush (2006) +studied the impact of school retention on later academic performance, assuming that other students’ +retention may affect their own performance depending only on whether their school has a higher or +lower retention rate. As another example, Leung (2020) considered a treatment spillover model in which +other individuals’ treatments affect one’s potential outcome only according to the number of treated +neighbors and the total size of the neighbors. +For the obtained spillover effects to be valid, we must justify the specification of the exposure map- +ping. However, there is no general theoretical guidance on what exposure mapping should be used, and +if the chosen specification is inappropriate, the resulting causal inference may be misleading (e.g., failure +to detect treatment spillovers).1 An approach to directly address this issue is to statistically test the +specification, where the null hypothesis of interest is whether a given exposure mapping is a “correct” +choice (a more formal argument will be given later). This is the aim of the present study. Specifi- +cally, with a focus on experimental situations where the treatment assignment mechanism is known +to researchers, we develop new randomization tests for testing the specification of general exposure +mappings. +Unlike in the standard Fisher randomization test, the null hypothesis of interest here is not “sharp” +in general in the sense that only a subset of the potential outcome values are imputable from the +observed outcomes. Consequently, to perform randomization tests, we must carefully select the ap- +propriate subsets of units and treatment assignments, which we call the focal subpopulation and focal +assignments, respectively. We are not the first to consider this type of “conditional” randomization +test. In the literature, Aronow (2012), Athey et al. (2018), Basse et al. (2019), and Puelz et al. (2022) +have considered similar conditional randomization tests for testing the structure of treatment spillovers +(they did not characterize their methods as a “specification test” though). However, they did not dis- +cuss a unified framework for constructing the appropriate test statistics for general exposure mappings; +instead, they focused on several specific nulls or specific experimental setups. +The main contributions of this study are three-fold: First, introducing the notion of coarseness of +exposure mappings, we propose a novel randomization testing approach that can test virtually any +null exposures and automatically produce model-free test statistics equipped with reasonable power in +1Some recent studies have investigated under what conditions one can estimate the meaningful causal parameters +even when the exposure mapping is misspecified or not explicitly specified (Aronow and Samii, 2017; S¨avje et al., 2021; +Hoshino and Yanagi, 2022; Leung, 2022). A common finding in these studies is that only if the network dependence +is sufficiently weak can we identify some composite causal parameters. +Nevertheless, for a general form of network +interference, knowledge of the true exposure is still essential. +2 + +most situations. Second, we prove the validity of our testing procedure under relatively mild condi- +tions. Lastly, we apply our specification test of exposure mappings to two prominent social network +experiments in the literature: one on farmers’ insurance adoption by Cai et al. (2015) and the other +on anti-conflict intervention school programs by Paluck et al. (2016). In both applications, our tests +provide statistical evidence for the existence of spillover effects and, in particular, suggest that having +at least one treated peer may serve as a good summary statistic for the treatment spillovers. +The rest of this paper is organized as follows. In Section 2, we introduce our randomization test. +In Section 3, we report the results of the Monte Carlo experiments. Section 4 presents two empirical +case studies. Finally, Section 5 concludes the paper. The proofs are presented in Appendix A. +2 +Randomization Test +2.1 +Setup +Consider a finite population of size n. Each unit is indexed as i P rns, where, for a positive integer +a, we denote ras “ t1, . . . , au. Throughout this paper, we focus on the experimental setup in which +an experimenter assigns binary treatments Z “ pZ1, . . . , Znq P Z to n units according to a known +assignment probability PZ. +Here, Z :“ tz P t0, 1un : PZpzq ą 0u denotes the set of all possible +assignment patterns. +In this section, we mainly consider the case wherein all units fully comply with their assigned +treatments. Thus, for all i, we do not distinguish between the assigned treatment Zi and the actual +treatment take-up of i, which we denote as Di. The case where noncompliance is allowed such that +Zi ‰ Di for some units will be discussed in Subsection 2.5. +The outcome variable is Yi P Y Ď R. In the most general treatment spillover model, each Yi may +be affected by all elements of Z. Denoting as Yipzq the potential outcome when Z “ z, we have +Yi ” YipZq. We assume that the potential outcomes are non-stochastic and that any random variation +arises only from the randomness of the treatment assignment (i.e., the design-based approach).2 +Let E : rns ˆ Z Ñ E denote an exposure mapping, where E is a finite set. For notational simplicity, +we often write Eipzq ” Epi, zq, whose range is Ei. By construction, Ei Ď E. In general, E may depend +on n and it is possible that Eipzq and Ejpzq have different functional forms. +Definition 2.1 (Correct exposure mapping). We say that an exposure mapping E is correct if, for all +i P rns and z, z1 P Z, Eipzq “ Eipz1q ùñ Yipzq “ Yipz1q. +A correct exposure mapping always exists and is not necessarily unique. In an extreme case, the +identity mapping Eipzq “ z is always correct. In another example, when the stable unit treatment +value assumption is fulfilled, Eipzq “ zi (i.e., no interference) and Eipzq “ pzi, zjq for any j are both +correct specifications. Fisher’s sharp null of no treatment effect can also be viewed as a special case +where the exposure mapping Eipzq is a constant function independent of z. +2Alternatively, one may assume that the potential outcomes are random and view the analysis as being conditioned on +all potential outcomes. +3 + +For an exposure mapping E and z P Z, we define the level set Lipz | Eq :“ tz1 P Z : Eipz1q “ Eipzqu. +If E is correct, then all treatment assignments in Lipz | Eq are associated with the same potential +outcome value for i. +In other words, when E is correct, we can define a corresponding potential +outcome function yi : Ei Ñ Y such that +yipEipzqq “ Yipz1q +for all z1 P Lipz | Eq. +In particular, Yi “ yipEipZqq holds under a correct E. +Once an exposure mapping is given, it induces a partition of Z specific to each i, where each block is +given by the corresponding Lipz | Eq. In particular, when E is correct, Lipz | Eq provides a partition by +the equivalence class in terms of the potential outcome value. Based on the coarseness of the partition, +we define the coarseness of the exposure mappings as follows: +Definition 2.2 (Coarseness). For exposure mappings E : rns ˆ Z Ñ E and E1 : rns ˆ Z Ñ E1, we say +that E is coarser than E1 if there is a surjective mapping c : E1 Ñ E such that cpE1pi, z1qq “ Epi, zq for +all i P rns, z P Z, and z1 P Lipz | Eq. When this holds true, we write E1 Ą E. +A concept similar to Definition 2.2 can be found in Vazquez-Bare (2022). The dimensions of E and +E1 are generally different. For example, when the data are composed of n{2 pairs, we may consider +Eipzq “ zi and E1 +ipzq “ pzi, zi1q, where i1 denotes i’s partner. By definition, the “finest” exposure +mapping is the identity mapping Eipzq “ z and the coarsest one is Eipzq “ a for some constant a +independent of z, which corresponds to the case of no treatment effect whatsoever. +2.2 +Testing procedure +We would like to test whether an exposure mapping E0 : rns ˆ Z Ñ E0 is correct: +H0 : E0 is a correct exposure mapping. +Let E1 : rns ˆ Z Ñ E1 denote another exposure mapping such that E1 Ą E0. Then, there exists a +mapping c1Ñ0 : E1 Ñ E0 that satisfies c1Ñ0pE1 +i pz1qq “ E0 +i pzq for z1 P Lipz | E0q. If H0 is true, both E0 +and E1 are correct, ensuring the existence of potential outcome functions y0 +i : E0 +i Ñ Y and y1 +i : E1 +i Ñ Y +satisfying Yi “ y0 +i pE0 +i pZqq “ y1 +i pE1 +i pZqq. We define +rE1 +i ” rE1 +i pZq :“ te1 P E1 +i : c1Ñ0pe1q “ E0 +i pZqu, +namely, the set of E1 +i values that map to E0 +i pZq through c1Ñ0. Then, under H0, +Yi “ y0 +i pE0 +i pZqq “ y1 +i pe1q +for all e1 P rE1 +i . +Thus, the values of all ty1 +i pe1q : e1 P rE1 +i u are identically imputable as Yi. By construction, E1 +i pZq P rE1 +i +is always satisfied, implying that |rE1 +i | ě 1 uniformly in i, where for a generic set A, |A| denotes the +cardinality of A. +4 + +Our key idea is to test whether the following equality under H0 is true for all focal units: +y1 +i pe1 +jq “ y1 +i pe1 +kq +for all e1 +j, e1 +k P rE1 +i . +(2.1) +More specifically, letting Npκq :“ ti P rns : |rE1 +i | “ κu for some κ ě 2, we choose a set of focal units as +S Ď Npκq, namely, the focal subpopulation. How S is constructed in practice will be discussed later. +Although the construction of S can be stochastic in general, the following analysis treats S as given. +For a focal subpopulation S, we define the set of focal assignments as +CS ” CSpZq :“ tz P Z : E1 +i pzq P rE1 +i +for all i P Su. +As long as z’s are taken from CS, for any i P S, it is satisfied that E0 +i pzq “ E0 +i pZq for all such z, +whereas we can generate variations in E1 +i values within rE1 +i . Our randomization test computes the null +distribution of a test statistic by randomly sampling the assignments z’s from CS and checking whether +(2.1) holds true for all i P S. In practice, when CS is too vast to compute, one may impose additional +conditions on CS to reduce its size, which might reduce the power of the test, but does not lose its +validity. +We generally cannot use the entire Npκq as the focal subpopulation, but need to form S as a subset +of Npκq to retain sufficient variations in the focal assignments in CS. The following example would be +helpful in understanding this. +Example 2.1. Suppose that the population is composed of n{2 couples. Let E0 +i pzq “ zi and E1 +i pzq “ +pzi, zi1q, where i1 indicates i’s partner. Trivially, E1 Ą E0 with c1Ñ0 being a function that selects the +first element of E1 +i pzq. If E0 is a correct exposure, then E1 is also correct, implying that +Yi “ y0 +i pZiq “ y1 +i pe1q for all e1 P rE1 +i “ tpZi, 0q, pZi, 1qu and i P rns. +Thus, Np2q coincides with the entire population rns if all individuals are treatment eligible. If we use +the entire Np2q as the focal subpopulation S, we must shuffle the treatment assignments while keeping +pzi, zi1q “ pZi, Zi1q for all pairs (otherwise, the potential outcomes are not imputable for both partners +in each pair). However, Z is clearly the only treatment assignment that satisfies such a constraint and +the randomization test is infeasible. In this example, the most reasonable focal subpopulation would +be obtained by randomly selecting one unit from each pair, such that |S| “ n{2. The corresponding +set of focal assignments is CS “ tz P Z : zi “ Zi for all i P Su. +As shown in the next example, our framework encompasses the Fisher randomization test of no +treatment effect as a special case. In this special case, one can use the entire population as the focal +units. +Example 2.2. Let E0 +i pzq “ a, where a is independent of z, and E1 +i pzq “ zi, with c1Ñ0 being a constant +function that always returns a. Then, if E0 is correct, we have +Yi “ y0 +i paq “ y1 +i pe1q for all e1 P rE1 +i “ t0, 1u and i P rns. +5 + +Thus, Np2q “ rns holds if all individuals are treatment eligible. In this case, the entire rns can be used +as the focal units and the corresponding focal assignment set is simply given by CS “ Z. +Now, let Tpz, YSpzqq be some predetermined test statistic for each given z P Z, where YSpzq “ +pYipzqqiPS. The choice of T will be discussed later. Recall that, under H0, YSpzq is imputable from +YS ” YSpZq as long as z P CS. Then, the p-value for Tpz, YSpzqq conditional on CS under H0 is the +probability that the realization of the test statistic under the conditional randomization distribution is +at least as extreme as its actual value: +ppZ, CSq :“ PrrTpz˚, YSpz˚qq ě TpZ, YSq | z˚ P CSs, +(2.2) +where the probability is with respect to z˚ „ PZ|ZPCS. In practice, it is difficult to exactly compute +(2.2) because |CS| is typically very large. Thus, we propose to approximate the p-value using the Monte +Carlo method: +Procedure 1 Randomization Test +Input: Z, YS, PZ|ZPCS +Output: the estimated p-value: ppR +1: Compute TpZ, YSq +2: for r “ 1 to R do +3: +Draw zprq independently from PZ|ZPCS +4: +Compute Tpzprq, YSpzprqqq under H0 +5: end for +6: Compute ppR :“ 1 +R +R +ÿ +r“1 +1tTpzprq, YSpzprqqq ě TpZ, YSqu +The next theorem provides the validation of this testing procedure.3 +Theorem 2.1. +(i) PrrppZ, CSq ď α | Z P CSs ď α for any α P p0, 1q under H0. +(ii) |ppR ´ ppZ, CSq| “ OP +` +R´1{2˘ +. +In Theorem 2.1(i), the probability of a type I error is generally not precisely the nominal level +α P p0, 1q. This is a common feature of the randomization approach owing to the discrete nature of +Z. Theorem 2.1(ii) shows that the stochastic order of the Monte Carlo approximation error is R´1{2, +where the probability is with respect to tzprqu „ PZ|ZPCS. Note that this result is independent of the +size of the focal subpopulation. Because R can be freely chosen by researchers, the p-value can be +estimated with arbitrary precision. +Remark 2.1 (Sampling from PZ|ZPCS). Procedure 1 requires repeatedly sampling new z’s from +PZ|ZPCS. In certain special cases, for example, when E0 +i pzq “ zi and PZ is given by Bernoulli tri- +als, we can draw directly from PZ|ZPCS relatively easily. Even when E0 is of a more general form, +3Here, we implicitly assume that |S| ě κ; otherwise, the test statistic may not be well defined. +6 + +noting that PZ|ZPCSpzq 9 1tz P CSu ¨ PZpzq, sampling from PZ|ZPCS can be done manually by prelim- +inarily drawing z from PZ, and if it satisfies z P CS, we keep this z and move on to the computation +of the test statistic; otherwise, we re-draw a new z from PZ. Although this approach is technically +simple, it has a drawback in that, if Z is a huge set and CS is small relative to Z, the probability of +observing z satisfying z P CS can be extremely small (e.g., one in several thousands), which makes it +computationally very inefficient. This computational issue is left for future work. +Remark 2.2 (Choice of E1). In practice, there may be a large number of possible candidates for E1. +As long as E1 Ą E0, the selection of E1 can be arbitrary and it does not have to be a correct exposure +in terms of size control; however, it may significantly affect the power of the test. In general, it is better +to employ a coarser E1 to secure the size of the focal subpopulation. However, note that, if E1 is too +“similar” to E0, the test may not exhibit sufficient power. For example, when we test for the presence +of treatment spillovers (E0 +i pzq “ zi), using E1 +i pzq “ pzi, z1q would result in very low power to detect +the spillover effect because only those affected by unit 1 can contribute to the detection. Thus, ideally, +we would like to choose E1 such that it can nicely capture the true interference pattern in a way that +E0 cannot, while maintaining its coarseness. How to find such an ideal E1 in practice is also left as an +important open question. +2.3 +Construction of the focal subpopulation in a social network framework +If the structure of the population is as simple as that in Example 2.1, the construction of the focal +subpopulation S is straightforward. However, when one deals with a more general network structure, +one finds that forming an appropriate S is a challenging task. To address this issue, Athey et al. (2018) +proposed several approaches that systematically or randomly choose focal units based on the shape +of the interaction network of each unit, independent of the actual treatment assignment. However, as +pointed out by Basse et al. (2019) and Puelz et al. (2022), constructing the focal subpopulation without +utilizing the observed treatment assignment may result in the loss of the power of the test. In this +subsection, we discuss this issue further in an empirically common social network setup. +Suppose that the individuals are connected through social networks. Let A “ pAijqi,jPrns be the +adjacency matrix, where Aij P t0, 1u represents whether j affects i (directed networks are allowed). We +set Aii “ 0 for all i. For each i, the set of interacting peers is denoted as Pi :“ tj P rns : Aij “ 1u +and individual i’s neighborhood is denoted as Pi :“ tiu Y Pi. For simplicity, assume that the exposure +mapping of interest depends only on the individual’s own and peers’ treatments: E0 +i pzq “ E0 +i ppzjqjPPiq. +Example 2.3. Suppose we have n “ 8 individuals in the population and that they form an undirected +social network, as shown in Figure 1. In the figure, the treated individuals are grayed and the controls +are white. We would like to test whether E0 +i pzq “ maxjPPi zj is correct, which claims that the only +thing important is having at least one treated unit in one’s own neighborhood. As a finer counterpart +of this, let E1 +i pzq “ pzi, maxjPPi zjq. +Suppose that we have observed p4, 3, 7, 8, 2, 3, 5, 1q for the outcomes of the units. Then, the potential +outcomes schedule under H0 can be summarized as in Table 1. In the table, the blank cells are those +not imputable from the observed outcomes. +Np3q comprises the individuals excluding ID 8, with +rE1 +i “ tp1, 0q, p0, 1q, p1, 1qu. Then, the “observed” y1 +i p1, 0q’s are t4, 5u. Similarly, we obtain t3, 2, 3u and +7 + +1 +2 +3 +4 +5 +6 +7 +8 +Note: Gray and white nodes represent the treatment and control units, respectively. +Figure 1: Undirected social network A +Table 1: Potential outcomes schedule under H0: E0 is correct +ID +Yi +Zi +E0 +i pZq +E1 +i pZq +y0 +i p0q +y0 +i p1q +y1 +i p0, 0q +y1 +i p1, 0q +y1 +i p0, 1q +y1 +i p1, 1q +Np3q +1 +4 +1 +1 +(1, 0) +4 +4 +4 +4 +✓ +2 +3 +0 +1 +(0, 1) +3 +3 +3 +3 +✓ +3 +7 +1 +1 +(1, 1) +7 +7 +7 +7 +✓ +4 +8 +1 +1 +(1, 1) +8 +8 +8 +8 +✓ +5 +2 +0 +1 +(0, 1) +2 +2 +2 +2 +✓ +6 +3 +0 +1 +(0, 1) +3 +3 +3 +3 +✓ +7 +5 +1 +1 +(1, 0) +5 +5 +5 +5 +✓ +8 +1 +0 +0 +(0, 0) +1 +1 +Note: The underlined y1’s are observed potential outcomes. +t7, 8u as the observed values of y1 +i p0, 1q and y1 +i p1, 1q, respectively. When H0 is true, these three samples +should have been drawn from the same distribution, which is exactly the argument we are trying to +test with our approach. +In this social network framework, we propose two approaches for constructing S. +Maximum independent set +A practical approach that generally works for any social network data +is to construct S Ď Npκq such that Pi X Pj “ H holds for any i, j P S. This is conceptually similar +to the ϵ-net approach in Athey et al. (2018), but differs in that it first selects Npκq according to the +observed treatments, which potentially results in an improvement in the power of the test. +Finding such an S can be translated into a well-known problem in graph theory. Let G “ pNpκq, Eq +be the “common-friend” graph with vertex set Npκq and edge set E “ tpi, jq P Npκq : Pi X Pj ‰ Hu. +Then, the independent set of G, which is a set of vertices such that no two vertices in the set are +adjacent, can be a valid candidate for S. In particular, we would like to find a maximum independent +set (MIS) of G. +Figure 2 shows the common-friend graph for the network data in Example 2.3. +We have two +MISs, namely, t1, 6, 7u and t2, 6, 7u. When we set S “ t2, 6, 7u, the admissible assignment vectors are +characterized by CS “ tz P Z : E0 +2pzq “ E0 +6pzq “ E0 +7pzq “ 1u. +Biclique method +We can extend the biclique method in Puelz et al. (2022) to our situation. To +this end, we define the null exposure graph and its biclique in our context. Let Z0 Ď Z denote a +predetermined set of treatment assignments such that Z P Z0. For example, one may construct Z0 by +drawing from PZ sufficiently many times. The null exposure graph of H0 with respect to Z0 is defined +8 + +1 +2 +3 +4 +5 +6 +7 +Figure 2: Common-friend graph G (Example 2.3) +as a bipartite graph G “ pNpκq Y Z0, Eq, where E “ tpi, zq P Npκq Y Z0 : c1Ñ0pE1 +i pzqq “ E0 +i pZqu. +That is, there exists an edge between i P Npκq and z P Z0 when Yi “ y0 +i pE0 +i pZqq “ y1 +i pE1 +i pzqq holds +under H0. Then, a biclique Bb “ pNb, Zbq in G is defined as a pair of sets Nb Ď Npκq and Zb Ď Z0 +such that pi, zq P E holds for all i P Nb and z P Zb. In general, we can find multiple bicliques for G +(b “ 1, 2, . . .) and it is typically desirable to have a larger biclique. By construction, if H0 is true, then +we have Yi “ y0 +i pE0 +i pZqq “ y1 +i pE1 +i pzqq for all pi, zq P Bb. Once a biclique Bb is obtained, we can simply +set S “ Nb and CS “ Zb. +2.4 +Choice of test statistic +There is certain freedom in the choice of test statistics. In this study, we consider the following three +types of statistics: Kruskal-Wallis (KW), average cross difference (ACD), and ordinary least squares +(OLS). +To define these statistics, we introduce additional notations. For each i P S, we order the elements +of rE1 +i as e1 +1, e1 +2, . . . , e1 +κ based on some rule. For example, in the case of Example 2.3, we can consider an +increasing order in terms of the value of 2zi ` maxjPPi zj, leading to pe1 +1, e1 +2, e1 +3q “ pp0, 1q, p1, 0q, p1, 1qq. +Note that, because rE1 +i may be heterogeneous among the individuals, the compositions of the ordered +elements pe1 +1, e1 +2, . . . , e1 +κq are also generally different among these individuals. When such a heterogeneity +is present, what sorting rule is adopted is a factor that affects the power of the test. +For a given +treatment assignment z, we partition the focal subpopulation S into κ groups: Sjpzq :“ ti P S : +E1 +i pzq “ e1 +ju for j P rκs. Now, we have κ potential outcomes py1 +i pe1 +1q, y1 +i pe1 +2q, . . . , y1 +i pe1 +κqq for each i, +which should take the same value under H0. +Noting that our task can be viewed as testing the equivalence of κ different treatments, we consider +the use of the KW statistic as standard, as in Keele et al. (2012) and Wang et al. (2020). First, we +rank all pYiqiPS from 1 to |S|. Let vi be the rank of Yi and Vjpzq be the summation of the ranks +for group Sjpzq: Vjpzq :“ ř +iPSjpzq vi. The KW statistic compares the average rank for each group j, +Vjpzq{|Sjpzq|, with the average rank for the entire |S|, p|S| ` 1q{2: +Tpz, YSq “ +12 +|S|p|S| ` 1q +κÿ +j“1 +|Sjpzq| +ˆ Vjpzq +|Sjpzq| ´ |S| ` 1 +2 +˙2 +. +9 + +The ACD statistic is defined simply as the average of the absolute average differences for all com- +binations of treatment pairs: +Tpz, YSq “ +2 +κpκ ´ 1q +ÿ +1ďjăkďκ +ˇˇˇˇˇ +ř +iPSjpzq Yi +|Sjpzq| +´ +ř +iPSkpzq Yi +|Skpzq| +ˇˇˇˇˇ . +It is also possible to consider a “model-based” test statistic, as in Athey et al. (2018). Suppose we +have some E0 and E1, where E0 might be vector-valued, and let Xipzq be a vector of variables whose +values are determined only through E1 +i pzq but not through E0 +i pzq. For example, when E0 +i pzq “ zi and +E1 +i pzq “ pzi, maxjPPi zjq, one may use Xipzq “ maxjPPi zj. Then, by fitting the following regression +model to the data in S, +Yi “ β0 ` E0 +i pzqJβ1 ` XipzqJβ2 ` errori, +(2.3) +we can use the F-statistic for the significance of pβ2 as the test statistic Tpz, YSq, where pβ2 denotes the +OLS estimate. +For another example, when we have E0 +i pzq “ maxjPPi zj and E1 +i pzq “ pzi, maxjPPi zjq, as in Exam- +ple 2.3, we may consider using Xipzq “ pzi, maxjPPi zjq. Note that, when one adopts this model-based +approach, the presumed model does not have to perfectly reflect the true interference structure. How- +ever, if they are significantly different, it will lead to a substantial loss of the power, as numerically +demonstrated in Section 3. +Remark 2.3 (Multiplicity of test statistics). As shown above, we generally have multiple statistics for +testing H0. Furthermore, by considering different values of κ and E1, we can generate a large number of +additional test statistics. One simple way to utilize the information in all s different statistics altogether +is to combine them into a single test statistic, T comb “ gpT 1, T 2, . . . , T sq, as suggested in Imbens and +Rubin (2015). Another approach is to apply, for example, Simes’ correction for multiple testing: letting +the ordered p-values be denoted by ppp1q +R ď ¨ ¨ ¨ ď pppsq +R , reject H0 if pppiq +R ď iα{s for some i P t1, . . . , su. See +Simes (1986) and Subsection 9.2.2 of Lehmann and Romano (2022) for more details. +2.5 +Imperfect compliance +Thus far, we have assumed that all individuals comply with their initial treatment assignments. How- +ever, in certain realistic situations, they are allowed to self-select their own treatment status. Now, we +write D “ pD1, . . . , Dnq as the n-dimensional vector of the actual treatment take-ups. When noncom- +pliance is allowed (D ‰ Z), the probability distribution of D is generally unknown. Thus, in this case, +we cannot perform the test in Procedure 1 based on the actual treatments because it is infeasible to +resample independent copies dprq’s of D from a known distribution. +One empirically tractable approach to this problem is to resort to an intention-to-treat (ITT) type +of analysis. That is, we consider formulating the exposure mapping as a function not of d but of the +10 + +initial assignment z.4 For example, suppose we have the following treatment selection model: +Di “ 1 +# +γ0i ` γ1iZi ` γ2i +ÿ +jPPi +Zj ą 0 ++ +and there are no treatment spillovers in the outcome model. In this case, the exposure mapping of +interest would be E0 +i pzq “ pzi, ř +jPPi zjq. Then, if we can find an appropriate E1 that is finer than E0, +in exactly the same way as in Procedure 1, we can test the validity of this model specification. +3 +Numerical Simulations +In this section, we assess the small sample performance of our randomization test using Monte Carlo +simulations. +3.1 +Perfect compliance +First, we consider the case of perfect compliance. The network is created from a simple Erd¨os–R´enyi +model with a probability of p “ 3{n, where we set n “ 200. +We consider the following two data +generating processes (DGPs) for the outcome variables: +DGP 1: Yi “ Di ` τ +ÿ +jPPi +Dj ` ξi, +DGP 2: Yi “ Di ` τ ¨ g +˜ ÿ +jPPi +Dj +¸ +` ξi, +where ξi „ Np0, 1q, τ P r0, 2s, and gpaq “ 1ta ď 2u ¨ a ` 1ta ě 3u ¨ a´1. For the treatment assignment +mechanism, we employ a complete randomization, where randomly selected n{2 units receive Z “ 1. +Because perfect compliance is assumed here, Di “ Zi holds for all i. We set E0 +i pzq “ zi, and, hence, +E0 is correct when τ “ 0. For the choice of E1, the following two exposure mappings are used: +Exposure 1: E1 +i pzq “ +ˆ +zi, max +jPPi zj +˙ +, +Exposure 2: E1 +i pzq “ +˜ +zi, +ÿ +jPPi +zj +¸ +. +Note that Exposure 1 is coarser than Exposure 2 and that only Exposure 2 is correct when τ ą 0. For +Exposure 1, it is natural to set κ “ 2 such that rEi “ tpZi, 0q, pZi, 1qu, and Np2q “ ti P rns : |Pi| ą 0u. +For Exposure 2, we set κ “ 4 such that rEi “ tpZi, 0q, . . . , pZi, 3qu and Np4q “ ti P rns : |Pi| “ 3u. +To construct the focal subpopulation S, we consider the following three approaches: (i) MIS method, +(ii) random selection of |Npκq|{2 focal units, and (iii) biclique method. Here, note that finding the +largest independent set and finding the largest biclique are both NP-hard problems, and, thus, we +approximate their solutions using a greedy vertex coloring algorithm and the binary inclusion-maximal +biclustering method, respectively.5 For (i) and (ii), we set R “ 2000. For (iii), we draw treatment +assignments 9 million times from PZ to create Z0. +4This type of exposure mapping was also considered in Hoshino and Yanagi (2022) and is termed as instrumental +exposure mapping. +5Specifically, +in the Monte Carlo simulations and the empirical illustrations below, +we use the functions +greedy vertex coloring and BCBimax in the R packages igraph and biclust, respectively. +11 + +For each setup, we perform our randomization test using the KW, ACD, and OLS statistics under +the nominal significance level of 5%. The OLS statistic is obtained as the F-statistic for the OLS +estimate of β2 in (2.3), where we set Xipzq “ maxjPPi zj for Exposure 1 and Xipzq “ ř +jPPi zj for +Exposure 2. +In addition to these three tests, we also report the results from the Simes-corrected +p-value based on them. The following results are based on 1,000 Monte Carlo replications. +Figures 3 and 4 show the rejection frequency of each method for different τ values in DGPs 1 and +2, respectively. In each figure, panels (a) and (b) present the simulation results for Exposures 1 and 2, +respectively. When τ “ 0, for all methods and test statistics, the rejection frequencies are sufficiently +close to the nominal level in both DGPs, which is consistent with our theory. Particularly in DGP +1, the power of these tests quickly increases as τ increases, suggesting the consistency of our testing +procedure. However, in DGP 2, we find that the power of the OLS statistic based on Exposure 2 is +significantly reduced. This may be due to “model misspecification” in the OLS regression caused by the +mishandling of the nonlinearity of the g function in this DGP. Note that the OLS model with Exposure +1 is also a misspecified model; however, the magnitude of the misspecification is mild relative to that of +Exposure 2. Even when Exposure 2 is used in DGP 2, the KW statistic remains sufficiently powerful. +Comparing the three methods for constructing the focal subpopulation, we find that the MIS per- +forms the best and the biclique the least. The random selection approach is in-between of the two. +However, caution should be exercised when interpreting this result. To be clear, a large part of the dif- +ference in the performance of these methods is essentially due to the difference in the sizes of S and CS. +Finding a reasonably large biclique becomes more difficult when the null exposure graph is sparser, as +in this simulation setting (see Subsection 6.2 of Puelz et al. (2022) for a related discussion). In addition, +even with the above-mentioned simplified algorithm, finding a large biclique is still computationally +very demanding; for example, even though we have employed a fairly large Z0, the resulting size of +S was, on average, less than 20 or so after a long computation time. In a different setup where the +biclique method can easily identify relatively large S and CS, its performance would be substantially +improved. +For both DGPs, except for the biclique method, Exposure 1 tends to provide more powerful tests +than those of Exposure 2, even though Exposure 1 is incorrect when τ ą 0. This result is possibly +due to the fact that a coarser Exposure 1 generally induces a larger S than that of Exposure 2, while +retaining a strong correlation with the true interference structure. Specifically, for the MIS method, +the sizes of S generated from Exposures 1 and 2 are approximately 80 and 30, respectively. +3.2 +Imperfect compliance +Next, we discuss the experiments for the case of imperfect compliance. In particular, we consider a +one-sided compliance situation; that is, only when Zi “ 1 can i choose Di “ 1. The initial treatment +assignment Z is generated from a complete randomization such that n{2 units are eligible to take the +treatment. For treatment-eligible units, the compliance status follows Bernoullip0.8q uniformly (i.e., no +interference within the treatment choices). All other parts of the simulation design are the same as +those in the perfect compliance case. Note that E0 +i pzq “ zi is correct when τ “ 0. +The simulation results are shown in Figures 5 and 6. Overall, the same comments as in the previous +experiment apply to this experiment as well. Our randomization test works satisfactorily in terms of +12 + +(a) Exposure 1 +(b) Exposure 2 +Figure 3: Simulation results: DGP 1 under perfect compliance +both size control and power property, although the power of the test seems slightly worse than that in +the perfect compliance case. This is reasonable considering that, unlike in the previous case, neither +Exposure 1 nor Exposure 2 is correct when τ ą 0. For the construction of the focal subpopulation, it +seems desirable to use the MIS approach. An interesting finding is that, when Exposure 2 is employed +in DGP 2, it is the ACD statistic, not the OLS statistic, that loses its power significantly. We suspect +that this result is due to the current simulation setup and cannot be generalized any further. +13 + +power. +power +power +1.00 +1.00 +1.00 +0.80 +0.80 +0.80 +0.60 +0.60 +0.60 +ACD +KW +0.40 +0.40 +0.40 +OLS +Simes +0.20 +0.20 +0.20 +0.05 +0.05 +0.05 +0.00 +0.00 +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +1 +T +T +MIS +Random selection +Bicliquepower. +power +power +1.00 +1.00 +1.00 +0.80 +0.80 +0.80 +0.60 +0.60 +0.60 +ACD +KW +0.40 +0.40 +0.40 +OLS +Simes +0.20 +0.20 +0.20 +0.05 +0.05 +0.05 +0.00 +0.00 +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +T +T +T +MIS +Random selection +Biclique(a) Exposure 1 +(b) Exposure 2 +Figure 4: Simulation results: DGP 2 under perfect compliance +4 +Empirical Illustrations +As empirical illustrations, we apply our randomization test to the existing datasets from two well-known +social network experiments in the literature. The first one is the data on farmers’ insurance adoption +in Cai et al. (2015), and the other is the data on anti-conflict intervention school programs in Paluck +et al. (2016). +For both datasets, we investigate the same type of null hypotheses. Here, let +Ea +i pzq “ zi, +Eb +i pzq “ max +jPPi +zj, +Ec +i pzq “ +ˆ +zi, max +jPPi zj +˙ +, +Ed +i pzq “ +˜ +zi, +ÿ +jPPi +zj +¸ +, +14 + +power. +power +power +1.00 +1.00 +1.00 +0.80 +0.80 +0.80 +0.60 +0.60 +0.60 +ACD +KW +0.40 +0.40 +0.40 +OLS +Simes +0.20 +0.20 +0.20 +0.05 +0.05 +0.05 +0.00 +0.00 +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +1 +T +T +MIS +Random selection +Bicliquepower. +power +power +1.00 +1.00 +1.00 +0.80 +0.80 +0.80 +0.60 +0.60 +0.60 +ACD +KW +0.40 +0.40 +0.40 +OLS +Simes +0.20 +0.20 +0.20 +0.05 +0.05 +0.05 +0.00 +0.00 +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +T +T +T +MIS +Random selection +Biclique(a) Exposure 1 +(b) Exposure 2 +Figure 5: Simulation results: DGP 1 under imperfect compliance +such that Ea Ă Ec, Eb Ă Ec, and Ec Ă Ed hold. Table 2 summarizes the null hypotheses of interest, +the construction of Npκq, and rE1 +i considered in both empirical applications. +For example, the null hypothesis Ha claims that Ea is a correct exposure such that there are no +treatment spillovers. To test this null hypothesis, we employ Ec as E1, which contains the information +about one’s own treatment and whether he/she has at least one treated peer. This choice of E1 leads +to rE1 +i “ tpZi, 0q, pZi, 1qu with κ “ 2 and to Np2q “ ti P rns : |Pi| ą 0u. The other null hypotheses, Hb +and Hc, can be interpreted similarly. When testing Hc, we choose a value for κ such that the resulting +|S| is maximized. In the following, given the simulation results in Section 3, we report only the results +obtained using the MIS method. +15 + +power. +power +power +1.00 +1.00 +1.00 +0.80 +0.80 +0.80 +0.60 +0.60 +0.60 +ACD +KW +0.40 +0.40 +0.40 +OLS +Simes +0.20 +0.20 +0.20 +0.05 +0.05 +0.05 +0.00 +0.00 +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2 +1 +T +MIS +Random selection +Bicliquepower. +power +power +1.00 +1.00 +1.00 +0.80 +0.80 +0.80 +0.60 +0.60 +0.60 +ACD +KW +0.40 +0.40 +0.40 +OLS +Simes +0.20 +0.20 +0.20 +0.05 +0.05 +0.05 +0.00 +0.00 +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +T +T +MIS +Random selection +Biclique(a) Exposure 1 +(b) Exposure 2 +Figure 6: Simulation results: DGP 2 under imperfect compliance +Table 2: The null hypotheses and settings +H0 +E0 +E1 +rE1 +i +κ +Npκq +Ha +Ea +Ec +tpZi, 0q, pZi, 1qu +κ “ 2 +ti P rns : |Pi| ą 0u +Hb +Eb +Ec +tp0, 1q, p1, 0q, p1, 1qu +κ “ 3 +ti P rns : |Pi| ą 0, Eb +i pZq “ 1u +Hc +Ec +Ed +tpZi, 1q, . . . , pZi, κqu +κ ě 2 +ti P rns : |Pi| “ κ, Ec +i pZq “ pZi, 1qu +4.1 +Social networks and farmers’ insurance decisions +Cai et al. (2015) conducted a field experiment to estimate the effect of providing intensive information +16 + +power. +power +power +1.00 +1.00 +1.00 +0.80 +0.80 +0.80 +0.60 +0.60 +0.60 +ACD +KW +0.40 +0.40 +0.40 +OLS +Simes +0.20 +0.20 +0.20 +0.05 +0.05 +0.05 +0.00 +0.00 +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +1 +T +T +MIS +Random selection +Bicliquepower. +power +power +1.00 +1.00 +1.00 +0.80 +0.80 +0.80 +0.60 +0.60 +0.60 +ACD +KW +0.40 +0.40 +0.40 +OLS +Simes +0.20 +0.20 +0.20 +0.05 +0.05 +0.05 +0.00 +0.00 +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2 +T +T +MIS +Random selection +Bicliquesessions about the weather insurance on farmers’ insurance take-up decisions. The authors demon- +strated the presence of significant treatment spillovers among the farmers by employing a series of +regression models, where the main explanatory variable was the fraction (or number) of friends who +were assigned to the intensive information sessions in advance of the focal farmers. +In this experiment, four types of sessions were conducted: first-round simple sessions, first-round +intensive sessions, second-round simple sessions, and second-round intensive sessions. In each round, +the simple sessions briefly described the insurance contract, whereas the intensive sessions explained the +insurance contract and the expected benefits of the insurance in detail. To randomly assign the farmers +to each session, the authors performed a stratified randomization with four strata constructed in each +village according to the household size and rice production acreage. All rice-producing households were +invited to participate in one of the four sessions, and almost 90% of them attended. Thus, because the +probability of non-attendance was low, we ignore the treatment non-compliance for simplicity. +In this analysis, the outcome variable of interest is Yi P t0, 1u, which indicates whether farmer i +decides to buy the weather insurance after attending the session. Let inti P t0, 1u denote whether i is +assigned to an intensive session, and let seci P t0, 1u denote whether i is assigned to the second-round +session. Because the treatment spillovers matter only for the participants in the second-round session, +as they can receive information from the first-round participants, we create the focal units using only +the farmers assigned to the second round. For the definition of the treatment variable, for a focal unit +i, we set zi “ 1 if inti “ 1. For a nonfocal unit j, we set zj “ 1 if both intj “ 1 and secj “ 0 are true. +When performing our randomization tests, we randomize both inti and seci following the protocol of +the original experiment. To obtain a certain number of focal units in each village, we only use villages +with at least 50 farmers as our sample. +The results of our randomization tests are summarized in Table 3. First, we can see that all p- +values for testing Ha are smaller than 5%, which indicates the presence of information spillovers among +the farmers. For the null hypothesis Hb, none of the test statistics reject this hypothesis at the 5% +significance level. Lastly, for Hc, we cannot reject this under any reasonable significance level with any +of the test statistics considered. In summary, these results suggest the existence of spillover effects in +farmers’ insurance purchasing decisions and that having at least one friend assigned to the intensive +session might be an important factor that accounts for the spillovers rather than the number of such +friends.6 +4.2 +Spillover effects of the anti-conflict intervention programs +In the second empirical case study, we apply the proposed test to the data from Paluck et al. (2016), +who investigated the impact of anti-conflict intervention programs on adolescents’ norms and attitudes +through a large-scale experiment in 56 American middle schools. Half of these schools were randomly +selected to host the programs. Within each selected school, a group of students (called seed-eligible +students) were non-randomly selected, and half of these students (called seed students) were chosen +through a stratified randomization and invited to join the program. The seed-eligible students’ strata +were determined by their individual characteristics, such as gender, grade, and friendship network +6Given this result, one might want to adopt Ec as the final model and reanalyze the data. However, note that doing +so raises another issue of “inference after model selection”, which is beyond the scope of this paper. +17 + +Table 3: Empirical results: farmers’ insurance decisions +p-values +Simes’ correction +κ +KW +ACD +OLS +10% +5% +1% +|S| +R +Testing for Ha +2 +0.021 +0.049 +0.021 +✓ +✓ +610 +100,000 +Testing for Hb +3 +0.054 +0.100 +0.054 +✓ +542 +100,000 +Testing for Hc +6 +0.949 +0.478 +0.979 +145 +100,000 +variables. The students’ friendship networks were measured by simply asking them to nominate up +to 10 friends in their school. Participating in the program was not mandatory for the seed students, +and, in fact, the compliance rate was approximately 40%, which corresponds to the case of imperfect +compliance. For more details on the experimental design, see the Supplementary Appendix of Paluck +et al. (2016). +The purpose of this experiment is to examine how the seed students who participated in the inter- +vention program could influence other students through their social networks to improve the climate +of the school. In each intervention meeting, the seed students were encouraged to identify common +conflict behaviors in their schools and discuss behavioral strategies to mitigate the conflicts. As an +important role of the seed students, they were allowed to hand out a program wristband as a reward +to students for their engagement in friendly or conflict-mitigating behaviors. Let Yi P t0, 1u be an +indicator of whether student i wears a program wristband, which is the outcome variable of interest +in this empirical analysis. Let Zi P t0, 1u indicate the treatment eligibility (i.e., whether i is a seed +student). +We perform a randomization test to examine the spillover effects of being selected as a seed student +on wearing wristbands, which can be viewed as the ITT-type analysis discussed in Subsection 2.5. When +performing our tests, we exclude the schools where fewer than three wristbands were distributed. In +addition, in line with Paluck et al. (2016), we construct the focal subpopulation from the seed-eligible +students who had at least one social referent peer (i.e., a student whose indegree was ranked in the top +10% of his/her school). +The results are summarized in Table 4. We can see from the table that all p-values for Ha and Hb are +sufficiently small to reject them at the 5% significance level. The rejection of these hypotheses together +suggests that the intervention program has strong spillover effects through the students’ networks. By +contrast, Hc is not rejected even under the significance level of 10%. Thus, like in the first empirical +study, we might conclude that the presence of even just one seed student friend, rather than the number +of treated friends, can reasonably explain the students’ anti-conflict activities. +18 + +Table 4: Empirical results: anti-conflict education program +p-values +Simes’ correction +κ +KW +ACD +OLS +10% +5% +1% +|S| +R +Testing for Ha +2 +0.019 +0.020 +0.011 +✓ +✓ +413 +100,000 +Testing for Hb +3 +0.001 +0.006 +0.001 +✓ +✓ +✓ +358 +100,000 +Testing for Hc +2 +0.212 +0.150 +0.214 +105 +100,000 +5 +Conclusion +In this study, we developed a novel randomization testing approach for the specification of general +exposure mappings in treatment effect models with interference. Based on the concept of coarseness +of exposure mappings, our proposed approach has a fairly broad empirical applicability and enables +us to construct model-free test statistics with a good power property. As empirical illustrations, we +have revisited two existing social network experiments in the literature: one is the data on farmers’ +insurance adoption studied in Cai et al. (2015) and the other is the data on anti-conflict education +programs studied in Paluck et al. (2016). From the results of the experiments on both datasets, we +found that the exposure mapping Eipzq “ pzi, maxjPPi zjq has a certain capability to account for the +spillover effects. These results indicate the usefulness of the proposed method. +Acknowledgments +We thank Jing Cai for kindly instructing us on how to recover the randomization strata from the +replication data of Cai et al. (2015). We also thank seminar participants at Keio University, Kwansei +Gakuin University, Tokyo University, and Osaka University for their helpful comments discussions. +This work was supported by JSPS KAKENHI grant numbers 19H01473 and 20K01597. The datasets +used in the empirical illustrations are available from the Interuniversity Consortium for Political and +Social Research (Cai et al., 2019; Paluck et al., 2020). +19 + +A +Appendix: Proof of Theorem 2.1 +(i) By the definition of CS, for any z˚ P CS, we have YSpz˚q “ YS under H0. Thus, we can write +ppZ, CSq “ PrrTpz˚, YSq ě TpZ, YSq | z˚ P CSs, where the probability is with respect to z˚ „ PZ|ZPCS. +Let FT|CS denote the conditional distribution function of ´Tpz˚, YSq given z˚ P CS induced from +PZ|ZPCS. Then, ppZ, CSq “ FT|CSp´TpZ, YSqq, and, thus, +PrrppZ, CSq ď α | Z P CSs “ PrrFT|CSp´TpZ, YSqq ď α | Z P CSs ď α, +where the inequality follows from the fact that ´TpZ, YSq is distributed as FT|CS given Z P CS. +(ii) Let ppzprqq :“ 1tTpzprq, YSq ě TpZ, YSqu, such that ppR “ R´1 řR +r“1 ppzprqq. Because zprq’s are +identically drawn from PZ|ZPCS, we have Etzprqu„PZ|ZPCS ppR “ Ez„PZ|ZPCS ppzq “ ppZ, CSq. Furthermore, +by the independence of the draws, +Etzprqu„PZ|ZPCS +` +ppR ´ ppZ, CSq +˘2 “ +Ez„PZ|ZPCS pppzq ´ ppZ, CSqq2 +R +“ ppZ, CSqp1 ´ ppZ, CSqq +R +. +Thus, the result follows from Chebyshev’s inequality. +20 + +References +Aronow, P.M., 2012. A general method for detecting interference between units in randomized experi- +ments, Sociological Methods & Research, 41 (1), 3–16. +Aronow, P.M., Eckles, D., Samii, C., and Zonszein, S., 2021. Spillover effects in experimental data, in: +J. Druckman and D.P. Green, eds., Advances in Experimental Political Science, Cambridge University +Press, chap. 16, 289–319. +Aronow, P.M. and Samii, C., 2017. Estimating average causal effects under general interference, with +application to a social network experiment, The Annals of Applied Statistics, 11 (4), 1912–1947. +Athey, S., Eckles, D., and Imbens, G.W., 2018. Exact p-values for network interference, Journal of the +American Statistical Association, 113 (521), 230–240. +Basse, G.W., Feller, A., and Toulis, P., 2019. Randomization tests of causal effects under interference, +Biometrika, 106 (2), 487–494. +Cai, J., De Janvry, A., and Sadoulet, E., 2015. Social networks and the decision to insure, American +Economic Journal: Applied Economics, 7 (2), 81–108. +Cai, J., De Janvry, A., and Sadoulet, E., 2019. Replication data for: Social networks and the decision +to insure, Inter-university Consortium for Political and Social Research [distributor], 2019-10-12. +https://doi.org/10.3886/E113593V1. +Hong, G. and Raudenbush, S.W., 2006. Evaluating kindergarten retention policy: A case study of +causal inference for multilevel observational data, Journal of the American Statistical Association, +101 (475), 901–910. +Hoshino, T. and Yanagi, T., 2022. Causal inference with noncompliance and unknown interference, +arXiv:2108.07455. +Imbens, G.W. and Rubin, D.B., 2015. Causal Inference in Statistics, Social, and Biomedical Sciences, +Cambridge University Press. +Keele, L., McConnaughy, C., and White, I., 2012. Strengthening the experimenter’s toolbox: Statistical +estimation of internal validity, American Journal of Political Science, 56 (2), 484–499. +Lehmann, E.L. and Romano, J.P., 2022. Testing Statistical Hypotheses, Fourth Edition, Springer. +Leung, M., 2020. Treatment and spillover effects under network interference, The Review of Economics +and Statistics, 102 (2), 368–380. +Leung, M., 2022. Causal inference under approximate neighborhood interference, Econometrica, 90 (1), +267–293. +Paluck, E.L., Shepherd, H., and Aronow, P.M., 2016. Changing climates of conflict: A social network +experiment in 56 schools, Proceedings of the National Academy of Sciences, 113 (3), 566–571. +21 + +Paluck, E.L., Shepherd, H.R., and Aronow, P., 2020. Changing climates of conflict: A social network +experiment in 56 schools, New Jersey, 2012-2013, Inter-university Consortium for Political and Social +Research [distributor], 2020-09-14. https://doi.org/10.3886/ICPSR37070.v2. +Puelz, D., Basse, G., Feller, A., and Toulis, P., 2022. A graph-theoretic approach to randomization +tests of causal effects under general interference, Journal of the Royal Statistical Society: Series B, +84 (1), 174–204. +S¨avje, F., Aronow, P.M., and Hudgens, M.G., 2021. Average treatment effects in the presence of +unknown interference, The Annals of Statistics, 49 (2), 673–701. +Simes, R.J., 1986. An improved bonferroni procedure for multiple tests of significance, Biometrika, +73 (3), 751–754. +Vazquez-Bare, G., 2022. Identification and estimation of spillover effects in randomized experiments, +Journal of Econometrics. +Wang, Y., Rosenberger, W.F., and Uschner, D., 2020. Randomization tests for multiarmed randomized +clinical trials, Statistics in Medicine, 39 (4), 494–509. +22 + diff --git a/QtE5T4oBgHgl3EQfZQ-4/content/tmp_files/load_file.txt b/QtE5T4oBgHgl3EQfZQ-4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dbf0a6030d12213d31640067ad7fd0ad54ef48fc --- /dev/null +++ b/QtE5T4oBgHgl3EQfZQ-4/content/tmp_files/load_file.txt @@ -0,0 +1,922 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf,len=921 +page_content='Randomization Test for the Specification of Interference Structure Tadao Hoshino∗ and Takahide Yanagi† January 2023 Abstract This study considers testing the specification of spillover effects in causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We focus on experimental settings in which the treatment assignment mechanism is known to researchers and develop a new randomization test utilizing a hierarchical relationship between different exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Compared with existing approaches, our approach is essentially applicable to any null exposure specifications and produces powerful test statistics without a priori knowledge of the true interference structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' As empirical illustrations, we revisit two existing social network experiments: one on farmers’ insurance adoption and the other on anti-conflict education programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Keywords: causal inference, exposure mapping, network interference, spillover effects, specification tests, treatment effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' ∗School of Political Science and Economics, Waseda University, 1-6-1 Nishi-waseda, Shinjuku-ku, Tokyo 169-8050, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Email: thoshino@waseda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' †Graduate School of Economics, Kyoto University, Yoshida Honmachi, Sakyo, Kyoto 606-8501, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Email: yanagi@econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='jp 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='05580v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='ME] 13 Jan 2023 1 Introduction Causal inference in the presence of cross-unit treatment interference has gained increasing attention in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Previous studies have highlighted the importance of accounting for potential treatment spillovers through empirical applications in many fields, including economics, education, epidemiology, and political science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Because individuals generally have different interaction networks, it is typically impossible to identify any meaningful causal parameters without some simplifying assumptions on the interference structure (Imbens and Rubin, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' A popular approach in the literature is to introduce some exposure mapping that summarizes the impacts from other individuals’ treatments into lower- dimensional sufficient statistics (Aronow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For example, Hong and Raudenbush (2006) studied the impact of school retention on later academic performance, assuming that other students’ retention may affect their own performance depending only on whether their school has a higher or lower retention rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' As another example, Leung (2020) considered a treatment spillover model in which other individuals’ treatments affect one’s potential outcome only according to the number of treated neighbors and the total size of the neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For the obtained spillover effects to be valid, we must justify the specification of the exposure map- ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' However, there is no general theoretical guidance on what exposure mapping should be used, and if the chosen specification is inappropriate, the resulting causal inference may be misleading (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', failure to detect treatment spillovers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1 An approach to directly address this issue is to statistically test the specification, where the null hypothesis of interest is whether a given exposure mapping is a “correct” choice (a more formal argument will be given later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' This is the aim of the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Specifi- cally, with a focus on experimental situations where the treatment assignment mechanism is known to researchers, we develop new randomization tests for testing the specification of general exposure mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Unlike in the standard Fisher randomization test, the null hypothesis of interest here is not “sharp” in general in the sense that only a subset of the potential outcome values are imputable from the observed outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Consequently, to perform randomization tests, we must carefully select the ap- propriate subsets of units and treatment assignments, which we call the focal subpopulation and focal assignments, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We are not the first to consider this type of “conditional” randomization test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In the literature, Aronow (2012), Athey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2018), Basse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2019), and Puelz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2022) have considered similar conditional randomization tests for testing the structure of treatment spillovers (they did not characterize their methods as a “specification test” though).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' However, they did not dis- cuss a unified framework for constructing the appropriate test statistics for general exposure mappings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' instead, they focused on several specific nulls or specific experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The main contributions of this study are three-fold: First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' introducing the notion of coarseness of exposure mappings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' we propose a novel randomization testing approach that can test virtually any null exposures and automatically produce model-free test statistics equipped with reasonable power in 1Some recent studies have investigated under what conditions one can estimate the meaningful causal parameters even when the exposure mapping is misspecified or not explicitly specified (Aronow and Samii,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' S¨avje et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Hoshino and Yanagi, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Leung, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' A common finding in these studies is that only if the network dependence is sufficiently weak can we identify some composite causal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Nevertheless, for a general form of network interference, knowledge of the true exposure is still essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 2 most situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Second, we prove the validity of our testing procedure under relatively mild condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Lastly, we apply our specification test of exposure mappings to two prominent social network experiments in the literature: one on farmers’ insurance adoption by Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2015) and the other on anti-conflict intervention school programs by Paluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In both applications, our tests provide statistical evidence for the existence of spillover effects and, in particular, suggest that having at least one treated peer may serve as a good summary statistic for the treatment spillovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In Section 2, we introduce our randomization test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In Section 3, we report the results of the Monte Carlo experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Section 4 presents two empirical case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Finally, Section 5 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The proofs are presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 2 Randomization Test 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1 Setup Consider a finite population of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Each unit is indexed as i P rns, where, for a positive integer a, we denote ras “ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' , au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Throughout this paper, we focus on the experimental setup in which an experimenter assigns binary treatments Z “ pZ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' , Znq P Z to n units according to a known assignment probability PZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Here, Z :“ tz P t0, 1un : PZpzq ą 0u denotes the set of all possible assignment patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In this section, we mainly consider the case wherein all units fully comply with their assigned treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Thus, for all i, we do not distinguish between the assigned treatment Zi and the actual treatment take-up of i, which we denote as Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The case where noncompliance is allowed such that Zi ‰ Di for some units will be discussed in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The outcome variable is Yi P Y Ď R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In the most general treatment spillover model, each Yi may be affected by all elements of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Denoting as Yipzq the potential outcome when Z “ z, we have Yi ” YipZq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We assume that the potential outcomes are non-stochastic and that any random variation arises only from the randomness of the treatment assignment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', the design-based approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2 Let E : rns ˆ Z Ñ E denote an exposure mapping, where E is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For notational simplicity, we often write Eipzq ” Epi, zq, whose range is Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' By construction, Ei Ď E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In general, E may depend on n and it is possible that Eipzq and Ejpzq have different functional forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1 (Correct exposure mapping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We say that an exposure mapping E is correct if, for all i P rns and z, z1 P Z, Eipzq “ Eipz1q ùñ Yipzq “ Yipz1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' A correct exposure mapping always exists and is not necessarily unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In an extreme case, the identity mapping Eipzq “ z is always correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In another example, when the stable unit treatment value assumption is fulfilled, Eipzq “ zi (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', no interference) and Eipzq “ pzi, zjq for any j are both correct specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Fisher’s sharp null of no treatment effect can also be viewed as a special case where the exposure mapping Eipzq is a constant function independent of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 2Alternatively, one may assume that the potential outcomes are random and view the analysis as being conditioned on all potential outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 3 For an exposure mapping E and z P Z, we define the level set Lipz | Eq :“ tz1 P Z : Eipz1q “ Eipzqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' If E is correct, then all treatment assignments in Lipz | Eq are associated with the same potential outcome value for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In other words, when E is correct, we can define a corresponding potential outcome function yi : Ei Ñ Y such that yipEipzqq “ Yipz1q for all z1 P Lipz | Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In particular, Yi “ yipEipZqq holds under a correct E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Once an exposure mapping is given, it induces a partition of Z specific to each i, where each block is given by the corresponding Lipz | Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In particular, when E is correct, Lipz | Eq provides a partition by the equivalence class in terms of the potential outcome value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Based on the coarseness of the partition, we define the coarseness of the exposure mappings as follows: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2 (Coarseness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For exposure mappings E : rns ˆ Z Ñ E and E1 : rns ˆ Z Ñ E1, we say that E is coarser than E1 if there is a surjective mapping c : E1 Ñ E such that cpE1pi, z1qq “ Epi, zq for all i P rns, z P Z, and z1 P Lipz | Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' When this holds true, we write E1 Ą E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' A concept similar to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2 can be found in Vazquez-Bare (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The dimensions of E and E1 are generally different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For example, when the data are composed of n{2 pairs, we may consider Eipzq “ zi and E1 ipzq “ pzi, zi1q, where i1 denotes i’s partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' By definition, the “finest” exposure mapping is the identity mapping Eipzq “ z and the coarsest one is Eipzq “ a for some constant a independent of z, which corresponds to the case of no treatment effect whatsoever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2 Testing procedure We would like to test whether an exposure mapping E0 : rns ˆ Z Ñ E0 is correct: H0 : E0 is a correct exposure mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Let E1 : rns ˆ Z Ñ E1 denote another exposure mapping such that E1 Ą E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Then, there exists a mapping c1Ñ0 : E1 Ñ E0 that satisfies c1Ñ0pE1 i pz1qq “ E0 i pzq for z1 P Lipz | E0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' If H0 is true, both E0 and E1 are correct, ensuring the existence of potential outcome functions y0 i : E0 i Ñ Y and y1 i : E1 i Ñ Y satisfying Yi “ y0 i pE0 i pZqq “ y1 i pE1 i pZqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We define rE1 i ” rE1 i pZq :“ te1 P E1 i : c1Ñ0pe1q “ E0 i pZqu, namely, the set of E1 i values that map to E0 i pZq through c1Ñ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Then, under H0, Yi “ y0 i pE0 i pZqq “ y1 i pe1q for all e1 P rE1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Thus, the values of all ty1 i pe1q : e1 P rE1 i u are identically imputable as Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' By construction, E1 i pZq P rE1 i is always satisfied, implying that |rE1 i | ě 1 uniformly in i, where for a generic set A, |A| denotes the cardinality of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 4 Our key idea is to test whether the following equality under H0 is true for all focal units: y1 i pe1 jq “ y1 i pe1 kq for all e1 j, e1 k P rE1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1) More specifically, letting Npκq :“ ti P rns : |rE1 i | “ κu for some κ ě 2, we choose a set of focal units as S Ď Npκq, namely, the focal subpopulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' How S is constructed in practice will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Although the construction of S can be stochastic in general, the following analysis treats S as given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For a focal subpopulation S, we define the set of focal assignments as CS ” CSpZq :“ tz P Z : E1 i pzq P rE1 i for all i P Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' As long as z’s are taken from CS, for any i P S, it is satisfied that E0 i pzq “ E0 i pZq for all such z, whereas we can generate variations in E1 i values within rE1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Our randomization test computes the null distribution of a test statistic by randomly sampling the assignments z’s from CS and checking whether (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1) holds true for all i P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In practice, when CS is too vast to compute, one may impose additional conditions on CS to reduce its size, which might reduce the power of the test, but does not lose its validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We generally cannot use the entire Npκq as the focal subpopulation, but need to form S as a subset of Npκq to retain sufficient variations in the focal assignments in CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The following example would be helpful in understanding this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Suppose that the population is composed of n{2 couples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Let E0 i pzq “ zi and E1 i pzq “ pzi, zi1q, where i1 indicates i’s partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Trivially, E1 Ą E0 with c1Ñ0 being a function that selects the first element of E1 i pzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' If E0 is a correct exposure, then E1 is also correct, implying that Yi “ y0 i pZiq “ y1 i pe1q for all e1 P rE1 i “ tpZi, 0q, pZi, 1qu and i P rns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Thus, Np2q coincides with the entire population rns if all individuals are treatment eligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' If we use the entire Np2q as the focal subpopulation S, we must shuffle the treatment assignments while keeping pzi, zi1q “ pZi, Zi1q for all pairs (otherwise, the potential outcomes are not imputable for both partners in each pair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' However, Z is clearly the only treatment assignment that satisfies such a constraint and the randomization test is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In this example, the most reasonable focal subpopulation would be obtained by randomly selecting one unit from each pair, such that |S| “ n{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The corresponding set of focal assignments is CS “ tz P Z : zi “ Zi for all i P Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' As shown in the next example, our framework encompasses the Fisher randomization test of no treatment effect as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In this special case, one can use the entire population as the focal units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Let E0 i pzq “ a, where a is independent of z, and E1 i pzq “ zi, with c1Ñ0 being a constant function that always returns a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Then, if E0 is correct, we have Yi “ y0 i paq “ y1 i pe1q for all e1 P rE1 i “ t0, 1u and i P rns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 5 Thus, Np2q “ rns holds if all individuals are treatment eligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In this case, the entire rns can be used as the focal units and the corresponding focal assignment set is simply given by CS “ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Now, let Tpz, YSpzqq be some predetermined test statistic for each given z P Z, where YSpzq “ pYipzqqiPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The choice of T will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Recall that, under H0, YSpzq is imputable from YS ” YSpZq as long as z P CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Then, the p-value for Tpz, YSpzqq conditional on CS under H0 is the probability that the realization of the test statistic under the conditional randomization distribution is at least as extreme as its actual value: ppZ, CSq :“ PrrTpz˚, YSpz˚qq ě TpZ, YSq | z˚ P CSs, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2) where the probability is with respect to z˚ „ PZ|ZPCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In practice, it is difficult to exactly compute (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2) because |CS| is typically very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Thus, we propose to approximate the p-value using the Monte Carlo method: Procedure 1 Randomization Test Input: Z, YS, PZ|ZPCS Output: the estimated p-value: ppR 1: Compute TpZ, YSq 2: for r “ 1 to R do 3: Draw zprq independently from PZ|ZPCS 4: Compute Tpzprq, YSpzprqqq under H0 5: end for 6: Compute ppR :“ 1 R R ÿ r“1 1tTpzprq, YSpzprqqq ě TpZ, YSqu The next theorem provides the validation of this testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='3 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (i) PrrppZ, CSq ď α | Z P CSs ď α for any α P p0, 1q under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (ii) |ppR ´ ppZ, CSq| “ OP ` R´1{2˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1(i), the probability of a type I error is generally not precisely the nominal level α P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' This is a common feature of the randomization approach owing to the discrete nature of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1(ii) shows that the stochastic order of the Monte Carlo approximation error is R´1{2, where the probability is with respect to tzprqu „ PZ|ZPCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Note that this result is independent of the size of the focal subpopulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Because R can be freely chosen by researchers, the p-value can be estimated with arbitrary precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1 (Sampling from PZ|ZPCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Procedure 1 requires repeatedly sampling new z’s from PZ|ZPCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In certain special cases, for example, when E0 i pzq “ zi and PZ is given by Bernoulli tri- als, we can draw directly from PZ|ZPCS relatively easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Even when E0 is of a more general form, 3Here, we implicitly assume that |S| ě κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' otherwise, the test statistic may not be well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 6 noting that PZ|ZPCSpzq 9 1tz P CSu ¨ PZpzq, sampling from PZ|ZPCS can be done manually by prelim- inarily drawing z from PZ, and if it satisfies z P CS, we keep this z and move on to the computation of the test statistic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' otherwise, we re-draw a new z from PZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Although this approach is technically simple, it has a drawback in that, if Z is a huge set and CS is small relative to Z, the probability of observing z satisfying z P CS can be extremely small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', one in several thousands), which makes it computationally very inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' This computational issue is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2 (Choice of E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In practice, there may be a large number of possible candidates for E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' As long as E1 Ą E0, the selection of E1 can be arbitrary and it does not have to be a correct exposure in terms of size control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' however, it may significantly affect the power of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In general, it is better to employ a coarser E1 to secure the size of the focal subpopulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' However, note that, if E1 is too “similar” to E0, the test may not exhibit sufficient power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For example, when we test for the presence of treatment spillovers (E0 i pzq “ zi), using E1 i pzq “ pzi, z1q would result in very low power to detect the spillover effect because only those affected by unit 1 can contribute to the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Thus, ideally, we would like to choose E1 such that it can nicely capture the true interference pattern in a way that E0 cannot, while maintaining its coarseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' How to find such an ideal E1 in practice is also left as an important open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='3 Construction of the focal subpopulation in a social network framework If the structure of the population is as simple as that in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1, the construction of the focal subpopulation S is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' However, when one deals with a more general network structure, one finds that forming an appropriate S is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' To address this issue, Athey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2018) proposed several approaches that systematically or randomly choose focal units based on the shape of the interaction network of each unit, independent of the actual treatment assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' However, as pointed out by Basse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2019) and Puelz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2022), constructing the focal subpopulation without utilizing the observed treatment assignment may result in the loss of the power of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In this subsection, we discuss this issue further in an empirically common social network setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Suppose that the individuals are connected through social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Let A “ pAijqi,jPrns be the adjacency matrix, where Aij P t0, 1u represents whether j affects i (directed networks are allowed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We set Aii “ 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For each i, the set of interacting peers is denoted as Pi :“ tj P rns : Aij “ 1u and individual i’s neighborhood is denoted as Pi :“ tiu Y Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For simplicity, assume that the exposure mapping of interest depends only on the individual’s own and peers’ treatments: E0 i pzq “ E0 i ppzjqjPPiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Suppose we have n “ 8 individuals in the population and that they form an undirected social network, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In the figure, the treated individuals are grayed and the controls are white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We would like to test whether E0 i pzq “ maxjPPi zj is correct, which claims that the only thing important is having at least one treated unit in one’s own neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' As a finer counterpart of this, let E1 i pzq “ pzi, maxjPPi zjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Suppose that we have observed p4, 3, 7, 8, 2, 3, 5, 1q for the outcomes of the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Then, the potential outcomes schedule under H0 can be summarized as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In the table, the blank cells are those not imputable from the observed outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Np3q comprises the individuals excluding ID 8, with rE1 i “ tp1, 0q, p0, 1q, p1, 1qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Then, the “observed” y1 i p1, 0q’s are t4, 5u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Similarly, we obtain t3, 2, 3u and 7 1 2 3 4 5 6 7 8 Note: Gray and white nodes represent the treatment and control units, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Figure 1: Undirected social network A Table 1: Potential outcomes schedule under H0: E0 is correct ID Yi Zi E0 i pZq E1 i pZq y0 i p0q y0 i p1q y1 i p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 0q y1 i p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 0q y1 i p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 1q y1 i p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 1q Np3q 1 4 1 1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 0) 4 4 4 4 ✓ 2 3 0 1 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 1) 3 3 3 3 ✓ 3 7 1 1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 1) 7 7 7 7 ✓ 4 8 1 1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 1) 8 8 8 8 ✓ 5 2 0 1 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 1) 2 2 2 2 ✓ 6 3 0 1 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 1) 3 3 3 3 ✓ 7 5 1 1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 0) 5 5 5 5 ✓ 8 1 0 0 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 0) 1 1 Note: The underlined y1’s are observed potential outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' t7, 8u as the observed values of y1 i p0, 1q and y1 i p1, 1q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' When H0 is true, these three samples should have been drawn from the same distribution, which is exactly the argument we are trying to test with our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In this social network framework, we propose two approaches for constructing S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Maximum independent set A practical approach that generally works for any social network data is to construct S Ď Npκq such that Pi X Pj “ H holds for any i, j P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' This is conceptually similar to the ϵ-net approach in Athey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2018), but differs in that it first selects Npκq according to the observed treatments, which potentially results in an improvement in the power of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Finding such an S can be translated into a well-known problem in graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Let G “ pNpκq, Eq be the “common-friend” graph with vertex set Npκq and edge set E “ tpi, jq P Npκq : Pi X Pj ‰ Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Then, the independent set of G, which is a set of vertices such that no two vertices in the set are adjacent, can be a valid candidate for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In particular, we would like to find a maximum independent set (MIS) of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Figure 2 shows the common-friend graph for the network data in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We have two MISs, namely, t1, 6, 7u and t2, 6, 7u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' When we set S “ t2, 6, 7u, the admissible assignment vectors are characterized by CS “ tz P Z : E0 2pzq “ E0 6pzq “ E0 7pzq “ 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Biclique method We can extend the biclique method in Puelz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2022) to our situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' To this end, we define the null exposure graph and its biclique in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Let Z0 Ď Z denote a predetermined set of treatment assignments such that Z P Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For example, one may construct Z0 by drawing from PZ sufficiently many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The null exposure graph of H0 with respect to Z0 is defined 8 1 2 3 4 5 6 7 Figure 2: Common-friend graph G (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='3) as a bipartite graph G “ pNpκq Y Z0, Eq, where E “ tpi, zq P Npκq Y Z0 : c1Ñ0pE1 i pzqq “ E0 i pZqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' That is, there exists an edge between i P Npκq and z P Z0 when Yi “ y0 i pE0 i pZqq “ y1 i pE1 i pzqq holds under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Then, a biclique Bb “ pNb, Zbq in G is defined as a pair of sets Nb Ď Npκq and Zb Ď Z0 such that pi, zq P E holds for all i P Nb and z P Zb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In general, we can find multiple bicliques for G (b “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=') and it is typically desirable to have a larger biclique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' By construction, if H0 is true, then we have Yi “ y0 i pE0 i pZqq “ y1 i pE1 i pzqq for all pi, zq P Bb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Once a biclique Bb is obtained, we can simply set S “ Nb and CS “ Zb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='4 Choice of test statistic There is certain freedom in the choice of test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In this study, we consider the following three types of statistics: Kruskal-Wallis (KW), average cross difference (ACD), and ordinary least squares (OLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' To define these statistics, we introduce additional notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For each i P S, we order the elements of rE1 i as e1 1, e1 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' , e1 κ based on some rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For example, in the case of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='3, we can consider an increasing order in terms of the value of 2zi ` maxjPPi zj, leading to pe1 1, e1 2, e1 3q “ pp0, 1q, p1, 0q, p1, 1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Note that, because rE1 i may be heterogeneous among the individuals, the compositions of the ordered elements pe1 1, e1 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' , e1 κq are also generally different among these individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' When such a heterogeneity is present, what sorting rule is adopted is a factor that affects the power of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For a given treatment assignment z, we partition the focal subpopulation S into κ groups: Sjpzq :“ ti P S : E1 i pzq “ e1 ju for j P rκs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Now, we have κ potential outcomes py1 i pe1 1q, y1 i pe1 2q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' , y1 i pe1 κqq for each i, which should take the same value under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Noting that our task can be viewed as testing the equivalence of κ different treatments, we consider the use of the KW statistic as standard, as in Keele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2012) and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' First, we rank all pYiqiPS from 1 to |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Let vi be the rank of Yi and Vjpzq be the summation of the ranks for group Sjpzq: Vjpzq :“ ř iPSjpzq vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The KW statistic compares the average rank for each group j, Vjpzq{|Sjpzq|, with the average rank for the entire |S|, p|S| ` 1q{2: Tpz, YSq “ 12 |S|p|S| ` 1q κÿ j“1 |Sjpzq| ˆ Vjpzq |Sjpzq| ´ |S| ` 1 2 ˙2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 9 The ACD statistic is defined simply as the average of the absolute average differences for all com- binations of treatment pairs: Tpz, YSq “ 2 κpκ ´ 1q ÿ 1ďjăkďκ ˇˇˇˇˇ ř iPSjpzq Yi |Sjpzq| ´ ř iPSkpzq Yi |Skpzq| ˇˇˇˇˇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' It is also possible to consider a “model-based” test statistic, as in Athey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Suppose we have some E0 and E1, where E0 might be vector-valued, and let Xipzq be a vector of variables whose values are determined only through E1 i pzq but not through E0 i pzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For example, when E0 i pzq “ zi and E1 i pzq “ pzi, maxjPPi zjq, one may use Xipzq “ maxjPPi zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Then, by fitting the following regression model to the data in S, Yi “ β0 ` E0 i pzqJβ1 ` XipzqJβ2 ` errori, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='3) we can use the F-statistic for the significance of pβ2 as the test statistic Tpz, YSq, where pβ2 denotes the OLS estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For another example, when we have E0 i pzq “ maxjPPi zj and E1 i pzq “ pzi, maxjPPi zjq, as in Exam- ple 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='3, we may consider using Xipzq “ pzi, maxjPPi zjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Note that, when one adopts this model-based approach, the presumed model does not have to perfectly reflect the true interference structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' How- ever, if they are significantly different, it will lead to a substantial loss of the power, as numerically demonstrated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='3 (Multiplicity of test statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' As shown above, we generally have multiple statistics for testing H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Furthermore, by considering different values of κ and E1, we can generate a large number of additional test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' One simple way to utilize the information in all s different statistics altogether is to combine them into a single test statistic, T comb “ gpT 1, T 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' , T sq, as suggested in Imbens and Rubin (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Another approach is to apply, for example, Simes’ correction for multiple testing: letting the ordered p-values be denoted by ppp1q R ď ¨ ¨ ¨ ď pppsq R , reject H0 if pppiq R ď iα{s for some i P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' , su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' See Simes (1986) and Subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2 of Lehmann and Romano (2022) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='5 Imperfect compliance Thus far, we have assumed that all individuals comply with their initial treatment assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' How- ever, in certain realistic situations, they are allowed to self-select their own treatment status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Now, we write D “ pD1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' , Dnq as the n-dimensional vector of the actual treatment take-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' When noncom- pliance is allowed (D ‰ Z), the probability distribution of D is generally unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Thus, in this case, we cannot perform the test in Procedure 1 based on the actual treatments because it is infeasible to resample independent copies dprq’s of D from a known distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' One empirically tractable approach to this problem is to resort to an intention-to-treat (ITT) type of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' That is, we consider formulating the exposure mapping as a function not of d but of the 10 initial assignment z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='4 For example, suppose we have the following treatment selection model: Di “ 1 # γ0i ` γ1iZi ` γ2i ÿ jPPi Zj ą 0 + and there are no treatment spillovers in the outcome model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In this case, the exposure mapping of interest would be E0 i pzq “ pzi, ř jPPi zjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Then, if we can find an appropriate E1 that is finer than E0, in exactly the same way as in Procedure 1, we can test the validity of this model specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 3 Numerical Simulations In this section, we assess the small sample performance of our randomization test using Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1 Perfect compliance First, we consider the case of perfect compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The network is created from a simple Erd¨os–R´enyi model with a probability of p “ 3{n, where we set n “ 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We consider the following two data generating processes (DGPs) for the outcome variables: DGP 1: Yi “ Di ` τ ÿ jPPi Dj ` ξi, DGP 2: Yi “ Di ` τ ¨ g ˜ ÿ jPPi Dj ¸ ` ξi, where ξi „ Np0, 1q, τ P r0, 2s, and gpaq “ 1ta ď 2u ¨ a ` 1ta ě 3u ¨ a´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For the treatment assignment mechanism, we employ a complete randomization, where randomly selected n{2 units receive Z “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Because perfect compliance is assumed here, Di “ Zi holds for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We set E0 i pzq “ zi, and, hence, E0 is correct when τ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For the choice of E1, the following two exposure mappings are used: Exposure 1: E1 i pzq “ ˆ zi, max jPPi zj ˙ , Exposure 2: E1 i pzq “ ˜ zi, ÿ jPPi zj ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Note that Exposure 1 is coarser than Exposure 2 and that only Exposure 2 is correct when τ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For Exposure 1, it is natural to set κ “ 2 such that rEi “ tpZi, 0q, pZi, 1qu, and Np2q “ ti P rns : |Pi| ą 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For Exposure 2, we set κ “ 4 such that rEi “ tpZi, 0q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' , pZi, 3qu and Np4q “ ti P rns : |Pi| “ 3u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' To construct the focal subpopulation S, we consider the following three approaches: (i) MIS method, (ii) random selection of |Npκq|{2 focal units, and (iii) biclique method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Here, note that finding the largest independent set and finding the largest biclique are both NP-hard problems, and, thus, we approximate their solutions using a greedy vertex coloring algorithm and the binary inclusion-maximal biclustering method, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='5 For (i) and (ii), we set R “ 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For (iii), we draw treatment assignments 9 million times from PZ to create Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 4This type of exposure mapping was also considered in Hoshino and Yanagi (2022) and is termed as instrumental exposure mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 5Specifically, in the Monte Carlo simulations and the empirical illustrations below, we use the functions greedy vertex coloring and BCBimax in the R packages igraph and biclust, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 11 For each setup, we perform our randomization test using the KW, ACD, and OLS statistics under the nominal significance level of 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The OLS statistic is obtained as the F-statistic for the OLS estimate of β2 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='3), where we set Xipzq “ maxjPPi zj for Exposure 1 and Xipzq “ ř jPPi zj for Exposure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In addition to these three tests, we also report the results from the Simes-corrected p-value based on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The following results are based on 1,000 Monte Carlo replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Figures 3 and 4 show the rejection frequency of each method for different τ values in DGPs 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In each figure, panels (a) and (b) present the simulation results for Exposures 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' When τ “ 0, for all methods and test statistics, the rejection frequencies are sufficiently close to the nominal level in both DGPs, which is consistent with our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Particularly in DGP 1, the power of these tests quickly increases as τ increases, suggesting the consistency of our testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' However, in DGP 2, we find that the power of the OLS statistic based on Exposure 2 is significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' This may be due to “model misspecification” in the OLS regression caused by the mishandling of the nonlinearity of the g function in this DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Note that the OLS model with Exposure 1 is also a misspecified model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' however, the magnitude of the misspecification is mild relative to that of Exposure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Even when Exposure 2 is used in DGP 2, the KW statistic remains sufficiently powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Comparing the three methods for constructing the focal subpopulation, we find that the MIS per- forms the best and the biclique the least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The random selection approach is in-between of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' However, caution should be exercised when interpreting this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' To be clear, a large part of the dif- ference in the performance of these methods is essentially due to the difference in the sizes of S and CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Finding a reasonably large biclique becomes more difficult when the null exposure graph is sparser, as in this simulation setting (see Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2 of Puelz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2022) for a related discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In addition, even with the above-mentioned simplified algorithm, finding a large biclique is still computationally very demanding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' for example, even though we have employed a fairly large Z0, the resulting size of S was, on average, less than 20 or so after a long computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In a different setup where the biclique method can easily identify relatively large S and CS, its performance would be substantially improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For both DGPs, except for the biclique method, Exposure 1 tends to provide more powerful tests than those of Exposure 2, even though Exposure 1 is incorrect when τ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' This result is possibly due to the fact that a coarser Exposure 1 generally induces a larger S than that of Exposure 2, while retaining a strong correlation with the true interference structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Specifically, for the MIS method, the sizes of S generated from Exposures 1 and 2 are approximately 80 and 30, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2 Imperfect compliance Next, we discuss the experiments for the case of imperfect compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In particular, we consider a one-sided compliance situation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' that is, only when Zi “ 1 can i choose Di “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The initial treatment assignment Z is generated from a complete randomization such that n{2 units are eligible to take the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For treatment-eligible units, the compliance status follows Bernoullip0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='8q uniformly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', no interference within the treatment choices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' All other parts of the simulation design are the same as those in the perfect compliance case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Note that E0 i pzq “ zi is correct when τ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The simulation results are shown in Figures 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Overall, the same comments as in the previous experiment apply to this experiment as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Our randomization test works satisfactorily in terms of 12 (a) Exposure 1 (b) Exposure 2 Figure 3: Simulation results: DGP 1 under perfect compliance both size control and power property, although the power of the test seems slightly worse than that in the perfect compliance case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' This is reasonable considering that, unlike in the previous case, neither Exposure 1 nor Exposure 2 is correct when τ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For the construction of the focal subpopulation, it seems desirable to use the MIS approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' An interesting finding is that, when Exposure 2 is employed in DGP 2, it is the ACD statistic, not the OLS statistic, that loses its power significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We suspect that this result is due to the current simulation setup and cannot be generalized any further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 13 power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' power power 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='60 ACD KW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='40 OLS Simes 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='0 1 T T MIS Random selection Bicliquepower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' power power 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='60 ACD KW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='40 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='0 T T T MIS Random selection Biclique(a) Exposure 1 (b) Exposure 2 Figure 4: Simulation results: DGP 2 under perfect compliance 4 Empirical Illustrations As empirical illustrations, we apply our randomization test to the existing datasets from two well-known social network experiments in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The first one is the data on farmers’ insurance adoption in Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2015), and the other is the data on anti-conflict intervention school programs in Paluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For both datasets, we investigate the same type of null hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Here, let Ea i pzq “ zi, Eb i pzq “ max jPPi zj, Ec i pzq “ ˆ zi, max jPPi zj ˙ , Ed i pzq “ ˜ zi, ÿ jPPi zj ¸ , 14 power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' power power 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='60 ACD KW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='40 OLS Simes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='20 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For example, the null hypothesis Ha claims that Ea is a correct exposure such that there are no treatment spillovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' To test this null hypothesis, we employ Ec as E1, which contains the information about one’s own treatment and whether he/she has at least one treated peer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' This choice of E1 leads to rE1 i “ tpZi, 0q, pZi, 1qu with κ “ 2 and to Np2q “ ti P rns : |Pi| ą 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The other null hypotheses, Hb and Hc, can be interpreted similarly.' metadata={'source': 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weather insurance on farmers’ insurance take-up decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The authors demon- strated the presence of significant treatment spillovers among the farmers by employing a series of regression models, where the main explanatory variable was the fraction (or number) of friends who were assigned to the intensive information sessions in advance of the focal farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In this experiment, four types of sessions were conducted: first-round simple sessions, first-round intensive sessions, second-round simple sessions, and second-round intensive sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In each round, the simple sessions briefly described the insurance contract, whereas the intensive sessions explained the insurance contract and the expected benefits of the insurance in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' To randomly assign the farmers to each session, the authors performed a stratified randomization with four strata constructed in each village according to the household size and rice production acreage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' All rice-producing households were invited to participate in one of the four sessions, and almost 90% of them attended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Thus, because the probability of non-attendance was low, we ignore the treatment non-compliance for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In this analysis, the outcome variable of interest is Yi P t0, 1u, which indicates whether farmer i decides to buy the weather insurance after attending the session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Let inti P t0, 1u denote whether i is assigned to an intensive session, and let seci P t0, 1u denote whether i is assigned to the second-round session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Because the treatment spillovers matter only for the participants in the second-round session, as they can receive information from the first-round participants, we create the focal units using only the farmers assigned to the second round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For the definition of the treatment variable, for a focal unit i, we set zi “ 1 if inti “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For a nonfocal unit j, we set zj “ 1 if both intj “ 1 and secj “ 0 are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' When performing our randomization tests, we randomize both inti and seci following the protocol of the original experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' To obtain a certain number of focal units in each village, we only use villages with at least 50 farmers as our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The results of our randomization tests are summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' First, we can see that all p- values for testing Ha are smaller than 5%, which indicates the presence of information spillovers among the farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For the null hypothesis Hb, none of the test statistics reject this hypothesis at the 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Lastly, for Hc, we cannot reject this under any reasonable significance level with any of the test statistics considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In summary, these results suggest the existence of spillover effects in farmers’ insurance purchasing decisions and that having at least one friend assigned to the intensive session might be an important factor that accounts for the spillovers rather than the number of such friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='2 Spillover effects of the anti-conflict intervention programs In the second empirical case study, we apply the proposed test to the data from Paluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2016), who investigated the impact of anti-conflict intervention programs on adolescents’ norms and attitudes through a large-scale experiment in 56 American middle schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Half of these schools were randomly selected to host the programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Within each selected school, a group of students (called seed-eligible students) were non-randomly selected, and half of these students (called seed students) were chosen through a stratified randomization and invited to join the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The seed-eligible students’ strata were determined by their individual characteristics, such as gender, grade, and friendship network 6Given this result, one might want to adopt Ec as the final model and reanalyze the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' However, note that doing so raises another issue of “inference after model selection”, which is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 17 Table 3: Empirical results: farmers’ insurance decisions p-values Simes’ correction κ KW ACD OLS 10% 5% 1% |S| R Testing for Ha 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='021 ✓ ✓ 610 100,000 Testing for Hb 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='054 ✓ 542 100,000 Testing for Hc 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='979 145 100,000 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The students’ friendship networks were measured by simply asking them to nominate up to 10 friends in their school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Participating in the program was not mandatory for the seed students, and, in fact, the compliance rate was approximately 40%, which corresponds to the case of imperfect compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' For more details on the experimental design, see the Supplementary Appendix of Paluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The purpose of this experiment is to examine how the seed students who participated in the inter- vention program could influence other students through their social networks to improve the climate of the school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In each intervention meeting, the seed students were encouraged to identify common conflict behaviors in their schools and discuss behavioral strategies to mitigate the conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' As an important role of the seed students, they were allowed to hand out a program wristband as a reward to students for their engagement in friendly or conflict-mitigating behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Let Yi P t0, 1u be an indicator of whether student i wears a program wristband, which is the outcome variable of interest in this empirical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Let Zi P t0, 1u indicate the treatment eligibility (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', whether i is a seed student).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We perform a randomization test to examine the spillover effects of being selected as a seed student on wearing wristbands, which can be viewed as the ITT-type analysis discussed in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' When performing our tests, we exclude the schools where fewer than three wristbands were distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' In addition, in line with Paluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2016), we construct the focal subpopulation from the seed-eligible students who had at least one social referent peer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', a student whose indegree was ranked in the top 10% of his/her school).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The results are summarized in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We can see from the table that all p-values for Ha and Hb are sufficiently small to reject them at the 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The rejection of these hypotheses together suggests that the intervention program has strong spillover effects through the students’ networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' By contrast, Hc is not rejected even under the significance level of 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Thus, like in the first empirical study, we might conclude that the presence of even just one seed student friend, rather than the number of treated friends, can reasonably explain the students’ anti-conflict activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 18 Table 4: Empirical results: anti-conflict education program p-values Simes’ correction κ KW ACD OLS 10% 5% 1% |S| R Testing for Ha 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='011 ✓ ✓ 413 100,000 Testing for Hb 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='001 ✓ ✓ ✓ 358 100,000 Testing for Hc 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='214 105 100,000 5 Conclusion In this study, we developed a novel randomization testing approach for the specification of general exposure mappings in treatment effect models with interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Based on the concept of coarseness of exposure mappings, our proposed approach has a fairly broad empirical applicability and enables us to construct model-free test statistics with a good power property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' As empirical illustrations, we have revisited two existing social network experiments in the literature: one is the data on farmers’ insurance adoption studied in Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2015) and the other is the data on anti-conflict education programs studied in Paluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' From the results of the experiments on both datasets, we found that the exposure mapping Eipzq “ pzi, maxjPPi zjq has a certain capability to account for the spillover effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' These results indicate the usefulness of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Acknowledgments We thank Jing Cai for kindly instructing us on how to recover the randomization strata from the replication data of Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' We also thank seminar participants at Keio University, Kwansei Gakuin University, Tokyo University, and Osaka University for their helpful comments discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' This work was supported by JSPS KAKENHI grant numbers 19H01473 and 20K01597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' The datasets used in the empirical illustrations are available from the Interuniversity Consortium for Political and Social Research (Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Paluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 19 A Appendix: Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='1 (i) By the definition of CS, for any z˚ P CS, we have YSpz˚q “ YS under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Thus, we can write ppZ, CSq “ PrrTpz˚, YSq ě TpZ, YSq | z˚ P CSs, where the probability is with respect to z˚ „ PZ|ZPCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Let FT|CS denote the conditional distribution function of ´Tpz˚, YSq given z˚ P CS induced from PZ|ZPCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Then, ppZ, CSq “ FT|CSp´TpZ, YSqq, and, thus, PrrppZ, CSq ď α | Z P CSs “ PrrFT|CSp´TpZ, YSqq ď α | Z P CSs ď α, where the inequality follows from the fact that ´TpZ, YSq is distributed as FT|CS given Z P CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' (ii) Let ppzprqq :“ 1tTpzprq, YSq ě TpZ, YSqu, such that ppR “ R´1 řR r“1 ppzprqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Because zprq’s are identically drawn from PZ|ZPCS, we have Etzprqu„PZ|ZPCS ppR “ Ez„PZ|ZPCS ppzq “ ppZ, CSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Furthermore, by the independence of the draws, Etzprqu„PZ|ZPCS ` ppR ´ ppZ, CSq ˘2 “ Ez„PZ|ZPCS pppzq ´ ppZ, CSqq2 R “ ppZ, CSqp1 ´ ppZ, CSqq R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Thus, the result follows from Chebyshev’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 20 References Aronow, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' A general method for detecting interference between units in randomized experi- ments, Sociological Methods & Research, 41 (1), 3–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Aronow, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', Eckles, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', Samii, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', and Zonszein, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Spillover effects in experimental data, in: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Druckman and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Green, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', Advances in Experimental Political Science, Cambridge University Press, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 16, 289–319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Aronow, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' and Samii, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Estimating average causal effects under general interference, with application to a social network experiment, The Annals of Applied Statistics, 11 (4), 1912–1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Athey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', Eckles, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', and Imbens, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Exact p-values for network interference, Journal of the American Statistical Association, 113 (521), 230–240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Basse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', Feller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', and Toulis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Randomization tests of causal effects under interference, Biometrika, 106 (2), 487–494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Cai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', De Janvry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', and Sadoulet, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Social networks and the decision to insure, American Economic Journal: Applied Economics, 7 (2), 81–108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Cai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', De Janvry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', and Sadoulet, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Replication data for: Social networks and the decision to insure, Inter-university Consortium for Political and Social Research [distributor], 2019-10-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='3886/E113593V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Hong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' and Raudenbush, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Evaluating kindergarten retention policy: A case study of causal inference for multilevel observational data, Journal of the American Statistical Association, 101 (475), 901–910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Hoshino, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' and Yanagi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Causal inference with noncompliance and unknown interference, arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='07455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Imbens, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' and Rubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Causal Inference in Statistics, Social, and Biomedical Sciences, Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Keele, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', McConnaughy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', and White, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Strengthening the experimenter’s toolbox: Statistical estimation of internal validity, American Journal of Political Science, 56 (2), 484–499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Lehmann, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' and Romano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Testing Statistical Hypotheses, Fourth Edition, Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Leung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Treatment and spillover effects under network interference, The Review of Economics and Statistics, 102 (2), 368–380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Leung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Causal inference under approximate neighborhood interference, Econometrica, 90 (1), 267–293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Paluck, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', Shepherd, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', and Aronow, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Changing climates of conflict: A social network experiment in 56 schools, Proceedings of the National Academy of Sciences, 113 (3), 566–571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 21 Paluck, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', Shepherd, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', and Aronow, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Changing climates of conflict: A social network experiment in 56 schools, New Jersey, 2012-2013, Inter-university Consortium for Political and Social Research [distributor], 2020-09-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='3886/ICPSR37070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Puelz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', Basse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', Feller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', and Toulis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' A graph-theoretic approach to randomization tests of causal effects under general interference, Journal of the Royal Statistical Society: Series B, 84 (1), 174–204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' S¨avje, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', Aronow, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', and Hudgens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Average treatment effects in the presence of unknown interference, The Annals of Statistics, 49 (2), 673–701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Simes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' An improved bonferroni procedure for multiple tests of significance, Biometrika, 73 (3), 751–754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Vazquez-Bare, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Identification and estimation of spillover effects in randomized experiments, Journal of Econometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', Rosenberger, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', and Uschner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' Randomization tests for multiarmed randomized clinical trials, Statistics in Medicine, 39 (4), 494–509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE5T4oBgHgl3EQfZQ-4/content/2301.05580v1.pdf'} diff --git a/RNAyT4oBgHgl3EQf7vr2/content/tmp_files/2301.00846v1.pdf.txt b/RNAyT4oBgHgl3EQf7vr2/content/tmp_files/2301.00846v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..20cb30703aa8edb4ebbfec12dfefffb6d422d48d --- /dev/null +++ b/RNAyT4oBgHgl3EQf7vr2/content/tmp_files/2301.00846v1.pdf.txt @@ -0,0 +1,741 @@ +arXiv:2301.00846v1 [math-ph] 2 Jan 2023 +Higher order first integrals of autonomous dynamical systems in +terms of geometric symmetries +Antonios Mitsopoulos1,a) and Michael Tsamparlis1,b) +1Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics, +University of Athens, Panepistemiopolis, Athens 157 83, Greece +a)Author to whom correspondence should be addressed: antmits@phys.uoa.gr +b)Email: mtsampa@phys.uoa.gr +Abstract +In general, a system of differential equations is integrable if there exist ‘sufficiently many’ first integrals +(FIs) so that its solution can be found by means of quadratures. Therefore, the determination of the FIs is +an important issue in order to establish the integrability of a dynamical system. In this work, we consider +holonomic autonomous dynamical systems defined by equations ¨qa = −Γa +bc(q) ˙qb ˙qc − Qa(q) where Γa +bc(q) are +the coefficients of a symmetric (possibly non-metrical) connection and −Qa(q) are the generalized forces. +We prove a theorem which produces the FIs of any order of such systems in terms of the ‘symmetries’ of +the geometry defined by the quantities Γa +bc(q). We apply the theorem to compute quadratic and cubic FIs +of various dynamical systems. +1 +Introduction +A first integral (FI) of a second order set of dynamical equations with generalized coordinates qa and generalized +velocities ˙qa ≡ dqa +dt is a function I(t, qa, ˙qa) satisfying the condition dI +dt = 0 along the dynamical equations. It is +important to have a systematic method for determining FIs because they can be used to reduce the order of the +dynamical equations and, if they are ‘enough’ in number [1], to find the solution of the system by quadrature +(Liouville integrability). +The standard method to compute the FIs is the method of Noether symmetries [2, 3, 4]. A different method +is the direct method [5, 6, 7, 8, 9] in which one assumes a functional form for the FI I and demands directly the +condition dI +dt = 0. This condition and the dynamical equations lead to a system of partial differential equations +(PDEs) whose solution provides the FIs. +In this work, we apply the direct method to autonomous holonomic dynamical systems in a space with a +symmetric connection Γa +bc(q) (not necessarily Riemannian) which is read from the dynamical equations. We +compute the resulting system of PDEs and solve it in terms of the ‘symmetries’ of Γa +bc(q). +The result is +stated as Theorem 1 and provides a systematic method to determine the FIs of any order, time-dependent +and autonomous, of these dynamical systems. In the special case where the symmetric connection Γa +bc(q) is +the Riemannian one defined in terms of the kinetic metric (kinetic energy) γab(q) of the system, the computed +FIs are directly related by means of the Inverse Noether Theorem [8, 10] to gauged generalized (i.e. velocity- +dependent) Noether symmetries. Finally, we apply Theorem 1 in order to find integrable and superintegrable +systems that admit quadratic (QFIs) and cubic FIs (CFIs). +2 +The conditions for higher order FIs +We consider autonomous holonomic dynamical systems of the form +¨qa = −Γa +bc(q) ˙qb ˙qc − Qa(q) +(1) +1 + +where Γa +bc(q) are the coefficients of a general connection and −Qa(q) are the generalized forces. Since only the +symmetric part Γa +(bc) contributes to the dynamical equations, without loss of generality, the quantities Γa +bc(q) +are assumed to be symmetric. +We look for mth-order FIs of the general form +I(m) = +m +� +r=0 +Mi1i2...ir(t, q) ˙qi1 ˙qi2... ˙qir = M + Mi1 ˙qi1 + Mi1i2 ˙qi1 ˙qi2 + ... + Mi1i2...im ˙qi1 ˙qi2... ˙qim +(2) +where Mi1...ir(t, q) are totally symmetric r-rank tensors and the index (m) denotes the order of the FI. +The FI condition +dI +dt = 0 +(3) +and the dynamical equations (1) result in the following system of PDEs: +M(i1i2...im|im+1) += +0 +(4) +Mi1i2...im,t + M(i1i2...im−1|im) += +0 +(5) +Mi1i2...ir,t + M(i1i2...ir−1|ir) − (r + 1)Mi1i2...irir+1Qir+1 += +0, r = 1, 2, ..., m − 1 +(6) +M,t − Mi1Qi1 += +0 +(7) +Mi1,tt − 2Mi1i2,tQi2 + (McQc),i1 += +0 +(8) +2 +� +M[i1≀c≀Qc� +|i2] − M[i1|i2],t += +0 +(9) +where | denotes the covariant derivative with respect to (wrt) the symmetric connection Γa +bc, a comma indi- +cates partial derivative wrt qa or t, round/square brackets indicate symmetrization/antisymmetrization of the +enclosed indices, and indices enclosed between wavy lines are overlooked by symmetrization or antisymmetriza- +tion symbols. +Equations (8) and (9) express the integrability conditions M,i1t = M,ti1 and M,[i1i2] = 0 of the scalar M, +respectively. +Equation (4) generalizes the concept of Killing tensors (KTs) to a non-metrical geometry with a symmetric +connection Γa +bc. In this context, Mi1i2...im is a generalized mth-order KT for Γa +bc. +The most general choice for the generalized mth-order KT Mi1i2...im in the case of an autonomous system is +Mi1...im(t, q) = C(0)i1...im(q) + +n +� +N=1 +C(N)i1...im(q)tN +N +(10) +where C(N)i1...im(q), N = 0, 1, ..., n, is a sequence of arbitrary mth-order generalized KTs of Γa +bc and n is the +degree of the considered polynomial. +The choice (10) and equation (5) indicate that we set +Mi1...ir(t, q) = +nr +� +Nr=0 +L(Nr)i1...ir(q)tNr, r = 1, 2, ..., m − 1 +(11) +where L(Nr)i1...ir(q), Nr = 0, 1, ..., nr, are arbitrary r-rank totally symmetric tensors and nr is the degree of the +considered polynomial. +We note that the degrees n, nr of the above polynomial expressions of t may be infinite. +Substituting (10) and (11) in the system of equations (4) - (9) (eq. (4) is identically zero since C(N)i1...im +are assumed to be generalized KTs), we end up with a system of five PDEs. The solution of this system is +lengthy and requires the consideration of many cases and subcases. We state the solution below as Theorem 1. +3 +Theorem for mth-order FIs of an autonomous holonomic dynam- +ical system +Theorem 1 The independent mth-order FIs of the dynamical system (1) are the following: +2 + +Integral 1. +I(m) +n += +� +− +n +� +N=1 +tN +N L(N−1)(i1...im−1|im) + C(0)i1...im +� +˙qi1... ˙qim + +m−1 +� +r=1 +� +n +� +N=0 +tNL(N)i1...ir +� +˙qi1... ˙qir + ++s tn+1 +n + 1 + +n +� +N=1 +L(N−1)cQc tN +N + G(q) +where C(0)i1...im, L(N)(i1...im−1|im) for N = 0, 1, ..., n − 1 are mth-order generalized KTs, L(n)i1...im−1 is an +(m − 1)th-order generalized KT, s is an arbitrary constant defined by the condition +L(n)i1Qi1 = s +(12) +while the vectors L(N)i1 and the totally symmetric tensors L(A)i1...ir, A = 0, 1, ..., n, r = 2, 3, ..., m−2 satisfy +the conditions: +L(n)(i1...im−2|im−1) += +−m +n L(n−1)(i1...im−1|im)Qim +(13) +L(k−1)(i1...im−2|im−1) += +− +m +k − 1L(k−2)(i1...im−1|im)Qim − kL(k)i1...im−1, k = 2, 3, ..., n +(14) +L(0)(i1...im−2|im−1) += +mC(0)i1...im−1imQim − L(1)i1...im−1 +(15) +L(n)(i1...ir−1|ir) += +(r + 1)L(n)i1...irir+1Qir+1, r = 2, 3, ..., m − 2 +(16) +L(k−1)(i1...ir−1|ir) += +(r + 1)L(k−1)i1...irir+1Qir+1 − kL(k)i1...ir, k = 1, 2, ..., n, r = 2, 3, ..., m − 2 (17) +� +L(n−1)cQc� +,i1 += +2nL(n)i1i2Qi2 +(18) +� +L(k−2)cQc� +,i1 += +2(k − 1)L(k−1)i1i2Qi2 − k(k − 1)L(k)i1, k = 2, 3, ..., n +(19) +G,i1 += +2L(0)i1i2Qi2 − L(1)i1. +(20) +Integral 2. +I(m) +e += eλt +λ +� +−L(i1...im−1|im) ˙qi1... ˙qim + λ +m−1 +� +r=1 +Li1...ir ˙qi1... ˙qir + Li1Qi1 +� +where λ ̸= 0, L(i1...im−1|im) is an mth-order generalized KT and the remaining totally symmetric tensors satisfy +the conditions: +L(i1...im−2|im−1) += +−m +λ L(i1...im−1|im)Qim − λLi1...im−1 +(21) +L(i1...ir−1|ir) += +(r + 1)Li1...irir+1Qir+1 − λLi1...ir, r = 2, 3, ..., m − 2 +(22) +(LcQc),i1 += +2λLi1i2Qi2 − λ2Li1. +(23) +We note that Theorem 1 for m = 2 and a Riemannian connection reduces to Theorem 1 of [7] and to +Theorem 3 of [8] for the case of QFIs. +Moreover, we have the following minor results. +Proposition 2 The independent mth-order FIs I(m) +n +and I(m) +e +satisfy the following recursion formulae: +a. I(k) +n +< I(k+1) +n +, that is, each kth-order FI I(k) +n +is a subcase of the next (k + 1)th-order FI I(k+1) +n +with the same +degree n of time-dependence for all k ∈ N. +b. I(m) +ℓ +< I(m) +ℓ+1, that is, the mth-order FI I(m) +ℓ +with time-dependence fixed by ℓ is a subcase of the mth-order FI +I(m) +ℓ+1 with time-dependence ℓ + 1 for all ℓ ∈ N. +c. I(k) +e +< I(k+1) +e +, that is, each kth-order FI I(k) +e +is a subcase of the next (k + 1)th-order FI I(k+1) +e +for all k ∈ N. +Proposition 3 The mth-order FI I(m) +n +consists of the following two independent FIs: +a. +The FI J(m,1) +ℓ +whose coefficients are polynomials of t containing even powers of t for even products of +velocities and odd powers of t for odd products of velocities. +b. The FI J(m,2) +ℓ +whose coefficients are polynomials of t containing even powers of t for odd products of velocities +and odd powers of t for even products of velocities. +3 + +For even orders m = 2ν (ν ∈ N) the independent FIs of Proposition 3 are computed by the formulae (ℓ ∈ N): +a. +J(m=2ν,1) +ℓ += +� +−t2ℓ +2ℓ L(2ℓ−1)(i1...im−1|im) − ... − t2 +2 L(1)(i1...im−1|im) + C(0)i1...im +� +˙qi1... ˙qim + ++ +odd +� +1≤r≤m−1 +� +t2ℓ−1L(2ℓ−1)i1...ir + ... + t3L(3)i1...ir + tL(1)i1...ir +� +˙qi1... ˙qir + ++ +even +� +1≤r≤m−1 +� +t2ℓL(2ℓ)i1...ir + ... + t2L(2)i1...ir + L(0)i1...ir +� +˙qi1... ˙qir + ++t2ℓ +2ℓ L(2ℓ−1)cQc + ... + t2 +2 L(1)cQc + G(q) +(24) +where C(0)i1...im, L(N)(i1...im−1|im) for N = 1, 3, ..., 2ℓ − 1 are mth-order generalized KTs and the following +conditions are satisfied: +L(2ℓ)(i1...im−2|im−1) += +− m +2ℓL(2ℓ−1)(i1...im−1|im)Qim +(25) +L(k−1)(i1...im−2|im−1) += +− +m +k − 1L(k−2)(i1...im−1|im)Qim − kL(k)i1...im−1, k = 3, 5, ..., 2ℓ − 1 +(26) +L(0)(i1...im−2|im−1) += +mC(0)i1...im−1imQim − L(1)i1...im−1 +(27) +L(2ℓ)(i1...ir−1|ir) += +(r + 1)L(2ℓ)i1...irir+1Qir+1, r = 3, 5, ..., m − 3 +(28) +L(k−1)(i1...ir−1|ir) += +(r + 1)L(k−1)i1...irir+1Qir+1 − kL(k)i1...ir, k = 1, 3, ..., 2ℓ − 1, r = 3, 5, ..., m − 3 +(29) +L(k−1)(i1...ir−1|ir) += +(r + 1)L(k−1)i1...irir+1Qir+1 − kL(k)i1...ir, k = 2, 4, ..., 2ℓ, r = 2, 4, ..., m − 2 +(30) +� +L(2ℓ−1)cQc� +,i1 += +4ℓL(2ℓ)i1i2Qi2 +(31) +� +L(k−2)cQc� +,i1 += +2(k − 1)L(k−1)i1i2Qi2 − k(k − 1)L(k)i1, k = 3, 5, ..., 2ℓ − 1 +(32) +G,i1 += +2L(0)i1i2Qi2 − L(1)i1. +(33) +b. +J(m=2ν,2) +ℓ += +� +− t2ℓ+1 +2ℓ + 1L(2ℓ)(i1...im−1|im) − ... − t3 +3 L(2)(i1...im−1|im) − tL(0)(i1...im−1|im) +� +˙qi1... ˙qim + ++ +odd +� +1≤r≤m−1 +� +t2ℓL(2ℓ)i1...ir + ... + t2L(2)i1...ir + L(0)i1...ir +� +˙qi1... ˙qir + ++ +even +� +1≤r≤m−1 +� +t2ℓ+1L(2ℓ+1)i1...ir + ... + t3L(3)i1...ir + tL(1)i1...ir +� +˙qi1... ˙qir + ++ t2ℓ+1 +2ℓ + 1L(2ℓ)cQc + ... + t3 +3 L(2)cQc + tL(0)cQc +(34) +where L(N)(i1...im−1|im) for N = 0, 2, ..., 2ℓ are mth-order generalized KTs and the following conditions are +satisfied: +L(2ℓ+1)(i1...im−2|im−1) += +− +m +2ℓ + 1L(2ℓ)(i1...im−1|im)Qim +(35) +L(k−1)(i1...im−2|im−1) += +− +m +k − 1L(k−2)(i1...im−1|im)Qim − kL(k)i1...im−1, k = 2, 4, ..., 2ℓ +(36) +L(2ℓ+1)(i1...ir−1|ir) += +(r + 1)L(2ℓ+1)i1...irir+1Qir+1, r = 3, 5, ..., m − 3 +(37) +L(k−1)(i1...ir−1|ir) += +(r + 1)L(k−1)i1...irir+1Qir+1 − kL(k)i1...ir, k = 1, 3, ..., 2ℓ + 1, r = 2, 4, ..., m − 2 +(38) +L(k−1)(i1...ir−1|ir) += +(r + 1)L(k−1)i1...irir+1Qir+1 − kL(k)i1...ir, k = 2, 4, ..., 2ℓ, r = 3, 5..., m − 3 +(39) +4 + +� +L(2ℓ)cQc� +,i1 += +2(2ℓ + 1)L(2ℓ+1)i1i2Qi2 +(40) +� +L(k−2)cQc� +,i1 += +2(k − 1)L(k−1)i1i2Qi2 − k(k − 1)L(k)i1, k = 2, 4, ..., 2ℓ. +(41) +Moreover, for even order FIs, it holds that +I(2ν) +2k += +J(2ν,1) +k ++ J(2ν,2) +k +� +L(2k)(i1...im−1|im) = 0; L(2k+1)(i1...ir) = 0, 1 ≤ r ≤ m − 1, r = even +� +I(2ν) +2k+1 += +J(2ν,1) +k+1 +� +L(2k+1)(i1...im−1|im) = 0; L(2k+2)(i1...ir) = 0, 1 ≤ r ≤ m − 1, r = even +� ++ J(2ν,2) +k +where m = 2ν, while for odd order FIs +I(2ν+1) +2k += +J(2ν+2,1) +k +(Mi1...im = 0) + J(2ν+2,2) +k +� +Mi1...im = 0; L(2k+1)(i1...ir) = 0, 1 ≤ r ≤ m − 1, r = even +� +I(2ν+1) +2k+1 += +J(2ν+2,1) +k+1 +� +Mi1...im = 0; L(2k+2)(i1...ir) = 0, 1 ≤ r ≤ m − 1, r = even +� ++ J(2ν+2,2) +k +(Mi1...im = 0) +where m = 2ν + 2. For completeness, we may introduce the notation J(2ν+1,1) +ℓ +≡ J(2ν+2,1) +ℓ +(Mi1...im = 0) and +J(2ν+1,2) +ℓ +≡ J(2ν+2,2) +ℓ +(Mi1...im = 0). +In the case of a Riemannian connection, the general mth-order FIs (2) are related to the generalized gauged +weak Noether symmetry +� +ξ = 0, ηi1 = −∂I(m) +∂ ˙qi1 , φa, f = I(m) − ∂I(m) +∂ ˙qi1 ˙qi1 +� +such that φa ˙qa + F a ∂I(m) +∂ ˙qa += 0 +(42) +by means of the Inverse Noether Theorem [8, 10]. In (42), we have +∂I(m) +∂ ˙qi1 += Mi1 + 2Mi1i2 ˙qi2 + 3Mi1i2i3 ˙qi2 ˙qi3 + ... + mMi1i2...im ˙qi2... ˙qim = +m−1 +� +r=0 +(r + 1)Mi1i2...ir+1 ˙qi2... ˙qir+1 +F a(t, q, ˙q) are the non-conservative generalized forces, φa(t, q, ˙q) is an additional vector generator, f(t, q, ˙q) is +the Noether function, and X = ξ(t, q, ˙q)∂t + ηa(t, q, ˙q)∂qa is the Lie generator. In the following, we consider +applications which show the importance of Theorem 1. +4 +Applications +4.1 +Application 1: The QFIs of a non-Riemannian dynamical system +Consider the dynamical system: +¨u += +−8β +u3 +� +u ˙u ˙w − w ˙u2� +− 1 +u2 +(43) +¨w += +−4β +u3 +� +u ˙w2 − 4w ˙u ˙w +� ++ 2w +u3 +(44) +where β is an arbitrary real constant. This system is autonomous holonomic of the form (1) with variables +qa = +� u +w +� +, Qa = 1 +u2 +� +1 +− 2w +u +� +. +The symmetric connection coefficients are read from the the dynamical equations: +Γ1 +22 = Γ2 +11 = 0, Γ1 +11 = Γ2 +12 = −8β w +u3 , Γ1 +12 = Γ2 +22 = 4β +u2 . +(45) +The curvature tensor Rabcd = Γa +bd,c − Γa +bc,d + Γa +scΓs +bd − Γa +sdΓs +bc is computed to be +R1112 = R2221 = −R2212 = −R1121 = −32b2w +u5 +, R2112 = −R2121 = 24bw +u4 . +5 + +Solving the generalized KT condition C(ab|c) = 0, we find that the connection (45) admits only the second +order generalized KT +Cab = ke +12βw +u2 +� +0 +1 +1 +0 +� +(46) +where k is an arbitrary constant. +Solving the generalized Killing vector (KV) condition L(a|b) = 0, we find La = 0; therefore, generalized KVs +do not exist. +Moreover, it can be shown that non-zero vectors Ba which generate reducible generalized KTs of the form +B(a|b) do not exist as well. +Applying Theorem 1 for m = 2, we find that the system admits only one QFI which is the +J(2,1) +1 += e +12βw +u2 +� +˙u ˙w + +1 +12β +� +. +(47) +To prove that the given system is integrable, we need one more independent FI of higher order in involution. +4.2 +Application 2: CFIs of a class of autonomous conservative dynamical systems. +Consider the two-dimensional (2d) autonomous conservative dynamical systems with potential V = F(x2 +νy2) +where ν is an arbitrary constant. Determine the potentials which admit CFIs. +In [11], the authors establish an isomorphism between the autonomous QFIs/CFIs of Hamilton’s equations +of an autonomous conservative dynamical system and the admissible Lie-B¨acklund symmetries of the Hamilton- +Jacobi equation. Using this result, they found the following three such potentials: +V(1a) = 1 +2x2 + 9 +2y2, V(1b) = 1 +2x2 + 1 +18y2, V(1c) = (x2 − y2)−2/3. +(48) +Applying Theorem 1, we find two more potentials and show that the results of [11] are just special cases. +These potentials are the following: +a. The new superintegrable potential +V1 = c0(x2 + 9y2) + c1y +(49) +where c0 and c1 are arbitrary constants, which admits the associated CFI +J1 = (x ˙y − y ˙x) ˙x2 − +c1 +18c0 +˙x3 + c1 +3 x2 ˙x + 6c0x2y ˙x − 2c0 +3 x3 ˙y. +(50) +b. The integrable potential +V2 = k(x2 − y2)−2/3 +(51) +where k is an arbitrary constant, which admits the CFI +J2 = (x ˙y − y ˙x) +� +˙y2 − ˙x2� ++ 4V2(y ˙x + x ˙y). +(52) +The potentials (48) are special cases of V1 and V2 for the following values of the parameters: +V(1a) = V1 +� +c1 = 0, c0 = 1 +2 +� +, V(1b) = V1 +� +x ↔ y; c1 = 0, c0 = 1 +18 +� +, V(1c) = V2(k = 1). +4.3 +Application 3: New integrable/superintegrable potentials that admit autonomous +and time-dependent CFIs. +In [12], the authors using the Jacobi metric approach found integrable and superintegrable potentials that admit +autonomous CFIs. +6 + +Applying Theorem 1, we find the new integrable potential +V = +k1 +(a2y − a5x)2 + k2 +r + k3(a2x + a5y) +r(a2y − a5x)2 +(53) +where k1, k2, k3, a2, a5 are arbitrary constants and r = +� +x2 + y2, which admits the CFI +J1 += +(x ˙y − y ˙x)2(a2 ˙x + a5 ˙y) + +2k1r2 +(a2y − a5x)2 (a2 ˙x + a5 ˙y) − k2(a2y − a5x) +r +(x ˙y − y ˙x) + ++ +k3r +a2y − a5x(a2 ˙y − a5 ˙x) − k3(a2x + a5y) +r(a2y − a5x) (x ˙y − y ˙x) + 2k3(a2x + a5y)r +(a2y − a5x)2 +(a2 ˙x + a5 ˙y). +We note that for k2 = 0, the special potential +V (k2 = 0) = +k1 +(a2y − a5x)2 + k3(a2x + a5y) +r(a2y − a5x)2 +(54) +admits also the additional time-dependent CFI +J2 += +−tJ1(k2 = 0) + (a2x + a5y)(x ˙y − y ˙x)2 + 2k1r2(a2x + a5y) +(a2y − a5x)2 ++ ++2k3r(a2x + a5y)2 +(a2y − a5x)2 ++ k3r. +We conclude that (54) is a new superintegrable potential. This result illustrates the importance of the time- +dependent FIs in the determination of the integrability/superintegrability. +5 +Conclusions +We draw the following conclusions: +a) We have developed a direct systematic method to compute the mth-order FIs of the autonomous holonomic +dynamical systems (1) in terms of the ‘symmetries’ of the geometric objects (symmetric connection or kinetic +metric, depending on the case) defined by the dynamical equations. +b) This method applies to non-Riemanian geometries with a symmetric connection. It has been shown that the +mth-order FIs require the generalized KTs and KVs defined by the symmetric connection Γa +bc. The case of a +Riemannian connection is just a special case. +c) The system of PDEs (4) - (9) resulting from the condition dI/dt = 0 and the dynamical equations consists +of two parts: A geometric part (eqs. (4), (5) ) common to all systems which share the same connection; and a +dynamical part (eqs. (6) - (9) ) which includes the generalized forces Qa of the specific system. +d) All mth-order FIs of an autonomous holonomic dynamical system in a Riemannian background geometry +via the Inverse Noether Theorem are associated to a gauged weak Noether symmetry. +References +[1] V.I. Arnold, ‘Mathematical Methods of Classical Mechanics’, Springer, (1989), proof in pp. 272-284. +[2] P.A. Damianou and C. Sophocleous, ‘Classification of Noether Symmetries for Lagrangians with Three +Degrees of Freedom’, Nonlin. Dyn. 36, 3 (2004). +[3] M. Tsamparlis and A. Paliathanasis, ‘Two-dimensional dynamical systems which admit Lie and Noether +symmetries’, J. Phys. A: Math. Theor. 44, 175202 (2011). +[4] A.K. Halder, A. Paliathanasis and P.G.L. Leach, ‘Noether’s Theorem and Symmetry’, Symmetry 10(12), +744 (2018). +7 + +[5] G.H. Katzin and J. Levine, ‘Geodesic first integrals with explicit path-parameter dependence in Riemannian +space-times’, J. Math. Phys. 22(9), 1878 (1981). +[6] J.T. Horwood, ‘Higher order first integrals in classical mechanics’, J. Math. Phys 48, 102902 (2007). +[7] M. Tsamparlis and A. Mitsopoulos, ‘Quadratic first integrals of autonomous conservative dynamical sys- +tems’, J. Math. Phys. 61, 072703 (2020). +[8] M. Tsamparlis and A. Mitsopoulos, ‘First integrals of holonomic systems without Noether symmetries’, J. +Math. Phys. 61, 122701 (2020). +[9] A. Mitsopoulos and M. Tsamparlis, ‘Higher order first integrals of autonomous dynamical systems’, J. +Geom. Phys. 170, 104383 (2021). +[10] D.S. Djukic and B.D. Vujanovic, ‘Noether’s Theory in Classical Nonconservative Mechanics’, Acta Me- +chanica 23, 17 (1975). +[11] A.S. Fokas and P.A. Lagerstrom, ‘Quadratic and Cubic Invariants in Classical Mechanics’, J. Math. Anal. +Appl. 74, 325 (1980). +[12] M. Karlovini and K. Rosquist, ‘A unified treatment of cubic invariants at fixed and arbitrary energy’, J. +Math. Phys. 41(1), 370 (2000). +8 + diff --git a/RNAyT4oBgHgl3EQf7vr2/content/tmp_files/load_file.txt b/RNAyT4oBgHgl3EQf7vr2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..706a90355f71b9e6385e5deacc4f32e9afec1ddc --- /dev/null +++ b/RNAyT4oBgHgl3EQf7vr2/content/tmp_files/load_file.txt @@ -0,0 +1,511 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf,len=510 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='00846v1 [math-ph] 2 Jan 2023 Higher order first integrals of autonomous dynamical systems in terms of geometric symmetries Antonios Mitsopoulos1,a) and Michael Tsamparlis1,b) 1Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics, University of Athens, Panepistemiopolis, Athens 157 83, Greece a)Author to whom correspondence should be addressed: antmits@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='uoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='gr b)Email: mtsampa@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='uoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='gr Abstract In general, a system of differential equations is integrable if there exist ‘sufficiently many’ first integrals (FIs) so that its solution can be found by means of quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Therefore, the determination of the FIs is an important issue in order to establish the integrability of a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' In this work, we consider holonomic autonomous dynamical systems defined by equations ¨qa = −Γa bc(q) ˙qb ˙qc − Qa(q) where Γa bc(q) are the coefficients of a symmetric (possibly non-metrical) connection and −Qa(q) are the generalized forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' We prove a theorem which produces the FIs of any order of such systems in terms of the ‘symmetries’ of the geometry defined by the quantities Γa bc(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' We apply the theorem to compute quadratic and cubic FIs of various dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 1 Introduction A first integral (FI) of a second order set of dynamical equations with generalized coordinates qa and generalized velocities ˙qa ≡ dqa dt is a function I(t, qa, ˙qa) satisfying the condition dI dt = 0 along the dynamical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' It is important to have a systematic method for determining FIs because they can be used to reduce the order of the dynamical equations and, if they are ‘enough’ in number [1], to find the solution of the system by quadrature (Liouville integrability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' The standard method to compute the FIs is the method of Noether symmetries [2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' A different method is the direct method [5, 6, 7, 8, 9] in which one assumes a functional form for the FI I and demands directly the condition dI dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' This condition and the dynamical equations lead to a system of partial differential equations (PDEs) whose solution provides the FIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' In this work, we apply the direct method to autonomous holonomic dynamical systems in a space with a symmetric connection Γa bc(q) (not necessarily Riemannian) which is read from the dynamical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' We compute the resulting system of PDEs and solve it in terms of the ‘symmetries’ of Γa bc(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' The result is stated as Theorem 1 and provides a systematic method to determine the FIs of any order, time-dependent and autonomous, of these dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' In the special case where the symmetric connection Γa bc(q) is the Riemannian one defined in terms of the kinetic metric (kinetic energy) γab(q) of the system, the computed FIs are directly related by means of the Inverse Noether Theorem [8, 10] to gauged generalized (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' velocity- dependent) Noether symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Finally, we apply Theorem 1 in order to find integrable and superintegrable systems that admit quadratic (QFIs) and cubic FIs (CFIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 2 The conditions for higher order FIs We consider autonomous holonomic dynamical systems of the form ¨qa = −Γa bc(q) ˙qb ˙qc − Qa(q) (1) 1 where Γa bc(q) are the coefficients of a general connection and −Qa(q) are the generalized forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Since only the symmetric part Γa (bc) contributes to the dynamical equations, without loss of generality, the quantities Γa bc(q) are assumed to be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' We look for mth-order FIs of the general form I(m) = m � r=0 Mi1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir(t, q) ˙qi1 ˙qi2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qir = M + Mi1 ˙qi1 + Mi1i2 ˙qi1 ˙qi2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' + Mi1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im ˙qi1 ˙qi2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qim (2) where Mi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir(t, q) are totally symmetric r-rank tensors and the index (m) denotes the order of the FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' The FI condition dI dt = 0 (3) and the dynamical equations (1) result in the following system of PDEs: M(i1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im|im+1) = 0 (4) Mi1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im,t + M(i1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) = 0 (5) Mi1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir,t + M(i1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir−1|ir) − (r + 1)Mi1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='irir+1Qir+1 = 0, r = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', m − 1 (6) M,t − Mi1Qi1 = 0 (7) Mi1,tt − 2Mi1i2,tQi2 + (McQc),i1 = 0 (8) 2 � M[i1≀c≀Qc� |i2] − M[i1|i2],t = 0 (9) where | denotes the covariant derivative with respect to (wrt) the symmetric connection Γa bc, a comma indi- cates partial derivative wrt qa or t, round/square brackets indicate symmetrization/antisymmetrization of the enclosed indices, and indices enclosed between wavy lines are overlooked by symmetrization or antisymmetriza- tion symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Equations (8) and (9) express the integrability conditions M,i1t = M,ti1 and M,[i1i2] = 0 of the scalar M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Equation (4) generalizes the concept of Killing tensors (KTs) to a non-metrical geometry with a symmetric connection Γa bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' In this context, Mi1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im is a generalized mth-order KT for Γa bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' The most general choice for the generalized mth-order KT Mi1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im in the case of an autonomous system is Mi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im(t, q) = C(0)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im(q) + n � N=1 C(N)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im(q)tN N (10) where C(N)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im(q), N = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', n, is a sequence of arbitrary mth-order generalized KTs of Γa bc and n is the degree of the considered polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' The choice (10) and equation (5) indicate that we set Mi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir(t, q) = nr � Nr=0 L(Nr)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir(q)tNr, r = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', m − 1 (11) where L(Nr)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir(q), Nr = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', nr, are arbitrary r-rank totally symmetric tensors and nr is the degree of the considered polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' We note that the degrees n, nr of the above polynomial expressions of t may be infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Substituting (10) and (11) in the system of equations (4) - (9) (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' (4) is identically zero since C(N)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im are assumed to be generalized KTs), we end up with a system of five PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' The solution of this system is lengthy and requires the consideration of many cases and subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' We state the solution below as Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 3 Theorem for mth-order FIs of an autonomous holonomic dynam- ical system Theorem 1 The independent mth-order FIs of the dynamical system (1) are the following: 2 Integral 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' I(m) n = � − n � N=1 tN N L(N−1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) + C(0)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im � ˙qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qim + m−1 � r=1 � n � N=0 tNL(N)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir � ˙qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qir + +s tn+1 n + 1 + n � N=1 L(N−1)cQc tN N + G(q) where C(0)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im, L(N)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) for N = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', n − 1 are mth-order generalized KTs, L(n)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1 is an (m − 1)th-order generalized KT, s is an arbitrary constant defined by the condition L(n)i1Qi1 = s (12) while the vectors L(N)i1 and the totally symmetric tensors L(A)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir, A = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', n, r = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', m−2 satisfy the conditions: L(n)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−2|im−1) = −m n L(n−1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im)Qim (13) L(k−1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−2|im−1) = − m k − 1L(k−2)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im)Qim − kL(k)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1, k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', n (14) L(0)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−2|im−1) = mC(0)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1imQim − L(1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1 (15) L(n)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir−1|ir) = (r + 1)L(n)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='irir+1Qir+1, r = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', m − 2 (16) L(k−1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir−1|ir) = (r + 1)L(k−1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='irir+1Qir+1 − kL(k)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', n, r = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', m − 2 (17) � L(n−1)cQc� ,i1 = 2nL(n)i1i2Qi2 (18) � L(k−2)cQc� ,i1 = 2(k − 1)L(k−1)i1i2Qi2 − k(k − 1)L(k)i1, k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', n (19) G,i1 = 2L(0)i1i2Qi2 − L(1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' (20) Integral 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' I(m) e = eλt λ � −L(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) ˙qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qim + λ m−1 � r=1 Li1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir ˙qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qir + Li1Qi1 � where λ ̸= 0, L(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) is an mth-order generalized KT and the remaining totally symmetric tensors satisfy the conditions: L(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−2|im−1) = −m λ L(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im)Qim − λLi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1 (21) L(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir−1|ir) = (r + 1)Li1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='irir+1Qir+1 − λLi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir, r = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', m − 2 (22) (LcQc),i1 = 2λLi1i2Qi2 − λ2Li1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' (23) We note that Theorem 1 for m = 2 and a Riemannian connection reduces to Theorem 1 of [7] and to Theorem 3 of [8] for the case of QFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Moreover, we have the following minor results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Proposition 2 The independent mth-order FIs I(m) n and I(m) e satisfy the following recursion formulae: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' I(k) n < I(k+1) n , that is, each kth-order FI I(k) n is a subcase of the next (k + 1)th-order FI I(k+1) n with the same degree n of time-dependence for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' I(m) ℓ < I(m) ℓ+1, that is, the mth-order FI I(m) ℓ with time-dependence fixed by ℓ is a subcase of the mth-order FI I(m) ℓ+1 with time-dependence ℓ + 1 for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' I(k) e < I(k+1) e , that is, each kth-order FI I(k) e is a subcase of the next (k + 1)th-order FI I(k+1) e for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Proposition 3 The mth-order FI I(m) n consists of the following two independent FIs: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' The FI J(m,1) ℓ whose coefficients are polynomials of t containing even powers of t for even products of velocities and odd powers of t for odd products of velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' The FI J(m,2) ℓ whose coefficients are polynomials of t containing even powers of t for odd products of velocities and odd powers of t for even products of velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 3 For even orders m = 2ν (ν ∈ N) the independent FIs of Proposition 3 are computed by the formulae (ℓ ∈ N): a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' J(m=2ν,1) ℓ = � −t2ℓ 2ℓ L(2ℓ−1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' − t2 2 L(1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) + C(0)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im � ˙qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qim + + odd � 1≤r≤m−1 � t2ℓ−1L(2ℓ−1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' + t3L(3)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir + tL(1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir � ˙qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qir + + even � 1≤r≤m−1 � t2ℓL(2ℓ)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' + t2L(2)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir + L(0)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir � ˙qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qir + +t2ℓ 2ℓ L(2ℓ−1)cQc + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' + t2 2 L(1)cQc + G(q) (24) where C(0)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im, L(N)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) for N = 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', 2ℓ − 1 are mth-order generalized KTs and the following conditions are satisfied: L(2ℓ)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−2|im−1) = − m 2ℓL(2ℓ−1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im)Qim (25) L(k−1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−2|im−1) = − m k − 1L(k−2)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im)Qim − kL(k)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1, k = 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', 2ℓ − 1 (26) L(0)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−2|im−1) = mC(0)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1imQim − L(1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1 (27) L(2ℓ)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir−1|ir) = (r + 1)L(2ℓ)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='irir+1Qir+1, r = 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', m − 3 (28) L(k−1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir−1|ir) = (r + 1)L(k−1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='irir+1Qir+1 − kL(k)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir, k = 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', 2ℓ − 1, r = 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', m − 3 (29) L(k−1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir−1|ir) = (r + 1)L(k−1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='irir+1Qir+1 − kL(k)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir, k = 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', 2ℓ, r = 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', m − 2 (30) � L(2ℓ−1)cQc� ,i1 = 4ℓL(2ℓ)i1i2Qi2 (31) � L(k−2)cQc� ,i1 = 2(k − 1)L(k−1)i1i2Qi2 − k(k − 1)L(k)i1, k = 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', 2ℓ − 1 (32) G,i1 = 2L(0)i1i2Qi2 − L(1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' (33) b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' J(m=2ν,2) ℓ = � − t2ℓ+1 2ℓ + 1L(2ℓ)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' − t3 3 L(2)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) − tL(0)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) � ˙qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qim + + odd � 1≤r≤m−1 � t2ℓL(2ℓ)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' + t2L(2)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir + L(0)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir � ˙qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qir + + even � 1≤r≤m−1 � t2ℓ+1L(2ℓ+1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' + t3L(3)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir + tL(1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir � ˙qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qir + + t2ℓ+1 2ℓ + 1L(2ℓ)cQc + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' + t3 3 L(2)cQc + tL(0)cQc (34) where L(N)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) for N = 0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', 2ℓ are mth-order generalized KTs and the following conditions are satisfied: L(2ℓ+1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−2|im−1) = − m 2ℓ + 1L(2ℓ)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im)Qim (35) L(k−1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−2|im−1) = − m k − 1L(k−2)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im)Qim − kL(k)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1, k = 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', 2ℓ (36) L(2ℓ+1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir−1|ir) = (r + 1)L(2ℓ+1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='irir+1Qir+1, r = 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', m − 3 (37) L(k−1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir−1|ir) = (r + 1)L(k−1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='irir+1Qir+1 − kL(k)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir, k = 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', 2ℓ + 1, r = 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', m − 2 (38) L(k−1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir−1|ir) = (r + 1)L(k−1)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='irir+1Qir+1 − kL(k)i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir, k = 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', 2ℓ, r = 3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', m − 3 (39) 4 � L(2ℓ)cQc� ,i1 = 2(2ℓ + 1)L(2ℓ+1)i1i2Qi2 (40) � L(k−2)cQc� ,i1 = 2(k − 1)L(k−1)i1i2Qi2 − k(k − 1)L(k)i1, k = 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=', 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' (41) Moreover, for even order FIs, it holds that I(2ν) 2k = J(2ν,1) k + J(2ν,2) k � L(2k)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' L(2k+1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir) = 0, 1 ≤ r ≤ m − 1, r = even � I(2ν) 2k+1 = J(2ν,1) k+1 � L(2k+1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im−1|im) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' L(2k+2)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir) = 0, 1 ≤ r ≤ m − 1, r = even � + J(2ν,2) k where m = 2ν, while for odd order FIs I(2ν+1) 2k = J(2ν+2,1) k (Mi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im = 0) + J(2ν+2,2) k � Mi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' L(2k+1)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir) = 0, 1 ≤ r ≤ m − 1, r = even � I(2ν+1) 2k+1 = J(2ν+2,1) k+1 � Mi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' L(2k+2)(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir) = 0, 1 ≤ r ≤ m − 1, r = even � + J(2ν+2,2) k (Mi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im = 0) where m = 2ν + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' For completeness, we may introduce the notation J(2ν+1,1) ℓ ≡ J(2ν+2,1) ℓ (Mi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im = 0) and J(2ν+1,2) ℓ ≡ J(2ν+2,2) ℓ (Mi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' In the case of a Riemannian connection, the general mth-order FIs (2) are related to the generalized gauged weak Noether symmetry � ξ = 0, ηi1 = −∂I(m) ∂ ˙qi1 , φa, f = I(m) − ∂I(m) ∂ ˙qi1 ˙qi1 � such that φa ˙qa + F a ∂I(m) ∂ ˙qa = 0 (42) by means of the Inverse Noether Theorem [8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' In (42), we have ∂I(m) ∂ ˙qi1 = Mi1 + 2Mi1i2 ˙qi2 + 3Mi1i2i3 ˙qi2 ˙qi3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' + mMi1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='im ˙qi2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qim = m−1 � r=0 (r + 1)Mi1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='ir+1 ˙qi2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' ˙qir+1 F a(t, q, ˙q) are the non-conservative generalized forces, φa(t, q, ˙q) is an additional vector generator, f(t, q, ˙q) is the Noether function, and X = ξ(t, q, ˙q)∂t + ηa(t, q, ˙q)∂qa is the Lie generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' In the following, we consider applications which show the importance of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 4 Applications 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='1 Application 1: The QFIs of a non-Riemannian dynamical system Consider the dynamical system: ¨u = −8β u3 � u ˙u ˙w − w ˙u2� − 1 u2 (43) ¨w = −4β u3 � u ˙w2 − 4w ˙u ˙w � + 2w u3 (44) where β is an arbitrary real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' This system is autonomous holonomic of the form (1) with variables qa = � u w � , Qa = 1 u2 � 1 − 2w u � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' The symmetric connection coefficients are read from the the dynamical equations: Γ1 22 = Γ2 11 = 0, Γ1 11 = Γ2 12 = −8β w u3 , Γ1 12 = Γ2 22 = 4β u2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' (45) The curvature tensor Rabcd = Γa bd,c − Γa bc,d + Γa scΓs bd − Γa sdΓs bc is computed to be R1112 = R2221 = −R2212 = −R1121 = −32b2w u5 , R2112 = −R2121 = 24bw u4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 5 Solving the generalized KT condition C(ab|c) = 0, we find that the connection (45) admits only the second order generalized KT Cab = ke 12βw u2 � 0 1 1 0 � (46) where k is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Solving the generalized Killing vector (KV) condition L(a|b) = 0, we find La = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' therefore, generalized KVs do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Moreover, it can be shown that non-zero vectors Ba which generate reducible generalized KTs of the form B(a|b) do not exist as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Applying Theorem 1 for m = 2, we find that the system admits only one QFI which is the J(2,1) 1 = e 12βw u2 � ˙u ˙w + 1 12β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' (47) To prove that the given system is integrable, we need one more independent FI of higher order in involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='2 Application 2: CFIs of a class of autonomous conservative dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Consider the two-dimensional (2d) autonomous conservative dynamical systems with potential V = F(x2 +νy2) where ν is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Determine the potentials which admit CFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' In [11], the authors establish an isomorphism between the autonomous QFIs/CFIs of Hamilton’s equations of an autonomous conservative dynamical system and the admissible Lie-B¨acklund symmetries of the Hamilton- Jacobi equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Using this result, they found the following three such potentials: V(1a) = 1 2x2 + 9 2y2, V(1b) = 1 2x2 + 1 18y2, V(1c) = (x2 − y2)−2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' (48) Applying Theorem 1, we find two more potentials and show that the results of [11] are just special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' These potentials are the following: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' The new superintegrable potential V1 = c0(x2 + 9y2) + c1y (49) where c0 and c1 are arbitrary constants, which admits the associated CFI J1 = (x ˙y − y ˙x) ˙x2 − c1 18c0 ˙x3 + c1 3 x2 ˙x + 6c0x2y ˙x − 2c0 3 x3 ˙y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' (50) b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' The integrable potential V2 = k(x2 − y2)−2/3 (51) where k is an arbitrary constant, which admits the CFI J2 = (x ˙y − y ˙x) � ˙y2 − ˙x2� + 4V2(y ˙x + x ˙y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' (52) The potentials (48) are special cases of V1 and V2 for the following values of the parameters: V(1a) = V1 � c1 = 0, c0 = 1 2 � , V(1b) = V1 � x ↔ y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' c1 = 0, c0 = 1 18 � , V(1c) = V2(k = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='3 Application 3: New integrable/superintegrable potentials that admit autonomous and time-dependent CFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' In [12], the authors using the Jacobi metric approach found integrable and superintegrable potentials that admit autonomous CFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 6 Applying Theorem 1, we find the new integrable potential V = k1 (a2y − a5x)2 + k2 r + k3(a2x + a5y) r(a2y − a5x)2 (53) where k1, k2, k3, a2, a5 are arbitrary constants and r = � x2 + y2, which admits the CFI J1 = (x ˙y − y ˙x)2(a2 ˙x + a5 ˙y) + 2k1r2 (a2y − a5x)2 (a2 ˙x + a5 ˙y) − k2(a2y − a5x) r (x ˙y − y ˙x) + + k3r a2y − a5x(a2 ˙y − a5 ˙x) − k3(a2x + a5y) r(a2y − a5x) (x ˙y − y ˙x) + 2k3(a2x + a5y)r (a2y − a5x)2 (a2 ˙x + a5 ˙y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' We note that for k2 = 0, the special potential V (k2 = 0) = k1 (a2y − a5x)2 + k3(a2x + a5y) r(a2y − a5x)2 (54) admits also the additional time-dependent CFI J2 = −tJ1(k2 = 0) + (a2x + a5y)(x ˙y − y ˙x)2 + 2k1r2(a2x + a5y) (a2y − a5x)2 + +2k3r(a2x + a5y)2 (a2y − a5x)2 + k3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' We conclude that (54) is a new superintegrable potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' This result illustrates the importance of the time- dependent FIs in the determination of the integrability/superintegrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 5 Conclusions We draw the following conclusions: a) We have developed a direct systematic method to compute the mth-order FIs of the autonomous holonomic dynamical systems (1) in terms of the ‘symmetries’ of the geometric objects (symmetric connection or kinetic metric, depending on the case) defined by the dynamical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' b) This method applies to non-Riemanian geometries with a symmetric connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' It has been shown that the mth-order FIs require the generalized KTs and KVs defined by the symmetric connection Γa bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' The case of a Riemannian connection is just a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' c) The system of PDEs (4) - (9) resulting from the condition dI/dt = 0 and the dynamical equations consists of two parts: A geometric part (eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' (4), (5) ) common to all systems which share the same connection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' and a dynamical part (eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' (6) - (9) ) which includes the generalized forces Qa of the specific system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' d) All mth-order FIs of an autonomous holonomic dynamical system in a Riemannian background geometry via the Inverse Noether Theorem are associated to a gauged weak Noether symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Arnold, ‘Mathematical Methods of Classical Mechanics’, Springer, (1989), proof in pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 272-284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Damianou and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Sophocleous, ‘Classification of Noether Symmetries for Lagrangians with Three Degrees of Freedom’, Nonlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 36, 3 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Tsamparlis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Paliathanasis, ‘Two-dimensional dynamical systems which admit Lie and Noether symmetries’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 44, 175202 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Halder, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Paliathanasis and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Leach, ‘Noether’s Theorem and Symmetry’, Symmetry 10(12), 744 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 7 [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Katzin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Levine, ‘Geodesic first integrals with explicit path-parameter dependence in Riemannian space-times’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 22(9), 1878 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Horwood, ‘Higher order first integrals in classical mechanics’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Phys 48, 102902 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Tsamparlis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Mitsopoulos, ‘Quadratic first integrals of autonomous conservative dynamical sys- tems’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 61, 072703 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Tsamparlis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Mitsopoulos, ‘First integrals of holonomic systems without Noether symmetries’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 61, 122701 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Mitsopoulos and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Tsamparlis, ‘Higher order first integrals of autonomous dynamical systems’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 170, 104383 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Djukic and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Vujanovic, ‘Noether’s Theory in Classical Nonconservative Mechanics’, Acta Me- chanica 23, 17 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Fokas and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Lagerstrom, ‘Quadratic and Cubic Invariants in Classical Mechanics’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 74, 325 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Karlovini and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Rosquist, ‘A unified treatment of cubic invariants at fixed and arbitrary energy’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 41(1), 370 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} +page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQf7vr2/content/2301.00846v1.pdf'} diff --git a/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf b/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2ae2ab834c61b1185507412da4462a8cf7afda9e --- /dev/null +++ b/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf @@ -0,0 +1,3 @@ +version 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Plotkin,2,3† Roberto Soria,4,5,6 Amy E. Reines,7 Jenny E. Greene,8 +Gemma E. Anderson,9 Vivienne F. Baldassare,10 Milo G. Owens,2 Ryan T. Urquhart,1 Elena Gallo,11 +James C. A. Miller-Jones,9 Jeremiah D. Paul,2 and Alexandar P. Rollings2 +1Center for Data Intensive and Time Domain Astronomy, Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA +2Department of Physics, University of Nevada, Reno, NV 89557, USA +3Nevada Center for Astrophysics, University of Nevada, Las Vegas, NV 89154, USA +4College of Astronomy and Space Sciences, University of the Chinese Academy of Sciences, Beijing 100049, China +5INAF - Osservatorio Astrofisico di Torino, Strada Osservatorio 20, 10025, Pino Torinese, Italy +6Sydney Institute for Astronomy, School of Physics A28, The University of Sydney, Sydney, NSW 2006, Australia +7eXtreme Gravity Institute, Montana State University, Bozeman, MT, USA +8Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA +9International Centre for Radio Astronomy Research, Curtin University, GPO Box U1987, Perth, WA 6845, Australia +10Department of Physics and Astronomy, Washington State University, Pullman, WA 99163, USA +11Department of Astronomy, University of Michigan, 1085 S University, Ann Arbor, MI 48109, USA +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Owing to their quiet evolutionary histories, nearby dwarf galaxies (stellar masses 𝑀★ ≲ +3 × 109𝑀⊙) have the potential to teach us about the mechanism(s) that ‘seeded’ the growth of +supermassive black holes, and also how the first stellar mass black holes formed and interacted +with their environments. Here, we present high spatial-resolution observations of three dwarf +galaxies in the X-ray (Chandra), the optical/near-infrared (Hubble Space Telescope), and the +radio (Karl G. Jansky Very Large Array). These three galaxies were previously identified as +hosting candidate active galactic nuclei on the basis of lower resolution X-ray imaging. With our +new observations, we find that X-ray sources in two galaxies (SDSS J121326.01+543631.6 +and SDSS J122111.29+173819.1) are off nuclear and lack corresponding radio emission, +implying they are likely luminous X-ray binaries. The third galaxy (Mrk 1434) contains two +X-ray sources (each with 𝐿X ≈ 1040 erg s−1) separated by 2.′′8, has a low-metallicity (12 ++ log (O/H) = 7.8), and emits nebular He ii 𝜆4686 line emission. The northern source has +spatially coincident point-like radio emission at 9.0 GHz and extended radio emission at 5.5 +GHz. We discuss X-ray binary interpretations (where an ultraluminous X-ray source blows a +‘radio bubble’) and active galactic nucleus interpretations (where a ≈ 4 × 105𝑀⊙ black hole +launches a jet). In either case, we find that the He ii emission cannot be photoionised by the +X-ray source, unless the source was ≈30—90 times more luminous several hundred years ago. +Key words: galaxies: dwarf — stars: black holes — radio continuum: galaxies — X-rays: +galaxies +1 +INTRODUCTION +There is abundant evidence that supermassive black holes (SMBHs; +106 ≲ 𝑀BH ≲ 109 𝑀⊙) ubiquitously exist at the centres of large +galaxies (e.g., Kormendy & Ho 2013), some of which accrete and +shine as active galactic nuclei (AGNs). Some lower-mass dwarf +galaxies (which we define by stellar masses 𝑀★ ≲ 3 × 109𝑀⊙) are +★ E-mail: thygesen@msu.edu +† E-mail: rplotkin@unr.edu +known to host nuclear black holes (e.g., Filippenko & Ho 2003; +Barth et al. 2004; Reines et al. 2011, 2013; Schramm et al. 2013; +Moran et al. 2014; Sartori et al. 2015; Mezcua et al. 2016, 2018; +Pardo et al. 2016; Ho & Kim 2016; Chen et al. 2017; Chilingarian +et al. 2018; Nguyen et al. 2019; Baldassare et al. 2020; Martínez- +Palomera et al. 2020; Cann et al. 2021; Schutte & Reines 2022), with +some mass estimates as low as 𝑀BH ≈104 𝑀⊙ (e.g., Baldassare et al. +2015; Woo et al. 2019). These black holes represent the lower-mass +end of the SMBH population, and we refer to them here as ‘massive +© 2023 The Authors +arXiv:2301.01317v1 [astro-ph.HE] 3 Jan 2023 + +2 +E. Thygesen et al. +black holes’ (mBHs; 104 ≲ 𝑀BH ≲ 106 𝑀⊙). An actively accreting +mBH can affect how dwarf galaxies provide feedback to their larger +scale environments (e.g., Dashyan et al. 2018; Trebitsch et al. 2018; +Mezcua et al. 2019), and more generally, mBHs represent a phase +that nuclear black holes must pass through as they grow to SMBH +sizes over cosmological time scales (e.g., Volonteri 2010). Given +that dwarf galaxies have had relatively quiet evolutionary histories, +constraining the fraction of dwarf galaxies hosting mBHs in the +local Universe, along with the mBH mass distribution, may provide +constraints on the mechanism(s) that formed the first black holes in +the Universe (e.g., Ricarte & Natarajan 2018; Inayoshi et al. 2020; +Volonteri et al. 2021). The fraction of dwarf galaxies hosting an +mBH is still relatively unknown, with current empirical constraints +implying ≳ 30−50% occupation (Miller et al. 2015; Gallo & Sesana +2019; Greene et al. 2020). +Stellar mass black holes (𝑀BH ≈10 𝑀⊙) and neutron stars are +also observed within some dwarf galaxies in the form of X-ray bi- +naries (XRBs). XRBs serve as probes of stellar populations within +galaxies, with the number and/or luminosity of XRBs expected to +scale with the star formation rate, stellar mass, and metallicity of +the host galaxy (e.g., Grimm et al. 2003; Gilfanov 2004; Linden +et al. 2010; Mineo et al. 2014; Lehmer et al. 2021). Intriguingly, +lower-metallicity galaxies appear to contain an excess of luminous +XRBs compared to Solar-metallicity galaxies (Prestwich et al. 2013; +Brorby et al. 2014; Douna et al. 2015; Ponnada et al. 2020; Lehmer +et al. 2021), which may be a consequence of lower-metallicity pro- +genitor stars having weaker stellar winds, and therefore producing +black hole remnants that are more numerous and/or more massive +(e.g., Heger et al. 2003; Mapelli et al. 2010). Besides tracing stel- +lar populations, the energy output from XRBs can also provide +feedback to their host galaxies. For example, line emission from the +high-ionisation He ii 𝜆4686 line (𝜒ion = 54.4 eV) has been observed +from some ultraluminous X-ray sources (ULXs),1 which is often +interpreted as an X-ray photoionised nebula (Pakull & Angebault +1986; Moon et al. 2011). Extrapolating such ULX phenomenology +in the local Universe to higher redshifts, XRBs could have con- +tributed to the X-ray heating of the intergalactic medium during the +Epoch of Reionisation and Cosmic Dawn (e.g., Mirabel et al. 2011; +Ponnada et al. 2020). Thus, characterising both the XRB and mBH +populations in nearby dwarf galaxies, particularly as a function of +host galaxy metallicity, is important for understanding the formation +of the first black holes and galaxies in the Universe. +X-ray observations are commonly used to identify accreting +compact objects, since hard X-ray emission (≳1–2 keV) is a uni- +versal signature of accretion. However, in several cases, it is very +challenging to determine the mass of an accreting object via X- +ray observations alone. In particular, both a rapidly-accreting XRB +and a weakly-accreting mBH/SMBH can have comparable X-ray +luminosities in the 1039 − 1041 erg s−1 range, and they can also +display similar X-ray spectral shapes (below ≈50 keV). Folding +in multiwavelength information is therefore essential for differ- +entiating between rapidly accreting XRBs and weakly-accreting +mBHs/SMBHs. It is well established that weakly accreting SMBHs +(i.e., low-luminosity AGNs with 𝐿bol ≲ 0.01𝐿Edd, where 𝐿bol is the +bolometric luminosity and 𝐿Edd = 1.3×1038 [𝑀BH/𝑀⊙] erg s−1 is +the Eddington luminosity) emit compact, usually unresolved, radio +emission with a flat spectrum ( 𝑓𝜈 ∝ 𝜈−𝛼, where 𝑓𝜈 is the radio flux +1 We define ULXs as having X-ray luminosities 𝐿𝑋 > 1039 erg s−1. ULXs +are most commonly interpreted as super-Eddington neutron star or black +hole XRBs (see, e.g., Feng & Soria 2011; Kaaret et al. 2017). +Table 1. Properties of the three dwarf galaxies in our sample. Column +1: galaxy names. The full designations of the second and third galax- +ies are SDSS J121326.01+543631.6 and J122111.29+173819.1. Column +2: distances to each galaxy, assuming 𝐻0 = 73 km s−1 Mpc−1 for Mrk +1434 and SDSS J1213, and using the Tully-Fisher relation for SDSS +J1221(Kashibadze et al. 2020). Column 3: stellar masses, following the +methodology of Reines & Volonteri (2015). Column 4: logarithm of star +formation rates, based on far-ultraviolet and infrared luminosities (Hao et al. +2011; Kennicutt & Evans 2012). Column 5: metallicities when available in +the literature (taken from Shirazi & Brinchmann 2012 for Mrk 1434 and +Zhao et al. 2013 for SDSS J1221). +Name +D +log 𝑀★ +log SFR +12+log (𝑂/𝐻) +(Mpc) +(𝑀⊙) +(𝑀⊙ yr−1) +(1) +(2) +(3) +(4) +(5) +Mrk 1434 +30.7 +6.6 +−0.9 +7.8 +SDSS J1213 +32.7 +7.3 +−2.2 +. . . +SDSS J1221 +16.1 +8.0 +−1.5 +8.3 +density at frequency 𝜈, and the radio spectral index 𝛼 = 0 for a flat +spectrum; Ho 2008). Such unresolved, flat spectrum radio emission +is usually interpreted as a partially self-absorbed synchrotron jet +(Blandford & Königl 1979). Meanwhile, rapidly accreting XRBs +do not launch jets that would be detectable beyond distances of a +few Mpc (Fender et al. 2004). Thus, the presence of unresolved +radio emission has the potential to exclude hard X-ray sources as +rapidly accreting XRBs. +In this paper, we present high spatial-resolution X-ray (Chan- +dra), optical/near-infrared (Hubble Space Telescope; HST), and ra- +dio observations (Karl G. Jansky Very Large Array; VLA) of three +nearby dwarf galaxies that each host at least one hard X-ray point +source. These three targets were initially identified as AGN candi- +dates by Lemons et al. (2015), but with the caveat that the positions +of their X-ray sources were poorly determined. From the multiwave- +length data presented here, we better locate the positions of the X-ray +sources within these three galaxies, and we attempt to constrain the +nature of each source (i.e., XRB or mBH). In Section 2 we detail +our sample selection and data reduction. We outline our results in +Section 3, followed by a discussion in Section 4. Our conclusions +are presented in Section 5. Unless stated otherwise, uncertainties +are reported at the 68% confidence level. +2 +OBSERVATIONS AND DATA REDUCTION +2.1 +Sample +Our three targets were selected from the survey by Lemons et al. +(2015), who cross matched ∼44,000 nearby dwarf galaxies (𝑧 < +0.055) from the NASA-Sloan Atlas2 to the Chandra Source Catalog +(CSC Release 1.1; Evans et al. 2010). They identified 19 galaxies +with hard X-ray point sources (2–7 keV), of which 10 contained an +X-ray source positionally consistent with the galaxy optical centre +(given positional uncertainties, we note that not every galaxy has +a well defined nucleus). They presented these 10 galaxies as AGN +candidates.3 +2 http://www.nsatlas.org/ +3 Since publication of Lemons et al. (2015), there is new theoretical evi- +dence that mBHs do not need to reside in the nucleus (e.g., Bellovary et al. +2019). +MNRAS 000, 1–11 (2023) + +X-ray Sources in Dwarf Galaxies +3 +Chandra provides exquisite spatial resolution (≈0.′′4) for tar- +gets located at the telescope’s aimpoint, but the resolution degrades +for sources located farther away. Of the 10 AGN candidates in +Lemons et al. (2015), they found that four galaxies contain X- +ray sources that are far enough from the aimpoint to have large +positional uncertainties (>5′′, which is comparable to the pro- +jected size of the entire dwarf galaxy). Of these four galaxies, +three contained X-ray sources with hard X-ray luminosities >3𝜎 +(>1.2 dex) larger than expected from the galaxy-wide contribu- +tion from X-ray binaries, given the stellar mass and star forma- +tion rate of each galaxy (see Sections 4.3 and 4.4 of Lemons et al. +2015). These three galaxies include: Mrk 1434 (𝑧 = 0.00747), SDSS +J121326.01+543631.6 (𝑧 = 0.00797; hereafter SDSS J1213), and +SDSS J122111.29+173819.1 (𝑧 = 0.00699; hereafter SDSS J1221; +see Table 1). Of particular interest, Mrk 1434 is a metal-poor galaxy +(12+log (𝑂/𝐻) = 7.8; Shirazi & Brinchmann 2012) and its optical +spectrum from the Sloan Digital Sky Survey (SDSS; York et al. +2000) shows nebular He ii line emission (Shirazi & Brinchmann +2012). +To better constrain the locations of the X-ray sources relative +to their host galaxies, we obtained new Chandra X-ray and HST +optical/near-infrared observations for these three galaxies. We also +obtained new VLA radio observations for one target, SDSS J1213, +while archival VLA data were already available for the other two +sources. We adopt distances for each galaxy based on their redshifts, +using 𝐻0 = 73 km s−1 Mpc−1, except for SDSS J1221, which is a +member of the Virgo cluster (VCC 459). For this galaxy, we use a +distance of 16.1 Mpc based on the Tully-Fisher relation (Kashibadze +et al. 2020). For all three galaxies, we adopt star formation rates +from Lemons et al. (2015), which are based on (dust-corrected) far- +ultraviolet and infrared luminosities and the relationships from Hao +et al. (2011) and Kennicutt & Evans (2012). For SDSS J1221, we +scale the star formation rate from Lemons et al. (2015) to 16.1 Mpc. +For stellar mass estimates, following Reines & Volonteri (2015), +we use the colour-dependent mass-to-light ratios from Zibetti et al. +(2009). +2.2 +Chandra +We obtained new Chandra observations (Cycle 17; PI Plotkin) with +each galaxy centred at the aimpoint of the S3 chip of the Advanced +CCD Imaging Spectrometer (ACIS; Garmire et al. 2003). Data +were telemetered in VFAINT mode. Chandra data reduction was +carried out using the Chandra Interactive Analysis of Observations +(ciao) software version 4.13 (Fruscione et al. 2006) and caldb +v4.9.5. The Chandra data were reprocessed using chandra_repro +to create new level 2 event files and bad pixel files, and to apply the +latest calibration files. We then searched for background flares using +the deflare script, and we did not find any periods with elevated +background levels. +Next we aligned the event file astrometry to the SDSS reference +frame. We first excluded areas on each X-ray image occupied by +the dwarf galaxy, so that our astrometric corrections would not be +influenced by sources within each target galaxy. We then filtered +each Chandra image to 0.5-7.0 keV and ran wavdetect to identify +X-ray point sources, adopting wavelet scales of 1,2,4,8, and 16, +setting sigthresh to 10−6 (i.e., approximately one false positive +per chip), and using a point spread function map (at 2.3 keV) with +an enclosed count fraction (ecf) of 0.9. The relatively large ecf +was chosen to help filter out weak X-ray sources, which would not +have sufficient positional accuracy for astrometric alignment. We +then cross-matched X-ray sources identified by wavdetect to the +SDSS catalog using wcs_match. We found only two common X- +ray/optical sources for Mrk 1434, zero common sources for SDSS +J1213, and one common source for SDSS J1221. Thus, we applied a +translational astrometric correction for Mrk 1434 (Δ𝑥 = 0.97, Δ𝑦 = +1.32 pixels) and for SDSS J1221 (Δ𝑥 = 0.01, Δ𝑦 = 0.96 pixels) +using wcs_update. No astrometric correction was applied to SDSS +J1213. +We next re-ran wavdetect on the aligned event files (filtered +from 0.5-7 keV, now including each target dwarf galaxy) to de- +termine positions in the aligned reference frame of X-ray sources +hosted by each dwarf galaxy. We used the same wavdetect param- +eters as above, except we used ecf=0.3 when generating the point +spread function map to allow the detection of fainter point sources. +wavdetect identified two X-ray sources in Mrk 1434, one source in +SDSS J1213, and one source in SDSS J1221. The positions of each +X-ray source are listed in Table 2. We estimated 95% uncertainties +of each X-ray position based on the distance from the telescope aim- +point and the number of counts detected by wavdetect, following +Equation 5 in Hong et al. (2005). Note, this 95% positional uncer- +tainty represents the statistical error on each source. For SDSS J1213 +in particular, where we could not perform an astrometric alignment +of the Chandra image, there is an additional systematic uncertainty +that could be as large as 2′′ (although 0.′′8 is more typical).4 +We then measured the number of counts from each X-ray +source using srcflux. We adopted circular apertures centred at +each wavdetect position with radii of 5 pixels, except for Mrk +1434, which contains two X-ray sources, where we adopted radii of +2.5 pixels to avoid the regions from each X-ray source from over- +lapping. The number of background counts per pixel was estimated +from nearby source-free regions of each image. These measure- +ments were performed in both broad (0.5-7.0 keV) and hard (2.0-7.0 +keV) images, and we detected 19–73 counts from each source in the +broad band and 8–23 counts in the hard band. All X-ray detections +(in all bands) are significant at the >99% level according to the +confidence tables in Kraft et al. (1991). +Finally, spectra were extracted for each X-ray source us- +ing specextract and fit using an absorbed powerlaw model +(tbabs*powerlaw) in the Interactive Spectral Interpretation Sys- +tem v1.6.2 (ISIS; Houck & Denicola 2000), adopting Cash statis- +tics (Cash 1979) given the relatively low number of counts per +source. We initially left the column density as a free parameter. +However, for three X-ray sources 𝑁𝐻 converged to zero, in which +case we froze the value to the Galactic column density and refit the +spectrum. Model fluxes were calculated using the cflux convolu- +tion model. Spectral parameters and model fluxes are reported in +Table 3. +2.3 +Hubble Space Telescope +We observed each galaxy with the Wide Field Camera 3 (WFC3) +aboard HST for one orbit per galaxy (PI Plotkin; program 14356). +For Mrk 1434 and SDSS J1221 we observed in both the F110W +and F606W filters (with the IR and UVIS channels, respectively), +and for SDSS J1213, which is a fainter galaxy, we took observations +only in the F110W filter. Observations in each filter were taken over +four dither positions, and we used the IRSUB512 subarray for Mrk +1434 and SDSS J1221. Total exposure times in each filter are listed +in Table 4. Data were downloaded from the Mikulski Archive for +Space Telescopes (MAST), and individual exposures were aligned +4 https://cxc.harvard.edu/cal/ASPECT/celmon/ +MNRAS 000, 1–11 (2023) + +4 +E. Thygesen et al. +Table 2. Details of Chandra observations. Column 1: name of X-ray source. Column 2: Chandra obsID. Column 3: date of observation. Column 4: exposure +time. Columns 5 & 6: right ascension and declination of each X-ray source. Column 7: radius of the 95% positional uncertainty of each Chandra source, based +on Equation 5 of Hong et al. (2005). Column 8: aperture corrected net count rate (in counts per ks) in the broad X-ray band (0.5-7.0 keV). Aperture corrections +of 0.90, 0.95, and 0.96 were used for Mrk 1434, SDSS J1213, and SDSS J1221, respectively. Column 9: aperture corrected net count rate (in counts per ks) in +the hard band (2.0-7.0 keV). Aperture corrections of 0.87, 0.93, and 0.93 were used for Mrk 1434, SDSS J1213, and SDSS J1221, respectively. +Source +obsID +Date +Exp Time +Right Ascension +Declination +𝑝err +Net Rate (0.5-7.0 keV) +Net Rate (2.0-7.0 keV) +(ks) +(J2000) +(J2000) +(′′) +(ks−1) +(ks−1) +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +Mrk 1434 X-N +18059 +2016 Jan 26 +5.0 +10:34:10.19 ++58:03:49.0 +0.35 +8.00 ± 2.22 +2.98 ± 1.40 +Mrk 1434 X-S +18059 +2016 Jan 26 +5.0 +10:34:10.11 ++58:03:46.3 +0.36 +6.83 ± 2.04 +2.06+1.36 +−0.94 +SDSS J1213 +18060 +2016 Aug 04 +7.0 +12:13:26.12 ++54:36:34.1 +0.38 +2.78 ± 1.10 +1.15+0.89 +−0.60 +SDSS J1221 +18061 +2016 Feb 13 +7.0 +12:21:11.00 ++17:38:18.0 +0.33 +10.82 ± 2.11 +3.46 ± 1.24 +Table 3. Chandra spectral parameters, fluxes, and luminosities. Column 1: name of X-ray source. Column 2: column density. Column 3: best-fit photon index. +Column 4: best-fit Cash statistic/degrees of freedom. Columns 5 & 6: logarithms of the unabsorbed model X-ray flux and luminosity from 0.5-10 keV, estimated +using the cflux convolution model. Columns 7 & 8: logarithms of the unabsorbed model X-ray flux and luminosity from 2-10 keV, estimated using the cflux +convolution model. +Broad (0.5-10.0 keV) +Hard (2.0-10.0 keV) +Source +𝑁𝐻 +Γ +C-stat/d.o.f. +log Flux +log Luminosity +log Flux +log Luminosity +(1020 cm−2) +(erg s−1 cm−2) +(erg s−1) +(erg s−1 cm−2) +(erg s−1) +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +Mrk 1434 X-N +<56.9𝑎 +1.3 ± 0.4 +16.0/13 +−12.8 ± 0.1 +40.2 ± 0.1 +−13.0 ± 0.2 +40.1 ± 0.2 +Mrk 1434 X-S +0.6𝑏 +1.7 ± 0.4 +10.3/13 +−13.1 ± 0.1 +40.0 ± 0.1 +−13.3 ± 0.2 +39.8 ± 0.2 +SDSS J1213 +1.4𝑏 +1.3 ± 0.5 +5.8/12 +−13.3 ± 0.2 +39.8 ± 0.2 +−13.5 ± 0.2 +39.6 ± 0.2 +SDSS J1221 +2.7𝑏 +1.6 ± 0.3 +27.1/32 +−12.8 ± 0.1 +39.6 ± 0.1 +−13.0 ± 0.1 +39.5 ± 0.1 +𝑎Best-fit column density 𝑁𝐻 = 8.0 × 1020 cm−2, reported as an upper limit (95% confidence level) because the uncertainty on +the best-fit value extends down to the Galactic value of 0.6 × 1020 cm−2 . +𝑏Column density frozen to the Galactic value during fitting, taken from Dickey & Lockman (1990). +and combined using AstroDrizzle in the DrizzlePac software +(Hack et al. 2013).5 The F110W drizzled images were created with +plate scales 0.′′06 pix−1 for Mrk 1434 and SDSS J1221, and 0.′′09 +pix−1 for SDSS J1213. All F606W images have plate scales 0.′′03 +pix−1. +We aligned the HST astrometry to the Gaia Data Release +2 (Gaia Collaboration et al. 2018) reference frame using the +tweakreg task within AstroDrizzle (after excluding sources +falling within each galaxy).6 For Mrk 1434, the corrections re- +sulted in astrometric shifts by (Δ𝑥 = 1.8, Δ𝑦 = 0.0) pixels (from +two common sources) and (Δ𝑥 = 1.9, Δ𝑦 = 2.2) pixels (from nine +common sources) in the F110W and F606W filters, respectively. For +SDSS J1213, we shifted the F110W filter by (Δ𝑥 = 0.6, Δ𝑦 = 2.8) +pixels (five common sources). Finally, for SDSS J1221 we could not +identify enough common sources between the HST image and the +Gaia catalog in the F110W filter (which has a smaller field of view). +So, we only aligned the F606W filter to the Gaia frame, shifting by +(Δ𝑥 = 0.2, Δ𝑦 = 5.3) pixels (four common sources), and we then +5 https://hst-docs.stsci.edu/drizzpac +6 We note that we aligned HST images to the Gaia frame and the Chandra +X-ray images to the SDSS frame, because we generally found a larger num- +ber of common HST/Gaia sources vs. common HST/SDSS sources (and +vice-versa for Chandra). Compared to the statistical uncertainty on each +Chandra position (0.′′3–0.′′4), we do not expect a meaningful offset between +the absolute astrometry of SDSS vs. Gaia, such that systematic uncertainties +in our astrometric alignments are dominated by the small number of sources +used to apply the corrections. +Table 4. Summary of HST observations. Column 1: galaxy name. Column +2: date of observations. Column 3: filters used for observations. Column 4: +exposure times in the F110W/F606W filters, respectively, when both filters +were used. All observations were taken through HST Proposal ID 14356. +Source +Date +Filter +Exp. Time +(min) +(1) +(2) +(3) +(4) +Mrk 1434 +2016 Apr 16 +F110W/F606W +8.6/30.9 +SDSS J1213 +2016 Apr 16 +F110W +43.7 +SDSS J1221 +2016 Apr 9 +F110W/F606W +8.6/26.9 +aligned the F110W filter to the F606W filter (via three common +sources between the two HST filters). +2.4 +Very Large Array +Mrk 1434 and SDSS J1221 both had archival datasets (PI Satyapal, +14A-358) from the VLA, while new data were obtained for SDSS +J1213 for this study (PI Plotkin, SH0563). All three galaxies were +observed in the most extended A configuration. Both Mrk 1434 and +SDSS J1221 observations were from 4.5-6.5 GHz (C band) and +8-10 GHz (X band), while SDSS J1213 was observed only from +8-12 GHz. +The Common Astronomy Software Applications (CASA; CASA +Team et al. 2022) software package version 5.1 was used to carry +out standard data reduction. We used 3C 286 to perform delay +MNRAS 000, 1–11 (2023) + +X-ray Sources in Dwarf Galaxies +5 +and bandpass calibrations, and to set the flux density scale. Nearby +phase calibrators (see Table 5) were observed to solve for the time- +dependent complex gain solutions. Imaging was performed using +the task tclean, using two Taylor terms (nterms=2) to account for +the wide fractional bandwidth and natural weighting to maximise +sensitivity. We achieved root-mean-square (rms) sensitivities rang- +ing from 3.7 to 8.7 𝜇Jy bm−1 in each observing band (see Table 5). +The only X-ray source for which we found coincident radio +emission is Mrk 1434 X-N, where we found radio detections at both +5.5 and 9.0 GHz within the X-ray error circle. We used imfit to fit +two-dimensional Gaussians in the image plane (at each frequency) +to calculate the size of the radio structure, and to measure peak +and integrated flux densities. As discussed further in Section 3.1, +the 5.5 GHz emission is slightly extended (with integrated flux +density 𝑓int = 0.191 ± 0.036 mJy) while the 9.0 GHz is point-like +( 𝑓peak = 0.036 ± 0.009 mJy). The centroids of the radio emission +at each frequency are offset by 0.′′20 ± 0.′′07. For the other two +galaxies, we place 3𝜎rms limits on their radio flux densities. We +note that SDSS J1221 displays radio emission aligned with a likely +H ii region toward the eastern outskirts of the galaxy that is not +associated with X-ray emission, so we do not discuss that radio +emission in this paper. +3 +RESULTS +In the following subsections we present the multiwavelength results +for each galaxy, deferring discussions regarding the possible nature +of each X-ray source to Section 4. Composite HST images are shown +for each galaxy in Figure 1, including the locations of X-ray sources. +3.1 +Mrk 1434 +Mrk 1434 hosts two X-ray sources separated by 2.′′8 (see Figure 1a), +both of which are classified as ULXs: the northern source (Mrk +1434 X-N), which is located toward the galactic nucleus, has an +unabsorbed hard X-ray luminosity 𝐿2−10 keV = (1.2 ± 0.6) × 1040 +erg s−1, and the southern source (Mrk 1434 X-S) has 𝐿2−10 keV = +(5.8 ± 0.2) × 1039 erg s−1. The X-ray spectra of each source are +fit by powerlaw models with photon indices of Γ = 1.3 ± 0.4 for +Mrk 1434 X-N and Γ = 1.7 ± 0.4 for Mrk 1434 X-S. Neither source +shows evidence for significant intrinsic absorption. +It is unlikely that either hard X-ray source is a superposed fore- +ground/background object. Given the density and flux distribution +of hard X-ray sources in the cosmic X-ray background (see, e.g., +Equation 2 of Moretti et al. 2003), we expect to only find 0.001 and +0.003 hard X-ray sources with 2-10 keV fluxes similar (or brighter) +than Mrk 1434 X-N and Mrk 1434 X-S, respectively, within the +projected size of the galaxy (which we conservatively approximate +as a circle with a 20′′ radius). +Radio emission is detected only from the northern source, Mrk +1434 X-N. At 5.5 GHz, the emission is extended with major and +minor axis full width half maxima of 1.′′1×0.′′6 (160 pc × 90 pc), +respectively, covering ≈3.5 synthesised beams. The centroid of the +5.5 GHz emission is 0.′′16 from the X-ray position (for reference, +the 95% Chandra error circle is 0.′′35), and the integrated luminos- +ity is 𝐿5.5,int = (1.2 ± 0.2) × 1036 erg s−1. At 9.0 GHz we detect +a point source located 0.′′32 from the X-ray position, with a peak +luminosity 𝐿9.0,peak = (3.7 ± 0.8) × 1035 erg s−1. We do not detect +any extended radio structures at 9.0 GHz, thereby indicating that the +emission seen at 5.5 GHz has a steep radio spectrum (our 5.5 and +9.0 GHz radio maps have similar sensitivities; see Table 5). Note, +extended emission is not simply resolved out at the higher radio fre- +quency, since the smallest baselines of the VLA in A configuration +are sensitive to structures up to ≈5′′ at 9.0 GHz, which is larger +than the ≈1′′ angular size of the 5.5 GHz emission. +At 9.0 GHz, the chance of a random alignment of a background +radio point source falling within the Chandra error circle is very +small. Integrating the differential source counts tabulated by de +Zotti et al. (2010) at 8.4 GHz, and assuming a flat radio spectrum +(as expected if the 9.0 GHz emission is from a compact jet; see +Section 4.1.2), we expect only ≈3 × 10−5 sources with 𝑓peak > +0.036 mJy within the X-ray error circle. The chance of a statistical +fluctuation as large as 0.036 mJy (i.e., 4𝜎rms) within the X-ray error +circle (which contains ≈240 pixels in the radio map) is also very +small (𝑝 = 3 × 10−5). Thus, we believe the 9.0 GHz emission is +indeed physically associated with the galaxy. However, we note that +the radio source lies toward the edge of the X-ray error circle. Thus, +even though the radio source formally falls within the Chandra +positional uncertainty, its association specifically with Mrk 1434 +X-N is less clear, particularly after considering that the Chandra X- +ray astrometry of Mrk 1434 was aligned to the optical frame using +only two common X-ray/SDSS sources. +Finally, we note that towards the southwest of the 0.′′35 Chan- +dra X-ray error circle of Mrk 1434 X-N, there is an optical/near- +infrared source that appears red in the HST composite image (see the +zoom-in of Figure 1a). If that source is a background quasar it may +also be responsible for the X-ray and/or radio emission. However, +the random alignment of such a background quasar is very unlikely, +as described below. The AB magnitude of the HST source in the +F606W filter is 18.8, which we convert to SDSS i≈18.7 assuming +a typical quasar spectrum (Vanden Berk et al. 2001). We then con- +sider SDSS Type 1 quasar counts from 0.3 < 𝑧 < 3.5 (Richards +et al. 2006; Ross et al. 2013), and we find only a negligible number +of background quasars (≈ 6 × 10−7) are likely to fall within the +Chandra X-ray circle by random chance (note, the random align- +ment of a radio-loud or a Type 2 quasar would be even rarer). That +source is likely intrinsic to the galaxy. +3.2 +SDSS J1213 and SDSS J1221 +SDSS J1213 and SDSS J1221 each contain a single hard X-ray +point source near the outskirts of each galaxy (Figure 1b-c). The +hard (2-10 keV) X-ray luminosities of the sources are 𝐿2−10 keV = +(4.3 ± 2.4)×1039 and (2.9 ± 0.8)×1039 erg s−1, respectively (Table +2), such that both sources are classified as ULXs. The chance of a +superposed foreground/background object is negligible (we expect +only 0.005 hard X-ray background sources for SDSS J1213 and +0.001 sources for SDSS J1221; Moretti et al. 2003). Neither galaxy +contains radio emission within the Chandra X-ray circles to 3𝜎rms +upper limits of < 1.4 × 1035 erg s−1 at 10.0 GHz for SDSS J1213, +and to limits of < 3.0 × 1034 erg s−1 and < 4.9 × 1034 erg s−1 at +5.5 and 9.0 GHz, respectively, for SDSS J1221. +4 +DISCUSSION +In the following subsections we discuss possible interpretations for +the X-ray sources in our sample of three dwarf galaxies. We focus +primarily on Mrk 1434 since it exhibits the most complex phe- +nomenology (i.e., two X-ray sources, one of which is coincident +with radio emission). We provide arguments for/against XRB in- +terpretations in Section 4.1.1 and for/against AGN interpretations +in Section 4.1.2. In Section 4.1.3 we discuss whether the observed +MNRAS 000, 1–11 (2023) + +6 +E. Thygesen et al. +Table 5. Summary of VLA observations. Column 1: galaxy name. Column 2: VLA Program ID. Column 3: date of observation. Column 4: name of phase +calibrator. Column 5: central frequency of each observation. Column 6: bandwidth of each observation. Column 7: the time spent integrating on each galaxy. +Column 8: the size of the (elliptical) synthesised beam along the major and minor axes. Column 9: rms noise of each image. +Source +Program +Date +Phase Calibrator +𝜈 +Δ𝜈 +𝜏 +𝜃bm +𝜎rms +(J2000) +(GHz) +(GHz) +(min) +(′′ × ′′) +(𝜇Jy bm−1) +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +Mrk 1434𝑎 +14A-358 +2014 Feb 24 +1035+564 +5.5 +2.0 +8.5 +0.45×0.38 +8.7 +Mrk 1434𝑏 +14A-358 +2014 Feb 24 +1035+564 +9.0 +2.0 +8.5 +0.27×0.24 +8.6 +SDSS J1213 +SH0563 +2016 Sep 30 +1219+482 +10.0 +4.0 +39.5 +0.28×0.23 +3.7 +SDSS J1221 +14A-358 +2014 Feb 26 +1158+248 +5.5 +2.0 +25.8 +0.42×0.38 +5.9 +SDSS J1221 +14A-358 +2014 Feb 26 +1158+248 +9.0 +2.0 +26.0 +0.25×0.23 +5.8 +𝑎Extended radio emission detected near Mrk 1434 X-N at 5.5 GHz, with 𝑓int = 0.191 ± 0.036 mJy and 𝑓peak = +0.054±0.009 mJy bm−1. The centroid of emission is located at RA=10h34m10.1867s ± 0.0042𝑠, Dec=58◦03′49.′′1481 +± 0.′′0763. +𝑏Point-like radio emission detected near Mrk 1434 X-N at 9.0 GHz, with 𝑓peak = 0.036±0.009 mJy bm−1. The emission +is located at RA=10h34m10.2045s ± 0.0039s, Dec=58◦03′49.′′2883 ± 0.′′0460. +3" +5" +5" +0.5" +(a) Mrk 1434 +3 arcsec +0.5 arcsec +5 arcsec +5 arcsec +(b) SDSS J1213 +(c) SDSS J1221 +Figure 1. (a) Composite HST image of Mrk 1434 in the F606W (blue/green) and the F110W (red) filters. The locations of the two X-ray point sources are +shown as red cross hairs, with the dashed red circles illustrating the sizes of the 95% positional errors from Chandra. The Zoom-in of the centre of the galaxy +shows the location of Mrk 1434 X-N relative to the radio emission, where yellow contours show the extended 5.5 GHz radio emission (1.′′1 × 0.′′6; contours +drawn at 3, 4, 5×𝜎rms) and the magenta contours show the unresolved emission at 9.0 GHz (contours drawn at 3, 4×𝜎rms). The sizes of the VLA synthesised +beams are 0.′′45 × 0.′′38 (5.5 GHz) and 0.′′27 × 0.′′24 (9.0 GHz), respectively. Note, the SDSS spectroscopic fibre, from which the nebular He ii emission is +detected, has a diameter of 3′′ and is placed at the centre of the galaxy. (b) HST image of SDSS J1213 in the F110W filter, with the location of the X-ray source +marked by the red cross hair and dashed circle. (c) HST composite image of SDSS J1221 in the F606W (blue/green) and the F110W (red) filters, with the +location of the X-ray source marked by the red cross hair and dashed circle. In all images, north is up and east is to the left. +MNRAS 000, 1–11 (2023) + +X-ray Sources in Dwarf Galaxies +7 +X-ray flux is sufficient to explain He ii line emission observed in +the SDSS spectrum of Mrk 1434. A discussion on the nature of the +X-ray sources in the other two galaxies is presented in Section 4.2. +4.1 +Mrk 1434 +4.1.1 +XRB Interpretations +As shown in Section 3.1, both X-ray sources in Mrk 1434 are +physically associated with the galaxy and luminous enough to be +classified as ULXs. The observed X-ray luminosity, however, is +higher than expected from the luminous tail of the galaxy’s XRB +population. The luminosities of both X-ray sources are above the +cutoff of the low-mass XRB luminosity function (e.g., Gilfanov +2004), so in the following we only consider high-mass XRBs using +the metallicity-dependent luminosity function from Lehmer et al. +(2021). For Mrk 1434, with 12 + log (𝑂/𝐻) = 7.8 and SFR=0.12 +𝑀⊙ yr−1, Lehmer et al. (2021) predict a total 0.5-8.0 keV X-ray +luminosity (i.e., from all X-ray point sources) of 𝐿0.5−8.0 keV = +(1.7 ± 0.15) ×1039 erg s−1 (where the error bar represents the 68% +confidence interval provided by Lehmer et al. 2021). They also +predict only 0.03+0.04 +−0.02 ULXs with 𝐿0.5−8.0 > 1040 erg s−1. For +reference, the unabsorbed 0.5–8.0 keV model luminosities of Mrk +1434 X-N and Mrk 1434 X-S are (1.3 ± 0.4)×1040 and (0.8 ± 0.2)× +1040 erg s−1, respectively. Thus, the combined X-ray luminosity of +both ULXs is ≈10 times higher than expected relative to the Lehmer +et al. (2021) luminosity function, which is significant even after +considering uncertainties and intrinsic scatter. +Even though the above suggests that it is statistically unlikely +for both sources to be XRBs, small number statistics could influence +the above arguments, and it is worth exploring XRB interpretations. +In particular, the extended 5.5 GHz radio emission from Mrk 1434 +X-N could represent a ‘ULX bubble’, as similar types of extended +radio structures have been observed from other ULXs, making the +radio emission a signature of a ULX outflow shocking the nearby +interstellar environment (e.g., Pakull et al. 2010; Soria et al. 2010, +2021; Cseh et al. 2012; Urquhart et al. 2019). If the 5.5 GHz radio +emission is indeed a ULX bubble, then with 𝐿5.5,int = (1.2 ± 0.2) × +1036 erg s−1 it would represent the most luminous ULX bubble yet +observed by a factor of ≈6 (Pakull et al. 2010; Soria et al. 2010, +2021). Meanwhile, the projected size of ≈160 pc × 90 pc (1.′′1×0.′′6) +in diameter is fairly typical compared to other ULX bubbles, where +diameters range from ≈25–350 pc (Soria et al. 2021; also see Table 1 +of Berghea et al. 2020 and references therein). Taking the peak +flux density of the 5.5 GHz structure, and extrapolating to 1 GHz +assuming a spectral index 𝛼 = 0.7, the intensity of the radio bubble +in Mrk 1434 X-N would be 𝐼1 GHz ≈ 6 × 10−16 erg s−1 cm−2 Hz−1 +sr−1, which is relatively large but reasonable compared to other +ULX radio bubbles with similar physical sizes (see Figure 5 of +Berghea et al. 2020). +Although a ULX bubble is one interpretation of the 5.5 GHz +emission, we stress that it is not a unique (or necessary) explanation. +Adopting SFR = 0.12 𝑀⊙ yr−1 for Mrk 1434 and the relation +between star formation rate and the 1.4 GHz specific luminosity +from Kennicutt & Evans (2012), we expect 𝐿5.5,SF ≈ 3.9×1036 erg +s−1 (we convert from 1.4 GHz to 5.5 GHz assuming a spectral index +𝛼 = 0.7). Considering that the intrinsic scatter on the conversion +between SFR and radio luminosity is on the order of ±0.3 dex +(Murphy et al. 2011), the observed extended structure at 5.5 GHz +could be produced entirely by star formation processes. Since the +extended radio structure at 5.5 GHz is not detected at 9.0 GHz, +the dominant radio emission mechanism in such a scenario would +most likely be synchrotron radiation with a steep spectrum from +supernova remnants. Note, our data exclude free-free radio emission +from an H ii region, which would produce a flat spectrum that would +be detectable at 9.0 GHz. +4.1.2 +AGN Interpretations +AGN can also produce extended radio emission, which is another +viable explanation for the 5.5 GHz radio structure. However, in light +of the discussion in the previous subsection that a super-Eddington +XRB is also capable of producing the observed extended emission +at 5.5 GHz, the resolved radio complex does not provide useful +diagnostics for attempting to discriminate between XRB vs. AGN. +Since the X-ray spectra of Mrk 1434 X-N and Mrk 1434 X-S (Γ = +1.3 ± 0.4 and Γ = 1.7 ± 0.4, respectively) are consistent with low- +luminosity AGNs (Younes et al. 2011; Yang et al. 2015), we focus +the following discussion on AGN scenarios with Eddington ratios +𝐿bol/𝐿Edd ≲ 0.01. For such weakly accreting AGN, we expect to +observe unresolved radio emission from a partially self-absorbed +compact jet (Ho 2008). By combining X-ray and radio luminosities, +we can then make crude estimates on black hole masses by appealing +to the fundamental plane of black hole activity (Merloni et al. 2003; +Falcke et al. 2004). For Mrk 1434 X-N, we then interpret the the +unresolved 9.0 GHz radio emission as arising from a compact jet, +and we utilise the fundamental plane regression by Gültekin et al. +(2019), +log +� +𝑀BH/108𝑀⊙ +� += (0.55 ± 0.22) + +(1.09 ± 0.10) log +� +𝐿5 GHz/1038 erg s−1� +− +(0.59 ± 0.16) log +� +𝐿2−10 keV/1040 erg s−1� +, +(1) +which has an intrinsic scatter ≈1 dex. We estimate that Mrk 1434 +X-N would have 𝑀BH ≈ 4×105 𝑀⊙ if powered by an mBH (see Ta- +ble 6). Note, we assume a flat radio spectrum to convert the observed +radio luminosity at 9.0 GHz to 5.0 GHz for use in the fundamen- +tal plane (we cannot use our 5.5 GHz radio map to estimate the 5 +GHz luminosity because we do not have enough signal-to-noise to +attempt to decompose a point source embedded within the extended +radio emission observed at 5.5 GHz). Similarly, the lack of radio +emission from Mrk 1434 X-S implies 𝑀BH ≲ 4 × 105𝑀⊙ (where +we adopt a 3𝜎rms upper limit, based on the observed 𝜎rms near +Mrk 1434 X-S in our 5.5 GHz image). These mass estimates imply +Eddington ratios (𝐿2−10 keV/𝐿Edd) of ≈ 2×10−4 and ≳ 1×10−4 for +Mrk 1434 X-N and Mrk 1434 X-S, respectively, which, assuming +bolometric corrections of ≈10, are consistent with Eddington ratios +for which the fundamental plane can be applied (see, e.g., Plotkin +et al. 2012). +4.1.3 +On the Origin of Nebular He ii Emission +In the following we determine whether the X-ray emission from Mrk +1434 is a strong enough source of photoionisation to explain the +strength of the He ii emission in the SDSS spectrum of Mrk 1434. +The observed He ii line flux is 𝐹4686,obs = (7.5 ± 0.1) × 10−16 +erg s−1 cm−2, which translates to a photon flux of 𝑁4686,obs = +(1.8 ± 0.1) × 10−4 photons s−1 cm−2. Every photon emitted in the +He ii line requires 5.2 ionizing photons incident on singly ionised +helium (Pakull & Angebault 1986). Given the ionisation potential +of singly ionised helium (𝜒ion = 54.4 eV), and considering that +MNRAS 000, 1–11 (2023) + +8 +E. Thygesen et al. +Table 6. mBH mass estimates and limits. Column 1: galaxy name. Column +2: logarithm of the hard X-ray luminosity. Column 3: logarithm of the radio +luminosity at 5 GHz, assuming a flat radio spectrum. For Mrk 1434 X-N, +this luminosity is based on the unresolved emission detected at 9 GHz. For +all other X-ray sources, limits are placed as 3𝜎rms. Column 4: logarithm of +the black hole mass (or limit) if X-ray sources are weakly accreting mBHs, +based on the fundamental plane of black hole activity (Gültekin et al. 2019). +Uncertainties on log 𝑀BH are ≈1 dex. +Source +log 𝐿2−10 keV +log 𝐿5 GHz +log 𝑀BH +(erg s−1) +(erg s−1) +(𝑀⊙) +(1) +(2) +(3) +(4) +Mrk 1434 X-N +40.1 ± 0.4 +35.3 ± 0.1 +5.6 +Mrk 1434 X-S +39.8 ± 0.3 +<35.2 +<5.6 +SDSS J1213 +39.6 ± 0.4 +<34.9 +<5.3 +SDSS J1221 +39.5 ± 0.2 +<34.4 +<5.0 +the photoionisation cross section has a steep 𝐸−3 +ph dependence on +photon energy, 𝐸ph, then producing the observed SDSS He ii line +flux requires a photon flux in the extreme ultraviolet (54–300 eV) +of 𝑁54−300 eV = 5.2𝑁4686,obs = (9.1 ± 0.1) × 10−4 photons s−1 +cm−2. Note, this photon flux is an underestimate because we have +not corrected the observed SDSS line flux for extinction. +The 3′′ SDSS spectroscopic fibre is centred near Mrk 1434 +X-N, such that if the He ii emission arises from photoionisation by +the X-ray source, we expect the emission to be dominated by Mrk +1434 X-N. We do not have direct measurements on the extreme +ultraviolet flux from 54-300 eV, so we extrapolate the Chandra +X-ray spectrum into the extreme ultraviolet. Our best-fit powerlaw +model predicts a photon flux of 0.3+2.5 +−0.2 × 10−4 photons s−1 cm−2 +(note the large range in uncertainty because we are extrapolating +the model to energies lower than the Chandra X-ray band). Thus, +while high-energy radiation from Mrk 1434 X-N may contribute to +some of the He ii photoionisation, the observed X-ray source is too +faint, by a factor of ≈30, to supply all of the photoionising photons. +If we assume a thermal X-ray emission model (tbabs*diskbb), it +becomes even more difficult for the X-ray source to explain the He ii +photionisation, as the extrapolated 54-300 eV extreme ultraviolet +flux becomes ≈90 times too faint. Adding a contribution of photons +form Mrk 1434 X-S would only increase the above photon flux by +a factor of ≈2, for either spectral model. +There is currently no evidence for significant X-ray variabil- +ity from Mrk 1434 over the past 1–2 decades. Coincidentally, the +SDSS spectrum and the archival Chandra observation from Lemons +et al. (2015, Chandra obsID 3347) were both taken in May 2002 +(separated by ≈2 weeks). The archival data from 2002 show nearly +identical X-ray luminosities (log 𝐿2−10 keV = 40.1 and 39.9 erg s−1 +for Mrk 1434 X-N and Mrk 1434 X-S, respectively; see Table 2 of +Lemons et al. 2015) compared to the Chandra observations pre- +sented here, which were taken nearly 14 years later (see Table 2 of +this paper). There are also two X-ray detections of Mrk 1434 in the +third XMM-Newton serendipitous source catalog (3XMM; Rosen +et al. 2016) in 2007 and 2008. Both X-ray sources are blended to- +gether due to XMM-Newton’s poorer spatial resolution. Comparing +the XMM-Newton fluxes to the combined fluxes of both sources in +the Chandra observations, X-ray variability is smaller than a factor +of ≈2 over the four observations. However, considering the light +travel time between the X-ray source and the ionised medium, it is +feasible that Mrk 1434 X-N was more active in the past. The pro- +jected radius of the SDSS spectroscopic fibre is 730 light years, and +we cannot exclude the possibility that Mrk 1434 X-N was ≈30–90 +times more luminous several hundred years ago, which appears to +be on the only viable way for the He ii emission to be powered by +X-ray photoionisation. +If the extended radio emission is produced by an outflow shock- +ing the interstellar medium, then one must also consider the pos- +sibility of the He ii emission being produced by ionisation from a +radiative shock (e.g., Dopita & Sutherland 1995). According to the +MAPPINGS III libraries of line ratios for radiative shocks (Allen +et al. 2008), assuming a shock velocity of 300 km s−1, we expect the +luminosity of the He ii 𝜆4686 emission line 𝐿4686 ≈ 4 × 10−4𝐿rad, +where 𝐿rad is the total radiative luminosity of the shock.7 Assuming +that the kinetic power required to inflate a bubble 𝑃kin ≈ 77/27𝐿rad +(Weaver et al. 1977), then explaining the observed He ii line via +shock ionisation requires an outflow with 𝑃kin ≈ 6 × 1041 erg s−1. +We do not have a reliable method to independently estimate +𝑃kin (especially considering that other emission lines in the SDSS +spectrum are dominated by star formation). However, for an order of +magnitude estimate, we calculate the minimum synchrotron energy +of the 5.5 GHz radio emission, which is 𝑊min ≈ 2 × 1052 erg +(Longair 1994).8 A 300 km s−1 shock would take ≈ 3 × 105 yr to +inflate a bubble with a 160 pc diameter, such that the average power +stored in internal energies of the synchrotron emitting structure is +¯𝑃min ≈ 2 × 1039 erg s−1 (i.e., the average power in particles and +in the magnetic field). Thus, an outflow would need to carry ≳102 +times more power in order for a shock to be the sole ionisation +source of the observed He ii emission line. Of course, ¯𝑃min is a +minimum energy estimate, and the power in bubbles/cavities carved +out by kinetic outflows have sometimes been observed to be larger, +sometimes by factors of several hundreds (e.g., Ito et al. 2008), such +that the above does not exclude the possibility of shock ionisation. +For comparison, the ULX NGC 6946 MF16 (Roberts & Col- +bert 2003) has a luminous and compact radio bubble (Berghea et al. +2020), which suggests a relatively powerful outflow. Adopting the +NGC 6946 MF16 bubble line flux in the [Fe ii] 𝜆16440 emission +line (4.2 × 10−15 erg s−1 cm−2) and a distance of 7.8 Mpc (Long +et al. 2020), the MAPPINGS III libraries for a 300 km s−1 shock +(with Solar abundances) suggest a kinetic power of 𝑃kin ≈ 7 × 1040 +erg s−1. Thus, the kinetic power of NGC 6946 MF16 (i.e., one of the +most powerful known ULX radio bubbles) is an order of magnitude +lower than the power required for shock ionisation to be responsible +for the observed strength of the He ii emission line near Mrk 1434 +X-N. Thus, if the He ii line is powered by shock ionisation, then it +would represent one of the most powerful bubbles carved by a ULX +outflow yet observed. +Intriguingly, Mrk 1434 is one member of a population of 182 +star forming galaxies with nebular He ii emission that were identi- +fied by Shirazi & Brinchmann (2012). The ratios of He ii/H𝛽 relative +to [N ii] 𝜆6584/H𝛼 are inconsistent with AGN for these galaxies. +Typically, when an AGN is absent, Wolf-Rayet stars are considered +the primary stellar population capable of producing enough extreme +ultraviolet flux above the 54 eV He ii ionisation edge. However, Shi- +razi & Brinchmann (2012) inspected the SDSS spectra for broad +emission features indicative of Wolf-Rayet stars, and they found no +Wolf-Rayet signatures in the spectrum of Mrk 1434. Thus, without +7 Given the low metallicity of Mrk 1434, we adopt the MAPPINGS III +model grid with Small Magellanic Cloud abundances. We also assume an +interstellar medium density of 1 cm−3 and equipartition of magnetic and +thermal pressures. +8 We adopt 𝐿5.5 ≈ 1036 erg s−1, a bubble diameter of ≈160 pc, and an +ion to electron energy ratio of 𝜂 = 40. We note that 𝑊min ∝ 𝜂4/7, and the +proper value of 𝜂 is not well constrained. +MNRAS 000, 1–11 (2023) + +X-ray Sources in Dwarf Galaxies +9 +concrete evidence that Mrk 1434 X-N was indeed brighter several +hundred years ago to power the He ii emission via photoionisation, +and/or lacking a reliable estimate of the kinetic power of an outflow +for shock ionisation, the source of extreme ultraviolet photons in +Mrk 1434 remains a mystery. Another plausible explanation could +be photoionisation from extreme ultraviolet photons emitted by ex- +otic stellar populations (like rapidly rotating stars) in metal-poor +environments (see the discussion in Section 6 of Shirazi & Brinch- +mann 2012). It is very plausible that several of the above scenarios +contribute toward producing the He ii line, and Shirazi & Brinch- +mann (2012) recovered a heterogeneous population (multiple mech- +anisms may even contribute to producing the He ii emission within +a single galaxy). For example, Senchyna et al. (2020) conclude +that X-ray photoionisation cannot explain nebular He ii emission +across a sample of nearly a dozen metal-poor galaxies. Meanwhile, +there are several well-established examples of X-ray sources that +are indeed sufficient to power nebular He ii emission (e.g., Pakull & +Angebault 1986; Moon et al. 2011; Schaerer et al. 2019; Simmonds +et al. 2021). Further observational constraints, ideally via system- +atic X-ray surveys of metal-poor dwarf galaxies under high spatial +resolution, are required to understand the level to which ULXs con- +tribute extreme ultraviolet radiation in metal-poor galaxies, which +has implications for understanding sources of ionisation and heating +of the intergalactic medium in the early Universe. +4.2 +SDSS J1213 and SDSS J1221 +Our new Chandra observations confirm the conclusion of Lemons +et al. (2015) that both X-ray sources are more luminous than ex- +pected from the XRB populations in each galaxy, as described be- +low. Unlike for Mrk 1434, the luminosities of both X-ray sources +in SDSS J1213 and SDSS J1221 are low enough that we should +consider both high-mass and low-mass XRBs. Following Lemons +et al. (2015), we therefore adopt the relation from Lehmer et al. +(2010), which predicts the hard X-ray luminosity from low-mass +and high-mass XRBs as a function of stellar mass and star for- +mation rate: +� +𝐿XRB +2−10/erg s−1� += (9.05 ± 0.37) × 1028 (𝑀★/𝑀⊙) + +(1.62 ± 0.22) × 1039 � +𝑆𝐹𝑅/𝑀⊙ yr−1� +, with an intrinsic scatter of +±0.34 dex. The Lehmer et al. (2010) relation predicts 𝐿XRB +2−10 = +1.2×1037 and 5.6×1037 erg s−1 for SDSS J1213 and SDSS J1221, +respectively. The predicted luminosities are ≈3 times higher if we +instead adopt the calibrations in Lehmer et al. (2019). Thus, the ob- +served X-ray luminosities are ≈120–360 and ≈17–50 times higher +than expected, for SDSS J1213 and SDSS J1221, respectively.9 +In light of recent theoretical motivation for ‘wandering’ mBHs +(Bellovary et al. 2019, 2021, also see, e.g., Mezcua & Domínguez +Sánchez 2020; Reines et al. 2020; Greene et al. 2021; Sargent +et al. 2022 for observational searches), an X-ray source being ‘off- +nucleus’ does not on its own preclude the possibility of an accreting +mBH. It is possible that these sources are mBHs launching jets +that are either (a) beneath our radio detection limit or (b) that are +very extended and ‘resolved out’ by the VLA when it is in its most +9 The Lehmer et al. (2010) relation is calibrated to galaxies with approxi- +mately Solar metallicities. The metallicity of SDSS J1213 is unknown, and +the metallicity of SDSS J1221 is log (𝑂/𝐻) + 12 = 8.3 (Zhao et al. 2013). +If we adopt the metallicity-dependent Lehmer et al. (2021) relation for high- +mass XRBs, the X-ray luminosity of the X-ray source in SDSS J1221 is still +≈20 times higher than expected for a galaxy with its star formation rate and +metallicity. +extended A configuration. The largest angular scale to which the +VLA is sensitive to radio emission at our observing frequencies +(X-band) and configuration (A) is 5.′′3, such that our VLA obser- +vations would not detect flux from extended jets larger than ≈850 +and ≈410 pc for SDSS J1213 and SDSS J1221, respectively. On the +other hand, the radio cores of weakly accreting AGN (bolometric +luminosities 𝐿bol < 0.01𝐿Edd) have flat radio spectra and are com- +pact enough that their radio emission should not be ‘resolved out’ +at VLA resolutions (see, e.g., Orienti & Prieto 2010). Thus, if only +considering mBHs in the weak accretion regime, we can use our ra- +dio upper limits in conjunction with the fundamental plane to place +mass limits of 𝑀BH < 2 × 105 and <1 × 105 𝑀⊙ for SDSS J1213 +and SDSS J1221, respectively. Requiring 𝐿bol < 0.01𝐿Edd, and as- +suming X-ray bolometric corrections of 10, then places lower limits +on black hole masses of ≳ 3×104 (SDSS J1213) and ≳ 2×104 𝑀⊙ +(SDSS J1221). Thus, there is a relatively narrow range of mass +where our VLA observations could ‘miss’ the compact radio jet +from a weakly accreting mBH. Note, our radio limits do not place +useful constraints on the possibility of a more rapidly accreting +mBH with 𝐿bol > 0.01𝐿Edd, which would correspond to a mass +𝑀BH ≲ 104𝑀⊙ for both sources. Nevertheless, even though our +data do not exclude the possibility of mBHs, Occam’s razor proba- +bly suggests that the simplest and most likely scenario is that these +are luminous XRBs. +4.3 +An Update to Lemons et al. (2015) +After considering the above multiwavelength observations, all 10 of +the dwarf galaxy AGN candidates identified by Lemons et al. (2015) +(via hard X-ray emission) now have sufficient spatial resolution to +determine if the X-ray sources indeed reside in galactic nuclei. Our +study reduces their number of AGN candidates to 7–8 (adopting an +AGN definition that requires nuclear sources). It is very unlikely that +any of these 7–8 nuclear sources are chance alignments with fore- +ground/background X-ray emitting objects. Adopting the hard (2-10 +keV) X-ray fluxes and X-ray position error circles of the nuclear can- +didates from Table 2 of Lemons et al. (2015), and replacing the X-ray +flux and positional uncertainty of Mrk 1434 X-N with the values +presented here, the Moretti et al. (2003) cosmic X-ray background +predicts only 0.003 sources to fall within the nuclei of the eight +possible nuclear mBH candidates. Obtaining 7–8 viable AGN can- +didates is a significant result, considering that (a) the Lemons et al. +(2015) dwarf galaxy survey was archival and therefore serendipitous +in nature, and (b) the three dwarf galaxies with follow-up presented +here represent three of their most unlikely AGN candidates (given +the poor spatial resolution of their archival Chandra data). Lemons +et al. (2015) found X-ray sources in 19 galaxies total (i.e., the re- +maining 11–12 galaxies host off-nuclear X-ray sources, most likely +XRBs). Thus, if a luminous X-ray source is detected in a dwarf +galaxy, our study (very roughly) implies a 30–40% chance10 that +it could be a nuclear mBH, which supports the viability of using +X-ray surveys to identify mBHs in low-mass galaxies, as long as +the survey is performed with sufficient sensitivity and spatial res- +olution. We stress the importance of high spatial-resolution X-ray +observations. For example, Mrk 1434 was previously identified as +an AGN from an XMM-Newton survey (Birchall et al. 2020), while +our higher spatial-resolution Chandra observation clearly resolves +10 This number is an upper limit, and it neglects biases inherent to an +archival/serendipitous survey, which is out of the scope of this paper to +quantify. +MNRAS 000, 1–11 (2023) + +10 +E. Thygesen et al. +the ‘nuclear’ X-ray source into two distinct sources (and even then, +it remains unclear if either source is indeed an accreting mBH). +5 +SUMMARY AND CONCLUSIONS +We have presented a multiwavelength study of three nearby dwarf +galaxies that host ULXs. Two galaxies in our sample, SDSS J1213 +and SDSS J1221, each contain single off-nuclear X-ray sources +that we suspect are luminous XRBs. The third galaxy, Mrk 1434 +hosts two X-ray sources separated by 2.′′8. The northern source +(Mrk 1434 X-N) also displays extended radio emission at 5.5 GHz +and point-like radio emission at 9.0 GHz. It remains unclear if +the X-ray sources in Mrk 1434 are XRBs or AGNs (especially +Mrk 1434 X-N), although either scenario is intriguing. If XRBs, +then the combined X-ray luminosity of both sources is larger than +expected for a galaxy with Mrk 1434’s star formation rate and +(low) metallicity. Futhermore, the extended radio emission at 5.5 +GHz could then represent the most luminous ‘ULX bubble’ ever +observed in the radio, although we stress that the 5.5 GHz radio +emission can also be attributed entirely to star formation within the +galaxy, or to an AGN jet. Regardless of the correct scenario, we +find that the line emission from He ii in Mrk 1434 is inconsistent +with a nebula being powered by the central X-ray source, unless the +central source underwent a period of higher activity several hundred +years ago, or if the the nebula is shock ionised by an outflow that +is an order of magnitude more powerful than yet observed from +a ULX. If Mrk 1434 X-N is an AGN, then the 9.0 GHz radio +emission may represent a compact synchrotron jet from a low- +luminosity AGN power by an mBH with 𝑀BH ≈ 4 × 105𝑀⊙. +We conclude by stressing the importance of high spatial-resolution +observations when performing multiwavelength searches for mBHs +in dwarf galaxies. +ACKNOWLEDGEMENTS +We thank the anonymous referee for helpful comments that im- +proved this manuscript. Support for this work was provided by the +National Aeronautics and Space Administration through Chandra +Award Number GO6-17079X issued by the Chandra X-ray Center, +which is operated by the Smithsonian Astrophysical Observatory +for and on behalf of the National Aeronautics Space Administration +under contract NAS8-03060. This research is based on observa- +tions made with the NASA/ESA Hubble Space Telescope obtained +from the Space Telescope Science Institute, which is operated by +the Association of Universities for Research in Astronomy, Inc., +under NASA contract NAS 5–26555. These observations are as- +sociated with program HST-GO-14356. Support for Program No. +HST-GO-14356 was provided by NASA through a grant from the +Space Telescope Science Institute, which is operated by the As- +sociation of Universities for Research in Astronomy, Incorporated, +under NASA contract NAS5-26555. RMP and JDP acknowledge +support from the National Science Foundation under grant No. +2206123. RS acknowledges support from grant number 12073029 +from the National Natural Science Foundation of China (NSFC). +AER acknowledges support provided by NASA through EPSCoR +grant number 80NSSC20M0231. GEA is the recipient of an Aus- +tralian Research Council Discovery Early Career Researcher Award +(project number DE180100346) funded by the Australian Govern- +ment. This research made use of Astropy,11 a community-developed +core Python package for Astronomy (Astropy Collaboration et al. +2013, 2018). +DATA AVAILABILITY +The data underlying this article are available in the Chandra Data +Archve under ObsIDs 18059, 18060, and 18061 (https://cda. +harvard.edu/chaser/), in the Barbara A. Mikulski Archive +for Space Telescopes under program ID 14356 (dx.doi.org/ +10.17909/3bxp-zt07), and in the National Radio Astronomy +Observatory Data Archive under programs 14-358 and SH0563 +(data.nrao.edu). +REFERENCES +Allen M. G., Groves B. A., Dopita M. A., Sutherland R. S., Kewley L. J., +2008, ApJS, 178, 20 +Astropy Collaboration et al., 2013, A&A, 558, A33 +Astropy Collaboration et al., 2018, AJ, 156, 123 +Baldassare V. F., Reines A. E., Gallo E., Greene J. E., 2015, ApJ, 809, L14 +Baldassare V. F., Geha M., Greene J., 2020, ApJ, 896, 10 +Barth A. J., Ho L. C., Rutledge R. E., Sargent W. L. W., 2004, ApJ, 607, 90 +Bellovary J. M., Cleary C. E., Munshi F., Tremmel M., Christensen C. R., +Brooks A., Quinn T. R., 2019, MNRAS, 482, 2913 +Bellovary J. M., et al., 2021, MNRAS, 505, 5129 +Berghea C. T., Johnson M. C., Secrest N. J., Dudik R. P., Hennessy G. S., +El-khatib A., 2020, ApJ, 896, 117 +Birchall K. L., Watson M. G., Aird J., 2020, MNRAS, 492, 2268 +Blandford R. D., Königl A., 1979, ApJ, 232, 34 +Brorby M., Kaaret P., Prestwich A., 2014, MNRAS, 441, 2346 +CASA Team et al., 2022, arXiv e-prints, p. arXiv:2210.02276 +Cann J. M., et al., 2021, ApJ, 912, L2 +Cash W., 1979, ApJ, 228, 939 +Chen C.-T. J., et al., 2017, ApJ, 837, 48 +Chilingarian I. V., Katkov I. Y., Zolotukhin I. Y., Grishin K. A., Beletsky +Y., Boutsia K., Osip D. J., 2018, ApJ, 863, 1 +Cseh D., et al., 2012, ApJ, 749, 17 +Dashyan G., Silk J., Mamon G. A., Dubois Y., Hartwig T., 2018, MNRAS, +473, 5698 +Dickey J. M., Lockman F. J., 1990, ARA&A, 28, 215 +Dopita M. A., Sutherland R. S., 1995, ApJ, 455, 468 +Douna V. M., Pellizza L. J., Mirabel I. F., Pedrosa S. E., 2015, A&A, 579, +A44 +Evans I. N., et al., 2010, ApJS, 189, 37 +Falcke H., Körding E., Markoff S., 2004, A&A, 414, 895 +Fender R. P., Belloni T. M., Gallo E., 2004, MNRAS, 355, 1105 +Feng H., Soria R., 2011, New Astron. Rev., 55, 166 +Filippenko A. V., Ho L. C., 2003, ApJ, 588, L13 +Fruscione A., et al., 2006, in Society of Photo-Optical Instrumentation En- +gineers (SPIE) Conference Series. p. 62701V, doi:10.1117/12.671760 +Gaia Collaboration et al., 2018, A&A, 616, A1 +Gallo E., Sesana A., 2019, ApJ, 883, L18 +Garmire G. P., Bautz M. W., Ford P. G., Nousek J. A., Ricker Jr. G. R., +2003, in Truemper J. E., Tananbaum H. D., eds, Proc. SPIEVol. 4851, +X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy.. pp +28–44, doi:10.1117/12.461599 +Gilfanov M., 2004, MNRAS, 349, 146 +Greene J. E., Strader J., Ho L. C., 2020, ARA&A, 58, 257 +Greene J. E., et al., 2021, ApJ, 917, 17 +Grimm H. J., Gilfanov M., Sunyaev R., 2003, MNRAS, 339, 793 +11 http://www.astropy.org +MNRAS 000, 1–11 (2023) + +X-ray Sources in Dwarf Galaxies +11 +Gültekin K., King A. L., Cackett E. M., Nyland K., Miller J. M., Di Matteo +T., Markoff S., Rupen M. P., 2019, ApJ, 871, 80 +Hack W. J., Dencheva N., Fruchter A. S., 2013, in Friedel D. N., ed., Astro- +nomical Society of the Pacific Conference Series Vol. 475, Astronomical +Data Analysis Software and Systems XXII. p. 49 +Hao C.-N., Kennicutt R. C., Johnson B. D., Calzetti D., Dale D. A., Mous- +takas J., 2011, ApJ, 741, 124 +Heger A., Fryer C. L., Woosley S. E., Langer N., Hartmann D. H., 2003, +ApJ, 591, 288 +Ho L. C., 2008, ARA&A, 46, 475 +Ho L. C., Kim M., 2016, ApJ, 821, 48 +Hong J., van den Berg M., Schlegel E. M., Grindlay J. E., Koenig X., Laycock +S., Zhao P., 2005, ApJ, 635, 907 +Houck J. C., Denicola L. A., 2000, in Manset N., Veillet C., Crabtree D., +eds, Astronomical Society of the Pacific Conference Series Vol. 216, +Astronomical Data Analysis Software and Systems IX. p. 591 +Inayoshi K., Visbal E., Haiman Z., 2020, ARA&A, 58, 27 +Ito H., Kino M., Kawakatu N., Isobe N., Yamada S., 2008, ApJ, 685, 828 +Kaaret P., Feng H., Roberts T. P., 2017, ARA&A, 55, 303 +Kashibadze O. G., Karachentsev I. D., Karachentseva V. E., 2020, A&A, +635, A135 +Kennicutt R. C., Evans N. J., 2012, ARA&A, 50, 531 +Kormendy J., Ho L. C., 2013, ARA&A, 51, 511 +Kraft R. P., Burrows D. N., Nousek J. A., 1991, ApJ, 374, 344 +Lehmer B. D., Alexander D. M., Bauer F. E., Brandt W. N., Goulding A. D., +Jenkins L. P., Ptak A., Roberts T. P., 2010, ApJ, 724, 559 +Lehmer B. D., et al., 2019, ApJS, 243, 3 +Lehmer B. D., et al., 2021, ApJ, 907, 17 +Lemons S. M., Reines A. E., Plotkin R. M., Gallo E., Greene J. E., 2015, +ApJ, 805, 12 +Linden T., Kalogera V., Sepinsky J. F., Prestwich A., Zezas A., Gallagher +J. S., 2010, ApJ, 725, 1984 +Long K. S., Blair W. P., Winkler P. F., Lacey C. K., 2020, ApJ, 899, 14 +Longair M. S., 1994, High energy astrophysics. Vol.2: Stars, the galaxy and +the interstellar medium. Vol. 2 +Mapelli M., Ripamonti E., Zampieri L., Colpi M., Bressan A., 2010, MN- +RAS, 408, 234 +Martínez-Palomera J., Lira P., Bhalla-Ladd I., Förster F., Plotkin R. M., +2020, ApJ, 889, 113 +Merloni A., Heinz S., di Matteo T., 2003, MNRAS, 345, 1057 +Mezcua M., Domínguez Sánchez H., 2020, ApJ, 898, L30 +Mezcua M., Civano F., Fabbiano G., Miyaji T., Marchesi S., 2016, ApJ, 817, +20 +Mezcua M., Civano F., Marchesi S., Suh H., Fabbiano G., Volonteri M., +2018, MNRAS, 478, 2576 +Mezcua M., Suh H., Civano F., 2019, MNRAS, 488, 685 +Miller B. P., Gallo E., Greene J. E., Kelly B. C., Treu T., Woo J.-H., Baldas- +sare V., 2015, ApJ, 799, 98 +Mineo S., Gilfanov M., Lehmer B. D., Morrison G. E., Sunyaev R., 2014, +MNRAS, 437, 1698 +Mirabel I. F., Dijkstra M., Laurent P., Loeb A., Pritchard J. R., 2011, A&A, +528, A149 +Moon D.-S., Harrison F. A., Cenko S. B., Shariff J. A., 2011, ApJ, 731, L32 +Moran E. C., Shahinyan K., Sugarman H. R., Vélez D. O., Eracleous M., +2014, AJ, 148, 136 +Moretti A., Campana S., Lazzati D., Tagliaferri G., 2003, ApJ, 588, 696 +Murphy E. J., et al., 2011, ApJ, 737, 67 +Nguyen D. D., et al., 2019, ApJ, 872, 104 +Orienti M., Prieto M. A., 2010, MNRAS, 401, 2599 +Pakull M. W., Angebault L. P., 1986, Nature, 322, 511 +Pakull M. W., Soria R., Motch C., 2010, Nature, 466, 209 +Pardo K., et al., 2016, ApJ, 831, 203 +Plotkin R. M., Markoff S., Kelly B. C., Körding E., Anderson S. F., 2012, +MNRAS, 419, 267 +Ponnada S., Brorby M., Kaaret P., 2020, MNRAS, 491, 3606 +Prestwich A. H., Tsantaki M., Zezas A., Jackson F., Roberts T. P., Foltz R., +Linden T., Kalogera V., 2013, ApJ, 769, 92 +Reines A. E., Volonteri M., 2015, ApJ, 813, 82 +Reines A. E., Sivakoff G. R., Johnson K. E., Brogan C. L., 2011, Nature, +470, 66 +Reines A. E., Greene J. E., Geha M., 2013, ApJ, 775, 116 +Reines A. E., Condon J. J., Darling J., Greene J. E., 2020, ApJ, 888, 36 +Ricarte A., Natarajan P., 2018, MNRAS, 481, 3278 +Richards G. T., et al., 2006, AJ, 131, 2766 +Roberts T. P., Colbert E. J. M., 2003, MNRAS, 341, L49 +Rosen S. R., et al., 2016, A&A, 590, A1 +Ross N. P., et al., 2013, ApJ, 773, 14 +Sargent A. J., Johnson M. C., Reines A. E., Secrest N. J., van der Horst A. J., +Cigan P. J., Darling J., Greene J. E., 2022, ApJ, 933, 160 +Sartori L. F., Schawinski K., Treister E., Trakhtenbrot B., Koss M., Shirazi +M., Oh K., 2015, MNRAS, 454, 3722 +Schaerer D., Fragos T., Izotov Y. I., 2019, A&A, 622, L10 +Schramm M., et al., 2013, ApJ, 773, 150 +Schutte Z., Reines A. E., 2022, Nature, 601, 329 +Senchyna P., Stark D. P., Mirocha J., Reines A. E., Charlot S., Jones T., +Mulchaey J. S., 2020, MNRAS, 494, 941 +Shirazi M., Brinchmann J., 2012, MNRAS, 421, 1043 +Simmonds C., Schaerer D., Verhamme A., 2021, A&A, 656, A127 +Soria R., Pakull M. W., Broderick J. W., Corbel S., Motch C., 2010, MNRAS, +409, 541 +Soria R., Pakull M. W., Motch C., Miller-Jones J. C. A., Schwope A. D., +Urquhart R. T., Ryan M. S., 2021, MNRAS, 501, 1644 +Trebitsch M., Volonteri M., Dubois Y., Madau P., 2018, MNRAS, 478, 5607 +Urquhart R., et al., 2019, MNRAS, 482, 2389 +Vanden Berk D. E., et al., 2001, AJ, 122, 549 +Volonteri M., 2010, A&ARv, 18, 279 +Volonteri M., Habouzit M., Colpi M., 2021, Nature Reviews Physics, 3, 732 +Weaver R., McCray R., Castor J., Shapiro P., Moore R., 1977, ApJ, 218, 377 +Woo J.-H., Cho H., Gallo E., Hodges-Kluck E., Le H. A. N., Shin J., Son +D., Horst J. C., 2019, Nature Astronomy, 3, 755 +Yang Q.-X., Xie F.-G., Yuan F., Zdziarski A. A., Gierliński M., Ho L. C., +Yu Z., 2015, MNRAS, 447, 1692 +York D. G., et al., 2000, AJ, 120, 1579 +Younes G., Porquet D., Sabra B., Reeves J. N., 2011, A&A, 530, A149 +Zhao Y., Gao Y., Gu Q., 2013, ApJ, 764, 44 +Zibetti S., Charlot S., Rix H.-W., 2009, MNRAS, 400, 1181 +de Zotti G., Massardi M., Negrello M., Wall J., 2010, A&ARv, 18, 1 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–11 (2023) + diff --git a/SNAzT4oBgHgl3EQfXPz4/content/tmp_files/load_file.txt b/SNAzT4oBgHgl3EQfXPz4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c2317e823a42c7a6766c9427dc2aa5ff03b0c58 --- /dev/null +++ b/SNAzT4oBgHgl3EQfXPz4/content/tmp_files/load_file.txt @@ -0,0 +1,1753 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf,len=1752 +page_content='MNRAS 000, 1–11 (2023) Preprint 5 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 Multiwavelength Scrutiny of 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Australia 10Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Washington State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Pullman,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' WA 99163,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' USA 11Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 1085 S University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' MI 48109,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' USA Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Owing to their quiet evolutionary histories, nearby dwarf galaxies (stellar masses 𝑀★ ≲ 3 × 109𝑀⊙) have the potential to teach us about the mechanism(s) that ‘seeded’ the growth of supermassive black holes, and also how the first stellar mass black holes formed and interacted with their environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Here, we present high spatial-resolution observations of three dwarf galaxies in the X-ray (Chandra), the optical/near-infrared (Hubble Space Telescope), and the radio (Karl G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Jansky Very Large Array).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' These three galaxies were previously identified as hosting candidate active galactic nuclei on the basis of lower resolution X-ray imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' With our new observations, we find that X-ray sources in two galaxies (SDSS J121326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='01+543631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 and SDSS J122111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='29+173819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1) are off nuclear and lack corresponding radio emission, implying they are likely luminous X-ray binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The third galaxy (Mrk 1434) contains two X-ray sources (each with 𝐿X ≈ 1040 erg s−1) separated by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′8, has a low-metallicity (12 + log (O/H) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8), and emits nebular He ii 𝜆4686 line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The northern source has spatially coincident point-like radio emission at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz and extended radio emission at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We discuss X-ray binary interpretations (where an ultraluminous X-ray source blows a ‘radio bubble’) and active galactic nucleus interpretations (where a ≈ 4 × 105𝑀⊙ black hole launches a jet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' In either case, we find that the He ii emission cannot be photoionised by the X-ray source, unless the source was ≈30—90 times more luminous several hundred years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Key words: galaxies: dwarf — stars: black holes — radio continuum: galaxies — X-rays: galaxies 1 INTRODUCTION There is abundant evidence that supermassive black holes (SMBHs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 106 ≲ 𝑀BH ≲ 109 𝑀⊙) ubiquitously exist at the centres of large galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Kormendy & Ho 2013), some of which accrete and shine as active galactic nuclei (AGNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Some lower-mass dwarf galaxies (which we define by stellar masses 𝑀★ ≲ 3 × 109𝑀⊙) are ★ E-mail: thygesen@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='edu † E-mail: rplotkin@unr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='edu known to host nuclear black holes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Filippenko & Ho 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Barth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Reines et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2011, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Schramm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Moran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Sartori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Mezcua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2016, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Pardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Ho & Kim 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Chilingarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Baldassare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Martínez- Palomera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Cann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Schutte & Reines 2022), with some mass estimates as low as 𝑀BH ≈104 𝑀⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Baldassare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Woo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' These black holes represent the lower-mass end of the SMBH population, and we refer to them here as ‘massive © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='01317v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='HE] 3 Jan 2023 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thygesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' black holes’ (mBHs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 104 ≲ 𝑀BH ≲ 106 𝑀⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' An actively accreting mBH can affect how dwarf galaxies provide feedback to their larger scale environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Dashyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Trebitsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Mezcua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2019), and more generally, mBHs represent a phase that nuclear black holes must pass through as they grow to SMBH sizes over cosmological time scales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Volonteri 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Given that dwarf galaxies have had relatively quiet evolutionary histories, constraining the fraction of dwarf galaxies hosting mBHs in the local Universe, along with the mBH mass distribution, may provide constraints on the mechanism(s) that formed the first black holes in the Universe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Ricarte & Natarajan 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Inayoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Volonteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The fraction of dwarf galaxies hosting an mBH is still relatively unknown, with current empirical constraints implying ≳ 30−50% occupation (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Gallo & Sesana 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Greene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Stellar mass black holes (𝑀BH ≈10 𝑀⊙) and neutron stars are also observed within some dwarf galaxies in the form of X-ray bi- naries (XRBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' XRBs serve as probes of stellar populations within galaxies, with the number and/or luminosity of XRBs expected to scale with the star formation rate, stellar mass, and metallicity of the host galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Grimm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Gilfanov 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Linden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Mineo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Intriguingly, lower-metallicity galaxies appear to contain an excess of luminous XRBs compared to Solar-metallicity galaxies (Prestwich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Brorby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Douna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Ponnada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2021), which may be a consequence of lower-metallicity pro- genitor stars having weaker stellar winds, and therefore producing black hole remnants that are more numerous and/or more massive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Heger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Mapelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Besides tracing stel- lar populations, the energy output from XRBs can also provide feedback to their host galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For example, line emission from the high-ionisation He ii 𝜆4686 line (𝜒ion = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 eV) has been observed from some ultraluminous X-ray sources (ULXs),1 which is often interpreted as an X-ray photoionised nebula (Pakull & Angebault 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Extrapolating such ULX phenomenology in the local Universe to higher redshifts, XRBs could have con- tributed to the X-ray heating of the intergalactic medium during the Epoch of Reionisation and Cosmic Dawn (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Mirabel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Ponnada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, characterising both the XRB and mBH populations in nearby dwarf galaxies, particularly as a function of host galaxy metallicity, is important for understanding the formation of the first black holes and galaxies in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' X-ray observations are commonly used to identify accreting compact objects, since hard X-ray emission (≳1–2 keV) is a uni- versal signature of accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' However, in several cases, it is very challenging to determine the mass of an accreting object via X- ray observations alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' In particular, both a rapidly-accreting XRB and a weakly-accreting mBH/SMBH can have comparable X-ray luminosities in the 1039 − 1041 erg s−1 range, and they can also display similar X-ray spectral shapes (below ≈50 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Folding in multiwavelength information is therefore essential for differ- entiating between rapidly accreting XRBs and weakly-accreting mBHs/SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' It is well established that weakly accreting SMBHs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', low-luminosity AGNs with 𝐿bol ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='01𝐿Edd, where 𝐿bol is the bolometric luminosity and 𝐿Edd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3×1038 [𝑀BH/𝑀⊙] erg s−1 is the Eddington luminosity) emit compact, usually unresolved, radio emission with a flat spectrum ( 𝑓𝜈 ∝ 𝜈−𝛼, where 𝑓𝜈 is the radio flux 1 We define ULXs as having X-ray luminosities 𝐿𝑋 > 1039 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' ULXs are most commonly interpreted as super-Eddington neutron star or black hole XRBs (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Feng & Soria 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Kaaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Properties of the three dwarf galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 1: galaxy names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The full designations of the second and third galax- ies are SDSS J121326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='01+543631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 and J122111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='29+173819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 2: distances to each galaxy, assuming 𝐻0 = 73 km s−1 Mpc−1 for Mrk 1434 and SDSS J1213, and using the Tully-Fisher relation for SDSS J1221(Kashibadze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 3: stellar masses, following the methodology of Reines & Volonteri (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 4: logarithm of star formation rates, based on far-ultraviolet and infrared luminosities (Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Kennicutt & Evans 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 5: metallicities when available in the literature (taken from Shirazi & Brinchmann 2012 for Mrk 1434 and Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2013 for SDSS J1221).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Name D log 𝑀★ log SFR 12+log (𝑂/𝐻) (Mpc) (𝑀⊙) (𝑀⊙ yr−1) (1) (2) (3) (4) (5) Mrk 1434 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 SDSS J1213 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' SDSS J1221 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 density at frequency 𝜈, and the radio spectral index 𝛼 = 0 for a flat spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Ho 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Such unresolved, flat spectrum radio emission is usually interpreted as a partially self-absorbed synchrotron jet (Blandford & Königl 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Meanwhile, rapidly accreting XRBs do not launch jets that would be detectable beyond distances of a few Mpc (Fender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, the presence of unresolved radio emission has the potential to exclude hard X-ray sources as rapidly accreting XRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' In this paper, we present high spatial-resolution X-ray (Chan- dra), optical/near-infrared (Hubble Space Telescope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' HST), and ra- dio observations (Karl G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Jansky Very Large Array;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' VLA) of three nearby dwarf galaxies that each host at least one hard X-ray point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' These three targets were initially identified as AGN candi- dates by Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015), but with the caveat that the positions of their X-ray sources were poorly determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' From the multiwave- length data presented here, we better locate the positions of the X-ray sources within these three galaxies, and we attempt to constrain the nature of each source (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', XRB or mBH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' In Section 2 we detail our sample selection and data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We outline our results in Section 3, followed by a discussion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Our conclusions are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Unless stated otherwise, uncertainties are reported at the 68% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2 OBSERVATIONS AND DATA REDUCTION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 Sample Our three targets were selected from the survey by Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015), who cross matched ∼44,000 nearby dwarf galaxies (𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='055) from the NASA-Sloan Atlas2 to the Chandra Source Catalog (CSC Release 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' They identified 19 galaxies with hard X-ray point sources (2–7 keV), of which 10 contained an X-ray source positionally consistent with the galaxy optical centre (given positional uncertainties, we note that not every galaxy has a well defined nucleus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' They presented these 10 galaxies as AGN candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 2 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='nsatlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='org/ 3 Since publication of Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015), there is new theoretical evi- dence that mBHs do not need to reside in the nucleus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Bellovary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023) X-ray Sources in Dwarf Galaxies 3 Chandra provides exquisite spatial resolution (≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′4) for tar- gets located at the telescope’s aimpoint, but the resolution degrades for sources located farther away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Of the 10 AGN candidates in Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015), they found that four galaxies contain X- ray sources that are far enough from the aimpoint to have large positional uncertainties (>5′′, which is comparable to the pro- jected size of the entire dwarf galaxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Of these four galaxies, three contained X-ray sources with hard X-ray luminosities >3𝜎 (>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 dex) larger than expected from the galaxy-wide contribu- tion from X-ray binaries, given the stellar mass and star forma- tion rate of each galaxy (see Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 of Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' These three galaxies include: Mrk 1434 (𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='00747), SDSS J121326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='01+543631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 (𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='00797;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' hereafter SDSS J1213), and SDSS J122111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='29+173819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 (𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='00699;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' hereafter SDSS J1221;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Of particular interest, Mrk 1434 is a metal-poor galaxy (12+log (𝑂/𝐻) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Shirazi & Brinchmann 2012) and its optical spectrum from the Sloan Digital Sky Survey (SDSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2000) shows nebular He ii line emission (Shirazi & Brinchmann 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' To better constrain the locations of the X-ray sources relative to their host galaxies, we obtained new Chandra X-ray and HST optical/near-infrared observations for these three galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We also obtained new VLA radio observations for one target, SDSS J1213, while archival VLA data were already available for the other two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We adopt distances for each galaxy based on their redshifts, using 𝐻0 = 73 km s−1 Mpc−1, except for SDSS J1221, which is a member of the Virgo cluster (VCC 459).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For this galaxy, we use a distance of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 Mpc based on the Tully-Fisher relation (Kashibadze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For all three galaxies, we adopt star formation rates from Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015), which are based on (dust-corrected) far- ultraviolet and infrared luminosities and the relationships from Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2011) and Kennicutt & Evans (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For SDSS J1221, we scale the star formation rate from Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015) to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For stellar mass estimates, following Reines & Volonteri (2015), we use the colour-dependent mass-to-light ratios from Zibetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 Chandra We obtained new Chandra observations (Cycle 17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' PI Plotkin) with each galaxy centred at the aimpoint of the S3 chip of the Advanced CCD Imaging Spectrometer (ACIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Garmire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Data were telemetered in VFAINT mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Chandra data reduction was carried out using the Chandra Interactive Analysis of Observations (ciao) software version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='13 (Fruscione et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2006) and caldb v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The Chandra data were reprocessed using chandra_repro to create new level 2 event files and bad pixel files, and to apply the latest calibration files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We then searched for background flares using the deflare script, and we did not find any periods with elevated background levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Next we aligned the event file astrometry to the SDSS reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We first excluded areas on each X-ray image occupied by the dwarf galaxy, so that our astrometric corrections would not be influenced by sources within each target galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We then filtered each Chandra image to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 keV and ran wavdetect to identify X-ray point sources, adopting wavelet scales of 1,2,4,8, and 16, setting sigthresh to 10−6 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', approximately one false positive per chip), and using a point spread function map (at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 keV) with an enclosed count fraction (ecf) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The relatively large ecf was chosen to help filter out weak X-ray sources, which would not have sufficient positional accuracy for astrometric alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We then cross-matched X-ray sources identified by wavdetect to the SDSS catalog using wcs_match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We found only two common X- ray/optical sources for Mrk 1434, zero common sources for SDSS J1213, and one common source for SDSS J1221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, we applied a translational astrometric correction for Mrk 1434 (Δ𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='97, Δ𝑦 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='32 pixels) and for SDSS J1221 (Δ𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='01, Δ𝑦 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='96 pixels) using wcs_update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' No astrometric correction was applied to SDSS J1213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We next re-ran wavdetect on the aligned event files (filtered from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5-7 keV, now including each target dwarf galaxy) to de- termine positions in the aligned reference frame of X-ray sources hosted by each dwarf galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We used the same wavdetect param- eters as above, except we used ecf=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 when generating the point spread function map to allow the detection of fainter point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' wavdetect identified two X-ray sources in Mrk 1434, one source in SDSS J1213, and one source in SDSS J1221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The positions of each X-ray source are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We estimated 95% uncertainties of each X-ray position based on the distance from the telescope aim- point and the number of counts detected by wavdetect, following Equation 5 in Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Note, this 95% positional uncer- tainty represents the statistical error on each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For SDSS J1213 in particular, where we could not perform an astrometric alignment of the Chandra image, there is an additional systematic uncertainty that could be as large as 2′′ (although 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′8 is more typical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 We then measured the number of counts from each X-ray source using srcflux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We adopted circular apertures centred at each wavdetect position with radii of 5 pixels, except for Mrk 1434, which contains two X-ray sources, where we adopted radii of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 pixels to avoid the regions from each X-ray source from over- lapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The number of background counts per pixel was estimated from nearby source-free regions of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' These measure- ments were performed in both broad (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 keV) and hard (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 keV) images, and we detected 19–73 counts from each source in the broad band and 8–23 counts in the hard band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' All X-ray detections (in all bands) are significant at the >99% level according to the confidence tables in Kraft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Finally, spectra were extracted for each X-ray source us- ing specextract and fit using an absorbed powerlaw model (tbabs*powerlaw) in the Interactive Spectral Interpretation Sys- tem v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 (ISIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Houck & Denicola 2000), adopting Cash statis- tics (Cash 1979) given the relatively low number of counts per source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We initially left the column density as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' However, for three X-ray sources 𝑁𝐻 converged to zero, in which case we froze the value to the Galactic column density and refit the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Model fluxes were calculated using the cflux convolu- tion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Spectral parameters and model fluxes are reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 Hubble Space Telescope We observed each galaxy with the Wide Field Camera 3 (WFC3) aboard HST for one orbit per galaxy (PI Plotkin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' program 14356).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For Mrk 1434 and SDSS J1221 we observed in both the F110W and F606W filters (with the IR and UVIS channels, respectively), and for SDSS J1213, which is a fainter galaxy, we took observations only in the F110W filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Observations in each filter were taken over four dither positions, and we used the IRSUB512 subarray for Mrk 1434 and SDSS J1221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Total exposure times in each filter are listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Data were downloaded from the Mikulski Archive for Space Telescopes (MAST), and individual exposures were aligned 4 https://cxc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='edu/cal/ASPECT/celmon/ MNRAS 000, 1–11 (2023) 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thygesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Details of Chandra observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 1: name of X-ray source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 2: Chandra obsID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 3: date of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 4: exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Columns 5 & 6: right ascension and declination of each X-ray source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 7: radius of the 95% positional uncertainty of each Chandra source, based on Equation 5 of Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 8: aperture corrected net count rate (in counts per ks) in the broad X-ray band (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Aperture corrections of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='90, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='95, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='96 were used for Mrk 1434, SDSS J1213, and SDSS J1221, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 9: aperture corrected net count rate (in counts per ks) in the hard band (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Aperture corrections of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='87, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='93, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='93 were used for Mrk 1434, SDSS J1213, and SDSS J1221, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Source obsID Date Exp Time Right Ascension Declination 𝑝err Net Rate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 keV) Net Rate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 keV) (ks) (J2000) (J2000) (′′) (ks−1) (ks−1) (1) (2) (3) (4) (5) (6) (7) (8) (9) Mrk 1434 X-N 18059 2016 Jan 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 10:34:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='19 +58:03:49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='00 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='98 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='40 Mrk 1434 X-S 18059 2016 Jan 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 10:34:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='11 +58:03:46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='36 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='83 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='06+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='94 SDSS J1213 18060 2016 Aug 04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 12:13:26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='12 +54:36:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='78 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='60 SDSS J1221 18061 2016 Feb 13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 12:21:11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='00 +17:38:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='33 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='82 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='46 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='24 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Chandra spectral parameters, fluxes, and luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 1: name of X-ray source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 2: column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 3: best-fit photon index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 4: best-fit Cash statistic/degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Columns 5 & 6: logarithms of the unabsorbed model X-ray flux and luminosity from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5-10 keV, estimated using the cflux convolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Columns 7 & 8: logarithms of the unabsorbed model X-ray flux and luminosity from 2-10 keV, estimated using the cflux convolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Broad (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 keV) Hard (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 keV) Source 𝑁𝐻 Γ C-stat/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' log Flux log Luminosity log Flux log Luminosity (1020 cm−2) (erg s−1 cm−2) (erg s−1) (erg s−1 cm−2) (erg s−1) (1) (2) (3) (4) (5) (6) (7) (8) Mrk 1434 X-N <56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9𝑎 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0/13 −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 Mrk 1434 X-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6𝑏 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3/13 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 SDSS J1213 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4𝑏 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8/12 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 SDSS J1221 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7𝑏 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1/32 −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 𝑎Best-fit column density 𝑁𝐻 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 × 1020 cm−2, reported as an upper limit (95% confidence level) because the uncertainty on the best-fit value extends down to the Galactic value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 × 1020 cm−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 𝑏Column density frozen to the Galactic value during fitting, taken from Dickey & Lockman (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' and combined using AstroDrizzle in the DrizzlePac software (Hack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 The F110W drizzled images were created with plate scales 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′06 pix−1 for Mrk 1434 and SDSS J1221, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′09 pix−1 for SDSS J1213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' All F606W images have plate scales 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′03 pix−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We aligned the HST astrometry to the Gaia Data Release 2 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2018) reference frame using the tweakreg task within AstroDrizzle (after excluding sources falling within each galaxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 For Mrk 1434, the corrections re- sulted in astrometric shifts by (Δ𝑥 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8, Δ𝑦 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0) pixels (from two common sources) and (Δ𝑥 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9, Δ𝑦 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2) pixels (from nine common sources) in the F110W and F606W filters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For SDSS J1213, we shifted the F110W filter by (Δ𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6, Δ𝑦 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8) pixels (five common sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Finally, for SDSS J1221 we could not identify enough common sources between the HST image and the Gaia catalog in the F110W filter (which has a smaller field of view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' So, we only aligned the F606W filter to the Gaia frame, shifting by (Δ𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2, Δ𝑦 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3) pixels (four common sources), and we then 5 https://hst-docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='edu/drizzpac 6 We note that we aligned HST images to the Gaia frame and the Chandra X-ray images to the SDSS frame, because we generally found a larger num- ber of common HST/Gaia sources vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' common HST/SDSS sources (and vice-versa for Chandra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Compared to the statistical uncertainty on each Chandra position (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′3–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′4), we do not expect a meaningful offset between the absolute astrometry of SDSS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Gaia, such that systematic uncertainties in our astrometric alignments are dominated by the small number of sources used to apply the corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Summary of HST observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 1: galaxy name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 2: date of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 3: filters used for observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 4: exposure times in the F110W/F606W filters, respectively, when both filters were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' All observations were taken through HST Proposal ID 14356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Source Date Filter Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Time (min) (1) (2) (3) (4) Mrk 1434 2016 Apr 16 F110W/F606W 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6/30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9 SDSS J1213 2016 Apr 16 F110W 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 SDSS J1221 2016 Apr 9 F110W/F606W 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6/26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9 aligned the F110W filter to the F606W filter (via three common sources between the two HST filters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 Very Large Array Mrk 1434 and SDSS J1221 both had archival datasets (PI Satyapal, 14A-358) from the VLA, while new data were obtained for SDSS J1213 for this study (PI Plotkin, SH0563).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' All three galaxies were observed in the most extended A configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Both Mrk 1434 and SDSS J1221 observations were from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz (C band) and 8-10 GHz (X band), while SDSS J1213 was observed only from 8-12 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The Common Astronomy Software Applications (CASA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' CASA Team et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2022) software package version 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 was used to carry out standard data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We used 3C 286 to perform delay MNRAS 000, 1–11 (2023) X-ray Sources in Dwarf Galaxies 5 and bandpass calibrations, and to set the flux density scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Nearby phase calibrators (see Table 5) were observed to solve for the time- dependent complex gain solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Imaging was performed using the task tclean, using two Taylor terms (nterms=2) to account for the wide fractional bandwidth and natural weighting to maximise sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We achieved root-mean-square (rms) sensitivities rang- ing from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 𝜇Jy bm−1 in each observing band (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The only X-ray source for which we found coincident radio emission is Mrk 1434 X-N, where we found radio detections at both 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz within the X-ray error circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We used imfit to fit two-dimensional Gaussians in the image plane (at each frequency) to calculate the size of the radio structure, and to measure peak and integrated flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' As discussed further in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1, the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz emission is slightly extended (with integrated flux density 𝑓int = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='191 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='036 mJy) while the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz is point-like ( 𝑓peak = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='036 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='009 mJy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The centroids of the radio emission at each frequency are offset by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For the other two galaxies, we place 3𝜎rms limits on their radio flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We note that SDSS J1221 displays radio emission aligned with a likely H ii region toward the eastern outskirts of the galaxy that is not associated with X-ray emission, so we do not discuss that radio emission in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 3 RESULTS In the following subsections we present the multiwavelength results for each galaxy, deferring discussions regarding the possible nature of each X-ray source to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Composite HST images are shown for each galaxy in Figure 1, including the locations of X-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 Mrk 1434 Mrk 1434 hosts two X-ray sources separated by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′8 (see Figure 1a), both of which are classified as ULXs: the northern source (Mrk 1434 X-N), which is located toward the galactic nucleus, has an unabsorbed hard X-ray luminosity 𝐿2−10 keV = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6) × 1040 erg s−1, and the southern source (Mrk 1434 X-S) has 𝐿2−10 keV = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2) × 1039 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The X-ray spectra of each source are fit by powerlaw models with photon indices of Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 for Mrk 1434 X-N and Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 for Mrk 1434 X-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Neither source shows evidence for significant intrinsic absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' It is unlikely that either hard X-ray source is a superposed fore- ground/background object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Given the density and flux distribution of hard X-ray sources in the cosmic X-ray background (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Equation 2 of Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2003), we expect to only find 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='001 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='003 hard X-ray sources with 2-10 keV fluxes similar (or brighter) than Mrk 1434 X-N and Mrk 1434 X-S, respectively, within the projected size of the galaxy (which we conservatively approximate as a circle with a 20′′ radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Radio emission is detected only from the northern source, Mrk 1434 X-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' At 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz, the emission is extended with major and minor axis full width half maxima of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′1×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′6 (160 pc × 90 pc), respectively, covering ≈3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 synthesised beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The centroid of the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz emission is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′16 from the X-ray position (for reference, the 95% Chandra error circle is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′35), and the integrated luminos- ity is 𝐿5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5,int = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2) × 1036 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' At 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz we detect a point source located 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′32 from the X-ray position, with a peak luminosity 𝐿9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0,peak = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8) × 1035 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We do not detect any extended radio structures at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz, thereby indicating that the emission seen at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz has a steep radio spectrum (our 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz radio maps have similar sensitivities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Note, extended emission is not simply resolved out at the higher radio fre- quency, since the smallest baselines of the VLA in A configuration are sensitive to structures up to ≈5′′ at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz, which is larger than the ≈1′′ angular size of the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' At 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz, the chance of a random alignment of a background radio point source falling within the Chandra error circle is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Integrating the differential source counts tabulated by de Zotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2010) at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 GHz, and assuming a flat radio spectrum (as expected if the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz emission is from a compact jet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2), we expect only ≈3 × 10−5 sources with 𝑓peak > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='036 mJy within the X-ray error circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The chance of a statistical fluctuation as large as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='036 mJy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 4𝜎rms) within the X-ray error circle (which contains ≈240 pixels in the radio map) is also very small (𝑝 = 3 × 10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, we believe the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz emission is indeed physically associated with the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' However, we note that the radio source lies toward the edge of the X-ray error circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, even though the radio source formally falls within the Chandra positional uncertainty, its association specifically with Mrk 1434 X-N is less clear, particularly after considering that the Chandra X- ray astrometry of Mrk 1434 was aligned to the optical frame using only two common X-ray/SDSS sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Finally, we note that towards the southwest of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′35 Chan- dra X-ray error circle of Mrk 1434 X-N, there is an optical/near- infrared source that appears red in the HST composite image (see the zoom-in of Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' If that source is a background quasar it may also be responsible for the X-ray and/or radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' However, the random alignment of such a background quasar is very unlikely, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The AB magnitude of the HST source in the F606W filter is 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8, which we convert to SDSS i≈18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 assuming a typical quasar spectrum (Vanden Berk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We then con- sider SDSS Type 1 quasar counts from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 (Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2013), and we find only a negligible number of background quasars (≈ 6 × 10−7) are likely to fall within the Chandra X-ray circle by random chance (note, the random align- ment of a radio-loud or a Type 2 quasar would be even rarer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' That source is likely intrinsic to the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 SDSS J1213 and SDSS J1221 SDSS J1213 and SDSS J1221 each contain a single hard X-ray point source near the outskirts of each galaxy (Figure 1b-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The hard (2-10 keV) X-ray luminosities of the sources are 𝐿2−10 keV = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4)×1039 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8)×1039 erg s−1, respectively (Table 2), such that both sources are classified as ULXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The chance of a superposed foreground/background object is negligible (we expect only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='005 hard X-ray background sources for SDSS J1213 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='001 sources for SDSS J1221;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Neither galaxy contains radio emission within the Chandra X-ray circles to 3𝜎rms upper limits of < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 × 1035 erg s−1 at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz for SDSS J1213, and to limits of < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 × 1034 erg s−1 and < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9 × 1034 erg s−1 at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz, respectively, for SDSS J1221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 4 DISCUSSION In the following subsections we discuss possible interpretations for the X-ray sources in our sample of three dwarf galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We focus primarily on Mrk 1434 since it exhibits the most complex phe- nomenology (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', two X-ray sources, one of which is coincident with radio emission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We provide arguments for/against XRB in- terpretations in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 and for/against AGN interpretations in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 we discuss whether the observed MNRAS 000, 1–11 (2023) 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thygesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Summary of VLA observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 1: galaxy name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 2: VLA Program ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 3: date of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 4: name of phase calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 5: central frequency of each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 6: bandwidth of each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 7: the time spent integrating on each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 8: the size of the (elliptical) synthesised beam along the major and minor axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 9: rms noise of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Source Program Date Phase Calibrator 𝜈 Δ𝜈 𝜏 𝜃bm 𝜎rms (J2000) (GHz) (GHz) (min) (′′ × ′′) (𝜇Jy bm−1) (1) (2) (3) (4) (5) (6) (7) (8) (9) Mrk 1434𝑎 14A-358 2014 Feb 24 1035+564 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='45×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='38 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 Mrk 1434𝑏 14A-358 2014 Feb 24 1035+564 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='27×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 SDSS J1213 SH0563 2016 Sep 30 1219+482 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='28×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 SDSS J1221 14A-358 2014 Feb 26 1158+248 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='42×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='38 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9 SDSS J1221 14A-358 2014 Feb 26 1158+248 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='25×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 𝑎Extended radio emission detected near Mrk 1434 X-N at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz, with 𝑓int = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='191 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='036 mJy and 𝑓peak = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='054±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='009 mJy bm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The centroid of emission is located at RA=10h34m10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1867s ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0042𝑠, Dec=58◦03′49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′1481 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′0763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 𝑏Point-like radio emission detected near Mrk 1434 X-N at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz, with 𝑓peak = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='036±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='009 mJy bm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The emission is located at RA=10h34m10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2045s ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0039s, Dec=58◦03′49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′2883 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′0460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 3" 5" 5" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5" (a) Mrk 1434 3 arcsec 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 arcsec 5 arcsec 5 arcsec (b) SDSS J1213 (c) SDSS J1221 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (a) Composite HST image of Mrk 1434 in the F606W (blue/green) and the F110W (red) filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The locations of the two X-ray point sources are shown as red cross hairs, with the dashed red circles illustrating the sizes of the 95% positional errors from Chandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The Zoom-in of the centre of the galaxy shows the location of Mrk 1434 X-N relative to the radio emission, where yellow contours show the extended 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz radio emission (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' contours drawn at 3, 4, 5×𝜎rms) and the magenta contours show the unresolved emission at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz (contours drawn at 3, 4×𝜎rms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The sizes of the VLA synthesised beams are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′45 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′38 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′27 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′24 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Note, the SDSS spectroscopic fibre, from which the nebular He ii emission is detected, has a diameter of 3′′ and is placed at the centre of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (b) HST image of SDSS J1213 in the F110W filter, with the location of the X-ray source marked by the red cross hair and dashed circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (c) HST composite image of SDSS J1221 in the F606W (blue/green) and the F110W (red) filters, with the location of the X-ray source marked by the red cross hair and dashed circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' In all images, north is up and east is to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023) X-ray Sources in Dwarf Galaxies 7 X-ray flux is sufficient to explain He ii line emission observed in the SDSS spectrum of Mrk 1434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A discussion on the nature of the X-ray sources in the other two galaxies is presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 Mrk 1434 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 XRB Interpretations As shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1, both X-ray sources in Mrk 1434 are physically associated with the galaxy and luminous enough to be classified as ULXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The observed X-ray luminosity, however, is higher than expected from the luminous tail of the galaxy’s XRB population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The luminosities of both X-ray sources are above the cutoff of the low-mass XRB luminosity function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Gilfanov 2004), so in the following we only consider high-mass XRBs using the metallicity-dependent luminosity function from Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For Mrk 1434, with 12 + log (𝑂/𝐻) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 and SFR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='12 𝑀⊙ yr−1, Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2021) predict a total 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 keV X-ray luminosity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', from all X-ray point sources) of 𝐿0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 keV = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='15) ×1039 erg s−1 (where the error bar represents the 68% confidence interval provided by Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' They also predict only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='02 ULXs with 𝐿0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 > 1040 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For reference, the unabsorbed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 keV model luminosities of Mrk 1434 X-N and Mrk 1434 X-S are (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4)×1040 and (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2)× 1040 erg s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, the combined X-ray luminosity of both ULXs is ≈10 times higher than expected relative to the Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2021) luminosity function, which is significant even after considering uncertainties and intrinsic scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Even though the above suggests that it is statistically unlikely for both sources to be XRBs, small number statistics could influence the above arguments, and it is worth exploring XRB interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' In particular, the extended 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz radio emission from Mrk 1434 X-N could represent a ‘ULX bubble’, as similar types of extended radio structures have been observed from other ULXs, making the radio emission a signature of a ULX outflow shocking the nearby interstellar environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Pakull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Soria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2010, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Cseh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Urquhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' If the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz radio emission is indeed a ULX bubble, then with 𝐿5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5,int = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2) × 1036 erg s−1 it would represent the most luminous ULX bubble yet observed by a factor of ≈6 (Pakull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Soria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2010, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Meanwhile, the projected size of ≈160 pc × 90 pc (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′1×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′6) in diameter is fairly typical compared to other ULX bubbles, where diameters range from ≈25–350 pc (Soria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' also see Table 1 of Berghea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Taking the peak flux density of the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz structure, and extrapolating to 1 GHz assuming a spectral index 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7, the intensity of the radio bubble in Mrk 1434 X-N would be 𝐼1 GHz ≈ 6 × 10−16 erg s−1 cm−2 Hz−1 sr−1, which is relatively large but reasonable compared to other ULX radio bubbles with similar physical sizes (see Figure 5 of Berghea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Although a ULX bubble is one interpretation of the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz emission, we stress that it is not a unique (or necessary) explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Adopting SFR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='12 𝑀⊙ yr−1 for Mrk 1434 and the relation between star formation rate and the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 GHz specific luminosity from Kennicutt & Evans (2012), we expect 𝐿5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5,SF ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9×1036 erg s−1 (we convert from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 GHz to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz assuming a spectral index 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Considering that the intrinsic scatter on the conversion between SFR and radio luminosity is on the order of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 dex (Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2011), the observed extended structure at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz could be produced entirely by star formation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Since the extended radio structure at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz is not detected at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz, the dominant radio emission mechanism in such a scenario would most likely be synchrotron radiation with a steep spectrum from supernova remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Note, our data exclude free-free radio emission from an H ii region, which would produce a flat spectrum that would be detectable at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 AGN Interpretations AGN can also produce extended radio emission, which is another viable explanation for the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz radio structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' However, in light of the discussion in the previous subsection that a super-Eddington XRB is also capable of producing the observed extended emission at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz, the resolved radio complex does not provide useful diagnostics for attempting to discriminate between XRB vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Since the X-ray spectra of Mrk 1434 X-N and Mrk 1434 X-S (Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 and Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4, respectively) are consistent with low- luminosity AGNs (Younes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2015), we focus the following discussion on AGN scenarios with Eddington ratios 𝐿bol/𝐿Edd ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For such weakly accreting AGN, we expect to observe unresolved radio emission from a partially self-absorbed compact jet (Ho 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' By combining X-ray and radio luminosities, we can then make crude estimates on black hole masses by appealing to the fundamental plane of black hole activity (Merloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Falcke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For Mrk 1434 X-N, we then interpret the the unresolved 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz radio emission as arising from a compact jet, and we utilise the fundamental plane regression by Gültekin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2019), log � 𝑀BH/108𝑀⊙ � = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='22) + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='10) log � 𝐿5 GHz/1038 erg s−1� − (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='16) log � 𝐿2−10 keV/1040 erg s−1� , (1) which has an intrinsic scatter ≈1 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We estimate that Mrk 1434 X-N would have 𝑀BH ≈ 4×105 𝑀⊙ if powered by an mBH (see Ta- ble 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Note, we assume a flat radio spectrum to convert the observed radio luminosity at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz for use in the fundamen- tal plane (we cannot use our 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz radio map to estimate the 5 GHz luminosity because we do not have enough signal-to-noise to attempt to decompose a point source embedded within the extended radio emission observed at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Similarly, the lack of radio emission from Mrk 1434 X-S implies 𝑀BH ≲ 4 × 105𝑀⊙ (where we adopt a 3𝜎rms upper limit, based on the observed 𝜎rms near Mrk 1434 X-S in our 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' These mass estimates imply Eddington ratios (𝐿2−10 keV/𝐿Edd) of ≈ 2×10−4 and ≳ 1×10−4 for Mrk 1434 X-N and Mrk 1434 X-S, respectively, which, assuming bolometric corrections of ≈10, are consistent with Eddington ratios for which the fundamental plane can be applied (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Plotkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 On the Origin of Nebular He ii Emission In the following we determine whether the X-ray emission from Mrk 1434 is a strong enough source of photoionisation to explain the strength of the He ii emission in the SDSS spectrum of Mrk 1434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The observed He ii line flux is 𝐹4686,obs = (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1) × 10−16 erg s−1 cm−2, which translates to a photon flux of 𝑁4686,obs = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1) × 10−4 photons s−1 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Every photon emitted in the He ii line requires 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 ionizing photons incident on singly ionised helium (Pakull & Angebault 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Given the ionisation potential of singly ionised helium (𝜒ion = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 eV), and considering that MNRAS 000, 1–11 (2023) 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thygesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' mBH mass estimates and limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 1: galaxy name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 2: logarithm of the hard X-ray luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 3: logarithm of the radio luminosity at 5 GHz, assuming a flat radio spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For Mrk 1434 X-N, this luminosity is based on the unresolved emission detected at 9 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For all other X-ray sources, limits are placed as 3𝜎rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Column 4: logarithm of the black hole mass (or limit) if X-ray sources are weakly accreting mBHs, based on the fundamental plane of black hole activity (Gültekin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Uncertainties on log 𝑀BH are ≈1 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Source log 𝐿2−10 keV log 𝐿5 GHz log 𝑀BH (erg s−1) (erg s−1) (𝑀⊙) (1) (2) (3) (4) Mrk 1434 X-N 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 Mrk 1434 X-S 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 <35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 <5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 SDSS J1213 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 <34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9 <5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 SDSS J1221 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 <34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='4 <5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 the photoionisation cross section has a steep 𝐸−3 ph dependence on photon energy, 𝐸ph, then producing the observed SDSS He ii line flux requires a photon flux in the extreme ultraviolet (54–300 eV) of 𝑁54−300 eV = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2𝑁4686,obs = (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1) × 10−4 photons s−1 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Note, this photon flux is an underestimate because we have not corrected the observed SDSS line flux for extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The 3′′ SDSS spectroscopic fibre is centred near Mrk 1434 X-N, such that if the He ii emission arises from photoionisation by the X-ray source, we expect the emission to be dominated by Mrk 1434 X-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We do not have direct measurements on the extreme ultraviolet flux from 54-300 eV, so we extrapolate the Chandra X-ray spectrum into the extreme ultraviolet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Our best-fit powerlaw model predicts a photon flux of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 × 10−4 photons s−1 cm−2 (note the large range in uncertainty because we are extrapolating the model to energies lower than the Chandra X-ray band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, while high-energy radiation from Mrk 1434 X-N may contribute to some of the He ii photoionisation, the observed X-ray source is too faint, by a factor of ≈30, to supply all of the photoionising photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' If we assume a thermal X-ray emission model (tbabs*diskbb), it becomes even more difficult for the X-ray source to explain the He ii photionisation, as the extrapolated 54-300 eV extreme ultraviolet flux becomes ≈90 times too faint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Adding a contribution of photons form Mrk 1434 X-S would only increase the above photon flux by a factor of ≈2, for either spectral model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' There is currently no evidence for significant X-ray variabil- ity from Mrk 1434 over the past 1–2 decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Coincidentally, the SDSS spectrum and the archival Chandra observation from Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015, Chandra obsID 3347) were both taken in May 2002 (separated by ≈2 weeks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The archival data from 2002 show nearly identical X-ray luminosities (log 𝐿2−10 keV = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1 and 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9 erg s−1 for Mrk 1434 X-N and Mrk 1434 X-S, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' see Table 2 of Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2015) compared to the Chandra observations pre- sented here, which were taken nearly 14 years later (see Table 2 of this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' There are also two X-ray detections of Mrk 1434 in the third XMM-Newton serendipitous source catalog (3XMM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Rosen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2016) in 2007 and 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Both X-ray sources are blended to- gether due to XMM-Newton’s poorer spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Comparing the XMM-Newton fluxes to the combined fluxes of both sources in the Chandra observations, X-ray variability is smaller than a factor of ≈2 over the four observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' However, considering the light travel time between the X-ray source and the ionised medium, it is feasible that Mrk 1434 X-N was more active in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The pro- jected radius of the SDSS spectroscopic fibre is 730 light years, and we cannot exclude the possibility that Mrk 1434 X-N was ≈30–90 times more luminous several hundred years ago, which appears to be on the only viable way for the He ii emission to be powered by X-ray photoionisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' If the extended radio emission is produced by an outflow shock- ing the interstellar medium, then one must also consider the pos- sibility of the He ii emission being produced by ionisation from a radiative shock (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Dopita & Sutherland 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' According to the MAPPINGS III libraries of line ratios for radiative shocks (Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2008), assuming a shock velocity of 300 km s−1, we expect the luminosity of the He ii 𝜆4686 emission line 𝐿4686 ≈ 4 × 10−4𝐿rad, where 𝐿rad is the total radiative luminosity of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='7 Assuming that the kinetic power required to inflate a bubble 𝑃kin ≈ 77/27𝐿rad (Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 1977), then explaining the observed He ii line via shock ionisation requires an outflow with 𝑃kin ≈ 6 × 1041 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We do not have a reliable method to independently estimate 𝑃kin (especially considering that other emission lines in the SDSS spectrum are dominated by star formation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' However, for an order of magnitude estimate, we calculate the minimum synchrotron energy of the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz radio emission, which is 𝑊min ≈ 2 × 1052 erg (Longair 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 A 300 km s−1 shock would take ≈ 3 × 105 yr to inflate a bubble with a 160 pc diameter, such that the average power stored in internal energies of the synchrotron emitting structure is ¯𝑃min ≈ 2 × 1039 erg s−1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', the average power in particles and in the magnetic field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, an outflow would need to carry ≳102 times more power in order for a shock to be the sole ionisation source of the observed He ii emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Of course, ¯𝑃min is a minimum energy estimate, and the power in bubbles/cavities carved out by kinetic outflows have sometimes been observed to be larger, sometimes by factors of several hundreds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Ito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2008), such that the above does not exclude the possibility of shock ionisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For comparison, the ULX NGC 6946 MF16 (Roberts & Col- bert 2003) has a luminous and compact radio bubble (Berghea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020), which suggests a relatively powerful outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Adopting the NGC 6946 MF16 bubble line flux in the [Fe ii] 𝜆16440 emission line (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 × 10−15 erg s−1 cm−2) and a distance of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='8 Mpc (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020), the MAPPINGS III libraries for a 300 km s−1 shock (with Solar abundances) suggest a kinetic power of 𝑃kin ≈ 7 × 1040 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, the kinetic power of NGC 6946 MF16 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', one of the most powerful known ULX radio bubbles) is an order of magnitude lower than the power required for shock ionisation to be responsible for the observed strength of the He ii emission line near Mrk 1434 X-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, if the He ii line is powered by shock ionisation, then it would represent one of the most powerful bubbles carved by a ULX outflow yet observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Intriguingly, Mrk 1434 is one member of a population of 182 star forming galaxies with nebular He ii emission that were identi- fied by Shirazi & Brinchmann (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The ratios of He ii/H𝛽 relative to [N ii] 𝜆6584/H𝛼 are inconsistent with AGN for these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Typically, when an AGN is absent, Wolf-Rayet stars are considered the primary stellar population capable of producing enough extreme ultraviolet flux above the 54 eV He ii ionisation edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' However, Shi- razi & Brinchmann (2012) inspected the SDSS spectra for broad emission features indicative of Wolf-Rayet stars, and they found no Wolf-Rayet signatures in the spectrum of Mrk 1434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, without 7 Given the low metallicity of Mrk 1434, we adopt the MAPPINGS III model grid with Small Magellanic Cloud abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We also assume an interstellar medium density of 1 cm−3 and equipartition of magnetic and thermal pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 8 We adopt 𝐿5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 ≈ 1036 erg s−1, a bubble diameter of ≈160 pc, and an ion to electron energy ratio of 𝜂 = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We note that 𝑊min ∝ 𝜂4/7, and the proper value of 𝜂 is not well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023) X-ray Sources in Dwarf Galaxies 9 concrete evidence that Mrk 1434 X-N was indeed brighter several hundred years ago to power the He ii emission via photoionisation, and/or lacking a reliable estimate of the kinetic power of an outflow for shock ionisation, the source of extreme ultraviolet photons in Mrk 1434 remains a mystery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Another plausible explanation could be photoionisation from extreme ultraviolet photons emitted by ex- otic stellar populations (like rapidly rotating stars) in metal-poor environments (see the discussion in Section 6 of Shirazi & Brinch- mann 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' It is very plausible that several of the above scenarios contribute toward producing the He ii line, and Shirazi & Brinch- mann (2012) recovered a heterogeneous population (multiple mech- anisms may even contribute to producing the He ii emission within a single galaxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For example, Senchyna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2020) conclude that X-ray photoionisation cannot explain nebular He ii emission across a sample of nearly a dozen metal-poor galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Meanwhile, there are several well-established examples of X-ray sources that are indeed sufficient to power nebular He ii emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Pakull & Angebault 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Schaerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Simmonds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Further observational constraints, ideally via system- atic X-ray surveys of metal-poor dwarf galaxies under high spatial resolution, are required to understand the level to which ULXs con- tribute extreme ultraviolet radiation in metal-poor galaxies, which has implications for understanding sources of ionisation and heating of the intergalactic medium in the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2 SDSS J1213 and SDSS J1221 Our new Chandra observations confirm the conclusion of Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015) that both X-ray sources are more luminous than ex- pected from the XRB populations in each galaxy, as described be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Unlike for Mrk 1434, the luminosities of both X-ray sources in SDSS J1213 and SDSS J1221 are low enough that we should consider both high-mass and low-mass XRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Following Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015), we therefore adopt the relation from Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2010), which predicts the hard X-ray luminosity from low-mass and high-mass XRBs as a function of stellar mass and star for- mation rate: � 𝐿XRB 2−10/erg s−1� = (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='37) × 1028 (𝑀★/𝑀⊙) + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='22) × 1039 � 𝑆𝐹𝑅/𝑀⊙ yr−1� , with an intrinsic scatter of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='34 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2010) relation predicts 𝐿XRB 2−10 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2×1037 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='6×1037 erg s−1 for SDSS J1213 and SDSS J1221, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The predicted luminosities are ≈3 times higher if we instead adopt the calibrations in Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, the ob- served X-ray luminosities are ≈120–360 and ≈17–50 times higher than expected, for SDSS J1213 and SDSS J1221, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='9 In light of recent theoretical motivation for ‘wandering’ mBHs (Bellovary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2019, 2021, also see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Mezcua & Domínguez Sánchez 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Reines et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Greene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Sargent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2022 for observational searches), an X-ray source being ‘off- nucleus’ does not on its own preclude the possibility of an accreting mBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' It is possible that these sources are mBHs launching jets that are either (a) beneath our radio detection limit or (b) that are very extended and ‘resolved out’ by the VLA when it is in its most 9 The Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2010) relation is calibrated to galaxies with approxi- mately Solar metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The metallicity of SDSS J1213 is unknown, and the metallicity of SDSS J1221 is log (𝑂/𝐻) + 12 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' If we adopt the metallicity-dependent Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2021) relation for high- mass XRBs, the X-ray luminosity of the X-ray source in SDSS J1221 is still ≈20 times higher than expected for a galaxy with its star formation rate and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' extended A configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The largest angular scale to which the VLA is sensitive to radio emission at our observing frequencies (X-band) and configuration (A) is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′3, such that our VLA obser- vations would not detect flux from extended jets larger than ≈850 and ≈410 pc for SDSS J1213 and SDSS J1221, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' On the other hand, the radio cores of weakly accreting AGN (bolometric luminosities 𝐿bol < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='01𝐿Edd) have flat radio spectra and are com- pact enough that their radio emission should not be ‘resolved out’ at VLA resolutions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Orienti & Prieto 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, if only considering mBHs in the weak accretion regime, we can use our ra- dio upper limits in conjunction with the fundamental plane to place mass limits of 𝑀BH < 2 × 105 and <1 × 105 𝑀⊙ for SDSS J1213 and SDSS J1221, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Requiring 𝐿bol < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='01𝐿Edd, and as- suming X-ray bolometric corrections of 10, then places lower limits on black hole masses of ≳ 3×104 (SDSS J1213) and ≳ 2×104 𝑀⊙ (SDSS J1221).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, there is a relatively narrow range of mass where our VLA observations could ‘miss’ the compact radio jet from a weakly accreting mBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Note, our radio limits do not place useful constraints on the possibility of a more rapidly accreting mBH with 𝐿bol > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='01𝐿Edd, which would correspond to a mass 𝑀BH ≲ 104𝑀⊙ for both sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Nevertheless, even though our data do not exclude the possibility of mBHs, Occam’s razor proba- bly suggests that the simplest and most likely scenario is that these are luminous XRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='3 An Update to Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015) After considering the above multiwavelength observations, all 10 of the dwarf galaxy AGN candidates identified by Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015) (via hard X-ray emission) now have sufficient spatial resolution to determine if the X-ray sources indeed reside in galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Our study reduces their number of AGN candidates to 7–8 (adopting an AGN definition that requires nuclear sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' It is very unlikely that any of these 7–8 nuclear sources are chance alignments with fore- ground/background X-ray emitting objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Adopting the hard (2-10 keV) X-ray fluxes and X-ray position error circles of the nuclear can- didates from Table 2 of Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015), and replacing the X-ray flux and positional uncertainty of Mrk 1434 X-N with the values presented here, the Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2003) cosmic X-ray background predicts only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='003 sources to fall within the nuclei of the eight possible nuclear mBH candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Obtaining 7–8 viable AGN can- didates is a significant result, considering that (a) the Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015) dwarf galaxy survey was archival and therefore serendipitous in nature, and (b) the three dwarf galaxies with follow-up presented here represent three of their most unlikely AGN candidates (given the poor spatial resolution of their archival Chandra data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Lemons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' (2015) found X-ray sources in 19 galaxies total (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', the re- maining 11–12 galaxies host off-nuclear X-ray sources, most likely XRBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thus, if a luminous X-ray source is detected in a dwarf galaxy, our study (very roughly) implies a 30–40% chance10 that it could be a nuclear mBH, which supports the viability of using X-ray surveys to identify mBHs in low-mass galaxies, as long as the survey is performed with sufficient sensitivity and spatial res- olution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We stress the importance of high spatial-resolution X-ray observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' For example, Mrk 1434 was previously identified as an AGN from an XMM-Newton survey (Birchall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2020), while our higher spatial-resolution Chandra observation clearly resolves 10 This number is an upper limit, and it neglects biases inherent to an archival/serendipitous survey, which is out of the scope of this paper to quantify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023) 10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Thygesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' the ‘nuclear’ X-ray source into two distinct sources (and even then, it remains unclear if either source is indeed an accreting mBH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 5 SUMMARY AND CONCLUSIONS We have presented a multiwavelength study of three nearby dwarf galaxies that host ULXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Two galaxies in our sample, SDSS J1213 and SDSS J1221, each contain single off-nuclear X-ray sources that we suspect are luminous XRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The third galaxy, Mrk 1434 hosts two X-ray sources separated by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='′′8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' The northern source (Mrk 1434 X-N) also displays extended radio emission at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz and point-like radio emission at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' It remains unclear if the X-ray sources in Mrk 1434 are XRBs or AGNs (especially Mrk 1434 X-N), although either scenario is intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' If XRBs, then the combined X-ray luminosity of both sources is larger than expected for a galaxy with Mrk 1434’s star formation rate and (low) metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Futhermore, the extended radio emission at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz could then represent the most luminous ‘ULX bubble’ ever observed in the radio, although we stress that the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='5 GHz radio emission can also be attributed entirely to star formation within the galaxy, or to an AGN jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Regardless of the correct scenario, we find that the line emission from He ii in Mrk 1434 is inconsistent with a nebula being powered by the central X-ray source, unless the central source underwent a period of higher activity several hundred years ago, or if the the nebula is shock ionised by an outflow that is an order of magnitude more powerful than yet observed from a ULX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' If Mrk 1434 X-N is an AGN, then the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='0 GHz radio emission may represent a compact synchrotron jet from a low- luminosity AGN power by an mBH with 𝑀BH ≈ 4 × 105𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' We conclude by stressing the importance of high spatial-resolution observations when performing multiwavelength searches for mBHs in dwarf galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the anonymous referee for helpful comments that im- proved this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Support for this work was provided by the National Aeronautics and Space Administration through Chandra Award Number GO6-17079X issued by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' This research is based on observa- tions made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', under NASA contract NAS 5–26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' These observations are as- sociated with program HST-GO-14356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Support for Program No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' HST-GO-14356 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the As- sociation of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' RMP and JDP acknowledge support from the National Science Foundation under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2206123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' RS acknowledges support from grant number 12073029 from the National Natural Science Foundation of China (NSFC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' AER acknowledges support provided by NASA through EPSCoR grant number 80NSSC20M0231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' GEA is the recipient of an Aus- tralian Research Council Discovery Early Career Researcher Award (project number DE180100346) funded by the Australian Govern- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' This research made use of Astropy,11 a community-developed core Python package for Astronomy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2013, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article are available in the Chandra Data Archve under ObsIDs 18059, 18060, and 18061 (https://cda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='edu/chaser/), in the Barbara A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Mikulski Archive for Space Telescopes under program ID 14356 (dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='17909/3bxp-zt07), and in the National Radio Astronomy Observatory Data Archive under programs 14-358 and SH0563 (data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='nrao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' REFERENCES Allen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Groves B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Dopita M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Sutherland R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Kewley L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2008, ApJS, 178, 20 Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2013, A&A, 558, A33 Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2018, AJ, 156, 123 Baldassare V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Reines A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Gallo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2015, ApJ, 809, L14 Baldassare V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Geha M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2020, ApJ, 896, 10 Barth A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Ho L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Rutledge R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Sargent W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2004, ApJ, 607, 90 Bellovary J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Cleary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Munshi F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Tremmel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Christensen C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Brooks A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Quinn T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2019, MNRAS, 482, 2913 Bellovary J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2021, MNRAS, 505, 5129 Berghea C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Johnson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Secrest N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Dudik R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Hennessy G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', El-khatib A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2020, ApJ, 896, 117 Birchall K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Watson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Aird J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2020, MNRAS, 492, 2268 Blandford R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Königl A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 1979, ApJ, 232, 34 Brorby M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Kaaret P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Prestwich A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2014, MNRAS, 441, 2346 CASA Team et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='02276 Cann J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2021, ApJ, 912, L2 Cash W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 1979, ApJ, 228, 939 Chen C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2017, ApJ, 837, 48 Chilingarian I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Katkov I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Zolotukhin I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Grishin K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Beletsky Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Boutsia K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Osip D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2018, ApJ, 863, 1 Cseh D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2012, ApJ, 749, 17 Dashyan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Silk J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Mamon G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Dubois Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Hartwig T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2018, MNRAS, 473, 5698 Dickey J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Lockman F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 1990, ARA&A, 28, 215 Dopita M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Sutherland R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 1995, ApJ, 455, 468 Douna V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Pellizza L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Mirabel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Pedrosa S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2015, A&A, 579, A44 Evans I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2010, ApJS, 189, 37 Falcke H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Körding E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Markoff S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2004, A&A, 414, 895 Fender R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Belloni T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Gallo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2004, MNRAS, 355, 1105 Feng H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Soria R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2011, New Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 55, 166 Filippenko A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Ho L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2003, ApJ, 588, L13 Fruscione A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2006, in Society of Photo-Optical Instrumentation En- gineers (SPIE) Conference Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 62701V, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='671760 Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2018, A&A, 616, A1 Gallo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Sesana A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2019, ApJ, 883, L18 Garmire G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Bautz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Ford P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Nousek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Ricker Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2003, in Truemper J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Tananbaum H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', eds, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' SPIEVol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 4851, X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='. pp 28–44, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='461599 Gilfanov M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2004, MNRAS, 349, 146 Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Strader J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Ho L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2020, ARA&A, 58, 257 Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2021, ApJ, 917, 17 Grimm H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Gilfanov M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Sunyaev R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2003, MNRAS, 339, 793 11 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='astropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='org MNRAS 000, 1–11 (2023) X-ray Sources in Dwarf Galaxies 11 Gültekin K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', King A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Cackett E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Nyland K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Miller J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Di Matteo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Markoff S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Rupen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2019, ApJ, 871, 80 Hack W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Dencheva N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Fruchter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2013, in Friedel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Astro- nomical Society of the Pacific Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 475, Astronomical Data Analysis Software and Systems XXII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 49 Hao C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Kennicutt R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Johnson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Calzetti D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Dale D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Mous- takas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2011, ApJ, 741, 124 Heger A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Fryer C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Woosley S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Langer N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Hartmann D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2003, ApJ, 591, 288 Ho L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2008, ARA&A, 46, 475 Ho L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Kim M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2016, ApJ, 821, 48 Hong J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', van den Berg M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Schlegel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Grindlay J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Koenig X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Laycock S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Zhao P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2005, ApJ, 635, 907 Houck J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Denicola L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2000, in Manset N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Veillet C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Crabtree D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', eds, Astronomical Society of the Pacific Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 216, Astronomical Data Analysis Software and Systems IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 591 Inayoshi K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Visbal E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Haiman Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2020, ARA&A, 58, 27 Ito H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Kino M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Kawakatu N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Isobe N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Yamada S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2008, ApJ, 685, 828 Kaaret P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Feng H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Roberts T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2017, ARA&A, 55, 303 Kashibadze O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Karachentsev I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Karachentseva V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2020, A&A, 635, A135 Kennicutt R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Evans N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2012, ARA&A, 50, 531 Kormendy J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Ho L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2013, ARA&A, 51, 511 Kraft R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Burrows D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Nousek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 1991, ApJ, 374, 344 Lehmer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Alexander D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Bauer F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Brandt W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Goulding A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Jenkins L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Ptak A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Roberts T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2010, ApJ, 724, 559 Lehmer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2019, ApJS, 243, 3 Lehmer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2021, ApJ, 907, 17 Lemons S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Reines A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Plotkin R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Gallo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2015, ApJ, 805, 12 Linden T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Kalogera V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Sepinsky J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Prestwich A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Zezas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Gallagher J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2010, ApJ, 725, 1984 Long K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Blair W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Winkler P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Lacey C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2020, ApJ, 899, 14 Longair M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 1994, High energy astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='2: Stars, the galaxy and the interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' 2 Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Ripamonti E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Zampieri L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Colpi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Bressan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2010, MN- RAS, 408, 234 Martínez-Palomera J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Lira P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Bhalla-Ladd I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Förster F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Plotkin R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2020, ApJ, 889, 113 Merloni A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Heinz S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', di Matteo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2003, MNRAS, 345, 1057 Mezcua M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Domínguez Sánchez H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2020, ApJ, 898, L30 Mezcua M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Civano F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Fabbiano G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Miyaji T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Marchesi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2016, ApJ, 817, 20 Mezcua M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Civano F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Marchesi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Suh H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Fabbiano G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Volonteri M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2018, MNRAS, 478, 2576 Mezcua M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Suh H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Civano F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2019, MNRAS, 488, 685 Miller B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Gallo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Kelly B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Treu T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Woo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Baldas- sare V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2015, ApJ, 799, 98 Mineo S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Gilfanov M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Lehmer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Morrison G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Sunyaev R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2014, MNRAS, 437, 1698 Mirabel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Dijkstra M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Laurent P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Loeb A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Pritchard J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2011, A&A, 528, A149 Moon D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Harrison F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Cenko S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Shariff J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2011, ApJ, 731, L32 Moran E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Shahinyan K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Sugarman H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Vélez D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Eracleous M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2014, AJ, 148, 136 Moretti A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Campana S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Lazzati D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Tagliaferri G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2003, ApJ, 588, 696 Murphy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2011, ApJ, 737, 67 Nguyen D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2019, ApJ, 872, 104 Orienti M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Prieto M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2010, MNRAS, 401, 2599 Pakull M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Angebault L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 1986, Nature, 322, 511 Pakull M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Soria R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Motch C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2010, Nature, 466, 209 Pardo K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2016, ApJ, 831, 203 Plotkin R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Markoff S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Kelly B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Körding E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Anderson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2012, MNRAS, 419, 267 Ponnada S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Brorby M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Kaaret P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2020, MNRAS, 491, 3606 Prestwich A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Tsantaki M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Zezas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Jackson F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Roberts T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Foltz R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Linden T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Kalogera V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2013, ApJ, 769, 92 Reines A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Volonteri M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2015, ApJ, 813, 82 Reines A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Sivakoff G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Johnson K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Brogan C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2011, Nature, 470, 66 Reines A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Geha M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2013, ApJ, 775, 116 Reines A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Condon J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Darling J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2020, ApJ, 888, 36 Ricarte A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Natarajan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2018, MNRAS, 481, 3278 Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2006, AJ, 131, 2766 Roberts T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Colbert E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2003, MNRAS, 341, L49 Rosen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2016, A&A, 590, A1 Ross N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2013, ApJ, 773, 14 Sargent A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Johnson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Reines A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Secrest N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', van der Horst A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Cigan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Darling J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2022, ApJ, 933, 160 Sartori L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Schawinski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Treister E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Trakhtenbrot B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Koss M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Shirazi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Oh K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2015, MNRAS, 454, 3722 Schaerer D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Fragos T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Izotov Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2019, A&A, 622, L10 Schramm M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2013, ApJ, 773, 150 Schutte Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Reines A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2022, Nature, 601, 329 Senchyna P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Stark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Mirocha J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Reines A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Charlot S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Jones T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Mulchaey J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2020, MNRAS, 494, 941 Shirazi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Brinchmann J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2012, MNRAS, 421, 1043 Simmonds C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Schaerer D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Verhamme A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2021, A&A, 656, A127 Soria R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Pakull M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Broderick J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Corbel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Motch C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2010, MNRAS, 409, 541 Soria R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Pakull M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Motch C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Miller-Jones J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Schwope A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Urquhart R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Ryan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2021, MNRAS, 501, 1644 Trebitsch M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Volonteri M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Dubois Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Madau P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2018, MNRAS, 478, 5607 Urquhart R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2019, MNRAS, 482, 2389 Vanden Berk D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2001, AJ, 122, 549 Volonteri M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2010, A&ARv, 18, 279 Volonteri M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Habouzit M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Colpi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2021, Nature Reviews Physics, 3, 732 Weaver R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', McCray R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Castor J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Shapiro P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Moore R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 1977, ApJ, 218, 377 Woo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Cho H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Gallo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Hodges-Kluck E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Le H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Shin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Son D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Horst J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2019, Nature Astronomy, 3, 755 Yang Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Xie F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Yuan F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Zdziarski A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Gierliński M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Ho L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Yu Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2015, MNRAS, 447, 1692 York D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2000, AJ, 120, 1579 Younes G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Porquet D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Sabra B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Reeves J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2011, A&A, 530, A149 Zhao Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Gao Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Gu Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2013, ApJ, 764, 44 Zibetti S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Charlot S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Rix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2009, MNRAS, 400, 1181 de Zotti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Massardi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Negrello M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', Wall J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=', 2010, A&ARv, 18, 1 This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAzT4oBgHgl3EQfXPz4/content/2301.01317v1.pdf'} diff --git a/SNFGT4oBgHgl3EQfgygC/vector_store/index.pkl b/SNFGT4oBgHgl3EQfgygC/vector_store/index.pkl new file mode 100644 index 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source simulation is the task where +two (or more) parties share some correlated randomness and +use local operations and no communication to convert this +into some target correlation. Wyner’s seminal result showed +that asymptotically the rate of uniform shared randomness +needed for this task is given by a mutual information +induced measure, now referred to as Wyner’s common in- +formation. This asymptotic result was extended by Hayashi +in the quantum setting to separable states, the largest class +of states for which this task can be performed. In this +work we characterize this task in the one-shot setting using +the smooth entropy framework. We do this by introducing +one-shot operational quantities and correlation measures +that characterize them. We establish asymptotic equipartition +properties for our correlation measures thereby recovering, +and in fact strengthening, the aforementioned asymptotic +results. In doing so, we consider technical points in one- +shot network information theory and generalize the support +lemma to the classical-quantum setting. We also introduce +entanglement versions of the distributed source simulation +task and determine bounds in this setting via quantum +embezzling. +I. INTRODUCTION +At the core of information theory is the notion of +correlation. This is present even in Shannon’s initial +work, as one can view both source and channel coding +as the limits of establishing perfect correlation between +inputs and outputs [1]. Another task where correlation +plays a central role is that of distributed source simulation, +which asks how much correlation must be provided to +two spatially-separated and non-interacting parties so +that they can generate a target joint distribution pXY up +to some tolerated error ε (see Fig. 1). It was established +by Wyner that when the tolerated error is expressed in +terms of regularized relative entropy, the rate of gener- +ating i.i.d. copies of pXZ is given by +R = +min +qXYZ:qXZ=pXZ & X−Y−Z I(XZ : Y)q +:= C(X : Z)p , +(1) +where X − Y − Z denotes a short Markov chain [2]. +The correlation measure on the right hand side is of- +ten referred to as ‘Wyner’s common information.’ The +achievability of this result was established by Wyner’s +introduction of what is now referred to as a soft-covering +lemma. +Ian George and Eric Chitambar are with the Department of Elec- +trical and Computer Engineering, University of Illinois at Urbana- +Champaign, Urbana, Illinois, 61801, USA, email: igeorge3@illinois.edu. +Min-Hsiu Hsieh is with Hon Hai (Foxconn) Research Institute, +Taipei, Taiwan, email: min-hsiu.hsieh@foxconn.com. +�qy +Copy +ΦY→X +X +Z +ΨY′→Z +Y′ +Y +≈ε +pXZ +Fig. 1: The general structure of distributed classical +source simulation. The seed is copied and distributed, +forming a source of shared randomness between Al- +ice and Bob. Then Alice and Bob apply local maps +to construct an output distribution �qXZ, which should +approximate the target distribution pXZ. +Since Wyner’s initial work, which was inspired by +prior work by G´acs-K¨orner [3] and Witsenhausen [4], +many variations of common information and refine- +ments of distributed source simulation have been con- +sidered. Liu et al. extended distributed source simulation +to multipartite joint distributions [5]. Yu and Tan consid- +ered R´enyi divergences and total variation as measures +of error, which in particular led to them establishing +a strong converse under the total variation measure +[6], [7]. Winter extended to the case where there is an +eavesdropper, the adversarial setting, so that it relates +to key distillation [8]. Chitambar et al. compared this +adversarial setting to the collaborative alternative [9]. +Moreover, Chitambar et al. related the adversarial setting +to quantum entanglement manipulations [10], [11] at +one point using the G´acs-K¨orner common information +[3], which also is relevant in round complexity of state +transformations [12]. Cuff established a general tradeoff +region between Wyner common information and the +classical reverse Shannon theorem when simulating a +classical channel [13]. There is also the related problem +of exact common information introduced by Kumar et +al. which considers that the target state p⊗n +XY is exactly +constructed but allows for variable-length codes [14]. It +was established by Yu and Tan that the exact common +information corresponds to the common information +with the error measured in terms of the max-divergence +[15]. Furthermore, the soft-covering lemma used for +achievability was established for error measured in total +variation by Hayashi [16] and Cuff [17] and in the +one-shot setting for R´enyi divergence error by Yu and +arXiv:2301.04301v1 [quant-ph] 11 Jan 2023 + +2 +Tan [18]. We refer the reader to Yu and Tan’s recent +monograph for further details on the history of common +information in the classical setting [19]. +However, the bulk of this previous research has been +restricted to the classical common information. In the +quantum setting, there are fundamental differences. In- +deed, one of the key features of quantum mechanics, and +consequently resources of quantum information theory, +is quantum entanglement, which is a form of correlation +that classical systems do not admit [20]. One way entan- +glement has been presented is as a quantum analogue +of perfect correlation, the latter being the underlying +resource in distributed classical source simulation. How- +ever, it has been shown that one cannot freely transform +entanglement by local processing like one can with +classical shared randomness without communication be- +tween the distributed parties [21]. That is to say, the fully +entangled equivalent of distributed source simulation is +not possible. Moreover, distributed parties who can only +communicate classically cannot generate entanglement +from shared randomness [22], and so the task cannot be +extended to using the shared randomness to generate an +entangled state. +Nonetheless, there is still space for a quantum ex- +tension. Specifically, the set of quantum distributions +which are not entangled are the separable states, which +is a strict superset of classical distributions. Separable +states can be decomposed into a convex combination +of product (quantum) distributions and consequently +should be able to be simulated in a distributed manner. +Indeed, Hayashi extended Wyner’s result to all separable +states in terms of trace norm error [23]. In doing so, he +introduced a novel covering lemma for quantum states +that does not presume i.i.d. structure, i.e. is a one-shot +characterization, although a quantum covering lemma +with i.i.d. structure had been previously introduced by +Ahlswede and Winter for considering different tasks +[24]. +This has remained the state of the quantum extension +of Wyner’s common information for more than a decade. +However, very recently there has been improvements +upon the quantum one-shot soft-covering lemma. In +particular, its error exponents have been characterized +in terms of R´enyi mutual information measures [25] +and its second-order asymptotics were established via +a characterization in terms of hypothesis testing mutual +information [26]. This would suggest the possibility of +establishing a one-shot version of Wyner’s common in- +formation for separable states which recovers Hayashi’s +asymptotic extension. +Summary of Results +In this work, we extend distributed source simula- +tion and Wyner’s common information to the one-shot +quantum setting for separable states using the smooth +entropy framework. In doing so, we introduce new +measures of operational tasks Cε +F, Cε +U,F, �Cε +F, where Cε +U,F +is the one-shot version of Wyner’s quantity restricted to +uniform shared randomness, Cε +F relaxes the requirement +that the randomness be uniform, and �Cε +F allows one +to distribute entangled states that are indistinguishable +from a Markov chain. We introduce new one-shot corre- +lation measures to extend Wyner’s common information. +Specifically, we introduce measures based on the max +mutual information [27], [28] Cε +max, �Cε +max as well as one +induced by the hypothesis testing divergence, Cε +h. We +establish achievability and converse bounds on the one- +shot distributed-source simulation and related tasks in +terms of these measures which hold in general if the +state is separable. +Theorem. +(One-Shot +Distributed +Source +Simulation +Bounds) Let ε ∈ (0, 1). Let ρAC +∈ +D(A ⊗ C) such +that ∥ρ − �ρ∥1 ≤ ε for some separable state �ρAC. Let +ε1, ε2 ∈ (0, 1) satisfy 2ε1 + ε2 < ε. Then, +C +√ε +max(A : C)ρ ≤ Cε +F +≤ Cε +U,F ≤ Cε1 +max(A : C)ρ + κ(ε2) , +where κ(ε2) is a constant that scales as o(n) for ρ⊗n +AC. +Theorem. (Variations of Distributed Source Simulation +Bounds) Let ρAC be a separable state. If δ ∈ (0, ε) and +η ∈ ( 7 +8ε, ε), then +�C +√ε +max(A : C)ρ ≤ �Cε +F(A : C)ρ ≤ �C +√ε−η +max (A : C)ρ + o(n) , +or if δ′ ∈ (0, 1 − ε), +�C1−ε−δ′ +h +(A : C)ρ + o(n) ≤ �Cε +F(A : C)ρ +≤ �C1−ε/8 +h +(A : C)ρ + o(n) , +where o(n) always represents a term that scales as o(n) +for ρ⊗n +AC. +We also establish a (weak) asymptotic equipartition +property (AEP) for the correlation measures induced by +max divergence for separable states, which do not follow +from pre-existing asymptotic equipartition properties. +Theorem. (AEP for One-Shot Wyner Common Information) +Let ρAC be separable. Then, +lim +ε→0 lim +n→∞ +� 1 +nCε +max(An : Cn)ρ⊗n +� += C(A : C)ρ, +lim +ε→0 lim +n→∞ +� 1 +n +�Cε +max(An : Cn)ρ⊗n +� += C(A : C)ρ. +These AEPs allow us to not only recover Hayashi’s +asymptotic extension of the Wyner common information, +but establish something stronger which says that even +if the source were not uniform or we allowed entan- +glement assistance but restricted to be approximately +indistinguishable from a Markov chain, asymptotically +these all achieve the same rate if the error is required to +go to zero. +Theorem. (All Variations Have Same Vanishing Error Rate) +C(A : C)ρ = lim +ε→0 lim +n→∞ +� 1 +nCε +F(An : Cn)ρ⊗n +� + +3 += lim +ε→0 lim +n→∞ +� 1 +nCε +U,F(An : Cn)ρ⊗n +� += lim +ε→0 lim +n→∞ +� 1 +n +�Cε +F(An : Cn)ρ⊗n +� +. +We also show how all the stated results extend beyond +bipartite setting. +Finally, as these results cannot be extended to an en- +tangled setting, we present entangled equivalents: ‘em- +bezzling source simulation’ and a variation that allows +for shared randomness. Both versions use embezzlement +[29] to simulate the target state (See Fig. 2). In particular, +we establish nearly tight upper and lower bounds in the +case shared randomness is included. +Theorem. Let ρ ∈ D(A ⊗ C) and ε ∈ [0, 1). Then +log(Entε +A:C(ρ)) ≤Cε +SREE,S(A : C)ρ +≤1 +ε log(EntA:C(ρ)) . +Moreover, both the lower bound and upper bound can +be shown to be nearly tight while. +σA′C′ +A′ +C′ +ΦA′→AA′ +ΦC′→CC′ +A +C +A′ +C′ +ρAC +≈ε +⊗ +σA′C′ +(a) Embezzling Source Simulation +σA′C′ +A′ +C′ +ΦA′→AA′ +ΦC′→CC′ +A +C +A′ +C′ +ρAC +≈ε +(b) Entangled Source Simulation +Fig. 2: Entangled state versions of distributed source +simulation. Grey lines represent allowed correlations of +either classical or quantum mechanical nature. (b) Em- +bezzling source simulation where the auxiliary state is +required to be output approximately decoupled from the +simulated state. (c) Entangled Source Simulation where +the auxiliary system may be arbitrarily correlated with +the target distribution. This in particular allows for the +use of classical correlation. +In establishing these listed results, we establish tech- +nical tools which may be of independent interest or +use. We establish a generalization of the support lemma +(Lemma 6) so as to establish cardinality bounds, which +could be of use in other quantum settings with an auxil- +iary classical random variable. We discuss the difficulty +of using one-shot measures induced by hypothesis test- +ing when an auxiliary random variable is used, which +we expect is relevant in other quantum network settings +in the smooth entropy framework as well. We also +prove various properties of one-shot mutual informa- +tions which exemplify the importance in choosing which +one-shot mutual information one uses. In particular, +we establish a property of max mutual information as +originally defined in [27] that allows for straightforward +cardinality bounds of an auxiliary classical random vari- +able, but the alternatives discussed in [28] do not satisfy +the same property. +Organization of the Paper +The rest of the paper is organized as follows. In +Section II we establish basic notation used throughout +this work. In Section III we present the necessary back- +ground on one-shot information measures. In Section +IV we introduce one-shot distributed source simulation, +its variants, and its impossibility for entangled states. +In Section V, we introduce the one-shot correlation +measures to capture distributed source simulation, the +smooth max common information, and its variants. We +also establish basic properties of these measures. In +particular, we straightforwardly generalize the support +lemma so as to establish cardinality bounds on these +measures and show that there are cases where these +measures are NP-hard to compute. In Section VI we +establish achievability for distributed source simulation +and its variants in terms of their respective measures by +modifying the one-shot soft-covering results of [26] to +be in terms of max mutual information. In Section VII +we establish converses for distributed source simulation +and its variants in terms of their one-shot correlation +measures. This along with the previous section estab- +lishes tight (to first order) characterization of these tasks +in terms of smooth mutual information quantities. In +Section VIII, we establish weak asymptotic equipartition +properties (AEPs) for our correlation measures, which do +not simply follow from previous results. By establishing +this weak AEP, we are able to both recover Hayashi’s +asymptotic extension of Wyner common information as +well as generalize it. In Section IX, we explain how +these results are straightforward to generalize to source +simulation of more than 2 parties and clarify certain +properties in this setting noted in [5]. In Section X we +present the entangled state versions of distributed source +simulation and establish bounds on the resources for this +task. Finally, in Section XI, we re-summarize what we +have presented and discuss avenues for future work. +II. NOTATION +Our notation largely follows standard texts to which +we refer the reader for further details [22], [30]. We will +denote finite alphabets by calligraphic roman letters at +the end of the alphabet, e.g. X , Y, .... The probability +simplex over finite alphabet X is denoted P(X ). We +talk of complex Euclidean spaces (CESs), equivalently +finite Hilbert spaces, denoted by capital roman letters, + +4 +e.g. A ≡ C|X |. Given a CES A, we define the follow- +ing classes of operators. The space of endomorphisms +is denoted L(A). The space of Hermitian operators is +Herm(A) ≡ {X ∈ L(A) : X = X∗} where ·∗ is the +conjugate transpose. The space of positive semidefinite +operators is Pos(A) ≡ {X∗X : X ∈ L(A)}, where we +remind the reader X ∈ Pos(A) if and only if all of the +eigenvalues are non-negative. We will often use P, Q to +denote generic positive semidefinite operators. +a) Quantum States: The space of quantum states, +referred to as density matrices is D(A) ≡ {ρ ∈ Pos(A) : +Tr(ρ) = 1}, where Tr(·) is the trace. Often times we will +have density matrices defined on tensor product spaces, +so we will add subscripts to the state to specify, e.g. +ρAB ∈ D(A ⊗ B). We say a state ρA is pure if there exists +a vector |ψ⟩ ∈ A such that ρA = |ψ⟩⟨ψ|. A quantum +state is classical if it is diagonal in the standard basis, +e.g. ρ = ∑x∈Σ p(x)Ex,x where p ∈ P(Σ) and {Ex,y}x,y∈Σ +form the standard basis for L(X). We denote classical +registers with capital roman letters at the end of the +alphabet, e.g. X, Y, ... to help distinguish from quantum +states. A classical-quantum (CQ) state has the following +convenient decomposition: ρXB = ∑x∈X p(x) |x⟩⟨x| ⊗ ρx +B, +where p ∈ P(X ) and {ρx +B}x∈X are referred to as the +conditional states. The space of sub-normalized states is +given by D≤(A) ≡ {ρ ∈ Pos(A) : Tr(ρ) ≤ 1}. +As alluded to in the introduction, quantum states +can be partitioned into states that are and aren’t en- +tangled, which are known as separable. A ‘bipartite’ +quantum state ρ ∈ D(A ⊗ B) is separable if and only if +there exists a finite alphabet X , probability distribution +p ∈ P(X ), and sets of density matrices {σx +A}x∈X ⊂ +D(A), {τx +B}x∈X ⊂ D(B) such that +ρ = ∑ +x∈X +p(x)σx +A ⊗ τx +B . +(2) +Any state that is not separable is entangled. We denote +the space of separable states in D(A ⊗ C) as SepD(A : C). +b) Quantum Channels: A map E : L(A) → L(B) is +a quantum channel, E ∈ C(A, B), if it is a completely +positive (CP) and trace preserving (TP) map. A par- +ticularly important class of channels for this work are +the classical-to-quantum or preparation channels. Given +CESs A ≡ C|X |, B ≡ C|Y|, a preparation channel Eprep +may be defined by its action +Eprep(W) := ∑ +x∈X +⟨x| W |x⟩ ρx +B , +where W ∈ L(A) and {ρx +B}x∈X ⊂ D(B). This means the +channel projects the input into the standard basis and +then prepares a state dependent on the outcome, i.e. if +given |x⟩⟨x|, it prepares ρx +B, hence its name. +c) Metrics on States: Lastly, we consider two metrics +on states. The first is the trace distance, which is the +quantum generalization of the total variation in the sense +that it captures the distinguishability between the two +quantum states. +Definition 1. Given ρ, σ ∈ D(A), the trace distance is +TD(ρ, σ) := 1 +2∥X − Y∥1 , +where ∥ · ∥1 is the Schatten one-norm. +We will also consider the purified distance P(·, ·) +which we refer the reader to [31] for detailed informa- +tion. For our purposes it will be sufficient to note that +for ρ, σ ∈ D(A), +TD(ρ, σ) ≤ P(ρ, σ) ≤ +� +2TD(ρ, σ) . +(3) +For the reader’s intuition we note that the purified +distance is greater than the trace distance because it +operationally measures the maximal distinguishability +between purifications of ρ, σ rather than the distinguisha- +bility of the states themselves. We also define the follow- +ing equivalences for notational convenience: +P(ρ, σ) ≤ ε ⇔ ρ ≈ε σ +(4) +2TD(ρ, σ) ≤ ε ⇔ ρ ≈TD +ε +σ +(5) +Lastly, we note, as they are metrics, they act as good +distance measures on quantum states. As such, we can +use them to measure the distance between a state and +the set of separable states, which will be useful later. +Definition 2. For ρAB ∈ D(A ⊗ B), the trace distance of +entanglement is defined as +ET(A : B)ρ := +inf +σAB∈SepD(A : B) TD(ρ, σ) , +and the purified distance of entanglement, EP(A : C)ρ, +is defined identically with TD(·, ·) replaced with the +purified distance P(·, ·). +III. ONE-SHOT ENTROPIES AND INFORMATION +MEASURES +We now summarize the background on one-shot en- +tropies and their relation to asymptotic entropies as +necessary for this work. We note that a secondary aspect +of this work is to highlight what it means to determine +the ‘correct’ one-shot mutual information in our setting +as there are a myriad of them and because the previous +work [26] initiates such a discussion. For this reason, this +section is longer as it motivates why there are so many +to begin with. +For P, Q ∈ Pos(A), the relative entropy is defined as +D(P||Q) = Tr[P log P] − Tr[P log Q] +when Supp(P) ⊆ Supp(Q) and is otherwise infinite. +This recovers the KL divergence if P, Q are classical. +From this definition one can extend the standard classical +information quantities from the KL divergence to the + +5 +quantum setting [30]. In particular, one can recover the +many equivalent definitions of mutual information: +I(A : B)ρ :=D(ρAB||ρA ⊗ ρB) += +min +σB∈D(B) D(ρAB||ρA ⊗ σB) += +min +τA∈D(A) +σB∈D(B) +D(ρAB||τA ⊗ σB) . +(6) +However, in the one-shot setting there are more en- +tropic measures to use. In the classical setting, this is +predominantly handled by the information spectrum di- +vergence [32]. In the quantum setting there are multiple +options. While there is the extension of the information +spectrum divergence [33], +Dε +s(ρ||Q) := sup +γ∈R +{Tr[ρ{ρ ≤ exp(γ)Q}] ≤ ε} , +(7) +there +is +also +the +smooth +entropy +calculus +which +‘smooths’ entropic quantities defined in terms of the +quantum +max-relative +divergence +and +duality +(See +[31]).1 The max-relative divergence is defined as +Dmax(P||Q) := inf{λ ∈ R : P ≤ exp(γ)Q} . +(8) +One appealing property of the max-relative divergence +is that it benefits from a particularly general data pro- +cessing inequality (DPI). +Proposition 1. For any CP map E, Dmax(E(P)||E(Q)) ≤ +Dmax(P||Q). +Proof. Let γ⋆ be the optimizer for Dmax(P||Q). As E is +CP, P ≤ exp(γ⋆)Q implies E(P) ≤ exp(γ⋆)E(Q). As +Dmax is defined as an infimum and we have just shown +γ⋆ is feasible for Dmax(E(P)||E(Q)), this completes the +proof. +Just as mutual information is defined from relative +entropy, the max mutual information is defined from the +max divergence. However, in general the three equiva- +lent definitions given in (6) are inequivalent for Dmax and +thus there are three possible max mutual informations. +Definition 3. For ρAB ∈ D≤(A ⊗ B), +I↑↑ +max(A : B)ρ := Dmax(ρAB||ρA ⊗ ρB) +(9) +I↑ +max(A : B)ρ := +min +σB∈D(B) Dmax(ρAB||ρA ⊗ σB) +(10) +I↓ +max(A : B)ρ := +min +τA∈D(A) +σB∈D(B) +Dmax(ρAB||τA ⊗ σB) . +(11) +We note (10) was introduced in [35] while (9) and (11) +were introduced in [28]. Our notation differs from both +of these to make the relation +I↑↑ +max ≥ I↑ +max(A : B) ≥ I↓ +max(A : B) , +explicit and also to align with the notation of more recent +work, namely [36]. Moreover, we note that we could +1This framework was initially introduced in the classical setting [34], +but it has not gained the same level of popularity in the classical +setting, which is why we write as if it is only in the quantum setting. +have defined the one-shot mutual informations for any +R´enyi mutual information I(A : B) (any mutual infor- +mation defined using a R´enyi divergence [31]). Certain +results in this work are presented in such a manner for +generality. +It is presumably clear that in general I↑↑ +max, I↑ +max, and +I↓ +max could behave quite differently. Indeed, as we will +see, there are certain properties that make I↑ +max preferable +for our purposes. Nonetheless, Ciganovic et al. [28] +showed that when smoothed, over a large parameter +range of smoothing, these measures become asymptoti- +cally equivalent. We summarize this in sufficient detail +to introduce the notion of smoothed measures and in- +troduce the result we will need later. +Definition 4. Let ρAB ∈ D≤(A ⊗ B) and ε ∈ (0, 1). Then +Bε(ρ) := {�ρ ∈ D≤(A ⊗ B) : P(ρ, �ρ) ≤ ε} , +where P(·, ·) is the purified distance metric. +Definition 5. For x ∈ {↑↑, ↑, ↓}, the smoothed max- +mutual information is +Ix,ε +max(A : B)ρ := +min +�ρ∈Bε(ρ) Ix,ε +max(A : B)�ρ . +Lemma 1. ([28, Theorem 3]) Let ρAB ∈ D(A ⊗ B), ε > 0, +ε′ ≥ 0, then +I↑↑,ε+2 +√ +ε+ε′ +max +(A : B)ρ ≤ I↑,ε′ +max(A : B)ρ + g(ε) , +where +g(x) := log +� +2(1 − x) + 3 +(1 − x)(1 − +√ +1 − x2 +� +. +If the previous three mutual informations were not +enough already, there is the hypothesis testing mutual +information, which was used to establish the one-shot +achievability and converse for soft-covering in [26], +which we use later in this work. The ε-hypothesis testing +divergence is defined as +exp(−Dε +h(ρ||σ)) +:= +inf +0≤Λ≤1{Tr[Λσ] : Tr[Λρ] ≥ 1 − ε } . +(12) +We note that the hypothesis testing divergence has a +natural relationship with the Petz divergence of order +zero [37]: +lim +ε→0 Dε +h(ρ||σ) = D0(ρ||σ) = Tr +� +Πρσ +� +, +(13) +where Πρ is the projector onto the support of ρ. Like +Dmax, this divergence satisfies a data-processing inequal- +ity +Dε +h(ρ||σ) ≥ Dε +h(E(ρ)||E(σ)) +∀E ∈ C(A, B) . +Moreover, one can define the hypothesis testing mutual +information using it: +Ih(A : B)ρ := Dε +h(ρAB||ρA ⊗ ρB) . +(14) + +6 +We note that in principle this could be defined as the +I↑↑ +h (A : B) mutual information. However, it is the only +one necessary for this work and to the best of our +knowledge the only hypothesis testing mutual informa- +tion defined, so for simplicity we keep this notation. +This completes our introduction to one-shot mutual +informations. We note that one can convert between +Dε +s(ρ||σ), Dε +max(ρ||σ), and Dε +h(ρ||σ) up to constant cor- +rection terms [33]. This in particular means that they +all are asymptotically equivalent for i.i.d. inputs ρ → +ρ⊗n, σ → σ⊗n [33], and so we should expect they can +roughly characterize the same operational tasks, though +perhaps through different proof methods. +a) One-Shot Entropies: Beyond the one-shot mutual +information, we will also need to make use of the follow- +ing standard entropies. The (conditional) von Neumann +entropy +H(A|B)ρ := − D(ρAB||1A ⊗ ρB) += max +σB∈D(B) −D(ρAB||1A ⊗ σB) , +which is used for asymptotic results and makes up the +chain rule I(A : B) = H(A) − H(A|B) [30]. Another +entropy which we will use in various definitions is the +(unconditional) quantum Hartley entropy, also known as +the zero order Petz R´enyi entropy, +H0(A)ρ := log rank(ρ) . +(15) +This may be viewed as the number of qubits (resp. bits) +one may compress the state ρ (resp. classical distribution) +in the zero error setting. Beyond the quantum Hartley +entropy, we will also need the min-entropy and max- +entropies +Hmin(A|B)ρ +:= +sup +σB∈D(B) +sup{γ ∈ R : ρ ≤ exp(−λ)1A ⊗ σB} +Hmax(A|B)ρ +:= +sup +σB∈D(B) +log F(ρAB, IA ⊗ σB) , +(16) +where +F(·, ·) +is +the +fidelity. +These +entropies +have +smoothed versions for ε ∈ (0, 1), +Hε +min(A|B)ρ := +max +�ρ∈Bε(ρ) Hmin(A|B)�ρ +Hε +max(A|B)ρ := +min +�ρ∈Bε(ρ) Hmax(A|B)�ρ . +These entropies have been studied in great detail and +in particular are known to satisfy strong AEPs, which +means for any ε ∈ (0, 1), +lim +n→∞ +� +Hε +min(An|Bn)ρ⊗n +� += H(A|B) , +and likewise for Hε +max. We refer the reader to [31] for +proofs and further details with regards to this topic. +What is important to know for our purposes is that the +chain rule for mutual information was extended for I↑,ε +max +in [28]. In particular, Ciganovic et al. established over +some parameter range of smoothing, +H≈ε +min(A)ρ − H≈ε +min(A|B)ρ +≲ I↑,ε +max(A : B)ρ +≲ H≈ε +max(A) − H≈ε +min(A|B)ρ . +(17) +A straightforward modification of this result to the entire +parameter range implies a strong AEP for for I↑,ε +max from +the strong AEP for the smoothed entropies (see the +appendix and in particular Proposition 27). +IV. ONE-SHOT DISTRIBUTED SOURCE SIMULATION +We are now in a position to introduce one-shot dis- +tributed source simulation. We will also introduce natu- +ral variants and show these variants are incomparable. +We will also show how (to arbitrary error) distributed +source simulation can only hold for separable states due +to its locality constraints and how this relates to the +structure of (short) Markov chains. +In principle, one-shot distributed source simulation is +‘simply’ constructing some target state ρAC up to some +tolerated error ε from some shared randomness and local +operations. We note that we say shared randomness as +the original classical randomness, pX = ∑x p(x) |x⟩⟨x|, +is copied resulting in perfectly correlated randomness +which we denote +χ|p +XX′ = ∑ +x +p(x) |x⟩⟨x|X ⊗ |x⟩⟨x|X′ . +(18) +See Fig. 3 for visualization. +pX +Copy +ΦX→A +A +C +ΨX′→C +X′ +X +≈ε +ρAC +χ|p +XX′ +�ρAC +Fig. 3: Diagram of distributed source simulation of a +quantum state. After the copying procedure (at the light +blue line), the two parties share a perfectly correlated +state χ|p +XX′. After their local processing (at the dark blue +line), the parties share a state �ρAC which should be +approximately the target state ρAC. +Following Wyner, our interest is in how much shared +randomness, measured in number of bits, is necessary. +The measure of how much randomness is necessary we +refer to as one-shot correlation of formation, mirroring +terminology from resource theories, such as the entan- +glement of formation [38], [39]. Like Wyner, we consider +the case of uniform randomness, but we also consider +the case where we let the randomness be non-uniform. +We now formalize all of this, starting with the notion of +approximate simulation. + +7 +Definition 6. Let σB ∈ D(B) and D(X ⊗ B) ∋ ρXB = +∑x p(x) |x⟩⟨x| ⊗ ρx +B be a CQ state. We say ρXB is a +ε−simulation of σB if ∥ ∑x p(x)ρx +B − σB∥1 ≤ ε. We denote +this ρXB ∼ε σB. +Definition 7. Let σAC ∈ D(A ⊗ C) and D(AXC) ∋ +ρAXC = ∑x∈Σ p(x)ρx +A ⊗ |x⟩⟨x| ⊗ ρx +B where p ∈ P(Σ). We +say ρAXC is a ε-distributed source simulation of σAC if +∥ ∑x p(x)ρx +A ⊗ ρx +C − σAC∥1 ≤ ε. +We could now use the definition of ε-distributed +source simulation to define one-shot correlation of for- +mation. However, we believe it is clearer to reduce the +definition to being in terms of a Markov chain and the +definition of ε-simulation. +Proposition 2. A ε−distributed source simulation of σAC +is a A − X − C Markov chain that is a ε−simulation of +σAC. +To establish this, we need the following theorem which +will be relevant for much of this work. +Theorem 2. ([40]) The following are equivalent +1) ρABC is a (short) quantum Markov chain (QMC), +denoted A − B − C or ρA−B−C. +2) There exists a CPTP map R : B → B ⊗ C such that +(idA ⊗ R)(ρAB) = ρABC. +3) There exists a CPTP map R : B → A ⊗ B such that +(R ⊗ idC)(ρBC) = ρABC. +4) I(A : C|B) = 0, where I(A : C|B) is the conditional +mutual information. +5) There exists a finite alphabet J such that there exists +a decomposition of B = ⊕j∈J bL +j ⊗ bR +j such that +ρABC = +� +j∈J +ρAbL +j ⊗ ρbR +j C . +All of these results in effect say that the A space and +C space are independent so long as one has access to the +B space. It is then trivial to prove Proposition 2. +Proof of Proposition 2. There are various ways to prove +ρAXC is a QMC. For our case, note R : |x⟩⟨x|B �→ +|x⟩⟨x|B ⊗ ρx +C and same idea for R. Then letting σAC act +as σB, and ρAXC as ρXB in Definition 6 completes the +proof. +With this established, we can define our correlation of +formation measures. We note if we write a minimization +over �ρA−X−C, this means we restrict to optimizing over +QMC with a classical register X, and if we write π as a +register, it means that the marginal on that register is the +uniform distribution on a (classical) space X. Later, we +will write minimization over A − X − C without being +a superscript when it is clear with respect to what state +the QMC is being considered. Lastly, we always define +the minimization over an empty set to be +∞. +Fig. 4: Feasible sets for correlation of formation opera- +tional quantities. For Cε +F, one smooths the initial state, +depicted by the purple ball, and then considers the +Markov chain extensions, depicted by black dots, of any +state in the smoothed ball. For �Cε +F, one considers all the +Markov chain extensions of ρ and then smooths each +of them. For �Cε +F, one considers all the Markov chains +extensions of ρ, smooths each, and then restricts to the +Markov chains in each ball resulting in a non-convex set, +depicted by the green jagged set. +Definition 8. Let ε ∈ [0, 1] and ρAC ∈ D(AC). The +correlation of formation is +Cε +F(A : C)ρ +:= +min +�ρA−X−C +� +H0(X)�ρ : ∥�ρAC − ρAC∥1 ≤ ε +� +. +(19) +Moreover, the one-shot uniform correlation of formation +is defined as +Cε +U,F(A : C)ρ +:= +min +�ρA−π−C +� +H0(X)�ρ : ∥�ρAC − ρAC∥1 ≤ ε +� +. +(20) +Note that by construction, the one-shot uniform cor- +relation of formation may be viewed as one-shot dis- +tributed source simulation. This can be seen as follows. +By definition we are considering QMC �ρA−π−C, so the +X register is uniform and there exist local channels R, R +that prepare the A and C registers from X. This means + +AXC +AC +B(p) +C(A:C)p +AXC +Quantum Markov +Extensions +AC +C(A :C)p +AXC +QuantumMarkov +Extensions +AC +C(A: C))8 +X is the uniform randomness input, R, R are the local +channels in the distributed source simulation, and by +definition of H0, given in (15), H0(X) is measuring the +minimum number of bits of uniform randomness neces- +sary. Thus we have defined our operational quantity of +primary interest. +We also define the related operational quantities of +“entanglement-assisted correlation of formation” and +“private correlation of formation.” +Definition +9. Let ε +∈ +(0, 1). The one-shot secret +entanglement-assisted correlation of formation is defined +as +�Cε +F(A : C)ρ +:= min +�ρAXC +� +H0(X)�ρ : ∃ρA−X−C : +∥�ρAXC − ρA−X−C∥1 ≤ ε +� +. +(21) +Definition 10. The one-shot private correlation of for- +mation is defined as +�Cε +F(A : C)ρ +:= +min +�ρA−X−C +� +H0(X)�ρ : ∃ρA−X−C : +∥�ρA−X−C − �ρA−X−C∥1 ≤ ε +� +. +(22) +First, we explain the choice of names. The secret +entanglement-assisted correlation of formation is named +such because �ρAXC is not restricted to being a Markov +chain, so it is of the general form �ρAXC = ∑x∈X �p(x)�ρx +AC, +which in general is distributing possibly entangled states +�ρx +AC according to distribution �p ∈ P(X ). Moreover, this +entanglement-assistance can also be viewed as ‘secret’ in +the following sense. The quantity measures the necessary +resources to, except with probability ε, win a ‘game’ +where the distinguisher tries to discriminate the output +� +rhoAXC and the set of Markov chain extensions of ρAC. +In other words, it means from a black box perspective +it is the minimal amount of classical information to leak +so that �ρAXC is ε-indistinguishable from a zero-error dis- +tributed source simulation with leaked classical register, +ρA−X−C. The one-shot private correlation of formation +captures the black box indistinguishability while also +requiring it be an actual distributed source simulation +implementation. These operational quantities have the +difference in their respective feasible sets depicted in +Fig. 4 and their corresponding tasks depicted in Fig. +5.We note the operational quantities �Cε +F and �Cε +F and +their corresponding tasks are somewhat less natural +than distributed source simulation. They however will +be characterized by a natural correlation measure and +so serve as a useful comparison to the correlation of +formation and standard distributed source simulation as +we will now begin to show. +We briefly note some relationships between these op- +erational quantities. +pX +Copy +ΦX→A +A +C +ΨX′→C +≈ε +ρAC +(a) Correlation of Formation, Cε +F +pX +ΞX→ACX +A +C +X +≈ε ρA−X−C +(b) Secret EA Correlation of Formation, �Cε +F +pX +Copy +ΦX→A +A +C +ΨX′→C +≈ε ρA−X−C +X +(c) Private Correlation of Formation, �Cε +F +Fig. 5: The three correlation of formations and their cor- +responding tasks: (a) Correlation of formation captures +the amount of randomness for distributed source simu- +lation. (b) Secret entanglement-assisted correlation of for- +mation captures the amount of broadcasted randomness +needed such that there exists a set of (possibly entangled) +states {�ρx +AC}x∈X to distribute so that the entire output +is approximately indistinguishable from a distributed +source simulation implementation of the target state. (c) +Private correlation of formation measures the amount of +randomness needed for a distributed source simulation +protocol so that if the randomness were to be leaked it +would be approximately indistinguishable from an exact +distributed source simulation of the target state with +leaked classical information. +Proposition 3. For ρAB ∈ Sep(A : B) and ε ∈ [0, 1), +{Cε +F, �Cε +F} ≤ �CF +& +Cε +F ≤ Cε +U,F . +Proof. That Cε +F ≤ Cε +U,F follows from it being a more +restricted optimization. That �Cε +F ≤ �Cε +F is also because it is +a more restricted optimization. That Cε +F ≤ �Cε +F is because +if �ρ ≈TD +ε +ρ, then as trace norm monotonically decreases +under partial trace, +ε ≥∥ TrX(�ρ − ρA−X−C)∥1 +≥∥∑ +x +p(x)�ρx +A ⊗ �ρx +C − ρAC∥1 ⇒ �ρ ∼ε ρ +where the implication is by definition of ε-simulation. +This completes the proof. +We stress in particular the following two points im- +plicit in the above proposition. First, one would expect +there are cases where the inequality Cε +F ≤ �Cε +F is large + +9 +because in �Cε +F the distinguisher has access to the X +register. That is to say, in general one expects there to +be a non-trivial difference between +min +�ρA−X−C +∥TrX(�ρA−X−C) − ρAC∥1 +and +min +�ρA−X−C +ρA−X−C +∑ +x +∥�p(x)�ρx +A ⊗ �ρx +C − p(x)ρx +A ⊗ ρx +C∥1 . +Second, intuitively Cε +F and �Cε +F do not seem comparable +as the resources allowed are fundamentally different. We +now show this to be formally true by establishing the +quantities are finite under different conditions. We begin +with the following lemma that will be used extensively +in this work. +Lemma 3. A quantum state ρAC ∈ D(A ⊗ C) has a +Markov chain extension ρA−B−C : TrB ρA−B−C = ρAC +if and only if ρAC ∈ SepD(A : C). +Proof. We prove both directions. +(⇒) Let ρAC ∈ SepD(A : C). Then by definition, ρAC = +∑x p(x)ρx +A ⊗ ρx +C for some finite alphabet X , p ∈ P(X ), +and sets of quantum states {ρx +A}x∈X , {ρx +C}x∈X . It follows +∑x p(x)ρx +A ⊗ |x⟩⟨x| ⊗ ρx +C is a QMC extension as R(·) := +∑x Tr[|x⟩⟨x| ·]ρx +C ⊗ |x⟩⟨x| and same idea for R. +(⇐) If one has a QMC ρA−B−C, then by Theorem 2, +ρA−B−C = � +j∈J p(j)ρAbL +j ⊗ ρbR +j C where the ρAbL +j , ρbR +j C +are density matrices on the respective subspaces. It +follows +TrB(ρA−B−C) = +� +j∈J +p(j) TrbL +j (ρAbL +j ) ⊗ TrbR +j (ρbR +j C) += ∑ +j∈J +p(j)ρj +A ⊗ ρj +C ∈ Sep(A : C) , +where we have used that for B = ⊕j∈J bL +j ⊗ bR +j , we may +decompose TrB = ⊕j∈J TrbL +j ⊗bR +j and then TrbL +j ⊗ TrbR +j will +distribute across the tensor product of the states decom- +position as is normal. This completes the proof. +We also will make use of the following straightforward +proposition. +Proposition 4. For all δ ∈ (0, 1), if there exists a QMC +ρA−X−C, there exists �ρA−π−C such that ρAC ≈TD +δ +�ρAC. +Proof. Let ρA−X−C = ∑x∈X p(x)ρx +A ⊗ |x⟩⟨x| ⊗ ρx +C. Then +there exists finite alphabet X ′ and {kx} ⊂ {0, ..., |X ′|} +such that ∑x kx = |X ′| and ∑x∈X |kx/|X ′| − p(x)| ≤ δ +since the rationals are dense in [0, 1]. Define �ρA−π−C = +1 +X′ ∑x′∈X ′ ρ f (x′) +A +⊗ |x′⟩⟨x′| ρ f (x′) +C +where f : X ′ → X is such +that for each x ∈ X , it maps kx of the elements of X ′ +to that x. Note this means �ρx′ +A = ρ f (x′) +A +for all x′ ∈ X ′ +not that we are actually indexing by f (x′) which would +erase information we need to preserve the Markov chain +condition. It follows by our construction +∥TrX′(�ρA−π−C) − TrX(ρA−X−C)∥1 += +����� ∑ +x∈X +kx +|X ′|ρx +A ⊗ ρx +C − ∑ +x∈X +p(x)ρx +A ⊗ ρx +C +����� +1 +≤ +����� ∑ +x∈X +kx +|X ′| |x⟩⟨x|⊗2 − ∑ +x +p(x) |x⟩⟨x|⊗2 +����� +1 +=∑ +x +���� +kx +|X ′| − p(x) +���� ≤ δ , +the first inequality uses the data-processing inequality +for the one-norm and the preparation channels Φ1(·) = +Tr[|x⟩⟨x| ·]ρx +A and Φ2(·) = Tr[|x⟩⟨x| ·]ρx +C. +Theorem 4. Let ρ ∈ D(A ⊗ C). +1) For ε ∈ [0, 1], �Cε +F(A : C), �Cε +F(A : C) are finite if and +only if ρAC ∈ SepD(A : C). +2) In contrast, Cε +F(A : C)ρ +is finite if and only if +ET(A : C)ρ ≤ 2ε. Likewise, Cε +U,F(A : C)ρ if and only if +there exists δ ∈ (0, ε) such that ET(A : C)ρ ≤ 2(ε − δ). +Proof. We begin with Item 1. The only if direction is +immediate because if there exists an appropriate QMC +extension, then the state is separable by Lemma 3. Like- +wise, by Lemma 3, if ρAC ∈ SepD(A : C), there exists a +QMC extension of ρAC, ρA−X−C = ∑x p(x)ρx +A ⊗ |x⟩⟨x| ⊗ +ρx +C for some finite alphabet X . This state is then feasible +for �Cε +F, �Cε +F. Note none of this has relied on the choice of +ε. +For Cε +F, note that if ET(A : C)ρ ≤ 2ε, there exists +�ρ ∈ SepD(A : C) such that �ρ ≈TD +ε +ρ. Then there exists +a QMC extension of �ρAC by Lemma 3 and this QMC +extension satisfies �ρA−X−C ∼ε ρAC. If no such state +exists, then there is no QMC extension by Lemma 3. For +the uniformity claim, note that if such a δ > 0 exists then +by Proposition 4 we may do the same argument again. +This completes the proof. +Therefore we see Cε +F is a fundamentally distinct mea- +sure from �Cε +F, �Cε +F as when they are finite is not even in +agreement. We will discuss this further after we have +introduced the one-shot correlation measures. +V. ONE-SHOT CORRELATION MEASURES +As is standard in information theory, the ultimate goal +is to establish bounds in terms of entropic quantities. +In our case we would particularly like to bound the +correlations of formations with entropic quantities that +recover Wyner’s common information in the asymptotic +limit. That is, our hope is to construct bounds in terms +of an entropic quantity C? that on i.i.d. inputs satisfies +lim +n→∞ +� 1 +nC?(An : Cn)ρ⊗n +� += +min +A−X−C I(AC : X)ρ , +where we note the right hand side is Wyner’s common +information as per (1). Motivated by this end goal, we +introduce the max common information, its smoothed +versions, and establish certain properties of it on sin- +gle copies of a state. In doing so we will present the +generalized support lemma. Moreover, we introduce the +hypothesis testing common information and show it + +10 +satisfies the same wanted properties. We end the section +by establishing that these introduced correlation mea- +sures act as one-shot converses to the distributed source +simulation tasks. +We claim the natural one-shot extension of Wyner’s +common information is the following. +Definition 11. Given ρAC ∈ D(A : C), the Max Common +information is +Cmax(A : C)ρ := +min +A−X−C I↑ +max(AC : X)ρ . +(23) +There are three choices we should justify: (i) the +restriction to a classical register, (ii) the minimization, +and (iii) the choice of I↑ +max rather than another version. +First we justify why we restrict to A − X − C Markov +chains in the above definition. We could of course argue +it is because we are interested in an operational interpre- +tation that will require the classical register. However, +this restriction can be made without loss of generality +for any mutual information that satisfies DPI as we now +show. +Lemma 5. Let ρABC be a A − B − C Markov chain. Then +there always exists a A − X − C Markov chain ρ′ +AXC such +that ρ′ +AC = ρAC and Ix(AC : B)ρ ≥ Ix(AC : X)ρ′, where +I is any mutual information measure that satisfies data- +processing and x ∈ {↑↑, ↑, ↓} following Definition 3. +Proof. We prove it for I↓(AC : X) case as it is then +straightforward to see the same proof method will hold +for the other cases. Let ρABC be a A − B − C Markov +chain. Then ρB = ⊕xp(x)ρbLx ⊗ ρbRx . Define the map +E : B → BX as E(·) = ∑j ΠB +x · ΠB +x ⊗ |x⟩⟨x|X where +{Πx} are the mutually orthogonal projectors onto the +subspaces bL +x ⊗ bR +x . Define ρ′ +ABXC := E(ρABC). Then it +follows +I↓(AC : B)ρ =D(ρABC||τAC ⊗ σB) +≥D(ρ′ +ABXC||τAC ⊗ σBX) +≥D(ρ′ +AXC||τAC ⊗ σX) +≥ +min +τ′∈D(AC) +σ′∈D(X) +D(ρ′ +AXC||τ′ +AC ⊗ σ′ +X) +=I↓(AC : X)ρ′ , +where the first inequality is DPI using E, the second +inequality is DPI using the partial trace on the B space, +and in both cases the map only acts on one side of +conditioned tensor product, and the final is just re- +minimizing. We can also guarantee σ′ +X is classical by +DPI and pinching on the computational basis. Moreover +ρ′ +AXC is A − X − C Markov chain extension of ρAC +trivially as E never acted on the AC spaces and its +recovery maps are just state preparations maps, e.g. +x �→ TrbLx (ρAbLx ). This completes the proof. +Remark 1. It is worth noting why the above isn’t proven +to be an equality. When one converts the B register to +a classical X register, they destroy any entanglement +between A (resp. C) and bL +x (resp. bR +x ). To recover this +entanglement, one needs to apply the recovery map, e.g. +ρX → ρB +R +−→ ρBC +R +−→ ρABC . +However, to preserve the form of mutual information, +you can only act on the B space, so it is not possible to +evaluate this directly. +This justifies the restriction to a classical register. To +address the second and third question, we will need to +introduce the generalized support lemma. +A. The Generalized Support Lemma +The convex cover method using the support lemma +is a standard method in classical information theory for +bounding the cardinality of an auxiliary variable [41, Ap- +pendix C]. This bounding is useful as then the space be- +ing optimized over is finite-dimensional and thus closed. +Effectively the support lemma implies that if all the +relevant constraints may be written as the expectation +of a function over conditional distributions according to +the auxiliary variable, then the auxiliary variable can be +made finite. Our extension states the same but replacing +conditional distributions with conditional states from an +appropriate state space. This generalization is necessary +for our purposes as we will need to consider a function +of conditional states, appearing in Lemma 7, that does +not reduce to functions of their spectra. +We begin by stating Carath´eodory’s theorem of which +the support lemma may be viewed as a corollary. +Proposition 5. (Fenchel-Eggleston-Carath´eodory) Any +point in a convex closure of a connected compact set +R ∈ Rd can be represented as a convex combination of +at most d points in R. +Now we present the general lemma. +Lemma 6. (Generalized Support Lemma) Let W be an +arbitrary set. Let the generalized state space S (A) be a +connected, compact subset of Pos(A) and {ρw}w∈W ⊆ +S (A) be a set of generalized conditional states. Let +{ fj}j∈[k] ⊂ S (A) → R be continuous. Then for any +Borel measure µ of W, there exists q ∈ P(W′) where +|W′| ≤ k and {σw′}w′∈W′ ⊂ S (A) such that +� +W fj(ρw)µ(dW) = ∑ +w′∈W′ +fj(σw′)q(w′) . +Proof. Our proof is a direct extension of the proof given +for the traditional support lemma by Csiszar and K¨orner +[42, Lemma 15.4]. By assumption, S (A) is a compact, +connected subset. By assumption each fj is continuous, +so the image of fj(ρ) is both connected and compact. De- +fine F(ρ) := ( f1(ρ), ..., fk(ρ)) and the set R := {F(ρ), ρ ∈ +S (A)}, which is connected and compact as product +preserves these properties. Moreover, defining +rj ≡ +� +W fj(ρw)µ(dW) +∀j ∈ [k] , + +11 +we have (r1, ..., rk) is an element of the convex closure +of R. Therefore, applying Proposition 5, there exist k +points of R, which we denote {F(σj)}j∈K, along with +a distribution q ∈ P([k]) such that +(r1, ..., rk) = ∑ +j∈[k] +q(j)F(σj) . +By definition of F(ρ), we can conclude rj = ∑j q(i) f (σi) +for all i ∈ [k]. Letting W′ = [k] completes the proof. +First, we note the reason we talk in terms of gen- +eralized state spaces that are subsets of the positive +density matrices is that, for example, this would allow +for cardinality bounds on subnormalized states which +may be of use given smoothed measures. In fact, any +closed convex subset of the (possibly subnormalized) +density matrices would work, since it would be compact +and all convex sets are (path-)connected. Moreover, the +generalized state space may be the product space of +closed convex subsets of the (possibly subnormalized) +density matrices, since the product of connected, com- +pact sets are also connected and compact, which is useful +for network settings.2 Note you can also restrict to the +support of some state space as needed. +We now can use this to bound the cardinality of the +the max common information which will also explain +why we chose I↑ +max. This relies on various technical +lemmas about mutual informations which we relegate +to an appendix and summarize here. +Lemma 7. Let ρABX be classical on X. Then, +exp +� +I↑ +max(AC : X)ρ +� += ∑ +x +px exp(Dmax(ρx +AC||ρAC)) , +where the right hand side is a continuous function over +the state space restricted to ρAC’s support. Moreover, +I↑↑, I↓ do not seem to satisfy such an averaging state- +ment. +Proof. One uses Corollary 2 in the appendix with the re- +placements A → AC, B → C. To simplify the RHS term, +note, as defined in the appendix, Imax(ρx +ACC||ρAC) = +Dmax(ρx +AC||ρAC) for each x. Finally taking an exponential +gets the form in the lemma. That I↑↑ +max(AC : X) and +I↓ +max(AC : X) do not seem to satisfy such an averaging +statement may be seen from Propositions 19 and 20 +respectively. +It is this previous lemma that justifies our choice +of I↑ +max in defining Cmax(A : C) as it is this property +of I↑ +max we now use to establish cardinality bounds +for minA−X−C Iε +max which in turn establishes cardinality +bounds for Cmax(A : C) in the case ε = 0. +Lemma 8. Let ρAC ∈ D(A ⊗ C) and ε ≥ 0. Then +without loss of generality minA−X−C Iε +max(AC : X)ρ may +2Formally, your state space is then S (A) × S (B) and functions +which are defined on Pos(A ⊗ B) would be extended to being on the +state space via composition with the map (ρ, σ) �→ ρ ⊗ σ. +be restricted such that |X| ≤ |A|2|C|2 + 3. Moreover, in +the case ρXZ is fully classical, then |Y| ≤ |X||Z| + 3. +Proof. We begin with the non-smooth case. Let {ρx +A|C := +ρx +A ⊗ ρx +C}x∈X ⊂ D≤(A) ⊗ D≤(C) with distribution pX +be a solution. Let {Mk}k∈K be the elements of a min- +imal informationally complete POVM on the space, +i.e. |K| = |A|2|C|2. Consider the following functions: +{ fk(·) := Tr +� +Mk · M∗ +k +� +for k ∈ [|K| − 1], fAC(·) := H(·), +fA(·) +:= +H(TrC(·)), fC(·) +:= +H(TrA(·)), fobj(·) +:= +exp(Dmax(·||ρAC)). Then, +Tr(MkρACM∗ +k) = ∑ +x +p(x) fk(ρx +A|C) = Pr[Outcome k] +∑ +x +p(x) fAC(ρx +A|C) = H(AC|X) +∑ +x +p(x) fA(ρx +A|C) = H(A|X) +∑ +x +p(x) fC(ρx +A|C) = H(C|X) +∑ +x +p(x) fobj(ρx +A|C) = exp +� +I↑ +max(AB : X)ρ +� +, +where we have used Lemma 7. Then by applying Lemma +6 for the state space D≤(A) ⊗ D≤(C) restricted to the +support of ρAB so that fobj is continuous, there exists a +distribution q ∈ P(X′) where |X′| ≤ |A|2|C|2 + 3 and +states {σx +A|C := σx +A ⊗ σx +C}x∈X ⊂ D≤(A) ⊗ D≤(C) that +equals the left hand side of each equation above and +thus the right hand side. As { fk}k are all but one POVM +element of an IC POVM, these constraints guarantee +that the output state is indeed ρAC. As I(A : C|X) = +H(A|X) + H(C|X) − H(AC|X), the next three guaran- +tee the Markov chain condition is satisfied. The last +constraint guarantees the max mutual information is +satisfied. This completes the proof for the non-smoothed +case. In the smoothed case, we know the optimizer will +be classical on the classical register (Proposition 17), so +we apply the non-smoothed proof to its optimizer. For +the classical case, we can replace the informationally +complete POVM with a measurement that only checks +|X||Z| − 1 elements of the joint distribution. This com- +pletes the proof. +Thus we have completed our formal justification for +our choice of definition of Cmax(A : C). +B. Smoothed Max Common Information +Having established the definition of max common in- +formation, we will want to define its smoothed version. +Note however we now reach a complication: there seems +to be two ways of smoothing Cmax. We could smooth +the state we start with, ρAC, or we could replace the +max mutual information with the smooth max mutual +information, which in effect is like smoothing the QMC +extension. We will define both. However, before doing +so, we recall that smoothing includes subnormalized +states and so in principle we need to generalize the +notion of QMC extensions, though ultimately we won’t +need to. + +12 +Definition 12. Let ρAC ∈ D≤(A ⊗ C). The set of its +quantum Markov chain extensions (QMC extensions) is +QMC(ρAC) +:={(R ◦ R)(ρB) : (TrB ◦R ◦ R)(ρB) = ρAC} , +(24) +where R ∈ C(B, BC), R ∈ C(B, AB). +With this established, we define our two versions of +smoothed max common information. +Definition 13. Let ρAC ∈ D(A ⊗ C). Let ε ∈ [0, 1]. We +define the ε-max common information as +Cε +max(A : C)ρ := +min +�ρ∈Bε(ρ) Cmax(A : C)�ρ , +(25) +where this may be expanded as +min +�ρ∈Bε(ρ) +min +σ∈QMC(�ρ)I↑ +max(AC : X)�ρ , +where the register being X follows from Lemma 5. +Definition 14. For ρAC ∈ D(A ⊗ C), we define the +alternative smoothed max common information as +�Cε +max(A : C)ρ = +min +A−X−C Iε +max(AC : X)�ρA−X−C , +where the restriction to a classical register is without loss +of generality as proven in Proposition 22 in the appendix. +We first need to justify the minimizations in these +definitions. For the alternative smoothed max common +information it immediately follows from our cardinality +bounds, Lemma 8. For the smoothed max common +information we need two steps: first, our cardinality +bounds relied on the entropic characterization of a QMC +but it is not clear the best way to generalize this to +subnormalized states, therefore we need the optimizer +of min�ρA−X−C Imax(AC : X)�ρ to be a normalized state. This +can be shown to always be the case. We establish this in +the appendix (Proposition 23) as this property is known +for I↑,ε +max and so the proof is effectively the same. The +second point is then just noting the smoothing ball is +compact. Therefore these two points combined justify the +minimizations in the definition of Cε +max. +Beyond these points, the reader may have noticed that +our notation for the two smoothed max common infor- +mations are suggestive of the notation for the correlation +of formation notations. Indeed, we will ultimately have +the correlation measures characterize the formation task +with the same notation. We can see this alignment very +easily by showing that Cε +max and �Cε +max are incomparable +in general in the same fashion as in Theorem 4 except +in terms of purified distance of entanglement due to our +definition of smoothing ball. +Proposition 6. Given ρAC ∈ D(A ⊗ C), Cε +max(A : C) is +finite if and only if EP(A : C)ρ ≤ ε. In contrast, �Cε +max is +finite if and only if ρAC ∈ SepD(A : C). +Proof. For the first claim, note that if EP(A : C)ρ ≤ ε, +then there exists a separable state �ρ such that �ρ ∈ Bε(ρ) +Fig. 6: The difference between SMCI and alternative +SMCI’s feasible sets visually. For SMCI, Cε +max, the +Markov chain extensions of the smoothed state are con- +sidered. In contrast, the alternative SMCI, �Cε +max considers +the smoothed state of each Markov chain extension. Note +these are effectively the same feasible sets as depicted for +two of the formation quantities, Cε +F and �Cε +F, in Fig. 4. +and by Lemma 3 it admits a QMC extension and thus +Cε +AC is finite. If EP(A : C)ρ > ε this argument doesn’t go +through so by Lemma 3, the value is infinite. For the +second claim, note �Cε +max is finite if and only if ρAC has +a QMC extension and by Lemma 3, this is only the case +if ρAC ∈ Sep(A : C). This completes the proof. +We also provide a visualization of the distinction of +the two measures in Fig. 6. +a) Hypothesis Testing Common Information: As noted +in the introduction, generally an alternative to results +in terms of max divergence are results from hypothesis +testing divergence. Motivated by this point, we define +the hypothesis testing common information. +Definition 15. Let ρAC ∈ D(A ⊗ C). The hypothesis +testing common information is given by +�Cε +h(A : C)ρ := +min +A−X−C Iε +h(AC : X)ρ . +The primary point to stress is that unlike with Dmax, +there is no freedom in how we smooth as the smooth- +ing comes from the definition of the measure. As the +notation would suggest, this means we can’t, at least +directly, use hypothesis testing mutual information to +characterize distributed source simulation. +Using that the zero order Petz-R´enyi divergence is a +limiting case of hypothesis testing divergence (13), we +could view Iε +h as the smoothed version of zero order +common information: +�C0(A : C)ρ +:= +min +A−X−C D0(ρA−X−C||ρAC ⊗ ρX) + +AXC +B(p) +AC +A:C) +AXC +Quantum Markov +Extensions +AC +ma13 += +min +A−X−C − log +� +� ∑ +x:p(x)>0 +p(x) Tr +� +Πρx +ACρAC +� +� +� , +where in the equality we have used (13). This can be +argued to be an intuitive correlation measure in the sense +that it intuitively measures how mutually orthogonal the +conditional states are, which will be a function of how +correlated the A and C spaces are. +Finally, we do note that in contrast to defining corre- +lation measures with I↑↑ +max, I↓ +max, we may get cardinality +bounds for �Cε +h, because it can be written as an expecta- +tion over the auxiliary random variable X: +exp +�−Iε +h(AC : X)ρACX +� = ∑ +x +p(x)Dε +h(ρx +AC||ρAC) , +which is an immediate corollary of Proposition 21 in the +appendix. +b) Data-Processing of Common Information Measures: +While we won’t need to apply it directly at any point in +this work, it is worth noting that (smoothed) common in- +formation degrades under local processing, which a cor- +relation measure should. Furthermore, thinking ahead, +it tells us that if it takes α amount of randomness to +distributed source simulate a target state ρAC and there +are local maps Φ, Ψ such that ρ′ +A′C′ = (Φ ⊗ Ψ)(ρAC), then +it can only require less randomness, i.e. ≤ α, to simulate +ρ′ +A′C′. +Proposition 7. For ε ∈ [0, 1), Cε +max, �Cε +max, �Cε +h are all +monotonic under local CPTP maps on both spaces. +Proof. We begin with Cε +max. We focus on a map being +applied on the A space, by symmetry of the argument +this will also establish the other case. Let �ρA−X−C be +the minimizer of Cε +max(A : C)ρ. Consider Φ(�ρ) where +ΦA→A′ ∈ C(A, A′). Note the resulting state is still a +QMC with recovery map Φ ◦ RX→AX and (TrX ◦Φ)(�ρ) ∈ +Bε(Φ(ρ)) by the DPI for purified distance. Then we +have, +Cε +max(A′ : C)Φ(ρ) ≤ Imax(A′C : X)Φ(�ρ) +≤ Imax(AC : B)�ρ = Cε +max(A : C)ρ , +where the first inequality is our choice of element in the +smoothed ball and QMC extension and the second is +the DPI. Note we needed local maps because we need +to preserve the QMC structure. +Similarly, let (�ρA−X−C, σAXC) ∈ QMC(ρ) × Bε(�ρ) be +the optimizers for �Cε +max(A : C)ρ. Then Φ(�ρ) is a QMC +for the same reason as earlier and Φ(σ) ∈ Bε(Φ(�ρ)) by +DPI of purified distance. Thus, +�Cε +max(A′ : C)Φ(ρ) ≤ Imax(A′C : X)Φ(σ) +≤ Imax(AC : B)σ = �Cε +max(A : C)ρ , +for the same reasons as before. The same argument holds +for �Cε +h as it also satisfies a DPI. +c) A Remark on Computability: One convenient prop- +erty of one-shot entropic quantities is that for small di- +mensions they can be solved easily as they form semidef- +inite programs [31]. However, here we also have the +constraint that we are optimizing over Markov chains. +This not only makes it hard to solve in general, but +actually means that there must be instances where it is +NP-hard to solve, because we can use whether or not the +solution is finite as a solution to the separability problem. +Proposition 8. There exist ρAC ∈ D(A ⊗ C) such that +computing Cε +max(A : C)ρ, �Cε +max(A : C)ρ is NP-hard. +Proof. As established in Proposition 6, �Cε +max is finite if +and only if ρAC is sufficiently close to the separable +state. Likewise for Cε +max but if it is within some distance +from the separable states. Therefore, if it is efficient to +compute always, then we have a method for the ability +to determine if the state is separable (or within some +distance from it). It is known determining membership +is strongly NP-hard [43], which means even with some +tolerated distance from the set of separable states β +determining whether it is in or out of the set is NP-hard. +Therefore this would be in contradiction with being able +to compute Cε +max, �Cε +max efficiently. +We note the above argument doesn’t say anything +about computational complexity when the target distri- +bution is fully classical and thus separable a priori. We +also note this is not a problem in terms of establishing +our results beyond that it means we cannot in general +compute the answer. +VI. ACHIEVABILITY FOR CORRELATION OF +FORMATIONS FROM ONE-SHOT SOFT-COVERING +LEMMA +In effect, Theorem 4 told us when the operational +task may be performed at all, i.e. when there is a finite +amount of randomness that allows distributed parties to +simulate the state (or the same idea for the related tasks). +However, it did not tell us how much randomness we +will need, which we will show the correlation measures +we have introduced will. As mentioned in the intro- +duction, the standard way of establishing the achievable +rate for distributed source simulation is a “soft-covering +lemma” and in this section we show how to relate a one- +shot soft-covering lemma to our correlation measures to +establish an achievable rate. +For intuition, we explain the name “soft-covering.” +Given the n−fold CQ state ρ⊗n +XB, by typicality only a +fraction of the conditional states ρ +xn +1 +B⊗n are necessary to +approximate ρ⊗n +B +well. Finding these xn +1 is non-trivial +however, and so a random coding approach is useful. +One can draw Mn strings xn +1 ∈ X ×n according to p⊗n +X +to consider an ensemble {ρ +xn +1 +B⊗n} where each state has +probability |Mn|−1. The soft-covering lemma says that +asymptotically if |Mn| ≥ I(X : B)ρ, this ensemble will +also approximate ρ⊗n +B +with vanishing error. A one-shot + +14 +soft-covering lemma would then be the same conceptual +idea except you draw codewords from x ∈ X rather than +xn +1 ∈ X ×n, and so the correlation measure presumably +would need to be larger than mutual information. +Recently, a one-shot soft-covering lemma for quantum +states that is optimal to second-order was established +[26]. The authors do this by establishing achievability +in terms of a mutual-information-like information spec- +trum divergence quantity and then converting to the +hypothesis testing mutual information as defined in (14). +As remarked upon in the correlation measure section, we +do not expect to capture distributed source simulation +from the hypothesis testing divergence. However, as +noted in Section III, it is equally valid to convert to +smooth max divergence from the information spectrum +divergence [33], and we suspect the authors of [26] omit- +ted this as their converse proof is less natural to convert +to smooth max mutual information. As our interest is +in a task whose converse is ultimately characterized in +terms of a max mutual information induced measure, +we will explain how to obtain the smooth max mutual +information version of the result from [26] as well as +present the hypothesis testing result. To state this, we +need three preliminaries: the definition of ‘minimal ran- +dom codebook size for ε−covering’ from [26] except we +will need to alter it up to a factor of a half.3 We will also +need to establish a few mutual information measures +that will allows us to express all bounds in a notationally +similar fashion. +Definition +16. +([26] +Altered) +Let +ρXB += +∑x∈X p(x) |x⟩⟨x| ⊗ ρx +B. The minimal random codebook +size for ε−covering, denoted by Mε(B : X)ρ is given by +inf +M∈N +� +|C| ≤ M & EC∼pX +����� +1 +|C| ∑ +x∈C +ρx +B − ρB +����� +1 +≤ ε +� +. +Note that taking the logarithm of Mε(B : X)ρ is the +number of bits needed to describe an element of the +random code. +Proposition 9. For ρ ∈ D≤(A ⊗ B), the following two +properties hold: +1) Iε +h(A : B)ρ = Iε +h(B : A)ρ. +2) Dε +s(ρAB||ρA ⊗ ρB) = Dε +s(ρBA||ρB ⊗ ρA). +Proof. 1) Q is feasible for Dε +h(ρAB||ρA ⊗ ρB) if and only +if SWAPA↔B(Q) is feasible for Dε +h(ρBA||ρB ⊗ ρA), where +SWAPA↔B is the CPTP map that re-orders the spaces. +This is because +Tr[QρA ⊗ ρB] += Tr[SWAPA↔B(Q)SWAPA↔B(ρA ⊗ ρB)] += Tr[SWAPA↔B(Q)ρB ⊗ ρA] , +(26) +3This necessity is more of a technicality. We use the purified distance +smoothing so if we consider ρ ≈TD +ε +�ρ, then ρ ≈√ε �ρ whereas if we use +trace distance directly there is a factor of 2. This is a problem as in the +converse we must convert to purified distance and √x : [0, 1] → [0, 1] +but +√ +2x : [0, 1] → [0, +√ +2] which doesn’t work with smoothing. +and the same argument for ρAB. As swapping the or- +dering of the spaces preserves positivity, we have the +feasible set for Iε +h(B : A)ρ is the same as Iε +h(A : B)ρ up +to a swapping of spaces and moreover (26) tells us the +objective function values are the same. This completes +the proof of the first claim. +2) Recall from (7) that Dε +s(ρAB||ρA ⊗ ρB) = sup{R ∈ +R| Tr +� +ρAB{ρAB ≤ 2RρA ⊗ ρB} +� ≤ ε}. Note for any R, +{ρAB ≤ 2RρA ⊗ ρB} ≡ Π is the projector onto the +non-negative eigenspace of 2RρA ⊗ ρB − ρAB. It follows +SWAPA↔B(Π) is the projector onto the non-negative +eigenspace of 2RρB ⊗ ρA − ρBA. Thus, for all R ∈ R, +Tr +� +ρAB{ρAB ≤ 2RρA ⊗ ρB} +� += Tr +� +ρBA{ρBA ≤ 2RρB ⊗ ρA} +� +. +This implies the claim as the feasible R for fixed ε is the +same for both. +With these addressed, we state the one-shot soft- +covering lemma and provide the part not proven in the +cited paper. +Lemma 9. ([26, Theorem 13 Achievability + A Little +More]) Let ρXB ∈ D(X ⊗ B) and ε ∈ (0, 1). If δ ∈ (0, ε) +and η ∈ ( 7 +8ε, ε), then +log +� +Mε(B : X)ρ +� +≤ I↑,√ε−η +max +(B : X)ρ + log(ν) + g(ε)+ +− log(1/ε − 1/8) + 3 log 3 + 7 , +(27) +where ν ≡ |spec(ρB)|, the number of distinct eigenvalues +of ρB, ε = 2 + �ε − 2 +√ +1 − �ε, and �ε = √ +ε/8 − √ε − η > 0. +Similarly, +log +� +Mε(B : X)ρ +� +≤ I1−ε/8 +h +(B : X)ρ + log +� +ν · ν′� − 2 , +(28) +where ν′ = |spec(ρX ⊗ ρB)|. +Remark 2. We note our spectral terms for hypothesis +testing mutual information differ from [26]. Our scaling +is made explicit in the proof. +Proof. The proof for Iε +h is effectively found in [26] except +that we changed the factor of half of which one has to +keep track. How to address that is shown equivalently +in proving the bound in terms of I↑↑,ε +max, so we don’t show +it explicitly. To establish the bound in terms of I↑↑,ε +max, the +proof is still primarily found in [26], we just start from +[26, Equation 4.2] where we multiply both sides by 2: +EC +����� +1 +|C| ∑ +x∈C +ρx +B − ρB +����� +1 +≤ 2 Tr +� +ρXB{PρB(ρXB) > cρX ⊗ ρB} +� ++ 2 +� +|spec(ρB)|c +|C| +, +(29) + +15 +where PρB is the pinching map on B according to ρB and +c > 0. Then let δ ∈ (0, ε) and choose +c ≡ exp +� +D1−(ε−δ)/2 +s +(PρB(ρXB)||ρX ⊗ ρB) + ζ} +� +for some small ζ > 0. The reason to define c in this man- +ner is because, by definition of ε−information spectrum +divergence (7) we obtain: +Tr +� +ρXB{PρB(ρXB) > cρX ⊗ ρB} +� += Tr +� +PρB(ρXB){PρB(ρXB) > cρX ⊗ ρB} +� +=1 − Tr +� +PρB(ρXB){PρB(ρXB) ≤ cρX ⊗ ρB} +� +<1 − (1 − (ε − δ)/2) +=(ε − δ)/2 +where the first equality is because the projected space +doesn’t change, and the strict inequality is using our +definition of c with the definition of ε−information +spectrum along with including small ζ and . Letting +|C| = ⌈|spec(ρB)|c(2/δ)−2⌉ and plugging these bounds +into (29) gets the target EC +��� 1 +|C| ∑x∈C ρx +B − ρB +��� +1 ≤ ε. +We remark so far the only change from the original +proof is that we scaled ε, δ by two. To now establish +bounds in terms of max mutual information, we will +pick ε ∈ (0, 1/2) as explained at the start of the proof +and we will let δ = ε/2. We define ε′ ≡ 1 − ε/8, δ′ ≡ ε/8. +By our choice of |C|, +log +� +Mε(B : X)ρ +� +≤ log |C| +≤ D1−(ε−δ)/2 +s +(PρB(ρXB)||ρX ⊗ ρB) + ζ ++ log |spec(ρB)| + 2 log(δ) += D1−(ε−δ)/2 +s +(PρB(ρBX)||ρB ⊗ ρX) + ζ ++ log(ν) + 2 log(δ) += Dε′−δ′ +s +(PρB(ρBX)||ρB ⊗ ρX) + ζ ++ log(ν) + 2 log(δ) +≤ D +√ +1−ε′ +max +(ρBX||ρB ⊗ ρX) + log(ν) ++ 2 log(δ) − 3 log +� +δ′� + log +� +ε′� ++ 3 log(3) + ζ += I↑↑,√ +ε/8 +max +(B : X)ρ + log(ν) − log(1/ε − 1/8) ++ 3 log 3 + 7 + ζ , +where the first inequality is by definition, the second is +our definition of |C|, the first equality is by Item 2 of +Proposition 9, the second equality is our definitions as +1 − ε/2 + ε/4 = 1 − ε/4 = ε′ − δ′, the third inequality +is [33, Eqn. 17] and where we have used PρB(ρXB) = +PρX⊗ρB(ρXB) as ρXB is classical on the X register, the +fifth is by definition and merging the logarithm terms. +As ζ > 0 was arbitrary, we let ζ tend to zero so that it +goes away. +Lastly, we wish to upper bound I↑↑,√ +ε/8 +max +by I↑,√ε−η +max +by applying Lemma 1. This means we need √ε − η < +√ +ε/8, so η ∈ ( 7 +8, 1) · ε. Then we need to solve 0 < �ε := +√ +ε/8 − √ε − η = ε + 2 +√ +ε for ε. The equation a = x + +2√x holds for x = 2 + a − √ +1 + a. Thus, ε = 2 + �ε − +2 +√ +1 + �ε. Plugging this value into Lemma 1 gets the max +mutual information bound. +To get the hypothesis testing bound, it is effectively +the same as in [26]. One starts from the second equality +in the chain of inequalities above except ε′ ≡ 1 − 3ε +16 and +δ′ ≡ ε/16. Now we will bound the spectrum divergence +term where we again use PρB(ρBX) = PρB⊗ρX(ρBX), +Dε′−δ′ +s +(PρB⊗ρX(ρBX)||ρB ⊗ ρX) +≤Dε′ +s (P||Q) − log +� +δ′� +=Dε′+δ′−δ′ +s +(P||Q) − log +� +δ′� +≤Dε′+δ′ +h +(ρXB||ρX ⊗ ρB) + log +� +ν′� − 2 log +� +δ′� +=I1−ε/8 +h +(B : X)ρ + log +� +ν′/ε2� ++ 8 , +where the first inequality is [33, Eqn. 19] and P, Q are the +Nussbaum-Szkoła distributions as discussed in [33] and +the second inequality is [33, Eqn. 28] as the inequality +always holds for θ(σ) replaced with ν′. The equalities +just use definitions of ε′, δ′. We combine this with the +bound we started from and this completes the proof. +We are now ready to prove achievability of distributed +source simulation for quantum states from the above +one-shot soft covering lemma. The idea is that if the state +ρAC can be sufficiently approximated by a separable state +�ρAC, then there is a QMC extension �ρA−X−C with recov- +ery maps R, R which will allow us to source simulate +using the random codebook rate. Note in particular this +means that for separable states, one can source simulate +to arbitrary non-zero error, but for an entangled state, +there is a fundamental limit. +Lemma 10. Let ε ∈ (0, 1). Let ρAC ∈ D(A ⊗ C) such that +2ET(A : C)ρ ̸= ε. Let ε1, ε2 ∈ (0, 1) such that 2ε1 + ε2 < ε. +Then, +Cε +U,F(A : C)ρ ≤ Cε1 +max(A : C)ρ + κ(ε2) , +where the κ(x) := log(ν) + g(x) − log(1/x − 1/8) + +3 log 3 + 7. Moreover, there exist choices of ε1, ε2 such +that this is finite whenever 2ET(A : C)ρ < ε, so it holds +for all separable states. +Furthermore, if ρAC ∈ SepD(A : C), and η ∈ (7/8ε, ε), +{ �Cε +F(A : C)ρ, �Cε +F(A : C)ρ} +≤ min{ �C +√ε−η +max (A : C)ρ + κ(ε), +�C1−ε/8 +h +(A : C)ρ + log +� +ν · ν′� − 2} . +Proof. Define 2ET(A : C)ρ = δE < ε. Let ε1 ∈ (δE/2, ε/2). +Let ε2 ∈ (0, ε − 2ε1). By our choice of ε1, it follows +that there is at least one separable state contained in +Bε1(ρ). Denote an arbitrary choice �ρAC. By Lemma 3, +we can think of �ρAC as the marginal of the CQ state +ρA−X−C = ∑x∈X p(x) |x⟩⟨x| ⊗ ρA +x ⊗ ρC +x . By definition, + +16 +Mε2(AC : X)�ρA−X−C is the minimal size such that the ex- +pectation over random codebooks is a ε−covering. It fol- +lows for this size of codebook, there must exist a a code- +book C′ with size Mε2(AC : X)ρ that is a ε2−covering. +Fix this codebook C′ and let | �X| +:= +Mε2(AC : X)�ρ. +Note that using the Markov chain extension, we have +that the ensemble is of the form {�ρA +x ⊗ �ρC +x }x∈C′ with +each element equiprobable and by Lemma 9 it satisfies +∥ 1 +|C′| ∑x∈C′ �ρA +x ⊗ �ρC +x − �ρAC∥1 ≤ ε2 . Using our definition of +ε1, ε2 along with the fact that (4) tells us purified distance +upper bounds trace distance, we have +����� +1 +|C′| ∑ +x∈C′ +�ρA +x ⊗ �ρC +x − ρAC +����� +1 +≤ 2ε1 + ε2 < ε . +(30) +We will now build the strategy using this. Let the +distributed source produce χ|π +�X �X′ and distribute the �X +register to Alice and �X′ to Bob. Now note that the +recovery maps’ actions are of the form R : |x⟩⟨x| �→ +�ρx +C ⊗ |x⟩⟨x|, R : |x⟩⟨x| �→ �ρx +A ⊗ |x⟩⟨x|. Therefore, upon +receiving their copies of |x⟩⟨x|, they may apply their +recovery maps. Ignoring their local copies of |x⟩⟨x|, the +joint state is then +1 +|C′| ∑ +x∈C′ +�ρA +x ⊗ �ρC +x . +It follows from (30) that this generated state is a +distributed source simulation to error +ε +and thus +log +� +Mε2(AC : X)�ρA−X−C +� += H0( �X)χ is an upper bound +on Cε +U,F(A : C)ρ. As the choice of Markov chain exten- +sion we picked was arbitrary, we can infimize over the +choices. However, as we established in Lemma 8, the +dimension of X may be bounded, and so this becomes +a minimization. Moreover, we could minimize over a +Bε1(ρ) as every state σ contained satisfies ∥σ − ρ∥1 ≤ +2ε1. Therefore, we have +Cε +U,F(A : C)ρ ≤ +min +�ρ∈Bε1(ρ) min +A−X−C I +√ε2−η +max +(AC : X)�ρ + κ += +min +�ρ∈Bε1(ρ) +�Cε2−η +max (A : C)�ρ + κ(ε2) , +where we used Lemma 9. To get the second upper bound +in the lemma, we note the following. First, Cmax(A : C) = +�C0 +max(A : C). Second, �Cε +max monotonically decreases as +smoothing parameter ε increases. Therefore, +Cε1 +max(A : C)ρ = +min +�ρ∈Bε1(ρ) +�C0 +max(A : C)�ρ +≥ +min +�ρ∈Bε1(ρ) +�C +√ε2−η +max +(A : C)�ρ , +which establishes this second upper bound. +For the moreover statements, note we could do the +same argument on Markov chain extensions of ρAC so +long as ρAC is separable by Lemma 3, and then we only +have a single smoothing parameter. +VII. CONVERSES FOR CORRELATION OF FORMATIONS +With the achievability established from the one-shot +cover lemma, we stress why the one-shot converse for +random coding is insufficient. The converse for the +minimal random codebook size for ε−covering gets the +minimal size to achieve an ε−covering with respect +to expectation over random codebooks. What we are +interested in is the minimal size for any codebook +to achieve distributed source simulation. As these are +distinct settings, we turn our attention to establishing +a converse to the one relevant for our purposes. This +will follow from the DPI of the measures that induce +the common informations and the relationship between +Cmax and H0 for perfectly correlated classical states χ, +which we now establish. +Proposition 10. For any distribution p ∈ P(X ), let +χ|p +XX′ = ∑x p(x) |x⟩⟨x| ⊗ |x⟩⟨x|. Then Cmax(X : X′)χ = +H0(X)χ. +Proof. First note that the optimal Markov chain is +pXX′ �X += +∑x p(x) |x⟩⟨x|⊗3 whose recovery maps are +merely copying. While intuitive, one also may make this +rigorous in the following manner. The seed will have to +be ∑y p′(y). By symmetry, the recovery maps for both +parties will be conditional distributions {q(x|y)}y∈Y. +These conditional distributions will in fact have to +be deterministic as otherwise the X and X′ spaces +won’t be perfectly correlated. Thus ∑y p′(y) can be +partitioned into sets Yxi +:= +{y +∈ +Y +: +q(xi|y) += +1} and then we can apply a coarse graining map +that takes Φ that takes Yxi +→ +xi for all i. Note +Φ(pXX′Y) += +pXX′ �X +and as it is a coarse grain- +ing map, by data-processing, +Imax(XX′ : �X)pXX′ �X += +Imax(XX′ : �X)Φ(pXX′Y) ≤ Imax(XX′ : Y)pXX′Y. Thus, as we +want to minimize Imax, this establishes we have the +optimal Markov chain. +With this established, by Corollary 2, we have +Imax(XX′ : �X)pXX′ �X += log +� +∑ +x +p(x) exp(Imax(px +XX′||pX)) +� +. +(31) +Noting px +XX′ = |x⟩⟨x|⊗2, +Imax(px +XX′||pX) += +min +q∈P(X ′) +� +λ : |x⟩⟨x|⊗2 ≤ 2λpX ⊗ ∑ +x′′ +q(x′′) +��x′�� +x′�� +� += min +� +λ : |x⟩⟨x|⊗2 ≤ 2λp(x′) +��x′�� +x′�� ⊗ |x⟩⟨x| ∀x′� += min +� +λ : 1 ≤ 2λp(x) +� +⇒ λ = p(x)−1 , +where in the first line we have just used the definition +and Proposition 17, in the second we have used it is clear +that qX = |x⟩⟨x| will decrease λ and then we are dealing +with diagonal operators so the bound must hold entry- +wise, the third is because the L.H.S. only has support + +17 +on |x⟩⟨x|⊗2 and this completes the argument. Therefore, +plugging this into (31), we +Imax(XX′ : �X)pXX′ �X += log +� +∑ +x +p(x) exp(log(1/p(x))) +� += log +� +∑ +x +1 +� += log(|X |) = Hmax(X)χ . +This completes the proof. +We now present the one-shot converses. We note the +square root in the correlation measure is due to purified +distance smoothing rather than something fundamental +per se. We begin with establishing the converse for cor- +relation of formation, i.e. distributed source simulation. +We again stress that the results hold for the entire range +of smoothing parameters so long as the state is separable. +Lemma 11. Let ε ∈ (0, 1). Let ρAC ∈ D(A ⊗ C) such that +2ET(ρ) < ε. +C +√ε +max(A : C)ρ ≤ Cε +F(A : C)ρ ≤ Cε +U,F(A : C)ρ . +In +particular, +the +bound +holds +for +any +ρAC +∈ +SepD(A : C). +Proof. Let Cε +F(A : C)ρ = n < ∞ where n is finite because +ET(ρ) < ε so there is �ρAC +∈ Sep(A : C) such that +�ρAC ≈TD +ε +ρAC. This means, P(�ρAC, ρ) ≤ √ε. Moreover, by +Lemmas 3 and 4, this means there exists �ρA− �X−C such +that ∥�ρAC − ρAC∥1 ≤ ε and | �X| = n is finite. Define +X |p with respect to the distribution defining ρX. The +recovery maps are preparation channels so that R �X→ �XA +Define R �X→A ≡ TrX ◦R and likewise for R. It follows +�ρA− �X−C = (R� +X′→C ◦ R �X→A)(χ|p +X �XX′). Then we have +C +√ε +max(A : C)ρ += +min +�ρ∈B +√ε(ρ) +min +�ρA− �X−C +Imax(AC : �X)�ρ +≤Imax(AC : �X)�ρ +=Imax(AC : �X)(R◦R)(χ) +≤Imax(XX′ : �X)χ = H0(X)χ = n = Cε +F(A : C)ρ , +where the first equality is by definition, the first in- +equality is by our choice of �ρ being feasible, the second +equality is by the equivalence established previously, +the second inequality is by data processing, and the +last steps are by Proposition 10 and our assumption +respectively. This completes the proof. +We now present the converse for the entanglement- +assisted correlation of formation and the private corre- +lation of formation. +Lemma 12. Let ρAC ∈ SepD(A : C). Then +�C +√ε +max(A : C) ≤ �Cε +F(A : C) ≤ �Cε +F(A : C) . +Moreover, for δ ∈ (0, 1 − ε), +�C1−ε−δ +h +(A : C) + 3 log(δ) − 3 log 3 − log(1 − ε) +≤ �Cε +F(A : C) ≤ �Cε +F(A : C) . +Proof. As �Cε +F ≤ �Cε +F, one can focus on �Cε +F. The proof for �Cε +F +is effectively the same as the previous except we need +ρA−X−C ∈ SepD(A : C) for the value to be finite and +then we consider a preparation channels of the form Φ : +x �→ σx +AC where ∥σA �XC − ρA− �X−C∥1 ≤ ε and | �X| = n, +but as we consider DPI for Imax this does not conflict +with Proposition 7. Finally, to get the hypothesis testing +bound, one defines √ +1 − ε = ε and applies [33, Eqn. +22]. +Finally, we may combine the results into our one- +shot bound for distributed source simulation and its +entanglement-assisted equivalent. These were reported +in the summary of results, but for simplicity we re-state +them here with proof. +Theorem 13. Let ε ∈ (0, 1). Let ρAC ∈ D(A ⊗ C) with +2ET(A : C)ρ < ε. Let ε1, ε2 ∈ (0, 1) such that 2ε1 + ε2 < ε. +Then, +C +√ε +max(A : C)ρ ≤ Cε +F +≤ Cε +U,F ≤ Cε1 +max(A : C)ρ + κ , +where κ is defined in Lemma 10. +Proof. As 2ET(A : C)ρ < ε was the constraint in both +Lemmas 10 and 11, we’ve satisfied the conditions to +apply them both. +Theorem 14. Let ρAC ∈ SepD(A : C). If δ ∈ (0, ε) and +η ∈ ( 7 +8ε, ε), then +�C +√ε +max(A : C)ρ ≤{ �Cε +F(A : C)ρ, �Cε +F(A : C)ρ} +≤ �C +√ε−η +max (A : C)ρ + κ , +or if δ′ ∈ (0, 1 − ε), +�C1−ε−δ′ +h +(A : C)ρ + γ +≤ { �Cε +F(A : C)ρ, �Cε +F(A : C)ρ} +≤ �C1−ε/8 +h +(A : C)ρ + log +� +ν · ν′� − 2 , +where γ ≡ 3 log(δ′) − 3 log 3 − log(1 − ε). +Proof. The conditions stated mean we satisfy the condi- +tions of Lemmas 10 and Lemma 12 at the same time. +VIII. RECOVERING ASYMPTOTIC RESULTS VIA A WEAK +AEP +Theorems 13 and 14 provide one-shot rates of dis- +tributed source simulation and its entanglement-assisted +counterpart. A natural, and perhaps necessary, ques- +tion would be whether we can in fact recover Wyner’s +asymptotic result, and Hayashi’s extension, from our +one-shot bounds. There are a few reasons for this ques- +tion being so important. First, while we have established + +18 +one-shot achievable and converse bounds, it is not a pri- +ori obvious these bounds will asymptotically converge +properly, though it would be surprising if they did not +given previous work on the smooth entropy framework. +Second, we have actually established bounds for an +operational task more general than Wyner’s setting. That +is, we have established upper and lower bounds for +distributed source simulation where the randomness is +not uniform, i.e. Cε +F, as well as when it is uniform, Cε +U,F. +It would therefore be interesting to determine whether +this setting has the same asymptotic rate. Finally, as +was discussed in determining the correlation measures +themselves, there seem to be further nuances in how +these correlation measures work. Indeed, as noted in the +introduction, I↑,ε +max inherits a strong AEP from its chain +rule decomposition into smooth min- and max-entropies +and their AEPs. However, for this to hold, every register +must be n−fold, but as we optimize over an extension, +we lose this structure. Therefore, any asymptotic be- +haviour we can prove is new. In this section we establish +weak AEPs for our max mutual information induced +correlation measures, as we summarize in the following +theorem. +Definition 17. For ρAC ∈ D(A ⊗ C), the Wyner Common +information is defined as +C(A : C)ρ := +min +A−X−C I(AC : X) , +where the minimization is over QMC extensions of ρAC. +Theorem 15. Let ρAC ∈ SepD(A : C). Then, +lim +ε→0 lim +n→∞ +� 1 +nCε +max(An : Cn)ρ⊗n +� += C(A : C)ρ +lim +ε→0 lim +n→∞ +� 1 +n +�Cε +max(An : Cn)ρ⊗n +� += C(A : C)ρ +Proof. These follow from Lemmas 17,18, 20, and 21. +An immediate corollary of these is that, assuming the +error is required to go to zero and the state is separable, +the rate of distributed source simulation with or without +uniform randomness, and the entanglement-assisted and +private distributed source simulation all are given by the +common information. +Theorem 16. Let ρ ∈ SepD(A : C). The rates of dis- +tributed source simulation with or without uniform ran- +domness, and the entanglement-assisted and private dis- +tributed source simulation all are given by the common +information: +C(A : C)ρ += lim +ε→0 lim +n→∞ +� 1 +nCε +F(An : Cn)ρ⊗n +� += lim +ε→0 lim +n→∞ +� 1 +nCε +U,F(An : Cn)ρ⊗n +� += lim +ε→0 lim +n→∞ +� 1 +n +�Cε +F(An : Cn)ρ⊗n +� += lim +ε→0 lim +n→∞ +� 1 +n +�Cε +F(An : Cn)ρ⊗n +� +Proof. This follows from applying Theorem 15 to The- +orems 13 and 14 by considering ρ⊗n +AB, taking the limit +as n → ∞ and then the limit as ε → 0. Note this +is because the smoothing parameters in the one-shot +bounds will go to zero as well as ε → 0, and because +the spectra of an i.i.d. state scales polynomially in n and +log(O(poly(n)))/n → 0. +We note two points in particular about this result. +First, this generalizes Wyner’s result and Hayashi’s ex- +tension as it shows it does not matter if the seeded ran- +domness was restricted to being uniform. This is in some +sense intuitive as one would expect that asymptotically +one would only need the conditional states ρ +xn +1 +A ⊗ ρ +xn +1 +C +where xn +1 is typical, and the typical set is approximately +equiprobable. Indeed, this is the intuition that allows us +to establish achievability for the AEP for Cε +max. Second, +the above result shows, at least in the vanishing error +scenario, there is no advantage to using entanglement +nor disadvantage to leaking, or rather broadcasting, the +X register to everyone. +The rest of this section proves the AEPs given in +Theorem 15. Specifically, we first establish achievability +for each AEP, though for conciseness the actual proof +of achievability for Cε +max is provided in the Appendix. +In both cases the achievability holds for all ε ∈ (0, 1). +We then prove a weak converse for each measure’s AEP. +That is, our proof methods for the converse require the +limit ε → 0. We end the section with a discussion on +what it would take to establish a strong converse. +A. Achievability for Weak AEPs +We begin with the achievability for the alternative +smooth max hypothesis testing common informations +�Cε +max, �Cε +h as the nuance with the smooth max common +information is more easily seen in contrast to this proof. +The proof for the alternative common informations in +effect follows directly from well-known second-order +expansions on i.i.d. states [33]. +Lemma 17. Let ε ∈ (0, 1). Let ρAC ∈ D(A : C). Then, +lim +n→∞ +� 1 +n +�Cε +max(A : C)ρ⊗n +AC +� +≤ C(A : C)ρAC +Proof. We prove the �Cε +max version and then state why +the other quantity is effectively the same proof. If ρAC +is entangled the bound is trivial so we assume it is +separable and has a Markov chain extension. Let τA−X−C +be the minimizer of C(A : C) +�Cε +max(An : Cn)ρ⊗n += +min +ˆρ∈QMC(ρ⊗n) +min +�ρ∈Bε( ˆρ) I↑ +max(AnCn : �X)�ρ +≤ +min +�τ∈Bε(τ⊗n) I↑ +max(AnCn : Xn)�τ +≤ +min +�τ∈Bε(τ⊗n) Dmax(�τ⊗n||τ⊗n +AC ⊗ τ⊗n +X ) + +19 +=Dε +max(τ⊗n||τ⊗n +AC ⊗ τ⊗n +B ) +=D(τABC||τAC ⊗ τB) − O(√ +n) + O(log(n)) +where the first inequality is choosing τ⊗n, the second is +because Imax minimizes over σBn and we have set it to +τ⊗n +B , the next equality is by definition, and then we have +taken the second order expansion [33, Eqn. 35]. Now +dividing by n and taking the limit, +lim +n→∞ +� 1 +n +�Cε +max(An : Cn)ρ⊗n +� +≤ lim +n→∞ +� 1 +n Dε +max(τ⊗n +ABC||τ⊗n +AC ⊗ τ⊗n +B ) +� +=D(τABC||τAC ⊗ τB) = I(AC : B)τ . +This completes the proof for �Cε +max. The proof for �Cε +h +is effectively the same by choosing the n−fold copy +of τA−X−C for the minimization over quantum Markov +chain extensions and then using the second order expan- +sion for i.i.d. states for Dε +h given in [33, Eqn. 34]. +We now note why the proof strategy given above +won’t work for Cε +max. Recall +Cε +max(A : C)ρ := +min +�ρ∈Bε(ρ) min +A−X−C I↑ +max(AC : X)�ρA−X−C . +What was crucial in the above proof was that Imax +itself was smoothed. However, as Iε +max monotonically +decreases as ε increases, we cannot smooth the Imax +in the above equation as we want an upper bound. +Therefore, if we let �ρ be the optimizer, then we lose +smoothing. As such, it seems we actually need to appeal +to (strong conditional) typicality. The proof is tedious +with little intuition so we present the result here and +sketch the proof for the intuition. The actual proof is +presented in the appendix. +Lemma 18. Let ε ∈ (0, 1). Let ρAC ∈ D(A ⊗ C). Then, +lim +n→∞ +� 1 +nCε +max(A : C)ρ⊗n +AC +� +≤ C(A : C)ρAC . +Proof Sketch, See Appendix for Full Proof. If +ρAC +̸∈ +SepD(A : C), C(A : C)ρ = +∞ and it is trivial. Therefore +we can focus on ρAC ∈ SepD(A : C). Consider ρ⊗n +A−X−C +where ρA−X−C is the minimizer of C(A : C)ρ. We now +want to use typicality to achieve I(AC : X)ρA−X−C up +to some error which we can take the limit of to make +vanish. To do this, we construct a state τn +AnXnCn which +is the strongly typical sequences of ρ⊗n +Xn on the Xn space +and the strong conditionally typical ρ|xn +1 +An , ρ|xn +1 +Cn states on +the other spaces. One may use the chain rule [27] +Imax(AC : X)ρ = HR(AC)ρ − Hmin(AC|X)ρ , +where HR is defined in the appendix. By using properties +of strong typicality, this decomposition allows one to +establish +Cε +max(An : Cn)ρ⊗n +≤ nI(AC : X)ρ + nO(δ) + log(1 − ε) , +where δ ∈ (0, 1) is a parameter of strong typicality. By +dividing by n and taking the limits δ → 0, n → ∞, we +establish the result. +B. Weak Converse for AEPs +Having established achievability, all that is left is to +establish is the (weak) converse. Before doing so, we ex- +plain why it does not trivially follow from known prop- +erties of Imax. In effect a weak converse for Iε +max(A : B) +would intuitively follow from the fact Imax(A : B)ρ ≥ +I(A : B)ρ. Moreover, as noted in the background, one +may use chain rules to decompose Iε +max(A : B) into +smooth conditional min- and max-entropies (17) at +which point it inherits a strong AEP from the strong AEP +for these measures. However, all of these results do not +apply because the max common information involves +a non-i.i.d. auxiliary random variable. Instead, we will +need to find a way to get bounds which are independent +of the random variable. To do so, we in part follow the +original converse of Wyner’s result [2]. To present these +proofs, we will need the following definition and a well- +known lemma that is a direct corollary of (a form of) +strong subadditivity. +Definition 18. Let ε ∈ (0, 1). Let ρ ∈ D(A ⊗ C) The +smoothed common information is +Cε(A : C)ρ := +min +�ρ∈Bε(ρ) min +A−X−C I(AC : X)�ρ . +Lemma 19. H(An +1|B) ≤ ∑i=1 H(Ai|B), with saturation if +and only if there exists a labeling of [n] such that ρAn +i B +is a Ai − B − An +i+1 Markov chain for all i ∈ [n − 1]. In +other words, with saturation only if Ai can be generated +from B for all i ∈ [n]. +Lemma 20. Let ε ∈ (0, 1). For ρ ∈ D(A ⊗ C), +lim +ε→0 lim +n→∞ +� 1 +nCε +max(An : Cn)ρ⊗n +AC +� +≥ C(A : C)ρAC . +Proof. Let σAnCn +∈ +Bε(ρ⊗n +AC) be the minimizer of +Cε +max(An : Cn)ρ⊗n +AC. Then +Cε +max(An : Cn)ρ⊗n +AC +=Cmax(An : Cn)σ ≥ C(An : Cn)σ = I(AnCn : X)σ , +where we used that Imax(A : B) ≥ I(A : B) and that +σAnXCn is a An − X − Cn Markov chain by definition of +common information. Now we decompose the final right +hand side of this chain of inequalities. +I(AnCn : X)σ += [H(AnCn) − H(AnCn|X)] += +� +H(AnCn)σ − +n +∑ +i=1 +H(AiCi|X)σ +� += +� +H(AnCn)σ − +n +∑ +i=1 +H(AiCi)σ + +20 ++ +n +∑ +i=1 +H(AiCi)σ − +n +∑ +i=1 +H(AiCi|X)σ +� +=H(AnCn)σ − +n +∑ +i=1 +H(AiCi)σ + +n +∑ +i=1 +I(AiCi : X)σ , +where the first equality is a well-known chain rule, the +second equality is by Lemma 19 because we saturate +as a An − X − Cn Markov chain is also a Ai − X − Ci +Markov chain for all i ∈ [n] as you could trace off the +marginals, and the final identity is again using the same +chain rule as the first equality. Therefore, dividing this +by n we have +1 +nCε +max(An : Cn)ρ⊗n +AC += 1 +n H(AnCn)σ − 1 +n +n +∑ +i=1 +H(AiCi)σ ++ 1 +n +n +∑ +i=1 +I(AiCi : X)σ +(32) +We now aim to introduce Cε(A : C)ρ into the above +bound. Consider τA−X−C +≡ +σAk−X−Ck where k +:= +argmink∈[n]I(AiCi : X)σ. This is a Markov chain for the +reason explained above and σAkCk ∈ Bε(ρAC) as purified +distance only decreases under partial trace. It follows +that τ is a feasible point for Cε(A : C), so we have +Cε(A : C) ≤ I(AkCk : X)σ ≤ 1 +n +n +∑ +i=1 +I(AiCi : X)σ , +where the second inequality is the minimizer lower +bounds the average. Combining this with (32), +1 +nCε +max(An : Cn)ρ⊗n +AC +≥Cε(A : C)ρ + 1 +n H(AnCn)σ − 1 +n +n +∑ +i=1 +H(AiCi)σ . +Note by assumption σAnCn ∈ Bε(ρ⊗n +AC), and σAiCi ∈ +Bε(ρAC) for all i ∈ [n] as explained earlier. As purified +distance upper bounds trace distance, we may use the +Fannes-Audenaert inequality to conclude +H(AnCn)σ ≥ nH(AC)ρ + εn log(|AC|) + h2(ε) +H(AiCi)σ ≥ H(AC)ρ − ε log(|AC|) − h2(ε) , +where we have used that the von Neumann entropy +is additive over tensor products in the first inequality. +Plugging these in and cancelling the von Neumann +entropy terms, we obtain +1 +nCε +max(An : Cn)ρ⊗n +AC +≥ Cε(A : C)ρ + +� +1 + 1 +n +� +h2(ε) + 2ε log(|AC|) . +Then if one lets n → ∞ and then ε → 0 on both sides, +one obtains the result. Note that this held for all density +matrices because if Bε(ρ) contains a separable state, a +Markov chain exists and so both Cε +max, Cε are finite and +otherwise, both are infinite. +As mentioned, effectively the same proof establishes a +weak converse for �Cε +max. +Lemma 21. Let ε ∈ (0, 1). For ρ ∈ D(A ⊗ C), +lim +ε→0 lim +n→∞ +� 1 +n +�Cε +max(An : Cn)ρ⊗n +AC +� +≥ C(A : C)ρAC . +Proof. The proof is similar so we only note the major dif- +ferences. First, σAnXCn that optimizes �Cε +max(An : Cn) is not +necessarily a Markov chain, though it is classical on the +auxiliary register. This follows as the smoothing is done +on the choice of Markov chain along with Proposition +22. Nonetheless, using Lemma 19, we establish +1 +n I(AnCn : B) +≥ 1 +n H(AnCn)σ − 1 +n +n +∑ +i=1 +H(AiCi)σ ++ 1 +n +n +∑ +i=1 +I(AiCi : X)σ , +(33) +where the inequality is because we no longer have that +σ is a Markov chain. +For the second step, there is more to change as now +we don’t know if σAiXCi is ever a Markov chain. Let +�ρAn−B−Cn be a (in case it is not unique) Markov chain +that corresponds to the optimizer σ. First, this means +�ρAnCn = ρ⊗n +AC, so �ρAiCi = ρAC for all i ∈ [n]. It also means +�ρAiBCi is a Ai − B − Ci Markov chain for all i ∈ [n]. It +follows τABIC = 1 +n ∑i∈[n] |i⟩⟨i|I ⊗ �ρAiBCi is a A − IB − C +Markov chain as conditioned on I, the remaining state +is a Markov chain conditioned on X. Moreover, τAC = +1 +n ∑i∈[n] TrB �ρAiBCi = +1 +n ∑i∈[n] ρAC = ρAC. Thus, τ is a +Markov chain extension of ρAC. Therefore, we have +C(A : C)ρ +≤I(AC : XI)τ +=I(AC : I)τ + I(AC : X|I)τ +=I(AC : I)τ + 1 +n +n +∑ +i=1 +I(AiCi : X)�ρ +=H(AC)τ − H(AC|I)τ + 1 +n +n +∑ +i=1 +I(AiCi : X)�ρ +=H(AC)ρ − 1 +n +n +∑ +i=1 +H(AiCi)�ρ + 1 +n +n +∑ +i=1 +I(AiCi : X)�ρ , +which implies +− 1 +n +n +∑ +i=1 +H(AiCi)�ρ ≥C(A : C)ρ − H(AC)ρ +− 1 +n +n +∑ +i=1 +I(AiCi : X)�ρ . +where the issue is everything is in terms of σ and �ρ. +However, we can use the Alicki-Fannes-Winter (AFW) +inequalities in the following manner. As �ρ ≈ε σ, we have +�ρAiBCi ≈ε σAiBCi for all i ∈ [n] and likewise if we trace + +21 +off B. Therefore using the AFW inequalities (for both +unconditional entropy and mutual information [30]), +− 1 +n +n +∑ +i=1 +H(AiCi)σ ≥ C(A : C)ρ − H(AC)ρ +− 1 +n +n +∑ +i=1 +I(AiCi : X)σ − 4ε log |AC| − h2(ε) +− 2(ε + 1) log(ε + 1) + ε log(ε) . +Then we can plug this into (33) and use Fannes- +Audenaert inequality on H(AnCn)σ in the same fashion +as the previous proof to obtain +1 +n +�Cε +max(An : Cn)ρ⊗n +AC +≥C(A : C)ρ − 3ε log(|AC|) +− 2(ε + 1) log(ε + 1) + ε log(ε) . +Taking the limit n → ∞ followed by letting ε → 0 +completes the proof where we use 0 log(0) = 0. +Remark 3. In principle one could establish bounds in +terms of �C1−ε−δ +h +, however this would either require prov- +ing a new chain rule or simply use converting I1−ε−δ +h +to +Iε +max, neither of which would provide particular insight +for our purposes, so we do not do this. +C. In Regards to a Strong Converse for the AEP +We end this section with some remarks on what it +would take to establish a strong converse for the AEP +which would not depend on the smoothing parame- +ter ε ∈ (0, 1), which we believe to be an interesting +general problem. First note that how the converse is +proven currently, one would have to guarantee the opti- +mizer Cε +max(An : Cn)ρ⊗n +AC was the optimizer of C(A : C)ρAC, +which seems difficult. Any relaxation of either Cε +max or +C, such as to C → Cε will then require a continuity +result. However, this certainly does not imply there is no +strong AEP. Note at the start of the proof we immediately +relax from Imax(A : C) to I(A : C) to make it upper bound +Cε, which presumably is adding looseness. Moreover, +[6] established a strong converse in the classical setting +for distributed source simulation, which would at least +suggest there should be a strong converse for the AEP +when restricted to classical states. For this reason it is +worthwhile to discuss what it would take to establish a +strong converse for the AEP. +First, for the smooth min- and max-entropy, the strong +AEP is proven via duality [31]. However the duality of +mutual informations requires the inverse of a quantum +state and therefore isn’t physical in the same fashion [44]. +As such, it seems we cannot establish an AEP in this +manner. To the best of the authors’ knowledge this is +the only known way to prove a strong AEP for smooth +min- and max-entropy, so we cannot borrow a strategy +from there. +Before discussing other strategies, we reduce the prob- +lem to one pertaining to min-entropy rather than max- +mutual information for conceptual clarity. As mentioned +in the introduction, using a straightforward general- +ization of known chain rules for I↑,ε +max, one can estab- +lish a strong AEP for I↑,ε +max from the strong AEPs for +conditional min- and max-entropies (see the appendix +and Proposition 27). One can modify these chain rules +to establish chain rules for SMCI that show the issue +could be converted to a question regarding conditional +min-entropy. That is, one can show the following (see +appendix for proof). +Proposition 11. Let ρAC ∈ D(A ⊗ C). A strong AEP for +smooth max common information holds if for all ε ∈ +(0, 1), there exists ε ∈ (ε, 1) such that 0 < ε + δ(ε) < 1 +where δ(ε) := 2 +� +ε(1 − ε) so that +lim +n→∞ +� +1 +n +max +�ρ∈Bε+δ(ε)(ρ⊗n) +max +An−X−Cn Hmin(AnCn|X) +� +≤ +max +A−X−C H(AC|X)ρ . +Immediately we can see this sufficient condition is +a converse in the sense that we are looking for the +regularized smoothed quantity to be upper bounded by +the conditional von Neumman entropy term. However, it +is distinct from the smooth min-entropy strong converse +as we take the extension on the purified state and have +a non-i.i.d. random auxiliary variable to deal with still. +Moreover, it is unclear how to switch the smoothing +from before taking the extension to after, even if we +restrict to separable states where both resulting sets are +non-empty.4 +However, it does not seem moving the smoothing +through is sufficient to obtain a strong converse AEP. +We propose this is fundamentally because it is in some +sense fundamentally a parallel reduction, though even +in the fully classical case where it can be forced to be +sequential, it won’t trivially follow from current results +that include quantum side-information. To show this, +we present these problems in terms of what would be +sufficient for a strong converse for the alternative SCMI, +�Cε +max. By the same chain rule argument as before, its +strong converse would be implied by the inequality +lim +n→∞ +� 1 +n +max +An−X−Cn Hε +min(AnCn|X) +� +≤? +max +A−X−C H(AC|X) +being true, which makes the smoothing simple. Next, +one can convert the smooth min-entropy to a smooth +max-entropy in exchange for a correction term that goes +away in the regularization, as is done in establishing the +strong converse for the min-entropy AEP [31]. Thus one +is interested in establishing something of the form +lim +n→∞ +� 1 +n +max +An−X−Cn Hε +max(AnCn|X) +� +≤? +max +A−X−C H(AC|X) . +4If this switching of optimizations could be done, the open problem +would become a Markov chain extension AEP for partially smoothed +min-entropy [45], or partially smoothed mutual information if one did +not use the chain rules. + +22 +This looks rather similar to the max-entropy version of +the entropy accumulation theorem (EAT) or its gener- +alization [46], [47], which says that a sufficiently well- +behaved (as-if-sequential) process that outputs entropy- +generating registers per round converges to the i.i.d. +behaviour. We now show why this does not work in +our case. +Note we consider An − X − Cn, which technically is +a parallel process where maps only act on X once to +generate An,Cn. However, this structure implies Ai − +Ai−1 +1 +XCi−1 +1 +− Ci for all i ∈ [n] [16]. This means our +process can be forced to look sequential in such a way +that it satisfies the constraints of both versions of the EAT +[46], [47]. However, in altering the problem in this man- +ner, the issue is that the effective maps will need access +to Ai−1 +1 +Ci−1 +1 +as side-information for generating the next +round as well as being the previous entropy generating +registers. This means one needs to have copies of these +registers. It is not obvious that this can be done in the +separable state case.5 This may be viewed as the general +difficulty of applying the EAT to this parallel setup. +We do note that we could restrict to a classical Markov +chain problem, Xn − Y − Zn, where registers can always +be copied so we can properly define EAT maps Gi per +round. However, in this setting and by modifying the +notation of some of the side-information registers for +clarity, the generalised EAT [47, Appendix A] will be of +the form +Hε +max(XnZn|YE)ρ +≤ ∑ +i +max +|ω⟩ H(XiZi|Xi−1 +1 +Zi−1 +i +YEi)(Gi⊗idEi )(ω) , +where +ω +is +any +purification +of +an +input +to +Gi. +The problem then is the LHS conditions on quan- +tum +side-information +we +don’t +want +to +consider +and by strong subadditivity of smooth max-entropy, +Hε +max(XnZn|Y �E) ≤ Hε +max(XnZn|Y), so this would require +modifying the max-entropy version of the EAT to not +include the quantum side-information. +To summarize, even for the alternative common in- +formation, it appears that a strong converse for the +fully separable case would require an i.i.d. reduction +that takes into account the Markov chain structure. This +does not align with the EAT nor a traditional deFinetti +theorem where a purification is involved. To the best of +our knowledge, none of the problems for establishing a +strong converse that we have noted are addressed in the +smooth entropy framework. We do however note there +are certainly other ways of establishing strong converses +in the face of auxiliary variables even in the quantum +setting, e.g. [49]. +5Technically, this is because while we know An − X − Cn is an exten- +sion of ρ⊗n +AC, it is not obvious from the structure of the Petz recovery +map [30] nor the general structure of the recovery map for Markov +chains specifically [48] that the recovery map RCi−1 +1 +XAi−1 +1 +→Ci +1XAi−1 +1 +won’t entangle some of the quantum systems. +IX. EXTENDING TO MANY RECEIVERS +We have now established the general framework of +one-shot distributed source simulation and its relation +smooth max common information. We now explain that +it is straightforward to generalize beyond simulating +a bipartite distribution. In the classical case this was +addressed by Liu et al. [5]. In that work if one goes to +the appendix where they prove it, they just point out +it is in effect the same as proof as before. Indeed, this +observation lifts to our setting with one nuance: it is +not clear how to argue the minimizer for multipartite +systems should be classical. This is because the proof +of Proposition 17 uses the decomposition of a quantum +Markov chain and it is not clear how to generalize this +to more systems. Regardless, this is in effect irrelevant +with regards to the operational task at hand, so we focus +on the classical seed. +We begin with notation. A − X − C is very natural +as it shows the two registers splaying out from X. We +can’t do this for more than two registers. As such, if we +imagine we want ρA1, ..., ρAm each generated from seed +X independently, then we will write this as X Am +1 . With +this notation established we can define the following +operational tasks. +Definition 19. Let ρAm +1 ∈ D(Am +1 ). Let ε ∈ (0, 1). The one- +shot correlation formation is defined as +Cε +F( : Am +1 : ) += min +� +H0(X)�ρ : �ρX +Am +1 : �ρ ∼ε ρAm +1 +� +. +(34) +Moreover, the one-shot uniform correlation of formation, +which is the one-shot common information, is defined +as +Cε +U,F(A : B) += min +� +H0(X)�ρ : �ρπ +Am +1 : �ρ ∼ε ρAm +1 +� +, +(35) +where we remind the reader π means the register is +uniform. +We can define the entanglement-assisted cases in a +similar fashion, so we omit them. Then we can define +the extended smoothed correlation measures. +Cmax( : Am +1 : )ρAm +1 +:= +min +�ρ∈Bε(ρAm +1 ) min +�ρX Am +1 +Imax(Am +1 : X)�ρAmX +�Cmax( : Am +1 : )ρAm +1 +:= min +ρX Am +1 +min +�ρ∈Bε(ρAm +1 X) Imax(Am +1 : X)�ρAmX +We can establish the one-shot converse the same way +as before using data-processing by extending the χ|p to +more parties. We can establish achievability using the +one-shot random covering as before. +Similarly, the weak AEPs for these will follow the +same way. The achievability of �Cε +max( : Am +1 : ) can still +be proven using the AEP for Iε +max. The achievability of +Cmax( : Am +1 : ) can be proven using strong typicality in the + +23 +same fashion as before since we can still decompose the +optimizer of C( : Am +1 : ) as ∑x∈X p(x) |x⟩⟨x| � +i∈[n] ρAi +x and +construct a state using strong conditional typicality from +that. The converses are established the same as before +with the replacement An +1Cn +1 �→ An +1,1An +2,1...An +m,1 where the +first subscript here denotes the party label. Thus we have +extended all our results to multiple parties. +a) Monotonicity in Number of Parties: In [5] the au- +thors note, albeit in the classical setting, that given ρAm +1 +and ρAk +1 ≡ TrAm +k (ρAm +1 ), C( : Am +1 : ) ≥ C( : Ak +1 : ), i.e. that +common information can only decrease as you decrease +the number of parties. They suggest (1) this is surprising +and (2) no such property is known to hold for mutual +information. We wish to briefly address these points in +case they provide conceptual clarity for the reader. +First, the authors suggest this monotonicity is surpris- +ing because “if the information is common it ought to +be non-increasing when more random variables are in- +cluded.” This suggests the authors view common infor- +mation as measuring the intersection of the randomness +of the states (random variables). However, the common +information is measuring the randomness needed to +produce each random variable independently, i.e. to +generate a common variable, and this is like measuring +the union of the randomness of the states in some +sense. Whether or not one agrees this is what “common” +should denote, if one views it in this fashion, it is +clear that it must increase when you add more random +variables. +Second, that the common information has such a +monotonic property is an immediate corollary of the +mutual information having the same property, which in +the quantum information community is called the data- +processing inequality (not to be confused with the data- +processing inequality for a Markov chain as is common +in classical information theory). We show this in the +following generic manner. +Proposition 12. Given ρAm +1 +and ρAk +1 +≡ +TrAm +k (ρAm +1 ), +C( : Am +1 : ) ≥ C( : Ak +1 : ) where C is the Wyner common +information defined using any mutual information I +satisfying data-processing. +Proof. Let ρ⋆ +Am +1 X be the minimizer of C( : Am +1 : ). Then, +C( : Am +1 : ) =I(Am +1 : X)ρ⋆ +≥I(Ak +1 : X)TrAm +k (ρ⋆) +≥ min +ρX Ak +1 +I(Ak +1 : X)ρ +=C( : Ak +1 : )ρ , +where the first inequality is the data-processing in- +equality and the second is using minimizing TrAm +k (ρ⋆) +is one seeding option for ρAk +1 and so we can further +minimize. +X. ENTANGLED STATE SOURCE SIMULATION +We have now established the limits of distributed +source simulation in the one-shot and asymptotic setting +in terms of smooth entropic quantities. These results +however have only applied to separable states, and so it +is worthwhile to ask what can be said about entangled +states. It is known that one cannot convert entangled +states with zero communication and no auxilliary re- +source [21]. However, it is also known that there exists +a sufficiently large (pure) entangled state that can be +used to generate any (pure) entangled state up to small +error with zero communication [29]. Specifically, what +the authors show is the following. +Theorem 22. ([29]) For any ε > 0 and target bipartite +pure state |ϕ⟩AB with Schmidt rank m, the catalyst state +|µ(n)⟩A′B′ = +1 +√Hn ∑n +j=1 +1√ +j |j⟩A′ |j⟩B′ is such that for n > +m1/ε there exist unitaries UAA′, WBB′ so that +F(U ⊗ W(|µ(n)⟩A′B′ |0⟩A |0⟩B), |µ(n)⟩A′B′ ⊗ |ϕ⟩AB) +≥ 1 − ε , +where F(·, ·) is the fidelity and Hn is the Harmonic +number. +This means that Alice and Bob, when given the proper +seed state |µ(n)⟩, can prepare the target state |ϕ⟩AB +using local operations and zero communication. In this +sense it seems the natural extension of distributed source +simulation to the quantum setting. However, there are +technical distinctions. Specifically, while in both cases +the seed state is (at least approximately) preserved, in +embezzling, the remaining seed state is (approximately) +decoupled from the target state. This has the further +advantage of allowing the seed state to be used in further +protocols in exchange for further degrading the total +approximation, which we note none of the correlation +of formations could guarantee. These similarities and +distinctions are summarized in Fig 7. +With this established, we define the embezzleable +entanglement of simulation, which measures the amount +of entanglement (with respect to a specific choice of +measure) necessary to source simulate a state in a dis- +tributed manner and is named in such so as to avoid +any confusion with entanglement of formation. To define +this, we will need the definition of entanglement rank +[22]. +Definition 20. For all r ≥ 1, the set of all operators R ∈ +Pos(A ⊗ B) for which there exists a finite alphabet X and +collection of linear operators {Mx}x∈X ⊂ L(A, B) such +that rank(Mx) ≤ r for all x ∈ X and +R = ∑ +x∈X +vec(Mx) vec(Mx)∗ , +where vec : L(A, B) → A ⊗ B defined via vec(|i⟩ ⟨j|) = +|j⟩ |i⟩. We say a density matrix ρAB ∈ D(A ⊗ B) has +entanglement rank r′ if it is contained Entr′(A : B) but +not Entr′−1(A : B). For notational simplicity, we define + +24 +pX +Copy +ΦX→A +A +C +ΨX′→C +≈ε ρAC +X +(a) Distributed Source Simulation +σ �A �C +A′ +C′ +Φ �A→AA′ +Φ �C→CC′ +A +C +A′ +C′ +ρAC +≈ε +⊗ +σA′C′ +(b) Embezzling Source Simulation +σA′C′ +A′ +C′ +ΦA′→AA′ +ΦC′→CC′ +A +C +A′ +C′ +ρAC +≈ε +(c) Entangled Source Simulation +Fig. 7: Comparison between distributed source simula- +tion and the entangled state versions. Grey lines rep- +resent allowed correlations of either classical or quan- +tum mechanical nature. (a) Distributed source simulation +where the X register is possibly strongly correlated to +A and C. (b) Embezzling source simulation where the +auxiliary state is required to be output approximately +decoupled from the simulated state. (c) Entangled source +simulation where the input is an arbitrary quantum state +and an appropriate marginal of the output must achieve +the target state to tolerable error ε. +EntA : B : D(A ⊗ B) → N≥1 as the function that takes a +density matrix and returns its entanglement rank. The +subscript is because the partitioning is relevant as will +be shown in a following proposition. +Note that the set of separable operators is equivalent +to the set Ent1(A : B) and if A ∼= B ∼= Cd, then all positive +semidefinite operators are contained within Entd(A : B). +That is to say, the entanglement rank measures ‘how +entangled’ an operator is and may be viewed as a mixed +state extension of Schmidt rank. +Definition 21. Let ρAC ∈ D(A ⊗ C) and ε ∈ (0, 1). +The ε−embezzleable entanglement of simulation is the +logarithm of the minimal entanglement rank of a bi- +partite state such that under local operations it may +be converted to ρ up to ε− error in purified distance. +Formally, +Cε +EE,S(A : C)ρ := log min{r ≥ 1 : ∃σ ∈ Entr(A′ : C′) : +(Φ ⊗ Ψ)(σ) ≈F +ε σ ⊗ ρ} , +where Φ ∈ C(A′, A′ ⊗ A) and Ψ ∈ C(C′, C′ ⊗ C) and +σ ≈F +ε ρ means F(σ, ρ) ≥ 1 − ε. +It is worthwhile to relate this back to the correlation +of formation measure. Much like Cε +F, it measures the +logarithm of the ‘dimension’ of the resource, in this +case entanglement rank, but does not care about the +uniformity of the resource. Second, one way of viewing +distributed source simulation is that the channels are the +recovery maps R : X → X ⊗ C, R : X → A ⊗ X, and +the definitions of Φ, Ψ and the error condition mirror +this as the channels (approximately) preserve the input +resource. +Note that while it only measures the entanglement +rank, its demand on the ancillary state being (approx- +imately) unchanged and uncorrelated means that it is +not clear how one would make use of a classical ancillary +state, at least when ε is sufficiently small. This is because +if ρ is built conditionally on σ, it will not be uncorrelated. +For this reason it seems to properly capture the notion +of embezzling as being the strategy. +With this definition introduced, we establish achiev- +ability bounds and then argue that these bounds should +be approximately tight. +Lemma 23. Let ρAB ∈ D(AB). Then SR(AR : B), SR(A : +BR) is the same for all purifications |ψ⟩ABR. +Proof. By +isometric +equivalence +of +purifications, +|ψ⟩R′AB += +(VR→R′ ⊗ 1) |ψ⟩RAB. +Let +|ψ⟩RAB += +∑i∈[m] +√pi |φi⟩RA ⊗ |ϕi⟩B be its Schmidt decomposition. +Then |ψ⟩R′AB = ∑i∈[m] +√piVR→R′ |φi⟩RA ⊗ |ϕi⟩B is its +Schmidt decomposition as an isometry maps pure states +to pure states. An identical argument holds for the other +partitioning. +The following lemma shows it is necessary to take the +minimization in the previous lemma. +Proposition +13. For pure state |ψ⟩RAB, in general +SR(AR : B) ̸= SR(A : BR). Moreover, the exists |ψ⟩ABR +such that SR(A : BR) − SR(AR : B) = dB − 1, the +maximum possible difference. +Proof. We present an example. Let ρAB = |φ⟩⟨φ|A ⊗ πB. +Then |ψ⟩ABR = |φ⟩A ⊗ |Φ+⟩BB′ ⊗ |φ⟩A′ where R ≡ A′B′. +It follows SR(A : BR) = 1 as it is product across this +partitioning. On the other hand, +|ψ⟩ABR = |φ⟩A ⊗ +� +�d−1/2 +B +∑ +i∈[dB] +|i⟩B |i⟩B′ +� +� ⊗ |φ⟩A′ +=d−1/2 +B +∑ +i∈[dB] +|i⟩B ⊗ |φ⟩A ⊗ (|φ⟩ ⊗ |i⟩)R . +as {|φ⟩⊗2 ⊗ |i⟩}i form the Schmidt vectors for the AR +space, SR(A : BR) = dB. This completes the proof. + +25 +Proposition 14. For ε ∈ (0, 1), one can construct ρAB +to accuracy ε via embezzling using |µ(n)⟩ for n > m1/ε +where +m = min{SR(AR : B), SR(A : BR)} , +Proof. Embezzling is a function of the Schmidt rank. By +the previous lemma, we only need to consider purifi- +cation |ψ⟩ABR. To have a notion of locality, either Alice +or Bob must embezzle in the purifying space. By the +previous proposition, in general there is a difference in +Schmidt rank depending on who purifies the state, so +we take the minimum. +Corollary 1. Let ρAC ∈ D(A : C) and ε ∈ (0, 1), then +Cε +EE,S(A : C)ρ +≤ 1 +ε log(min{EntAR:C(ψ), EntA:CR(ψ)}) , +where ψACR is an arbitrary purification of ρAC. +Proof. This follows the definition of Cε +EE,S and the previ- +ous proposition where we have taken the logarithm. +There are two questions: the first would be if this +strategy, when no classical side-information is allowed, +is optimal. Roughly speaking, it is in the sense that +[29] showed that if one allowed LOCC and a state +dependent catalyst that the error is bounded below +by Ω(1/ log(n)), but the universal embezzling strategy +scales as O(1/ log(n)) where n is the Schmidt rank +of the seed state. Of course this scaling requires the +error demanded be small, i.e. if the error is sufficiently +large, it may be feasible to use less entanglement; this is +developed further by authors of this work in a separate +paper [50]. We also stress that Cε +EE,S seems to be captured +effectively by embezzling as we already argued why, in +general, a classical auxiliary state could not be useful. +The second question would be if this strategy has any +notion of compressibility in the sense that it requires less +resources for many copies of an i.i.d. state. This is not +so: we show this strategy scales in the number of copies. +Proposition 15. For ε ∈ (0, 1), one can construct ρ⊗k +AB +to accuracy ε via embezzling using |µ(n)⟩ for n > m1/ε +where +m = k · min{SR(AR : B), SR(A : BR)} . +Proof. Consider a purification of ρ⊗k +AB, |ψ⟩AkBkR′. Con- +sider a purification of ρAB, |φ⟩ABR. It follows |φ⟩⊗k is +a purification of ρ⊗k +AB. By the isometric equivalence of +purifications, there is an isometry or reversed isometry +taking R′ to Rk. As this is a local map, it can’t change +the Schmidt rank. Thus, SR(AkR′ : Bk) = SR(AkRk : Bk) +and likewise for the other partitioning. Finally, SR(AkRk : +Bk) = +k · SR(AR +: +B), and likewise for the other +partitioning. This completes the proof. +a) Entanglement of Simulation without Decoupling: In +the previous strategy, as already noted, the inclusion of +a classical register is effectively not feasible because we +require the catalyst to be decoupled from the output +state. However, distributed source simulation does not +require this decoupling condition as the X seed register +will be correlated with the A and C registers (See Fig. 7). +It follows one may argue the setting of Cε +EE,S is too re- +strictive. The question then becomes what is the natural +setup to correspond with distributed source simulation. +The most general setting would be local operations and +shared entanglement (LOSE) with no constraints, that is, +the input can be any state and a marginal of the output +is the target state up to some tolerated error. This aligns +with the notation of Fig. 7. +However, this unconstrained LOSE setting, if we mea- +sure the needed entanglement in terms of entanglement +rank, has a simple characterization as we briefly prove +following a few definitions. +Definition 22. Let ε ∈ (0, 1) and ρAC ∈ D(A ⊗ C). The +entanglement of simulation is given by +Cε +E,S := log min{EntA:C(σ) : TrA′C′(Φ ⊗ Ψ)(σ) ≈F +ε ρ} . +where Φ ∈ C( �A, A′ ⊗ A), Ψ ∈ C( �C, C′ ⊗ C) and ρ ≈F +ε +means F(ρ, σ) ≥ 1 − ε. +Definition 23. Let ρAB ∈ D(A ⊗ B). For r ≥ 1, the fidelity +of entanglement rank is defined as +EF,r(A : B)ρ := +sup +σAB∈EntrD(A:B) +F(ρ, σ) . +From this, we define the ε−approximate entanglement +rank as +Entε +A:B(ρ) := min{r ≥ 1 : EntF,r′(A : B)ρ ≥ 1 − ε} , +which measures the smallest entanglement rank for a +state σ to be ε-close to ρ under fidelity. +Proposition 16. For ε ∈ [0, 1], ρAC ∈ D(A ⊗ C), +Cε +E,S = log(Entε +A:C(ρ)) . +Proof. We prove this is a lower bound and then note +it can be achieved. It is known that local operations +can only decrease entanglement rank [22]. Note that +the throwing out of the A′, C′ spaces are also local +operations. Therefore, the input state σ must have an +entanglement rank that is lower bounded by Entε +A:C(ρ) +or else it will violate the error restriction. Moreover, the +state τAC that minimizes Entε +A:C(ρ) can be forwarded and +the local operations be trivial, so this is also achievable. +Taking the logarithm completes the proof. +We note part of the simplicity of this setting is our +choice of measure. [51] analyzed the quantum correlation +complexity, which is the same setting but measuring +the logarithm of the rank of the seed state rather than +its entanglement rank and the analysis in this case +is more arduous. However, given the triviality of the + +26 +unconstrained case under our choice of measure, we +choose to restrict the input state to aribtrary classical +correlation and embezzling states. +Definition 24. Let ε ∈ (0, 1) and ρAC ∈ D(A ⊗ C). +The entanglement of simulation restricted to embezzling +states is +Cε +E|Emb,S(A : C)ρ +:= log min{r ≥ 1 : TrA′C′ ◦(Φ ⊗ Ψ)(σ(r)) ≈F +ε ρ} , +where σ(r) �A �C := |µ(r)⟩ ⊗ σXAXC, �A ∼= A′ ⊗ XA, �C ∼= C′ ⊗ +XC, Φ ∈ C( �A, A′ ⊗ A), Ψ ∈ C( �C, C′ ⊗ C), and σ ≈F +ε ρ +means F(ρ, σ) ≥ 1 − ε. +We remark that one advantage of choosing the embez- +zling state is that we have a notion of a consistent +resource, which mirrors that the uniform correlation of +formation has a consistent resource.6 +The difference between embezzleable entanglement of +simulation, Cε +EE,S, and the entanglement of simulation +restricted to embezzling states, Cε +E|E,S is of course that +we allow for arbitrary classical assistance in the latter +which as addressed we cannot in general do with the +former. This in particular allows us to distribute a flag +state in the latter setting, which results in the following +theorem. +Theorem 24. Let ρ ∈ D(A ⊗ C) and ε ∈ [0, 1). Then +log(Entε +A:C(ρ)) ≤ Cε +SREE,S(A : C)ρ ≤ 1 +ε log(EntA:C(ρ)) . +Moreover, for sufficiently small ε ∈ (0, 1) the upper +bound is nearly optimal for pure state |ψ⟩ that is not +local unitarily equivalent to |µ(r′)⟩ for any r′, and the +lower bound is tight when ε = 0 and |ψ⟩ is local +unitarily equivalent to some |µ(r′)⟩. These two points +imply these bounds are approximately tight although +they don’t match. +Proof. We first prove the achievability and converse. +Then we provide examples where these are (nearly) tight +bounds. +(Achievability) The upper bound is trivial when ε = 0, +so we assume ε ∈ (0, 1). Consider the decomposi- +tion of ρAC = ∑x∈X p(x) |ψx⟩⟨ψx| such that the max +Schmidt rank of {|ψx⟩}x∈X is minimized over all pos- +sible decompositions of ρAC. Let m be this max Schmidt +rank. Let |µ(r)⟩ such that r > m1/ε. Then let σ(r) = +|µ(r)⟩⟨µ(r)|A′C′ ⊗ ∑x p(x) |x⟩⟨x|XA ⊗ |x⟩⟨x|XC. Let Alice +receive A′XA and Charlie receive C′XC. Conditioned +on x, Alice and Charlie embezzle in |ψx⟩ using µ(r). +For notational simplicity, we describe this in terms of +conditional isometries. Call the set of local isometries +conditioned on x ∈ X that do this {Ux∈X } and {Vx}x∈X . +These isometries include the local unitaries U∗ +x,W∗ +x from +Theorem 22 which we have pushed into the isometries +6One might note that maximally entangled states would be the +natural equivalent of uniform shared randomness, but [21] proves this +won’t work for LOSE. +using the isometric invariance of fidelity. Call the total +maps that implement this Φ and Ψ. Already tracing out +the classical registers and the purification registers, +F(|µ(r)⟩⟨µ(r)| ⊗ ρAC, TrA′C′(Φ ⊗ Ψ)(σ)) +≥ ∑ +x∈X +p(x)F(|µ(n)⟩ ⊗ |ψx⟩ , +(Ux ⊗ Vx) |µ(n)⟩ |0⟩ |0⟩) +≥ ∑ +x∈X +p(x)(1 − ε) +=1 − ε , +where the first inequality is going to the isometric picture +and using the joint concavity of fidelity and we’ve used +the fidelity guarantee of Theorem 22 in the second +inequality. Taking a logarithm of r due to Definition 24 +completes the achievability. +(Converse) Let r′ := Entε +A:C(ρ). By definition, the strategy +must be a product map Φ ⊗ Ψ. It follows it is a separable +map [22, Definition 6.17], and thus it cannot increase +the entanglement rank [22, Theorem 6.23]. Thus, the +entanglement rank of the seed state is at least this m′ +value. +(Near Tightness of Upper Bound) Consider the target state +ρAB is a pure φAB such that it is not local unitarily equiv- +alent to any member of {|µ(r)⟩}. There is no advantage +to the shared randomness setting as there is an optimal +pair of local maps from |µ(r′)⟩ to φ for any r′. Thus, we +have reduced the problem to embezzling, which is near +optimal as shown in [29]. +(Tightness of Lower Bound) Consider the target state ρAB is +local unitarily equivalent to |µ(r′)⟩. Then as F(ρ, σ) = 1 +if and only if ρ = σ. Thus if ε = 0, one must forward +|µ(r′)⟩ instead of any smaller embezzling state. +While the above result is approximately tight in certain +settings, we remark the lower bound is clearly loose +in general as the inability to convert one pure state to +another with no communication is not only a function of +the Schmidt rank, but the Schmidt coefficients [21], [50]. +For example, in [21], the authors show that the amount +of communication necessary for pure state transmission +depends on the ‘entanglement spread,’ which is effec- +tively a function of the maximum Schmidt coefficient +and the number of Schmidt coefficients. +XI. CONCLUSION AND OPEN PROBLEMS +In this paper we have considered the task of dis- +tributed source simulation in the one-shot setting for +fully quantum systems. We introduced various one-shot +operational quantities related to distributed source sim- +ulation and similar tasks. We then introduced one-shot +correlation measures and established that these correla- +tion measures bound these operational quantities and +thus characterize one-shot distributed source simulation +and its related tasks. In particular, this established a +one-shot version of Wyner’s common information result +[2] in the smooth entropy framework. In doing so, we + +27 +generalized the support lemma to preparation channels +and showed nuances about smoothing measures when +an auxiliary random variable is involved— ideas which +likely will have further use in one-shot quantum net- +work theory. One particular technical point of interest is +that we found it is important to not be smoothing the +auxiliary variable and this led to inducing the measure +via Dmax but one could not do the same using Dε +h as it +automatically smooths all argmuents. +We then proceeded to recover asymptotic results from +the one-shot results by establishing asymptotic equipar- +tition properties for our one-shot correlation measures. +In doing so, we perhaps intuitively extended Wyner’s +original result [2], and Hayashi’s extension to separable +states [16], by showing that asymptotically there is no +advantage in non-uniform shared randomness in the +task. We also showed that asymptotically the variations +on the one-shot distributed source simulation we had +considered, private entanglement-assisted and private +distributed source simulation, converge are given by +the same rate, at least in the case where we demand +asymptotically vanishing error. This is to say many +variations of the tasks that are clearly in general different +in the one shot setting become equivalent asymptotically +in the vanishing error setting. An open question would +be whether these results generalize to not requiring +vanishing error. Motivated by Yu and Tan’s recent result +strong converse in the classical setting [6], [7], it would +be our expectation they would. To remain in the smooth +entropy framework, this would require establishing a +strong converse for our asymptotic equipartition prop- +erty. We showed one option would be to establish a +new property for conditional min entropy (Proposition +11) that likely would require new tools as we explained +in that section why current methods do not seem sat- +isfactory. Alternatively, one could establish the strong +converse in some other fashion outside of the smooth +entropy calculus. +An open problem related to the establishment of a +strong converse would be the establishment of second- +order asymptotics, though this may be difficult with +the rate being a function of the auxiliary random vari- +able. Relatedly, we note that our one-shot upper and +lower bounds (Theorem 13) are not both linear in the +smoothing parameter ε. If one were to establish second- +order asymptotics in a similar fashion as is standard in +the smooth entropy framework [33], this would need +to be resolved. We note that the Renes-Renner one-shot +bounds for compression with quantum side-information +[52] have similar smoothing problems as our Theorem 13 +and this was resolved by Tomamichel and Hayashi using +they hypothesis testing entropy [33]. One hope would be +to do the same by establishing a converse for one-shot +distributed source simulation for only separable states +in terms of �Cε +h and then finding a method for a second- +order expansion of that quantity, though it’s not clear +why that would be easier. +After establishing the general framework of one-shot +distributed source simulation for bipartite states we con- +sidered two variations: more parties and an entangled +state variation. In the multipartite setting we explained +while all the proofs extend in a straightforward manner +and made some clarifications with regards to comments +in previous work [5]. In the quantum setting, we dis- +cussed the ability to use embezzling as a strategy for +the equivalent of distributed source simulation as con- +version of entangled states to arbitrary error with zero +communication is impossible. We considered both the +setting where the seed state cannot use any classical +correlation and when it can and gave tight upper and +lower bounds that in general do not match. +XII. ACKNOWLEDGMENTS +I.G. would like to thank Vincent Y. F. Tan for pro- +viding a draft of Yu’s and his monograph on common +information [19] prior to its publication. This work was +supported in part by the National Science Foundation +under Grant 2112890. I.G. acknowledges the support of +an Illinois Distinguished Fellowship. +REFERENCES +[1] C. E. Shannon, “A mathematical theory of communication,” Bell +System Technical Journal, vol. 27, no. 3, pp. 379–423, 1948. +[2] A. Wyner, “The common information of two dependent random +variables,” IEEE Transactions on Information Theory, vol. 21, no. 2, +pp. 163–179, 1975. +[3] P. G´acs and J. K¨orner, “Common information is far less than +mutual information,” Problems of Control and Information Theory, +vol. 2, no. 2, pp. 149–162, 1973. +[4] H. S. Witsenhausen, “On sequences of pairs of dependent random +variables,” SIAM Journal on Applied Mathematics, vol. 28, no. 1, +pp. 100–113, 1975. +[5] W. Liu, G. Xu, and B. Chen, “The common information of n +dependent random variables,” in 2010 48th Annual Allerton Confer- +ence on Communication, Control, and Computing (Allerton), pp. 836– +843, IEEE, 2010. +[6] L. Yu and V. Y. Tan, “Wyner’s common information under renyi +divergence measures,” in 2018 IEEE International Symposium on +Information Theory (ISIT), pp. 156–160, 2018. +[7] L. Yu and V. Y. F. Tan, “Corrections to “wyner’s common infor- +mation under r´enyi divergence measures” [may 18 3616-3632],” +IEEE Transactions on Information Theory, vol. 66, no. 4, pp. 2599– +2608, 2020. +[8] A. Winter, “Secret, public and quantum correlation cost of triples +of random variables,” in Proceedings. International Symposium on +Information Theory, 2005. ISIT 2005., pp. 2270–2274, IEEE, 2005. +[9] E. Chitambar, M.-H. Hsieh, and A. Winter, “The private and pub- +lic correlation cost of three random variables with collaboration,” +IEEE Transactions on Information Theory, vol. 62, no. 4, pp. 2034– +2043, 2016. +[10] E. Chitambar, B. Fortescue, and M.-H. Hsieh, “Quantum versus +classical advantages in secret key distillation (and their links to +quantum entanglement),” arXiv preprint arXiv:1510.07344, 2015. +[11] E. Chitambar, B. Fortescue, and M.-H. Hsieh, “Classical analog to +entanglement reversibility,” Physical review letters, vol. 115, no. 9, +p. 090501, 2015. +[12] E. Chitambar and M.-H. Hsieh, “Round complexity in the local +transformations of quantum and classical states,” Nature commu- +nications, vol. 8, no. 1, pp. 1–7, 2017. +[13] P. Cuff, “Communication requirements for generating correlated +random variables,” in 2008 IEEE International Symposium on Infor- +mation Theory, pp. 1393–1397, 2008. +[14] G. R. Kumar, C. T. Li, and A. El Gamal, “Exact common informa- +tion,” in 2014 IEEE International Symposium on Information Theory, +pp. 161–165, 2014. + +28 +[15] L. Yu and V. Y. F. Tan, “On exact and ∞-R´enyi common infor- +mations,” IEEE Transactions on Information Theory, vol. 66, no. 6, +pp. 3366–3406, 2020. +[16] M. Hayashi, “General nonasymptotic and asymptotic formulas in +channel resolvability and identification capacity and their appli- +cation to the wiretap channel,” IEEE Transactions on Information +Theory, vol. 52, no. 4, pp. 1562–1575, 2006. +[17] P. Cuff, “Distributed channel synthesis,” IEEE Transactions on +Information Theory, vol. 59, no. 11, pp. 7071–7096, 2013. +[18] L. Yu and V. Y. F. Tan, “R´enyi resolvability and its applications +to the wiretap channel,” IEEE Transactions on Information Theory, +vol. 65, no. 3, pp. 1862–1897, 2019. +[19] L. Yu and V. Y. F. Tan, “Common information, noise stability, and +their extensions,” Foundations and Trends® in Communications and +Information Theory, vol. 19, no. 2, pp. 107–389, 2022. +[20] J. S. Bell, “On the einstein podolsky rosen paradox,” Physics +Physique Fizika, vol. 1, no. 3, p. 195, 1964. +[21] P. Hayden and A. Winter, “Communication cost of entanglement +transformations,” Physical Review A, vol. 67, no. 1, p. 012326, 2003. +[22] J. Watrous, The Theory of Quantum Information. Cambridge Uni- +versity Press, 2018. +[23] M. Hayashi, Quantum Information: An Introduction. Springer, 2006. +[24] R. Ahlswede and A. Winter, “Strong converse for identification +via quantum channels,” IEEE Transactions on Information Theory, +vol. 48, no. 3, pp. 569–579, 2002. +[25] H.-C. Cheng and L. Gao, “Error exponent and strong converse +for quantum soft covering,” 2022. +[26] Y.-C. Shen, L. Gao, and H.-C. Cheng, “Optimal second-order +rates for quantum soft covering and privacy amplification,” arXiv +preprint arXiv:2202.11590, 2022. +[27] M. Berta, M. Christandl, and R. Renner, “The quantum reverse +shannon theorem based on one-shot information theory,” Com- +munications in Mathematical Physics, vol. 306, no. 3, pp. 579–615, +2011. +[28] N. Ciganovi´c, N. J. Beaudry, and R. Renner, “Smooth max- +information as one-shot generalization for mutual information,” +IEEE Transactions on Information Theory, vol. 60, no. 3, pp. 1573– +1581, 2013. +[29] W. van Dam and P. Hayden, “Universal entanglement transforma- +tions without communication,” Physical Review A, vol. 67, no. 6, +p. 060302, 2003. +[30] M. M. Wilde, “From classical to quantum shannon theory,” arXiv +preprint arXiv:1106.1445, 2011. +[31] M. Tomamichel, Quantum information processing with finite re- +sources: mathematical foundations, vol. 5. Springer, 2015. +[32] T. S. Han, Information-Spectrum Methods in Information Theory. +Springer, 2003. +[33] M. Tomamichel and M. Hayashi, “A hierarchy of information +quantities for finite block length analysis of quantum tasks,” IEEE +Transactions on Information Theory, vol. 59, no. 11, pp. 7693–7710, +2013. +[34] R. Renner and S. Wolf, “Smooth r´enyi entropy and applications,” +in International Symposium onInformation Theory, 2004. ISIT 2004. +Proceedings., p. 233, IEEE, 2004. +[35] M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and R. Renner, +“The uncertainty principle in the presence of quantum memory,” +Nature Physics, vol. 6, no. 9, pp. 659–662, 2010. +[36] A. McKinlay and M. Tomamichel, “Decomposition rules for quan- +tum r´enyi mutual information with an application to information +exclusion relations,” Journal of Mathematical Physics, vol. 61, no. 7, +p. 072202, 2020. +[37] S. Khatri and M. M. Wilde, “Principles of quantum communica- +tion theory: A modern approach,” 2020. +[38] W. K. Wootters, “Entanglement of formation of an arbitrary state +of two qubits,” Physical Review Letters, vol. 80, no. 10, p. 2245, +1998. +[39] E. Chitambar and G. Gour, “Quantum resource theories,” Reviews +of Modern Physics, vol. 91, no. 2, p. 025001, 2019. +[40] P. Hayden, R. Jozsa, D. Petz, and A. Winter, “Structure of states +which satisfy strong subadditivity of quantum entropy with +equality,” Communications in mathematical physics, vol. 246, no. 2, +pp. 359–374, 2004. +[41] A. El Gamal and Y.-H. Kim, Network Information Theory. Cam- +bridge University Press, 2011. +[42] I. Csisz´ar and J. K¨orner, Information theory: coding theorems for +discrete memoryless systems. Cambridge University Press, 2011. +[43] S. Gharibian, “Strong np-hardness of the quantum separability +problem,” 2008. +[44] M. Hayashi and M. Tomamichel, “Correlation detection and +an operational interpretation of the r´enyi mutual information,” +Journal of Mathematical Physics, vol. 57, no. 10, p. 102201, 2016. +[45] A. Anshu, M. Berta, R. Jain, and M. Tomamichel, “Partially +smoothed information measures,” IEEE Transactions on Information +Theory, vol. 66, no. 8, pp. 5022–5036, 2020. +[46] F. Dupuis, O. Fawzi, and R. Renner, “Entropy accumulation,” +Communications in Mathematical Physics, vol. 379, no. 3, pp. 867– +913, 2020. +[47] T. Metger, O. Fawzi, D. Sutter, and R. Renner, “Generalised +entropy accumulation,” 2022. +[48] D. Sutter, “Approximate quantum markov chains,” in Approximate +Quantum Markov Chains, pp. 75–100, Springer, 2018. +[49] H.-C. Cheng, N. Datta, and C. Rouz´e, “Strong converse bounds +in quantum network information theory,” IEEE Transactions on +Information Theory, vol. 67, no. 4, pp. 2269–2292, 2021. +[50] I. George and E. Chitambar, “Revisiting pure state transforma- +tions with zero communication.” In Preparation. +[51] R. Jain, Y. Shi, Z. Wei, and S. Zhang, “Correlation/communication +complexity of generating bipartite states,” 2012. +[52] J. M. Renes and R. Renner, “One-shot classical data compression +with quantum side information and the distillation of common +randomness or secret keys,” IEEE Transactions on Information +Theory, vol. 58, no. 3, pp. 1985–1991, 2012. +[53] M. +M¨uller-Lennert, +F. +Dupuis, +O. +Szehr, +S. +Fehr, +and +M. +Tomamichel, +“On +quantum +r´enyi +entropies: +A +new +generalization and some properties,” Journal of Mathematical +Physics, vol. 54, no. 12, p. 122203, 2013. +APPENDIX +MUTUAL INFORMATION LEMMAS +In this section of the appendix, we establish various +properties of mutual information measures. Remember +there are three versions of (smoothed) max mutual infor- +mation (Definition 11). In each result we specify which +mutual information we mean. Many of these results are +relatively straightforward variations of proofs from [53] +and/or [31, Chapter 6]. +The first establishes that when smoothing a quantum +state with classical registers, you can restrict to optimiz- +ers that are classical. +Proposition 17. Let ρAXBY ∈ Pos(AXBY) be classical on +X and Y. For ε ∈ [0, +� +Tr(ρ)), the smoothing ball Bε(ρ) +may be restricted to QCQC states and the optimal τ, τ +will be classical on the same registers when optimizing +over Iε,x(AX : BY)ρ for all x ∈ {↓, ↑, ↑↑} where I is any +mutual information defined on any R´enyi divergence D. +Proof. By definition, +Iε,x(AX : BY)ρ = +min +�ρ∈Bε(ρ) +min +τ∈SAX +|�ρ ,τ∈SBY +|�ρ +D(�ρ||τ ⊗ τ) , +where the sets S will also depend which x ∈ {↓, ↑ +, ↑↑} we are using. Next, by data-processing, D((PX ⊗ +PY)(�ρ)||PX(τ) ⊗ PY(τ)) ≤ D(�ρ||τ ⊗ τ), where PX, PY are +the pinching maps onto the computational bases for the +registers X and Y. As ρAXBY is classical on X and Y, +by data-processing of purified distance, we may restrict +minimizing Bε(ρ) to states that are classical on X as ρ is +invariant under pinching on X and Y. Furthermore then +SAX +|�ρ +may be restricted to being classical on X and same +idea for the other set with respect to Y as the optimizing + +29 +choice of �ρ, τ, τ will be classical on those spaces as we +showed via DPI. This completes the proof. +Next, we will need to establish that we can write terms +proportional to max mutual information in terms of an +expectation on the classical register. This will be broken +up into multiple steps. Each step has a conditional +entropic equivalent which can be found in [31]. +Proposition 18. Let ρABX ∈ D≤(ABX) such that it is +classical on X. That is, ρABX = ∑x p(x) |x⟩⟨x| ⊗ ρx +AB. Let +α′ := α − 1. Then for α ∈ (0, 1) ∪ (1, ∞) we have the +following +I↑↑ +α (A : BX) += 1 +α′ log +� +∑ +x +px exp +�(α′)Dα(ρx +AB||ρA ⊗ ρx +B) +� +� +I↑ +α(A : BX)ρ += +α +−α′ log +� +∑ +x +px exp +�α′ +α I↑ +α(ρx +AB||ρA) +�� +I↓ +α(A : BX) += +min +q∈P(X ) +τA,{τx +B}x +1 +−α′ log +� +∑ +x +pα +xq(x)−α′ +· exp +�(α′)Dα(ρx +AB||τA ⊗ ρx +B) +� +� +, +where Iα is any Petz or Sandwiched R´enyi mutual +information over the specified ranges and for the middle +quantity we define +I↑ +α(ρAB||σA) := min +τB Dα(ρAB||σA ⊗ τB) . +Proof. The proof is largely identical to that of [31, Propo- +sition 5.1], but we provide it for completeness. We begin +from the fact that for any CQ states ρXA, σXA, it holds +Dα(ρXA||σXA) += +1 +α − 1 log +� +∑ +x +pα +xq1−α +y +exp((α − 1)Dα(ρx +A||σx +A)) +� +. +As I↑↑ +α (A : BX) = Dmax(ρABX||ρA ⊗ ρBX), we have its +simplification by direct calculation. For I↓ +α(A : BX) we +have established the minimizers τBX can be restricted +to being classical in Proposition 17, thus Iα(A : BX)ρ = +minτA,τBX Dmax(ρ||τ ⊗ τBX) so again a direct calculation +can be established. +Lastly we establish the remaining case. First recall +I↑ +α(A : BX) = minτBX Dα(ρ||ρA ⊗ τBX). It follows from +above then that we have +I↑ +α(A : BX)ρ += 1 +α′ min +τBX log +� +∑ +x +pα +xq−α′ +x +exp +�(α′)Dα(ρx +AB||ρA ⊗ τx +B) +� +� += 1 +−α′ +min +q∈P(X ) log +� +∑ +x +pα +xq−α′ +x +exp((α − 1)Iα(ρx +AB||ρA)) +� +, +where the first is by definition and the second is by +defining I↑ +α(ρAB||σA) := minτB Dα(ρAB||σA ⊗ τB). The +reason this definition suffices in the second step is be- +cause minimizing over τBX = ∑x qx |x⟩⟨x| ⊗ τx +B is equiv- +alent to minimizing the set {τx +B}x and distribution qx +independently, so for each x we can move the choice of +minimizing τx +B in front of the relative entropy. Next, we +define rx := px exp +� +α−1 +α I↑ +α(ρx +AB||ρA) +� +for every x ∈ X . +This means rα +xp−α +x += exp +� +(α − 1)I↑ +α(A : B)ρx +AB +� +for each +x ∈ X . Thus we can plug this substitution back in to +obtain: +I↑ +α(A : BX)ρ += +1 +α − 1 +min +q∈P(X) log +� +∑ +x +pα +xq1−α +x +rα +xp−α +x +� += +1 +α − 1 +min +q∈P(X) log +� +∑ +x +q1−α +x +rα +x +� +. +The last optimization problem is a straightforward min- +imization over a simplex and thus can be solved using +KKT conditions. One could skip over this, but just to be +complete, it is provided at the end of the proof. It finds +the optimizer is qx = rx/(∑x rx) for all x, so we will plug +this in to get the answer: +I↑ +α(A : BX)ρ += +1 +α − 1 log +� +∑ +x +q1−α +x +rα +x +� += +1 +α − 1 log +� +�∑ +x +rα +x +� +∑ +x′ +rx′/rx +�α−1� +� += +1 +α − 1 log +� +�∑ +x +rx +� +∑ +x′ +rx′ +�α−1� +� , +where the second equality is by definition of qx and the +third is because rα +xr1−α +x += rx. Continuing onwards, += +α +α − 1 log +� +� +� +∑ +x +rx +�1/α � +(∑ +x′ +rx′) +�1−1/α� +� += +α +α − 1 log +� +∑ +x +rx +� +, +where the first line we have multiplied and divided by +α and then pulled the 1/α factor into the logarithm and +distributed. Plugging in the definition of rx completes +the proof. +(Solving the KKT Criteria) We can move the log out so +we are optimizing +min +� +∑ +x +q1−α +x +rα +x : ∑ +x +qx − 1 = 0 , −qx ≤ 0 ∀x ∈ X +� +. +Denote the objective function f (q), the equality con- +straint h(q), and the inequality constraints gi(q). Note +∂ +∂qx f (q) += +(1 − α)(rx/qx)α, +∂ +∂qx h(q) += +1, +and + +30 +∂ +∂qx gi(q) = −δx,i. Thus the Lagrangian constraint is +(1 − α) ∑x(rx/qx)αex + λ ∑x ex − ∑x µxex = 0. Since this +is effectively entry-wise, this means (α − 1)(rx/qx)α + +µx = λ for all x. Note if there is any x such that qx = 0, +then this would make λ = ∞, but to be primal feasible +there must exists qx ∈ (0, 1] which would make λ also +finite. This is a contradiction, therefore we can conclude +qx > 0 for all x. If qx > 0 for all x, then complementary +slackness requires µx = 0 for all x. This simplifies the +Lagrange constraint so that by moving things around +we conclude +λ = (α − 1)(rx/qx)α ⇒ qx = rx(α − 1)1/α/λ . +Then by primal feasibility condition +∑ +x +qx = 1 ⇒ λ = (α − 1)1/α ∑ +x +rx . +Finally, plugging this value of λ into qx, we have qx = +rx/(∑x rx). Note that the objective function is linear as α +is fixed as are the constraints so we have linear constraint +qualifications which tells us this is indeed a minimizer. +Note what the above shows us is that only I↑ +α can be +expressed as a mixture of the same information measure +over the conditional states. This gives us the following +nice corollary. +Corollary 2. Let ρABX +∈ D≤(ABX) such that it is +classical on X. Then +I↑ +max(A : BX)ρ = log +� +∑ +x +px exp +� +I↑ +max(ρx +AB||ρA) +�� +, +where +I↑ +max(ρx +AB||ρA) := +min +τB∈D(B) Dmax(ρx +AB||ρA ⊗ τB) . +Proof. By the previous proposition, we know this to hold +for α ∈ (1, ∞). Since I↑ +max := limα→∞ I↑ +α, we just need to +take the limit on the right hand side. This just means +we need to use the product law of limits. By L’Hopital’s +rule, α/(α − 1) and (α − 1)/α both go to one and as +limα→∞ Dα = Dmax, I↑ +α is Imax. This completes the proof. +We also show explicitly that I↑↑ +max and I↓ +max won’t sat- +isfy the property we need for applying the generalized +support lemma. +Proposition 19. Let ρAX ∈ Pos(AX) be classical on X. +Then we have +I↑↑ +max(A : X) = max +x +Dmax(ρx +A||ρA) . +Proof. We can just focus on ρAX ∈ D(AX) by nor- +malization property of Dmax. So we can write ρAX = +∑x p(x) |x⟩⟨x| ⊗ ρx +A. Then we have +I↑↑ +max(A : X)ρ +=Dmax(ρAX||ρA ⊗ ρX) += min{λ : ∑ +x +p(x) |x⟩⟨x| ⊗ ρx +A +≤ exp(λ)∑ +x +(|x⟩⟨x| ⊗ ρA)} += min{λ : ρx +A ≤ exp(λ)ρA} += max +x +Dmax(ρx +A||ρA) , +where the second equality is the definition of Dmax and +expanding the states, +Proposition 20. Let ρAX ∈ Pos(AX) be classical on X. +Then +I↓ +max(A : X)ρAX = +min +q∈P(X ) +τA∈D(A) +log +� p(x) +q(x) ∥τ−1/2 +A +ρx +Aτ−1/2 +A +∥∞ +� +. +Proof. By definition, +I↓ +max(A : X)ρAX += +min +τA∈D(A) +σX∈D(X) +Dmax(ρAX||τA ⊗ σX) += log +min +q∈P(X ) +τA∈D(A) +max +x +p(x) +q(x) ∥τ−1/2 +A +ρx +Aτ−1/2 +A +∥∞ , +where we have used that we can restrict σX to clas- +sical states by Proposition 17 and that Dmax(ρ||σ) = +log ∥σ−1/2ρσ−1/2∥ to simplify in the second line. +We also show that Iε +h(A : BX) may be written as an +expectation like D↑ +max. We will use the following well- +known lemma [37]. +Lemma 25. For Dε +h(ρ||σ), without loss of generality the +optimizer 0 ≤ Λ⋆ ≤ 1 satisfies the constraint with +equality. That is, Tr[Λ⋆ρ] = 1 − ε. +Proposition 21. Let ρABX ∈ D(A ⊗ B ⊗ X) be classical +on the X register. Then +Iε +h(A : BX)ρ = − log +� +∑ +x +p(x)Dε +h(ρx +AB||ρA ⊗ ρx +B) +� +. +Proof. By definition and using the preceding lemma, +exp +�−Iε +h(A : BX)ρ +� +:= +inf +0≤Λ≤1{Tr[Λ(ρA ⊗ ρBX)] : Tr[ΛρABX] = 1 − ε} . +Next, since ρA ⊗ ρBX and ρABX are both invariant under +dephasing onto the X register, without loss of generality +the optimizer must be of the form Λ = ∑x |x⟩⟨x| ⊗ Λx +where 0 ≤ Λx ≤ 1AB for all x ∈ X . Also note that we +can decompose ρABX = ∑x p(x) |x⟩⟨x| ⊗ ρx +AB. Therefore, +exp +�−Iε +h(A : BX)ρ +� += inf +{Λx}x +� +∑ +x +p(x) Tr[Λx +ABρA ⊗ ρx +B] : +∑ +x +p(x) Tr[Λx +ABρx +AB] = 1 − ε} . +Now if Tr +� +Λx +ABρx +AB +� ̸= 1 − ε for any x ∈ X , the constraint +will not be satisfied, so every Tr +� +Λxρx +AB +� = 1 − ε for each + +31 +x ∈ X . Denote Λx,⋆ := argmin(Dε +h(ρx +AB||ρA ⊗ ρx +B)). If the +constraint is satisfied but there is an x such that Λx ̸= +Λx,⋆, then, as Iε +h is a minimization, we could improve the +objective function by replacing Λx with Λx,⋆. Therefore, +the optimal is choosing Λx = Λx,⋆ for all x ∈ X . By +definition of the Λx,⋆’s, this completes the proof. +Smoothed Max Common Information Lemmas +We first establish the alternative smoothed max com- +mon information is always optimized by a classical +auxiliary random variable. +Proposition 22. +min +�ρ∈QMC(ρ) Iε +max(AC : B)�ρ = +min +A−X−C Iε +max(AC : X)�ρ +Proof. This follows by the same argument as Lemma 5, +which we briefly explain. Let (ρA−B−C, �ρABC, σB) be the +optimizers, i.e. +�Cε +max(A : C)ρ = Dmax(�ρABC||�ρAC ⊗ σB) . +Let E : B → BX be as defined in Lemma 5 with respect +to ρA−B−C. Then by data-processing of Dmax, +�Cε +max(A : C)ρ ≥ I↑ +max(AC : X)(TrB ◦E)(�ρ) . +However, as explained in Lemma 5, (TrB ◦E)(ρA−B−C) +is a Markov Chain extension of ρAC, ρA−X−C. As +purified distance decreases under quantum channels, +(TrB ◦E)(�ρ) ∈ Bε(ρA−X−C). Thus this was a minimizer +to begin with, which shows we can restrict to a classical +Markov chain extension. +Here we establish that Cε +max is always optimized by +a normalized state. The proof is effectively identical to +that of [28, Lemma 22]. We make use of the following +lemma. +Lemma 26. ([28, Lemma 21]) Let ρAB ∈ D(A ⊗ B) and +ε ≥ 0. If �ρ ∈ Bε(ρ), then �ρ/ Tr(�ρ) ∈ Bε(ρ). +Proposition 23. Let ρAC ∈ D(A ⊗ C) and ε ≥ 0. Then +there exists normalized quantum Markov chain state +�ρA−X−C such that �ρAC ∈ Bε(ρ) such that Cε +max(A : C)ρ = +I↑ +max(AC : X)�ρA−X−C. +Proof. Let �ρAC ∈ Bε(ρ) with QMC extension �ρA−X−C be +the optimizer of Cε +max. Then there exists σX ∈ D(X) such +that +exp +� +Cε +max(A : C)ρ +��ρAC ⊗ σX ≥ �ρA−X−C +⇒ exp +� +Cε +max(A : C)ρ +� +�ρAC +Tr(�ρAC) ⊗ σX ≥ �ρA−X−C +Tr(�ρAC) . +It follows +Cε +max(A : C)ρ ≥ Dmax +� +�ρAC +Tr(�ρAC) ⊗ σX|| �ρA−X−C +Tr(�ρAC) +� +, +but as this state is contained in the optimization by +Lemma 26 and Cε +max minimizes over states, this renor- +malized state would be the optimizer so long as it +were a QMC extension of �ρAC/ Tr(�ρAC), which we will +now show it is. By definition of QMC extensions on +subnormalized states we know Tr(�ρAC) = Tr(�ρX) so +�ρA−X−C/ Tr(�ρAC) = (R ◦ R)(�ρX)/ Tr(�ρX) ∈ D(A ⊗ X ⊗ +C) and is a QMC extension of �ρAC by the recoverability +characterization of QMCs in Theorem 2. This completes +the proof. +ACHIEVABILITY PROOF FOR SMOOTH MAX COMMON +INFORMATION AEP +In the section we establish the achievability for the +SMCI AEP. To avoid restating many standard results +about strong conditional typicality, we will refer to the +relevant results in [30] when needed. +Proof. If ρAC ̸∈ Sep(A : C), this is trivial as C(A : C) = ++∞. We therefore assume ρAC ∈ Sep(A : C) for the rest +of the proof. At a high level, the proof is as follows: +we bound Cε +max in terms of one shot entropies evaluated +on any MC extension of the smoothed initial state. We +then construct a MC extension for the n−fold case that +for sufficiently large n guarantees its marginal can be +arbitrarily close to ρ⊗n +AC. Lastly, we bound the one-shot +entropies of this specific MC extension using strong +conditional typicality. Taking the appropriate limits then +completes the proof. +We begin by bounding Cε +max(A : C). +Cε +max(A : C)ρ += +min +�ρ∈Bε(ρ) min +A−X−C Imax(AC : X)�ρA−X−C +≤ +min +�ρ∈Bε(ρ) min +A−X−C +� +HR(AC)�ρ − Hmin(AC|X)�ρ +� +(36) +Where the inequality follows from [27, Lemma B.11] and +HR(A)ρ := − log sup{γ ∈ R : γΠρA ≤ ρA} , +(37) +where ΠρA is the projector onto the support of ρA. In +particular, this means exp +�−HR(A)ρ +� = λmin(ρA). +Next we construct a Markov chain distribution whose +marginal is contained in Bε(ρ⊗n +AC) for sufficiently large n. +Let ρA−X−C be any Markov chain extension. It follows +it is of the form ∑x∈X p(x) |x⟩⟨x| ⊗ ρx +A ⊗ ρx +C. Then we +consider +τn +AnXnCn +:= Pr +� +xn ∈ Tδ +Xn +�−1 ∑ +xn∈Tδ +xn +p(xn) |xn⟩⟨xn| +⊗ τxn +An ⊗ τxn +Cn , +where Tδ +Xn is the strongly typical set for ρX, τxn +An := +Tr +� +ρxn +AnΠδ +An|xn +�−1 +Πδ +An|xnρxn +AnΠδ +An|xn where Πδ +An|xn is the +projector onto the strong conditionally typical subspace, +and similarly for τxn +Cn. This is a Markov chain by its +algebraic structure, so we just need to verify its purified +distance can be made arbitrarily small as n grows. We +will instead just use trace norm as purified distance goes +to zero as trace norm does by (4). +���τn +�A �X �C − ρ⊗n +A−X−C +��� +1 + +32 += +����� Pr +� +xn ∈ Tδ +Xn +�−1 ∑ +xn∈Tδ +Xn +p(xn) |xn⟩⟨xn| ⊗ τxn +An ⊗ τxn +Cn +− ∑ +xn∈X n +p(xn) |xn⟩⟨xn| ⊗ ρxn +An ⊗ ρxn +Cn +����� +1 += +����� ∑ +xn∈Tδ +Xn +� +Pr +� +xn ∈ Tδ +Xn +�−1 +p(xn) − p(xn) +� +|xn⟩⟨xn| +⊗ +� +τxn +An ⊗ τxn +Cn − ρxn +An ⊗ ρxn +Cn +� ����� +1 ++ Pr +� +xn ̸∈ Tδ +Xn +� += ∑ +xn∈Tδ +Xn +� +Pr +� +xn ∈ Tδ +Xn +�−1 +p(xn) − p(xn) +� +· +���τxn +An ⊗ τxn +Cn − ρxn +An ⊗ ρxn +Cn +��� +1 + Pr +� +xn ̸∈ Tδ +Xn +� +where everything so far is just expanding definitions. We +now bound the trace norm for a single xn. To simplify +notation, let Π := Πδ +An|xn ⊗ Πδ +Cn|xn and Π⊥ := (IAnCn − +Π). +���τxn +An ⊗ τxn +Cn − ρxn +An ⊗ ρxn +Cn +��� +1 += +���τxn +An ⊗ τxn +Cn − Πρxn +An ⊗ ρxn +CnΠ +− +� +ρxn +An ⊗ ρxn +Cn − Πρxn +An ⊗ ρxn +CnΠ +� ��� +1 +≤ +���τxn +An ⊗ τxn +Cn − Πρxn +An ⊗ ρxn +CnΠ +��� +1 ++ +���ρxn +An ⊗ ρxn +Cn − Πρxn +An ⊗ ρxn +CnΠ +��� +1 += +� +Tr +� +ρxn +AnΠδ +An|xn +�−1 +· Tr +� +ρxn +CnΠδ +Cn|xn +�−1 +− 1 +� ++ +� +1 − Tr +� +ρxn +AnΠδ +An|xn +� +· Tr +� +ρxn +CnΠδ +Cn|xn +�� +≤ +� +2ε′ − ε′2 +(1 − ε′)2 +� ++ 2ε′ − ε′2 +≤2 +ε′ +(1 − ε′)2 + 2ε′ +≤4ε′ + 2ε′ = 6ε′ +where the first inequality is the triangle inequality, the +following equality is using the definition of τxn +An ⊗ τxn +Cn, +the second inequality is for sufficiently large n using the +‘unit probability’ property of conditionally typical state, +and we have assumed that ε′ ≤ 1/2 so that ε′/(1 − ε′)2 ≤ +2ε′. Now we plug this back into our equation to get +���τn +�A �X �C − ρ⊗n +A−X−C +��� +1 +=6ε′ · ∑ +xn∈Tδ +Xn +(Pr +� +xn ∈ Tδ +Xn +�−1 +p(xn) − p(xn)) ++ Pr +� +xn ̸∈ Tδ +Xn +� +≤6ε′ · +� +Pr +� +xn ∈ Tδ +Xn +�−1 +Pr +� +xn ∈ Tδ +Xn +� +− Pr +� +xn ∈ Tδ +Xn +�� ++ Pr +� +xn ̸∈ Tδ +Xn +� +≤6ε′ · (1 − (1 − ε′)) + ε′ = 6ε′2 + ε′ ≤ 4ε′ , +where we again use that n must be sufficiently large +and in the final inequality we have used that we already +assumed ε′ ≤ 1/2, so 6ε′2 ≤ 3ε′. +It follows for sufficiently large n, for any ε ∈ (0, 1), +you can pick a ε′ small enough that τn ∈ Bε(ρ⊗n +A−X−C) as +purified distance is upper bounded by +� +2∥τn − ρ⊗n∥1. +Therefore, for sufficiently large n, using (36), +Cε +max(An : Cn)ρ +≤ HR(AC)τn − Hmin(AC|X)τn . +(38) +Now we just need to bound these terms. We start with +the Hmin term. First, by properties of strong typicality for +a classical system [30, Section 14.7.2], the strongly typical +sequences satisfy p(xn) ≤ 2−n(H(X)−cδ) and |Tδ +Xn| ≤ +2n(H(X)+cδ. Second, the min-entropy of a conditional +state ρxn +An is given by Hmin(An)ρxn +An = − log ∥ρxn +An∥∞ and +recall ∥ · ∥∞ = λmax(·) for a positive semidefinite opera- +tor. So we want to bound this. Note, using properties +of strong conditional quantum typicality [30, Section +15.2.4], +τxn +An = Tr +� +ρxn +AnΠδ +An|xn +�−1 +Πδ +An|xnρxn +AnΠδ +An|xn +≤(1 − ε)−12−n(H(A|X)−δ′′)Πδ +An|xn . +It follows +∥τxn +An∥∞ ≤ (1 − ε)−12−n(H(A|X)−δ′′) , +where we have used Πδ +An|xn is a projector. and by an +identical argument one can bound ∥τCn +|xn∥∞. +Combining these points, we have +− Hmin(AC|X) += log +� +� ∑ +x∈Tδ +Xn +p(xn) exp +� +−Hmin(ρAC +|xn ) +� +� +� += log +� +∑ +x∈Tδ +Xn +p(xn) exp +� +−Hmin(ρA +|xn) +� +· exp +� +−Hmin(ρC +|xn) +�� += log +� +� ∑ +x∈Tδ +Xn +p(xn)∥ρA +|xn∥∞∥ρC +|xn∥∞ +� +� +≤ log +� +2−n(H(X)−cδ)2n(H(X)+cδ)� ++ log +� +(1 − ε)−12−n(H(A|X)−δ′′)� ++ log +� +(1 − ε)−12−n(H(C|X)−δ′′)� += − n [H(A|X) + H(C|X)] + 2(cδ + δ′′) +− 2 log(1 − ε) += − nH(AC|X) + 2n(cδ + δ′′) − 2 log(1 − ε) , +(39) + +33 +where the first equality is from is the expansion of +min-entropy conditioned a classical register [31] and the +second equality is additivity of min-entropy over tensor +products. +Next we bound HR(AC)τn. First note that +τn +AnCn +=∑ +xn +�p(xn)τAn +|xn ⊗ τCn +|xn +≥(1 − ε)−3 ∑ +xn +p(xn)2−n(H(A|X)+δ′′)2−n(H(C|X)+δ′′) +· Πδ +An|xn ⊗ Πδ +Cn|xn +≥(1 − ε)−32−n(H(X)+δ)2−nH(AC|X)2−2nδ′′ +· ∑ +xn∈Tδ +Xn +Πδ +An|xn ⊗ Πδ +Cn|xn +=(1 − ε)−32−nH(AC)2−n(2δ′′−δ) ∑ +xn∈Tδ +Xn +Πδ +An|xn ⊗ Πδ +Cn|xn +≥(1 − ε)−32−nH(AC)2−n(2δ′′−δ)Πδ +An|xn ⊗ Πδ +Cn|xn , +where we have just used strong conditional typicality +properties again and at the end we have just picked an +arbitrary xn and its conditional state which is dominated +by itself and thus the sum. It is an immediate conse- +quence that +λmin(τn +AnCn) ≥ (1 − ε)−32−nH(AC)2−n(2δ′′−δ) . +(40) +Note by definition, (37), exp(−HR(AC)) = λmin(ρAC). +Therefore, using (40) +HR(AC) ≤ − log +� +(1 − ε)−32−nH(AC)2−n(2δ′′−δ)� +=nH(AC)ρ + n(2δ′′ − δ) + 3 log(1 − ε) . +We can plug this bound into the RHS of (38) along with +(39) to obtain +Cε +max(An : Cn)ρ +≤n +� +H(AC)ρ − H(AC|X)ρ +� + n +� +4δ′′ + (2c − 1)δ) +� ++ 1 log(1 − ε) +=nI(AC : X)ρ + n +� +4δ′′ + (2c − 1)δ) +� + 1 log(1 − ε) . +Note this holds for any choice of Markov Chain exten- +sion. Therefore, picking ρ⋆ as the Markov Chain that +optimizes C(A : C)ρ and dividing by n, we have +1 +nCε +max(An : Cn) +≤C(A : C) + +� +4δ′′ + (2c − 1)δ) +� + 1 +n log(1 − ε) , +so letting n → ∞ and δ → 0 completes the proof. +CHAIN RULES FOR SMOOTH MAX COMMON +INFORMATION +In this section we establish lemmas for how to reduce +the asymptotic behaviour of SMCI to properties of the +conditional min-entropy. To do so, we first show how +to generalize results of [27], [28] to get a strong AEP +for I↑,ε +max, and then just note how to modify these proofs +to establish chain rules for SMCI. We begin by proving +the lower bounds in detail as this requires the most +alteration from the previous proof [28]. We first refine +the basic trick for a lower bound. +Proposition 24. Let ε ∈ (0, 1) and ρ ∈ D(A). Then there +exists 0 ≤ Π ≤ 1 such that [Π, ρ] = 0, P(ρ, ΠρΠ) ≤ +2 +� +ε(1 − ε) and +Hε +min(A)ρ ≤ Hmin(A)ΠρΠ . +Proof. By [35, Lemma 18], there exists a Π as speci- +fied such that Tr +�(1 − Π2)ρ +� ≤ 2ε satisfying the min- +entropy bound given. Note 2ε +≥ +Tr +�(1 − Π2)ρ +� += +1 − Tr +� +Π2ρ +� +which +gives +us +Tr +� +Π2ρ +� +≥ +1 − +2ε, so Tr +� +Π2ρ +�2 +≥ +(1 − 2ε)2. Finally this means +� +1 − Tr(Π2ρ)2 ≤ +� +1 − (1 − 2ε)2. By [27, Lemma A.7], +we have P(ρ, ΠρΠ) ≤ Tr(ρ)−1/2 +� +Tr(ρ)2 − Tr(Π2ρ)2 = +� +1 − Tr(Π2ρ)2 ≤ +� +1 − (1 − 2ε)2 = 2 +� +ε(1 − ε). +Now we can use this refinement to extend the result +of [28] to the full parameter range. +Proposition 25. ([28, Lemma 6, Extended to Full Param- +eter Range]) Let ρAB ∈ D(A ⊗ B). Let 0 < ε < ε < 1 such +that such that ε + δ(ε) ∈ (0, 1) where δ(ε) := 2 +� +ε(1 − ε). +Then it holds +I↑,ε +max(A : B)ρ ≥ Hε−ε +min(A)ρ − Hε+δ(ε) +min +(A|B)ρ . +Proof. The proof is largely identical to the original except +that we have refined certain steps. By re-arranging [27, +Lemma B.13] and maximizing over the smoothing ball, +we have +Hε+δ((ε) +min +(A|B)ρ +≥ +max +�ρ∈Bε+δ(ε)(ρ) +� +Hmin(A)�ρ − Imax(A : B)�ρ +� +. +Then, +Hε+δ((ε) +min +(A|B)ρ +≥ +max +�ρ∈Bε+δ(ε)(ρ) +� +Hmin(A)�ρ − Imax(A : B)�ρ +� +≥ +max +ω∈Bε(ρ) +� +max +Π [Hmin(A)ΠωΠ − Imax(A : B)ΠωΠ] +� +, +where the new maximization is over 0 ≤ ΠA ≤ 1A +such that ΠωΠ ≈δ(ε) ω. This is a lower bound because +we restricted smoothing to Bε, so ω ≈ε ρ which using +purified distance is a metric implies ΠωΠ ≈ε+δ(ε) ρ. Let +ω⋆ ∈ Bε(ρ) ∩ D(A ⊗ B) be the optimizer of Iε +max(A : B)ρ +which is normalized without loss of generality [28, +Lemma 22]. Then, +Hε+δ(ε) +min +(A|B)ρ +≥ max +Π [Hmin(A)Πω⋆Π − Imax(A : B)Πω⋆Π] +≥ max +Π [Hmin(A)Πω⋆Π] − Imax(A : B)ω⋆ , + +34 +where the first we have chosen ω⋆ rather than maxi- +mizing and the second we have used that Dmax actually +satisfies data-processing for CPTNI maps. Then as ω⋆ is +normalized and we range over Π such that Πω⋆Π ≈δ(ε) +ω⋆, we know by Proposition 24 we can bound the state +Hε+δ(ε) +min +(A|B)ρ ≥ Hε +min(A)ω⋆ − Imax(A : B)ω⋆ +≥ Hε−ε +min(A)ρ − I↑,ε +max(A : B)ρ , +where the first line is using ω⋆ was the optimizer for +I↑,ε +max(A : B)ρ. The second line is because if �ρ ∈ Bε−ε(ρ), +as ω ≈ε +ρ, we can conclude �ρ ≈ε +ω and thus is +included in the previous line’s optimization. As smooth +min-entropy is maximized, this suffices. Re-ordering the +terms completes the proof. +As a corollary, we have the following lower bound on +SMCI. +Corollary 3. Let ρAC ∈ D(A ⊗ C). Let 0 < ε < ε < 1 such +that ε + δ(ε) where δ(ε) := 2 +� +ε(1 − ε). Then it holds, +Cε +max(A : C)ρ +≥Hε−ε +min(AC)ρ − +max +�ρ∈Bε+δ(ε)(ρ) +max +A−X−C Hmin(AC|X) , +Proof. The proof is effectively the same as the previous +proposition except one must keep track of the min- +imiziation over Markov chain extensions along with +the projection. One will need to minimize over the +Markov chain extension under the projection Π and +maintain the Markov chain property, i.e. consider a +min(ΠωΠ)A−B−C Imax(AC : B)ΠωΠ term where we still de- +mand ΠωΠ ≈δ(ε) ω. Note if the feasible set is empty, +then this term is infinite and so, as we subtract it, in this +setting the lower bound trivially holds, so the proof will +go in both when such a Π exists or does not. +The upper bound chain rule for Iε +max is already suffi- +cient, so we just state it. +Proposition 26. ([27, Lemma B.12]) Let ε ∈ (0, 1). Then, +I↑,ε +max(A : B)ρ +≤ Hε2/48 +max (A)ρ − Hε2/48 +min (A|B)ρ − 2 log +� +ε2/24 +� +. +Corollary 4. Let ε ∈ (0, 1) and ρAC ∈ D(A ⊗ C), then +Cε +max(A : C)ρ ≤ Hε2/48 +max (AC)ρ − max +A−X−C Hε2/48 +min (AC|X)ρ +− 2 log +� +ε2/24 +� +. +Proof. The proof is the same as the previous proposi- +tion except you push the minimization over Markov +extensions through and note the minus sign flips the +minimization into a maximization. +It is clear that the chain rules for I↑,ε +max can then be used +to establish a strong AEP for I↑,ε +max as we quickly show. +Proposition 27. Let ε ∈ (0, 1) and ρ ∈ D(A ⊗ B). Then, +lim +n→∞ +� 1 +n Iε +max(An : Bn)ρ⊗n +� += I(A : C)ρ . +Proof. Fix ε, δ(ε) that satisfy Proposition 25 to get +Hε−ε +min(A⊗n)ρ⊗n − Hε+δ(ε) +min +(An|Bn)ρ⊗n +≤ I↑,ε +max(An : Bn)ρ⊗n +≤ Hε2/48 +max (An)ρ⊗n − Hε2/48 +min (An|Bn)ρ⊗n +− 2 log +� +ε2/24 +� +. +Dividing by n and taking the limit as n → ∞, using the +AEP for smooth min and max-entropies [31], +H(A)ρ − H(A|B)ρ ≤ lim +n→∞ +� +I↑,ε +max(An : Bn)ρ⊗n +� +≤H(A)ρ − H(A|B)ρ . +Noting that H(A) − H(A|B) = I(A : B) by standard +chain rules completes the proof. +This general proof method then gets us the following. +Proposition 28. Let ρAC ∈ D(A ⊗ C). Let ε ∈ (0, 1), +δ ∈ (0, ε), ε ∈ (ε, 1) such that 0 < ε + δ((ε)) < 1 where +δ(ε) := 2 +� +ε(1 − ε). Then +H(AC)− +lim +n→∞ +� +1 +n +max +�ρ∈Bε+δ(ε)(ρ⊗n) +max +An−X−Cn Hmin(AnCn|X) +� +≤ lim +n→∞ +� 1 +nCε +max(An : Cn)ρ⊗n +� +≤H(AC) − lim +n→∞ +� +max +An−X−Cn +1 +n Hε−δ +min(AnCn|X)ρ +� +, +Proof. The proof method is the same as the previous +proposition. +Note that we don’t particularly care about the upper +bound as our achievability result held for all ε ∈ (0, 1) +(Lemma 18). Therefore, our interest is in if the lower +bound in the previous proposition can upper bound +the common information in the regularized limit. To get +Proposition 11, note that by a standard chain rule, +C(A : C)ρ = +min +A−X−C I(AC : X)ρ +=H(AC) − max +A−X−C H(AC|X)ρ , +so to acquire the asymptotic bound, it would suffice that +H(AC) − max +A−X−C H(AC|X)ρ +≤H(AC) +− lim +n→∞ +� +1 +n +max +�ρ∈Bε+δ(ε)(ρ⊗n) +max +An−X−Cn Hmin(AnCn|X) +� +, +which by cancelling and multiplying by negative one +gets us the term in Proposition 11. + diff --git a/VtE3T4oBgHgl3EQfFAlo/content/tmp_files/load_file.txt b/VtE3T4oBgHgl3EQfFAlo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f89ca027938fffcd856e3d47e145f089c149e2e3 --- /dev/null +++ b/VtE3T4oBgHgl3EQfFAlo/content/tmp_files/load_file.txt @@ -0,0 +1,1924 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf,len=1923 +page_content='1 One-Shot Distributed Source Simulation: As Quantum as it Can Get Ian George, Min-Hsiu Hsieh, and Eric Chitambar Abstract—Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Wyner’s seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner’s common in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This asymptotic result was extended by Hayashi in the quantum setting to separable states, the largest class of states for which this task can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In this work we characterize this task in the one-shot setting using the smooth entropy framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We do this by introducing one-shot operational quantities and correlation measures that characterize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We establish asymptotic equipartition properties for our correlation measures thereby recovering, and in fact strengthening, the aforementioned asymptotic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In doing so, we consider technical points in one- shot network information theory and generalize the support lemma to the classical-quantum setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also introduce entanglement versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' INTRODUCTION At the core of information theory is the notion of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This is present even in Shannon’s initial work, as one can view both source and channel coding as the limits of establishing perfect correlation between inputs and outputs [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Another task where correlation plays a central role is that of distributed source simulation, which asks how much correlation must be provided to two spatially-separated and non-interacting parties so that they can generate a target joint distribution pXY up to some tolerated error ε (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It was established by Wyner that when the tolerated error is expressed in terms of regularized relative entropy, the rate of gener- ating i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' copies of pXZ is given by R = min qXYZ:qXZ=pXZ & X−Y−Z I(XZ : Y)q := C(X : Z)p , (1) where X − Y − Z denotes a short Markov chain [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The correlation measure on the right hand side is of- ten referred to as ‘Wyner’s common information.’ The achievability of this result was established by Wyner’s introduction of what is now referred to as a soft-covering lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Ian George and Eric Chitambar are with the Department of Elec- trical and Computer Engineering, University of Illinois at Urbana- Champaign, Urbana, Illinois, 61801, USA, email: igeorge3@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Min-Hsiu Hsieh is with Hon Hai (Foxconn) Research Institute, Taipei, Taiwan, email: min-hsiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='hsieh@foxconn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' �qy Copy ΦY→X X Z ΨY′→Z Y′ Y ≈ε pXZ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1: The general structure of distributed classical source simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The seed is copied and distributed, forming a source of shared randomness between Al- ice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then Alice and Bob apply local maps to construct an output distribution �qXZ, which should approximate the target distribution pXZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Since Wyner’s initial work, which was inspired by prior work by G´acs-K¨orner [3] and Witsenhausen [4], many variations of common information and refine- ments of distributed source simulation have been con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' extended distributed source simulation to multipartite joint distributions [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Yu and Tan consid- ered R´enyi divergences and total variation as measures of error, which in particular led to them establishing a strong converse under the total variation measure [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Winter extended to the case where there is an eavesdropper, the adversarial setting, so that it relates to key distillation [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Chitambar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' compared this adversarial setting to the collaborative alternative [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, Chitambar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' related the adversarial setting to quantum entanglement manipulations [10], [11] at one point using the G´acs-K¨orner common information [3], which also is relevant in round complexity of state transformations [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Cuff established a general tradeoff region between Wyner common information and the classical reverse Shannon theorem when simulating a classical channel [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' There is also the related problem of exact common information introduced by Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' which considers that the target state p⊗n XY is exactly constructed but allows for variable-length codes [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It was established by Yu and Tan that the exact common information corresponds to the common information with the error measured in terms of the max-divergence [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Furthermore, the soft-covering lemma used for achievability was established for error measured in total variation by Hayashi [16] and Cuff [17] and in the one-shot setting for R´enyi divergence error by Yu and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='04301v1 [quant-ph] 11 Jan 2023 2 Tan [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We refer the reader to Yu and Tan’s recent monograph for further details on the history of common information in the classical setting [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, the bulk of this previous research has been restricted to the classical common information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In the quantum setting, there are fundamental differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In- deed, one of the key features of quantum mechanics, and consequently resources of quantum information theory, is quantum entanglement, which is a form of correlation that classical systems do not admit [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' One way entan- glement has been presented is as a quantum analogue of perfect correlation, the latter being the underlying resource in distributed classical source simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' How- ever, it has been shown that one cannot freely transform entanglement by local processing like one can with classical shared randomness without communication be- tween the distributed parties [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That is to say, the fully entangled equivalent of distributed source simulation is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, distributed parties who can only communicate classically cannot generate entanglement from shared randomness [22], and so the task cannot be extended to using the shared randomness to generate an entangled state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Nonetheless, there is still space for a quantum ex- tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Specifically, the set of quantum distributions which are not entangled are the separable states, which is a strict superset of classical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Separable states can be decomposed into a convex combination of product (quantum) distributions and consequently should be able to be simulated in a distributed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Indeed, Hayashi extended Wyner’s result to all separable states in terms of trace norm error [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In doing so, he introduced a novel covering lemma for quantum states that does not presume i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' is a one-shot characterization, although a quantum covering lemma with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' structure had been previously introduced by Ahlswede and Winter for considering different tasks [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This has remained the state of the quantum extension of Wyner’s common information for more than a decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, very recently there has been improvements upon the quantum one-shot soft-covering lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In particular, its error exponents have been characterized in terms of R´enyi mutual information measures [25] and its second-order asymptotics were established via a characterization in terms of hypothesis testing mutual information [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This would suggest the possibility of establishing a one-shot version of Wyner’s common in- formation for separable states which recovers Hayashi’s asymptotic extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Summary of Results In this work, we extend distributed source simula- tion and Wyner’s common information to the one-shot quantum setting for separable states using the smooth entropy framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In doing so, we introduce new measures of operational tasks Cε F, Cε U,F, �Cε F, where Cε U,F is the one-shot version of Wyner’s quantity restricted to uniform shared randomness, Cε F relaxes the requirement that the randomness be uniform, and �Cε F allows one to distribute entangled states that are indistinguishable from a Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We introduce new one-shot corre- lation measures to extend Wyner’s common information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Specifically, we introduce measures based on the max mutual information [27], [28] Cε max, �Cε max as well as one induced by the hypothesis testing divergence, Cε h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We establish achievability and converse bounds on the one- shot distributed-source simulation and related tasks in terms of these measures which hold in general if the state is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (One-Shot Distributed Source Simulation Bounds) Let ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C) such that ∥ρ − �ρ∥1 ≤ ε for some separable state �ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε1, ε2 ∈ (0, 1) satisfy 2ε1 + ε2 < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, C √ε max(A : C)ρ ≤ Cε F ≤ Cε U,F ≤ Cε1 max(A : C)ρ + κ(ε2) , where κ(ε2) is a constant that scales as o(n) for ρ⊗n AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (Variations of Distributed Source Simulation Bounds) Let ρAC be a separable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If δ ∈ (0, ε) and η ∈ ( 7 8ε, ε), then �C √ε max(A : C)ρ ≤ �Cε F(A : C)ρ ≤ �C √ε−η max (A : C)ρ + o(n) , or if δ′ ∈ (0, 1 − ε), �C1−ε−δ′ h (A : C)ρ + o(n) ≤ �Cε F(A : C)ρ ≤ �C1−ε/8 h (A : C)ρ + o(n) , where o(n) always represents a term that scales as o(n) for ρ⊗n AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also establish a (weak) asymptotic equipartition property (AEP) for the correlation measures induced by max divergence for separable states, which do not follow from pre-existing asymptotic equipartition properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (AEP for One-Shot Wyner Common Information) Let ρAC be separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, lim ε→0 lim n→∞ � 1 nCε max(An : Cn)ρ⊗n � = C(A : C)ρ, lim ε→0 lim n→∞ � 1 n �Cε max(An : Cn)ρ⊗n � = C(A : C)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' These AEPs allow us to not only recover Hayashi’s asymptotic extension of the Wyner common information, but establish something stronger which says that even if the source were not uniform or we allowed entan- glement assistance but restricted to be approximately indistinguishable from a Markov chain, asymptotically these all achieve the same rate if the error is required to go to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (All Variations Have Same Vanishing Error Rate) C(A : C)ρ = lim ε→0 lim n→∞ � 1 nCε F(An : Cn)ρ⊗n � 3 = lim ε→0 lim n→∞ � 1 nCε U,F(An : Cn)ρ⊗n � = lim ε→0 lim n→∞ � 1 n �Cε F(An : Cn)ρ⊗n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also show how all the stated results extend beyond bipartite setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Finally, as these results cannot be extended to an en- tangled setting, we present entangled equivalents: ‘em- bezzling source simulation’ and a variation that allows for shared randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Both versions use embezzlement [29] to simulate the target state (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In particular, we establish nearly tight upper and lower bounds in the case shared randomness is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρ ∈ D(A ⊗ C) and ε ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then log(Entε A:C(ρ)) ≤Cε SREE,S(A : C)ρ ≤1 ε log(EntA:C(ρ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, both the lower bound and upper bound can be shown to be nearly tight while.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' σA′C′ A′ C′ ΦA′→AA′ ΦC′→CC′ A C A′ C′ ρAC ≈ε ⊗ σA′C′ (a) Embezzling Source Simulation σA′C′ A′ C′ ΦA′→AA′ ΦC′→CC′ A C A′ C′ ρAC ≈ε (b) Entangled Source Simulation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2: Entangled state versions of distributed source simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Grey lines represent allowed correlations of either classical or quantum mechanical nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (b) Em- bezzling source simulation where the auxiliary state is required to be output approximately decoupled from the simulated state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (c) Entangled Source Simulation where the auxiliary system may be arbitrarily correlated with the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This in particular allows for the use of classical correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In establishing these listed results, we establish tech- nical tools which may be of independent interest or use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We establish a generalization of the support lemma (Lemma 6) so as to establish cardinality bounds, which could be of use in other quantum settings with an auxil- iary classical random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We discuss the difficulty of using one-shot measures induced by hypothesis test- ing when an auxiliary random variable is used, which we expect is relevant in other quantum network settings in the smooth entropy framework as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also prove various properties of one-shot mutual informa- tions which exemplify the importance in choosing which one-shot mutual information one uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In particular, we establish a property of max mutual information as originally defined in [27] that allows for straightforward cardinality bounds of an auxiliary classical random vari- able, but the alternatives discussed in [28] do not satisfy the same property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Organization of the Paper The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In Section II we establish basic notation used throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In Section III we present the necessary back- ground on one-shot information measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In Section IV we introduce one-shot distributed source simulation, its variants, and its impossibility for entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In Section V, we introduce the one-shot correlation measures to capture distributed source simulation, the smooth max common information, and its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also establish basic properties of these measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In particular, we straightforwardly generalize the support lemma so as to establish cardinality bounds on these measures and show that there are cases where these measures are NP-hard to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In Section VI we establish achievability for distributed source simulation and its variants in terms of their respective measures by modifying the one-shot soft-covering results of [26] to be in terms of max mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In Section VII we establish converses for distributed source simulation and its variants in terms of their one-shot correlation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This along with the previous section estab- lishes tight (to first order) characterization of these tasks in terms of smooth mutual information quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In Section VIII, we establish weak asymptotic equipartition properties (AEPs) for our correlation measures, which do not simply follow from previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By establishing this weak AEP, we are able to both recover Hayashi’s asymptotic extension of Wyner common information as well as generalize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In Section IX, we explain how these results are straightforward to generalize to source simulation of more than 2 parties and clarify certain properties in this setting noted in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In Section X we present the entangled state versions of distributed source simulation and establish bounds on the resources for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Finally, in Section XI, we re-summarize what we have presented and discuss avenues for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' NOTATION Our notation largely follows standard texts to which we refer the reader for further details [22], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We will denote finite alphabets by calligraphic roman letters at the end of the alphabet, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' X , Y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='. The probability simplex over finite alphabet X is denoted P(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We talk of complex Euclidean spaces (CESs), equivalently finite Hilbert spaces, denoted by capital roman letters, 4 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A ≡ C|X |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Given a CES A, we define the follow- ing classes of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The space of endomorphisms is denoted L(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The space of Hermitian operators is Herm(A) ≡ {X ∈ L(A) : X = X∗} where ·∗ is the conjugate transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The space of positive semidefinite operators is Pos(A) ≡ {X∗X : X ∈ L(A)}, where we remind the reader X ∈ Pos(A) if and only if all of the eigenvalues are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We will often use P, Q to denote generic positive semidefinite operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' a) Quantum States: The space of quantum states, referred to as density matrices is D(A) ≡ {ρ ∈ Pos(A) : Tr(ρ) = 1}, where Tr(·) is the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Often times we will have density matrices defined on tensor product spaces, so we will add subscripts to the state to specify, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ρAB ∈ D(A ⊗ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We say a state ρA is pure if there exists a vector |ψ⟩ ∈ A such that ρA = |ψ⟩⟨ψ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A quantum state is classical if it is diagonal in the standard basis, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ρ = ∑x∈Σ p(x)Ex,x where p ∈ P(Σ) and {Ex,y}x,y∈Σ form the standard basis for L(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We denote classical registers with capital roman letters at the end of the alphabet, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' X, Y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' to help distinguish from quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A classical-quantum (CQ) state has the following convenient decomposition: ρXB = ∑x∈X p(x) |x⟩⟨x| ⊗ ρx B, where p ∈ P(X ) and {ρx B}x∈X are referred to as the conditional states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The space of sub-normalized states is given by D≤(A) ≡ {ρ ∈ Pos(A) : Tr(ρ) ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As alluded to in the introduction, quantum states can be partitioned into states that are and aren’t en- tangled, which are known as separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A ‘bipartite’ quantum state ρ ∈ D(A ⊗ B) is separable if and only if there exists a finite alphabet X , probability distribution p ∈ P(X ), and sets of density matrices {σx A}x∈X ⊂ D(A), {τx B}x∈X ⊂ D(B) such that ρ = ∑ x∈X p(x)σx A ⊗ τx B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (2) Any state that is not separable is entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We denote the space of separable states in D(A ⊗ C) as SepD(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' b) Quantum Channels: A map E : L(A) → L(B) is a quantum channel, E ∈ C(A, B), if it is a completely positive (CP) and trace preserving (TP) map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A par- ticularly important class of channels for this work are the classical-to-quantum or preparation channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Given CESs A ≡ C|X |, B ≡ C|Y|, a preparation channel Eprep may be defined by its action Eprep(W) := ∑ x∈X ⟨x| W |x⟩ ρx B , where W ∈ L(A) and {ρx B}x∈X ⊂ D(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This means the channel projects the input into the standard basis and then prepares a state dependent on the outcome, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' if given |x⟩⟨x|, it prepares ρx B, hence its name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' c) Metrics on States: Lastly, we consider two metrics on states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The first is the trace distance, which is the quantum generalization of the total variation in the sense that it captures the distinguishability between the two quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Given ρ, σ ∈ D(A), the trace distance is TD(ρ, σ) := 1 2∥X − Y∥1 , where ∥ · ∥1 is the Schatten one-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We will also consider the purified distance P(·, ·) which we refer the reader to [31] for detailed informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For our purposes it will be sufficient to note that for ρ, σ ∈ D(A), TD(ρ, σ) ≤ P(ρ, σ) ≤ � 2TD(ρ, σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (3) For the reader’s intuition we note that the purified distance is greater than the trace distance because it operationally measures the maximal distinguishability between purifications of ρ, σ rather than the distinguisha- bility of the states themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also define the follow- ing equivalences for notational convenience: P(ρ, σ) ≤ ε ⇔ ρ ≈ε σ (4) 2TD(ρ, σ) ≤ ε ⇔ ρ ≈TD ε σ (5) Lastly, we note, as they are metrics, they act as good distance measures on quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As such, we can use them to measure the distance between a state and the set of separable states, which will be useful later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ρAB ∈ D(A ⊗ B), the trace distance of entanglement is defined as ET(A : B)ρ := inf σAB∈SepD(A : B) TD(ρ, σ) , and the purified distance of entanglement, EP(A : C)ρ, is defined identically with TD(·, ·) replaced with the purified distance P(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ONE-SHOT ENTROPIES AND INFORMATION MEASURES We now summarize the background on one-shot en- tropies and their relation to asymptotic entropies as necessary for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We note that a secondary aspect of this work is to highlight what it means to determine the ‘correct’ one-shot mutual information in our setting as there are a myriad of them and because the previous work [26] initiates such a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For this reason, this section is longer as it motivates why there are so many to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For P, Q ∈ Pos(A), the relative entropy is defined as D(P||Q) = Tr[P log P] − Tr[P log Q] when Supp(P) ⊆ Supp(Q) and is otherwise infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This recovers the KL divergence if P, Q are classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' From this definition one can extend the standard classical information quantities from the KL divergence to the 5 quantum setting [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In particular, one can recover the many equivalent definitions of mutual information: I(A : B)ρ :=D(ρAB||ρA ⊗ ρB) = min σB∈D(B) D(ρAB||ρA ⊗ σB) = min τA∈D(A) σB∈D(B) D(ρAB||τA ⊗ σB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (6) However, in the one-shot setting there are more en- tropic measures to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In the classical setting, this is predominantly handled by the information spectrum di- vergence [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In the quantum setting there are multiple options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' While there is the extension of the information spectrum divergence [33], Dε s(ρ||Q) := sup γ∈R {Tr[ρ{ρ ≤ exp(γ)Q}] ≤ ε} , (7) there is also the smooth entropy calculus which ‘smooths’ entropic quantities defined in terms of the quantum max-relative divergence and duality (See [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='1 The max-relative divergence is defined as Dmax(P||Q) := inf{λ ∈ R : P ≤ exp(γ)Q} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (8) One appealing property of the max-relative divergence is that it benefits from a particularly general data pro- cessing inequality (DPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For any CP map E, Dmax(E(P)||E(Q)) ≤ Dmax(P||Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let γ⋆ be the optimizer for Dmax(P||Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As E is CP, P ≤ exp(γ⋆)Q implies E(P) ≤ exp(γ⋆)E(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As Dmax is defined as an infimum and we have just shown γ⋆ is feasible for Dmax(E(P)||E(Q)), this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Just as mutual information is defined from relative entropy, the max mutual information is defined from the max divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, in general the three equiva- lent definitions given in (6) are inequivalent for Dmax and thus there are three possible max mutual informations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ρAB ∈ D≤(A ⊗ B), I↑↑ max(A : B)ρ := Dmax(ρAB||ρA ⊗ ρB) (9) I↑ max(A : B)ρ := min σB∈D(B) Dmax(ρAB||ρA ⊗ σB) (10) I↓ max(A : B)ρ := min τA∈D(A) σB∈D(B) Dmax(ρAB||τA ⊗ σB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (11) We note (10) was introduced in [35] while (9) and (11) were introduced in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Our notation differs from both of these to make the relation I↑↑ max ≥ I↑ max(A : B) ≥ I↓ max(A : B) , explicit and also to align with the notation of more recent work, namely [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, we note that we could 1This framework was initially introduced in the classical setting [34], but it has not gained the same level of popularity in the classical setting, which is why we write as if it is only in the quantum setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' have defined the one-shot mutual informations for any R´enyi mutual information I(A : B) (any mutual infor- mation defined using a R´enyi divergence [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Certain results in this work are presented in such a manner for generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It is presumably clear that in general I↑↑ max, I↑ max, and I↓ max could behave quite differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Indeed, as we will see, there are certain properties that make I↑ max preferable for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Nonetheless, Ciganovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [28] showed that when smoothed, over a large parameter range of smoothing, these measures become asymptoti- cally equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We summarize this in sufficient detail to introduce the notion of smoothed measures and in- troduce the result we will need later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAB ∈ D≤(A ⊗ B) and ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then Bε(ρ) := {�ρ ∈ D≤(A ⊗ B) : P(ρ, �ρ) ≤ ε} , where P(·, ·) is the purified distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For x ∈ {↑↑, ↑, ↓}, the smoothed max- mutual information is Ix,ε max(A : B)ρ := min �ρ∈Bε(ρ) Ix,ε max(A : B)�ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ([28, Theorem 3]) Let ρAB ∈ D(A ⊗ B), ε > 0, ε′ ≥ 0, then I↑↑,ε+2 √ ε+ε′ max (A : B)ρ ≤ I↑,ε′ max(A : B)ρ + g(ε) , where g(x) := log � 2(1 − x) + 3 (1 − x)(1 − √ 1 − x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If the previous three mutual informations were not enough already, there is the hypothesis testing mutual information, which was used to establish the one-shot achievability and converse for soft-covering in [26], which we use later in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The ε-hypothesis testing divergence is defined as exp(−Dε h(ρ||σ)) := inf 0≤Λ≤1{Tr[Λσ] : Tr[Λρ] ≥ 1 − ε } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (12) We note that the hypothesis testing divergence has a natural relationship with the Petz divergence of order zero [37]: lim ε→0 Dε h(ρ||σ) = D0(ρ||σ) = Tr � Πρσ � , (13) where Πρ is the projector onto the support of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Like Dmax, this divergence satisfies a data-processing inequal- ity Dε h(ρ||σ) ≥ Dε h(E(ρ)||E(σ)) ∀E ∈ C(A, B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, one can define the hypothesis testing mutual information using it: Ih(A : B)ρ := Dε h(ρAB||ρA ⊗ ρB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (14) 6 We note that in principle this could be defined as the I↑↑ h (A : B) mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, it is the only one necessary for this work and to the best of our knowledge the only hypothesis testing mutual informa- tion defined, so for simplicity we keep this notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes our introduction to one-shot mutual informations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We note that one can convert between Dε s(ρ||σ), Dε max(ρ||σ), and Dε h(ρ||σ) up to constant cor- rection terms [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This in particular means that they all are asymptotically equivalent for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' inputs ρ → ρ⊗n, σ → σ⊗n [33], and so we should expect they can roughly characterize the same operational tasks, though perhaps through different proof methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' a) One-Shot Entropies: Beyond the one-shot mutual information, we will also need to make use of the follow- ing standard entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The (conditional) von Neumann entropy H(A|B)ρ := − D(ρAB||1A ⊗ ρB) = max σB∈D(B) −D(ρAB||1A ⊗ σB) , which is used for asymptotic results and makes up the chain rule I(A : B) = H(A) − H(A|B) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Another entropy which we will use in various definitions is the (unconditional) quantum Hartley entropy, also known as the zero order Petz R´enyi entropy, H0(A)ρ := log rank(ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (15) This may be viewed as the number of qubits (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' bits) one may compress the state ρ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' classical distribution) in the zero error setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Beyond the quantum Hartley entropy, we will also need the min-entropy and max- entropies Hmin(A|B)ρ := sup σB∈D(B) sup{γ ∈ R : ρ ≤ exp(−λ)1A ⊗ σB} Hmax(A|B)ρ := sup σB∈D(B) log F(ρAB, IA ⊗ σB) , (16) where F(·, ·) is the fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' These entropies have smoothed versions for ε ∈ (0, 1), Hε min(A|B)ρ := max �ρ∈Bε(ρ) Hmin(A|B)�ρ Hε max(A|B)ρ := min �ρ∈Bε(ρ) Hmax(A|B)�ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' These entropies have been studied in great detail and in particular are known to satisfy strong AEPs, which means for any ε ∈ (0, 1), lim n→∞ � Hε min(An|Bn)ρ⊗n � = H(A|B) , and likewise for Hε max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We refer the reader to [31] for proofs and further details with regards to this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' What is important to know for our purposes is that the chain rule for mutual information was extended for I↑,ε max in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In particular, Ciganovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' established over some parameter range of smoothing, H≈ε min(A)ρ − H≈ε min(A|B)ρ ≲ I↑,ε max(A : B)ρ ≲ H≈ε max(A) − H≈ε min(A|B)ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (17) A straightforward modification of this result to the entire parameter range implies a strong AEP for for I↑,ε max from the strong AEP for the smoothed entropies (see the appendix and in particular Proposition 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ONE-SHOT DISTRIBUTED SOURCE SIMULATION We are now in a position to introduce one-shot dis- tributed source simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We will also introduce natu- ral variants and show these variants are incomparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We will also show how (to arbitrary error) distributed source simulation can only hold for separable states due to its locality constraints and how this relates to the structure of (short) Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In principle, one-shot distributed source simulation is ‘simply’ constructing some target state ρAC up to some tolerated error ε from some shared randomness and local operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We note that we say shared randomness as the original classical randomness, pX = ∑x p(x) |x⟩⟨x|, is copied resulting in perfectly correlated randomness which we denote χ|p XX′ = ∑ x p(x) |x⟩⟨x|X ⊗ |x⟩⟨x|X′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (18) See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 3 for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' pX Copy ΦX→A A C ΨX′→C X′ X ≈ε ρAC χ|p XX′ �ρAC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 3: Diagram of distributed source simulation of a quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' After the copying procedure (at the light blue line), the two parties share a perfectly correlated state χ|p XX′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' After their local processing (at the dark blue line), the parties share a state �ρAC which should be approximately the target state ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Following Wyner, our interest is in how much shared randomness, measured in number of bits, is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The measure of how much randomness is necessary we refer to as one-shot correlation of formation, mirroring terminology from resource theories, such as the entan- glement of formation [38], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Like Wyner, we consider the case of uniform randomness, but we also consider the case where we let the randomness be non-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We now formalize all of this, starting with the notion of approximate simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 7 Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let σB ∈ D(B) and D(X ⊗ B) ∋ ρXB = ∑x p(x) |x⟩⟨x| ⊗ ρx B be a CQ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We say ρXB is a ε−simulation of σB if ∥ ∑x p(x)ρx B − σB∥1 ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We denote this ρXB ∼ε σB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let σAC ∈ D(A ⊗ C) and D(AXC) ∋ ρAXC = ∑x∈Σ p(x)ρx A ⊗ |x⟩⟨x| ⊗ ρx B where p ∈ P(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We say ρAXC is a ε-distributed source simulation of σAC if ∥ ∑x p(x)ρx A ⊗ ρx C − σAC∥1 ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We could now use the definition of ε-distributed source simulation to define one-shot correlation of for- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, we believe it is clearer to reduce the definition to being in terms of a Markov chain and the definition of ε-simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A ε−distributed source simulation of σAC is a A − X − C Markov chain that is a ε−simulation of σAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To establish this, we need the following theorem which will be relevant for much of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ([40]) The following are equivalent 1) ρABC is a (short) quantum Markov chain (QMC), denoted A − B − C or ρA−B−C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2) There exists a CPTP map R : B → B ⊗ C such that (idA ⊗ R)(ρAB) = ρABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 3) There exists a CPTP map R : B → A ⊗ B such that (R ⊗ idC)(ρBC) = ρABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 4) I(A : C|B) = 0, where I(A : C|B) is the conditional mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 5) There exists a finite alphabet J such that there exists a decomposition of B = ⊕j∈J bL j ⊗ bR j such that ρABC = � j∈J ρAbL j ⊗ ρbR j C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' All of these results in effect say that the A space and C space are independent so long as one has access to the B space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It is then trivial to prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' There are various ways to prove ρAXC is a QMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For our case, note R : |x⟩⟨x|B �→ |x⟩⟨x|B ⊗ ρx C and same idea for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then letting σAC act as σB, and ρAXC as ρXB in Definition 6 completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' With this established, we can define our correlation of formation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We note if we write a minimization over �ρA−X−C, this means we restrict to optimizing over QMC with a classical register X, and if we write π as a register, it means that the marginal on that register is the uniform distribution on a (classical) space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Later, we will write minimization over A − X − C without being a superscript when it is clear with respect to what state the QMC is being considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lastly, we always define the minimization over an empty set to be +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 4: Feasible sets for correlation of formation opera- tional quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For Cε F, one smooths the initial state, depicted by the purple ball, and then considers the Markov chain extensions, depicted by black dots, of any state in the smoothed ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For �Cε F, one considers all the Markov chain extensions of ρ and then smooths each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For �Cε F, one considers all the Markov chains extensions of ρ, smooths each, and then restricts to the Markov chains in each ball resulting in a non-convex set, depicted by the green jagged set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ [0, 1] and ρAC ∈ D(AC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The correlation of formation is Cε F(A : C)ρ := min �ρA−X−C � H0(X)�ρ : ∥�ρAC − ρAC∥1 ≤ ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (19) Moreover, the one-shot uniform correlation of formation is defined as Cε U,F(A : C)ρ := min �ρA−π−C � H0(X)�ρ : ∥�ρAC − ρAC∥1 ≤ ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (20) Note that by construction, the one-shot uniform cor- relation of formation may be viewed as one-shot dis- tributed source simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This can be seen as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By definition we are considering QMC �ρA−π−C, so the X register is uniform and there exist local channels R, R that prepare the A and C registers from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This means AXC AC B(p) C(A:C)p AXC Quantum Markov Extensions AC C(A :C)p AXC QuantumMarkov Extensions AC C(A: C))8 X is the uniform randomness input, R, R are the local channels in the distributed source simulation, and by definition of H0, given in (15), H0(X) is measuring the minimum number of bits of uniform randomness neces- sary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus we have defined our operational quantity of primary interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also define the related operational quantities of “entanglement-assisted correlation of formation” and “private correlation of formation.” Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The one-shot secret entanglement-assisted correlation of formation is defined as �Cε F(A : C)ρ := min �ρAXC � H0(X)�ρ : ∃ρA−X−C : ∥�ρAXC − ρA−X−C∥1 ≤ ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (21) Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The one-shot private correlation of for- mation is defined as �Cε F(A : C)ρ := min �ρA−X−C � H0(X)�ρ : ∃ρA−X−C : ∥�ρA−X−C − �ρA−X−C∥1 ≤ ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (22) First, we explain the choice of names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The secret entanglement-assisted correlation of formation is named such because �ρAXC is not restricted to being a Markov chain, so it is of the general form �ρAXC = ∑x∈X �p(x)�ρx AC, which in general is distributing possibly entangled states �ρx AC according to distribution �p ∈ P(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, this entanglement-assistance can also be viewed as ‘secret’ in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The quantity measures the necessary resources to, except with probability ε, win a ‘game’ where the distinguisher tries to discriminate the output � rhoAXC and the set of Markov chain extensions of ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In other words, it means from a black box perspective it is the minimal amount of classical information to leak so that �ρAXC is ε-indistinguishable from a zero-error dis- tributed source simulation with leaked classical register, ρA−X−C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The one-shot private correlation of formation captures the black box indistinguishability while also requiring it be an actual distributed source simulation implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' These operational quantities have the difference in their respective feasible sets depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 4 and their corresponding tasks depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='We note the operational quantities �Cε F and �Cε F and their corresponding tasks are somewhat less natural than distributed source simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' They however will be characterized by a natural correlation measure and so serve as a useful comparison to the correlation of formation and standard distributed source simulation as we will now begin to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We briefly note some relationships between these op- erational quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' pX Copy ΦX→A A C ΨX′→C ≈ε ρAC (a) Correlation of Formation, Cε F pX ΞX→ACX A C X ≈ε ρA−X−C (b) Secret EA Correlation of Formation, �Cε F pX Copy ΦX→A A C ΨX′→C ≈ε ρA−X−C X (c) Private Correlation of Formation, �Cε F Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 5: The three correlation of formations and their cor- responding tasks: (a) Correlation of formation captures the amount of randomness for distributed source simu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (b) Secret entanglement-assisted correlation of for- mation captures the amount of broadcasted randomness needed such that there exists a set of (possibly entangled) states {�ρx AC}x∈X to distribute so that the entire output is approximately indistinguishable from a distributed source simulation implementation of the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (c) Private correlation of formation measures the amount of randomness needed for a distributed source simulation protocol so that if the randomness were to be leaked it would be approximately indistinguishable from an exact distributed source simulation of the target state with leaked classical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ρAB ∈ Sep(A : B) and ε ∈ [0, 1), {Cε F, �Cε F} ≤ �CF & Cε F ≤ Cε U,F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That Cε F ≤ Cε U,F follows from it being a more restricted optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That �Cε F ≤ �Cε F is also because it is a more restricted optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That Cε F ≤ �Cε F is because if �ρ ≈TD ε ρ, then as trace norm monotonically decreases under partial trace, ε ≥∥ TrX(�ρ − ρA−X−C)∥1 ≥∥∑ x p(x)�ρx A ⊗ �ρx C − ρAC∥1 ⇒ �ρ ∼ε ρ where the implication is by definition of ε-simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We stress in particular the following two points im- plicit in the above proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First, one would expect there are cases where the inequality Cε F ≤ �Cε F is large 9 because in �Cε F the distinguisher has access to the X register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That is to say, in general one expects there to be a non-trivial difference between min �ρA−X−C ∥TrX(�ρA−X−C) − ρAC∥1 and min �ρA−X−C ρA−X−C ∑ x ∥�p(x)�ρx A ⊗ �ρx C − p(x)ρx A ⊗ ρx C∥1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Second, intuitively Cε F and �Cε F do not seem comparable as the resources allowed are fundamentally different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We now show this to be formally true by establishing the quantities are finite under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We begin with the following lemma that will be used extensively in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A quantum state ρAC ∈ D(A ⊗ C) has a Markov chain extension ρA−B−C : TrB ρA−B−C = ρAC if and only if ρAC ∈ SepD(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We prove both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (⇒) Let ρAC ∈ SepD(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then by definition, ρAC = ∑x p(x)ρx A ⊗ ρx C for some finite alphabet X , p ∈ P(X ), and sets of quantum states {ρx A}x∈X , {ρx C}x∈X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows ∑x p(x)ρx A ⊗ |x⟩⟨x| ⊗ ρx C is a QMC extension as R(·) := ∑x Tr[|x⟩⟨x| ·]ρx C ⊗ |x⟩⟨x| and same idea for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (⇐) If one has a QMC ρA−B−C, then by Theorem 2, ρA−B−C = � j∈J p(j)ρAbL j ⊗ ρbR j C where the ρAbL j , ρbR j C are density matrices on the respective subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows TrB(ρA−B−C) = � j∈J p(j) TrbL j (ρAbL j ) ⊗ TrbR j (ρbR j C) = ∑ j∈J p(j)ρj A ⊗ ρj C ∈ Sep(A : C) , where we have used that for B = ⊕j∈J bL j ⊗ bR j , we may decompose TrB = ⊕j∈J TrbL j ⊗bR j and then TrbL j ⊗ TrbR j will distribute across the tensor product of the states decom- position as is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also will make use of the following straightforward proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For all δ ∈ (0, 1), if there exists a QMC ρA−X−C, there exists �ρA−π−C such that ρAC ≈TD δ �ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρA−X−C = ∑x∈X p(x)ρx A ⊗ |x⟩⟨x| ⊗ ρx C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then there exists finite alphabet X ′ and {kx} ⊂ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=', |X ′|} such that ∑x kx = |X ′| and ∑x∈X |kx/|X ′| − p(x)| ≤ δ since the rationals are dense in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Define �ρA−π−C = 1 X′ ∑x′∈X ′ ρ f (x′) A ⊗ |x′⟩⟨x′| ρ f (x′) C where f : X ′ → X is such that for each x ∈ X , it maps kx of the elements of X ′ to that x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note this means �ρx′ A = ρ f (x′) A for all x′ ∈ X ′ not that we are actually indexing by f (x′) which would erase information we need to preserve the Markov chain condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows by our construction ∥TrX′(�ρA−π−C) − TrX(ρA−X−C)∥1 = ����� ∑ x∈X kx |X ′|ρx A ⊗ ρx C − ∑ x∈X p(x)ρx A ⊗ ρx C ����� 1 ≤ ����� ∑ x∈X kx |X ′| |x⟩⟨x|⊗2 − ∑ x p(x) |x⟩⟨x|⊗2 ����� 1 =∑ x ���� kx |X ′| − p(x) ���� ≤ δ , the first inequality uses the data-processing inequality for the one-norm and the preparation channels Φ1(·) = Tr[|x⟩⟨x| ·]ρx A and Φ2(·) = Tr[|x⟩⟨x| ·]ρx C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρ ∈ D(A ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1) For ε ∈ [0, 1], �Cε F(A : C), �Cε F(A : C) are finite if and only if ρAC ∈ SepD(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2) In contrast, Cε F(A : C)ρ is finite if and only if ET(A : C)ρ ≤ 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Likewise, Cε U,F(A : C)ρ if and only if there exists δ ∈ (0, ε) such that ET(A : C)ρ ≤ 2(ε − δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We begin with Item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The only if direction is immediate because if there exists an appropriate QMC extension, then the state is separable by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Like- wise, by Lemma 3, if ρAC ∈ SepD(A : C), there exists a QMC extension of ρAC, ρA−X−C = ∑x p(x)ρx A ⊗ |x⟩⟨x| ⊗ ρx C for some finite alphabet X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This state is then feasible for �Cε F, �Cε F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note none of this has relied on the choice of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For Cε F, note that if ET(A : C)ρ ≤ 2ε, there exists �ρ ∈ SepD(A : C) such that �ρ ≈TD ε ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then there exists a QMC extension of �ρAC by Lemma 3 and this QMC extension satisfies �ρA−X−C ∼ε ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If no such state exists, then there is no QMC extension by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For the uniformity claim, note that if such a δ > 0 exists then by Proposition 4 we may do the same argument again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore we see Cε F is a fundamentally distinct mea- sure from �Cε F, �Cε F as when they are finite is not even in agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We will discuss this further after we have introduced the one-shot correlation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ONE-SHOT CORRELATION MEASURES As is standard in information theory, the ultimate goal is to establish bounds in terms of entropic quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In our case we would particularly like to bound the correlations of formations with entropic quantities that recover Wyner’s common information in the asymptotic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That is, our hope is to construct bounds in terms of an entropic quantity C?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' that on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' inputs satisfies lim n→∞ � 1 nC?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (An : Cn)ρ⊗n � = min A−X−C I(AC : X)ρ , where we note the right hand side is Wyner’s common information as per (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Motivated by this end goal, we introduce the max common information, its smoothed versions, and establish certain properties of it on sin- gle copies of a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In doing so we will present the generalized support lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, we introduce the hypothesis testing common information and show it 10 satisfies the same wanted properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We end the section by establishing that these introduced correlation mea- sures act as one-shot converses to the distributed source simulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We claim the natural one-shot extension of Wyner’s common information is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Given ρAC ∈ D(A : C), the Max Common information is Cmax(A : C)ρ := min A−X−C I↑ max(AC : X)ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (23) There are three choices we should justify: (i) the restriction to a classical register, (ii) the minimization, and (iii) the choice of I↑ max rather than another version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First we justify why we restrict to A − X − C Markov chains in the above definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We could of course argue it is because we are interested in an operational interpre- tation that will require the classical register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, this restriction can be made without loss of generality for any mutual information that satisfies DPI as we now show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρABC be a A − B − C Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then there always exists a A − X − C Markov chain ρ′ AXC such that ρ′ AC = ρAC and Ix(AC : B)ρ ≥ Ix(AC : X)ρ′, where I is any mutual information measure that satisfies data- processing and x ∈ {↑↑, ↑, ↓} following Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We prove it for I↓(AC : X) case as it is then straightforward to see the same proof method will hold for the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρABC be a A − B − C Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then ρB = ⊕xp(x)ρbLx ⊗ ρbRx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Define the map E : B → BX as E(·) = ∑j ΠB x · ΠB x ⊗ |x⟩⟨x|X where {Πx} are the mutually orthogonal projectors onto the subspaces bL x ⊗ bR x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Define ρ′ ABXC := E(ρABC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then it follows I↓(AC : B)ρ =D(ρABC||τAC ⊗ σB) ≥D(ρ′ ABXC||τAC ⊗ σBX) ≥D(ρ′ AXC||τAC ⊗ σX) ≥ min τ′∈D(AC) σ′∈D(X) D(ρ′ AXC||τ′ AC ⊗ σ′ X) =I↓(AC : X)ρ′ , where the first inequality is DPI using E, the second inequality is DPI using the partial trace on the B space, and in both cases the map only acts on one side of conditioned tensor product, and the final is just re- minimizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We can also guarantee σ′ X is classical by DPI and pinching on the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover ρ′ AXC is A − X − C Markov chain extension of ρAC trivially as E never acted on the AC spaces and its recovery maps are just state preparations maps, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' x �→ TrbLx (ρAbLx ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It is worth noting why the above isn’t proven to be an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' When one converts the B register to a classical X register, they destroy any entanglement between A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' C) and bL x (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' bR x ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To recover this entanglement, one needs to apply the recovery map, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ρX → ρB R −→ ρBC R −→ ρABC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, to preserve the form of mutual information, you can only act on the B space, so it is not possible to evaluate this directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This justifies the restriction to a classical register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To address the second and third question, we will need to introduce the generalized support lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The Generalized Support Lemma The convex cover method using the support lemma is a standard method in classical information theory for bounding the cardinality of an auxiliary variable [41, Ap- pendix C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This bounding is useful as then the space be- ing optimized over is finite-dimensional and thus closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Effectively the support lemma implies that if all the relevant constraints may be written as the expectation of a function over conditional distributions according to the auxiliary variable, then the auxiliary variable can be made finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Our extension states the same but replacing conditional distributions with conditional states from an appropriate state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This generalization is necessary for our purposes as we will need to consider a function of conditional states, appearing in Lemma 7, that does not reduce to functions of their spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We begin by stating Carath´eodory’s theorem of which the support lemma may be viewed as a corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (Fenchel-Eggleston-Carath´eodory) Any point in a convex closure of a connected compact set R ∈ Rd can be represented as a convex combination of at most d points in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Now we present the general lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (Generalized Support Lemma) Let W be an arbitrary set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let the generalized state space S (A) be a connected, compact subset of Pos(A) and {ρw}w∈W ⊆ S (A) be a set of generalized conditional states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let { fj}j∈[k] ⊂ S (A) → R be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then for any Borel measure µ of W, there exists q ∈ P(W′) where |W′| ≤ k and {σw′}w′∈W′ ⊂ S (A) such that � W fj(ρw)µ(dW) = ∑ w′∈W′ fj(σw′)q(w′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Our proof is a direct extension of the proof given for the traditional support lemma by Csiszar and K¨orner [42, Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By assumption, S (A) is a compact, connected subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By assumption each fj is continuous, so the image of fj(ρ) is both connected and compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' De- fine F(ρ) := ( f1(ρ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=', fk(ρ)) and the set R := {F(ρ), ρ ∈ S (A)}, which is connected and compact as product preserves these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, defining rj ≡ � W fj(ρw)µ(dW) ∀j ∈ [k] , 11 we have (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=', rk) is an element of the convex closure of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, applying Proposition 5, there exist k points of R, which we denote {F(σj)}j∈K, along with a distribution q ∈ P([k]) such that (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=', rk) = ∑ j∈[k] q(j)F(σj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By definition of F(ρ), we can conclude rj = ∑j q(i) f (σi) for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Letting W′ = [k] completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First, we note the reason we talk in terms of gen- eralized state spaces that are subsets of the positive density matrices is that, for example, this would allow for cardinality bounds on subnormalized states which may be of use given smoothed measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In fact, any closed convex subset of the (possibly subnormalized) density matrices would work, since it would be compact and all convex sets are (path-)connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, the generalized state space may be the product space of closed convex subsets of the (possibly subnormalized) density matrices, since the product of connected, com- pact sets are also connected and compact, which is useful for network settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='2 Note you can also restrict to the support of some state space as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We now can use this to bound the cardinality of the the max common information which will also explain why we chose I↑ max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This relies on various technical lemmas about mutual informations which we relegate to an appendix and summarize here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρABX be classical on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, exp � I↑ max(AC : X)ρ � = ∑ x px exp(Dmax(ρx AC||ρAC)) , where the right hand side is a continuous function over the state space restricted to ρAC’s support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, I↑↑, I↓ do not seem to satisfy such an averaging state- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' One uses Corollary 2 in the appendix with the re- placements A → AC, B → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To simplify the RHS term, note, as defined in the appendix, Imax(ρx ACC||ρAC) = Dmax(ρx AC||ρAC) for each x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Finally taking an exponential gets the form in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That I↑↑ max(AC : X) and I↓ max(AC : X) do not seem to satisfy such an averaging statement may be seen from Propositions 19 and 20 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It is this previous lemma that justifies our choice of I↑ max in defining Cmax(A : C) as it is this property of I↑ max we now use to establish cardinality bounds for minA−X−C Iε max which in turn establishes cardinality bounds for Cmax(A : C) in the case ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C) and ε ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then without loss of generality minA−X−C Iε max(AC : X)ρ may 2Formally, your state space is then S (A) × S (B) and functions which are defined on Pos(A ⊗ B) would be extended to being on the state space via composition with the map (ρ, σ) �→ ρ ⊗ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' be restricted such that |X| ≤ |A|2|C|2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, in the case ρXZ is fully classical, then |Y| ≤ |X||Z| + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We begin with the non-smooth case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let {ρx A|C := ρx A ⊗ ρx C}x∈X ⊂ D≤(A) ⊗ D≤(C) with distribution pX be a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let {Mk}k∈K be the elements of a min- imal informationally complete POVM on the space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' |K| = |A|2|C|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Consider the following functions: { fk(·) := Tr � Mk · M∗ k � for k ∈ [|K| − 1], fAC(·) := H(·), fA(·) := H(TrC(·)), fC(·) := H(TrA(·)), fobj(·) := exp(Dmax(·||ρAC)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, Tr(MkρACM∗ k) = ∑ x p(x) fk(ρx A|C) = Pr[Outcome k] ∑ x p(x) fAC(ρx A|C) = H(AC|X) ∑ x p(x) fA(ρx A|C) = H(A|X) ∑ x p(x) fC(ρx A|C) = H(C|X) ∑ x p(x) fobj(ρx A|C) = exp � I↑ max(AB : X)ρ � , where we have used Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then by applying Lemma 6 for the state space D≤(A) ⊗ D≤(C) restricted to the support of ρAB so that fobj is continuous, there exists a distribution q ∈ P(X′) where |X′| ≤ |A|2|C|2 + 3 and states {σx A|C := σx A ⊗ σx C}x∈X ⊂ D≤(A) ⊗ D≤(C) that equals the left hand side of each equation above and thus the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As { fk}k are all but one POVM element of an IC POVM, these constraints guarantee that the output state is indeed ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As I(A : C|X) = H(A|X) + H(C|X) − H(AC|X), the next three guaran- tee the Markov chain condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The last constraint guarantees the max mutual information is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof for the non-smoothed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In the smoothed case, we know the optimizer will be classical on the classical register (Proposition 17), so we apply the non-smoothed proof to its optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For the classical case, we can replace the informationally complete POVM with a measurement that only checks |X||Z| − 1 elements of the joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This com- pletes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus we have completed our formal justification for our choice of definition of Cmax(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Smoothed Max Common Information Having established the definition of max common in- formation, we will want to define its smoothed version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note however we now reach a complication: there seems to be two ways of smoothing Cmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We could smooth the state we start with, ρAC, or we could replace the max mutual information with the smooth max mutual information, which in effect is like smoothing the QMC extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We will define both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, before doing so, we recall that smoothing includes subnormalized states and so in principle we need to generalize the notion of QMC extensions, though ultimately we won’t need to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 12 Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D≤(A ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The set of its quantum Markov chain extensions (QMC extensions) is QMC(ρAC) :={(R ◦ R)(ρB) : (TrB ◦R ◦ R)(ρB) = ρAC} , (24) where R ∈ C(B, BC), R ∈ C(B, AB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' With this established, we define our two versions of smoothed max common information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We define the ε-max common information as Cε max(A : C)ρ := min �ρ∈Bε(ρ) Cmax(A : C)�ρ , (25) where this may be expanded as min �ρ∈Bε(ρ) min σ∈QMC(�ρ)I↑ max(AC : X)�ρ , where the register being X follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ρAC ∈ D(A ⊗ C), we define the alternative smoothed max common information as �Cε max(A : C)ρ = min A−X−C Iε max(AC : X)�ρA−X−C , where the restriction to a classical register is without loss of generality as proven in Proposition 22 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We first need to justify the minimizations in these definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For the alternative smoothed max common information it immediately follows from our cardinality bounds, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For the smoothed max common information we need two steps: first, our cardinality bounds relied on the entropic characterization of a QMC but it is not clear the best way to generalize this to subnormalized states, therefore we need the optimizer of min�ρA−X−C Imax(AC : X)�ρ to be a normalized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This can be shown to always be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We establish this in the appendix (Proposition 23) as this property is known for I↑,ε max and so the proof is effectively the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The second point is then just noting the smoothing ball is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore these two points combined justify the minimizations in the definition of Cε max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Beyond these points, the reader may have noticed that our notation for the two smoothed max common infor- mations are suggestive of the notation for the correlation of formation notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Indeed, we will ultimately have the correlation measures characterize the formation task with the same notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We can see this alignment very easily by showing that Cε max and �Cε max are incomparable in general in the same fashion as in Theorem 4 except in terms of purified distance of entanglement due to our definition of smoothing ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Given ρAC ∈ D(A ⊗ C), Cε max(A : C) is finite if and only if EP(A : C)ρ ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In contrast, �Cε max is finite if and only if ρAC ∈ SepD(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For the first claim, note that if EP(A : C)ρ ≤ ε, then there exists a separable state �ρ such that �ρ ∈ Bε(ρ) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 6: The difference between SMCI and alternative SMCI’s feasible sets visually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For SMCI, Cε max, the Markov chain extensions of the smoothed state are con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In contrast, the alternative SMCI, �Cε max considers the smoothed state of each Markov chain extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note these are effectively the same feasible sets as depicted for two of the formation quantities, Cε F and �Cε F, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' and by Lemma 3 it admits a QMC extension and thus Cε AC is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If EP(A : C)ρ > ε this argument doesn’t go through so by Lemma 3, the value is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For the second claim, note �Cε max is finite if and only if ρAC has a QMC extension and by Lemma 3, this is only the case if ρAC ∈ Sep(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also provide a visualization of the distinction of the two measures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' a) Hypothesis Testing Common Information: As noted in the introduction, generally an alternative to results in terms of max divergence are results from hypothesis testing divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Motivated by this point, we define the hypothesis testing common information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The hypothesis testing common information is given by �Cε h(A : C)ρ := min A−X−C Iε h(AC : X)ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The primary point to stress is that unlike with Dmax, there is no freedom in how we smooth as the smooth- ing comes from the definition of the measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As the notation would suggest, this means we can’t, at least directly, use hypothesis testing mutual information to characterize distributed source simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Using that the zero order Petz-R´enyi divergence is a limiting case of hypothesis testing divergence (13), we could view Iε h as the smoothed version of zero order common information: �C0(A : C)ρ := min A−X−C D0(ρA−X−C||ρAC ⊗ ρX) AXC B(p) AC A:C) AXC Quantum Markov Extensions AC ma13 = min A−X−C − log � � ∑ x:p(x)>0 p(x) Tr � Πρx ACρAC � � � , where in the equality we have used (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This can be argued to be an intuitive correlation measure in the sense that it intuitively measures how mutually orthogonal the conditional states are, which will be a function of how correlated the A and C spaces are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Finally, we do note that in contrast to defining corre- lation measures with I↑↑ max, I↓ max, we may get cardinality bounds for �Cε h, because it can be written as an expecta- tion over the auxiliary random variable X: exp �−Iε h(AC : X)ρACX � = ∑ x p(x)Dε h(ρx AC||ρAC) , which is an immediate corollary of Proposition 21 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' b) Data-Processing of Common Information Measures: While we won’t need to apply it directly at any point in this work, it is worth noting that (smoothed) common in- formation degrades under local processing, which a cor- relation measure should.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Furthermore, thinking ahead, it tells us that if it takes α amount of randomness to distributed source simulate a target state ρAC and there are local maps Φ, Ψ such that ρ′ A′C′ = (Φ ⊗ Ψ)(ρAC), then it can only require less randomness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ≤ α, to simulate ρ′ A′C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ε ∈ [0, 1), Cε max, �Cε max, �Cε h are all monotonic under local CPTP maps on both spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We begin with Cε max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We focus on a map being applied on the A space, by symmetry of the argument this will also establish the other case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let �ρA−X−C be the minimizer of Cε max(A : C)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Consider Φ(�ρ) where ΦA→A′ ∈ C(A, A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note the resulting state is still a QMC with recovery map Φ ◦ RX→AX and (TrX ◦Φ)(�ρ) ∈ Bε(Φ(ρ)) by the DPI for purified distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then we have, Cε max(A′ : C)Φ(ρ) ≤ Imax(A′C : X)Φ(�ρ) ≤ Imax(AC : B)�ρ = Cε max(A : C)ρ , where the first inequality is our choice of element in the smoothed ball and QMC extension and the second is the DPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note we needed local maps because we need to preserve the QMC structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Similarly, let (�ρA−X−C, σAXC) ∈ QMC(ρ) × Bε(�ρ) be the optimizers for �Cε max(A : C)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then Φ(�ρ) is a QMC for the same reason as earlier and Φ(σ) ∈ Bε(Φ(�ρ)) by DPI of purified distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus, �Cε max(A′ : C)Φ(ρ) ≤ Imax(A′C : X)Φ(σ) ≤ Imax(AC : B)σ = �Cε max(A : C)ρ , for the same reasons as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The same argument holds for �Cε h as it also satisfies a DPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' c) A Remark on Computability: One convenient prop- erty of one-shot entropic quantities is that for small di- mensions they can be solved easily as they form semidef- inite programs [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, here we also have the constraint that we are optimizing over Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This not only makes it hard to solve in general, but actually means that there must be instances where it is NP-hard to solve, because we can use whether or not the solution is finite as a solution to the separability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' There exist ρAC ∈ D(A ⊗ C) such that computing Cε max(A : C)ρ, �Cε max(A : C)ρ is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As established in Proposition 6, �Cε max is finite if and only if ρAC is sufficiently close to the separable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Likewise for Cε max but if it is within some distance from the separable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, if it is efficient to compute always, then we have a method for the ability to determine if the state is separable (or within some distance from it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It is known determining membership is strongly NP-hard [43], which means even with some tolerated distance from the set of separable states β determining whether it is in or out of the set is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore this would be in contradiction with being able to compute Cε max, �Cε max efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We note the above argument doesn’t say anything about computational complexity when the target distri- bution is fully classical and thus separable a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also note this is not a problem in terms of establishing our results beyond that it means we cannot in general compute the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ACHIEVABILITY FOR CORRELATION OF FORMATIONS FROM ONE-SHOT SOFT-COVERING LEMMA In effect, Theorem 4 told us when the operational task may be performed at all, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' when there is a finite amount of randomness that allows distributed parties to simulate the state (or the same idea for the related tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, it did not tell us how much randomness we will need, which we will show the correlation measures we have introduced will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As mentioned in the intro- duction, the standard way of establishing the achievable rate for distributed source simulation is a “soft-covering lemma” and in this section we show how to relate a one- shot soft-covering lemma to our correlation measures to establish an achievable rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For intuition, we explain the name “soft-covering.” Given the n−fold CQ state ρ⊗n XB, by typicality only a fraction of the conditional states ρ xn 1 B⊗n are necessary to approximate ρ⊗n B well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Finding these xn 1 is non-trivial however, and so a random coding approach is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' One can draw Mn strings xn 1 ∈ X ×n according to p⊗n X to consider an ensemble {ρ xn 1 B⊗n} where each state has probability |Mn|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The soft-covering lemma says that asymptotically if |Mn| ≥ I(X : B)ρ, this ensemble will also approximate ρ⊗n B with vanishing error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A one-shot 14 soft-covering lemma would then be the same conceptual idea except you draw codewords from x ∈ X rather than xn 1 ∈ X ×n, and so the correlation measure presumably would need to be larger than mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Recently, a one-shot soft-covering lemma for quantum states that is optimal to second-order was established [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The authors do this by establishing achievability in terms of a mutual-information-like information spec- trum divergence quantity and then converting to the hypothesis testing mutual information as defined in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As remarked upon in the correlation measure section, we do not expect to capture distributed source simulation from the hypothesis testing divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, as noted in Section III, it is equally valid to convert to smooth max divergence from the information spectrum divergence [33], and we suspect the authors of [26] omit- ted this as their converse proof is less natural to convert to smooth max mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As our interest is in a task whose converse is ultimately characterized in terms of a max mutual information induced measure, we will explain how to obtain the smooth max mutual information version of the result from [26] as well as present the hypothesis testing result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To state this, we need three preliminaries: the definition of ‘minimal ran- dom codebook size for ε−covering’ from [26] except we will need to alter it up to a factor of a half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='3 We will also need to establish a few mutual information measures that will allows us to express all bounds in a notationally similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ([26] Altered) Let ρXB = ∑x∈X p(x) |x⟩⟨x| ⊗ ρx B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The minimal random codebook size for ε−covering, denoted by Mε(B : X)ρ is given by inf M∈N � |C| ≤ M & EC∼pX ����� 1 |C| ∑ x∈C ρx B − ρB ����� 1 ≤ ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note that taking the logarithm of Mε(B : X)ρ is the number of bits needed to describe an element of the random code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ρ ∈ D≤(A ⊗ B), the following two properties hold: 1) Iε h(A : B)ρ = Iε h(B : A)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2) Dε s(ρAB||ρA ⊗ ρB) = Dε s(ρBA||ρB ⊗ ρA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1) Q is feasible for Dε h(ρAB||ρA ⊗ ρB) if and only if SWAPA↔B(Q) is feasible for Dε h(ρBA||ρB ⊗ ρA), where SWAPA↔B is the CPTP map that re-orders the spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This is because Tr[QρA ⊗ ρB] = Tr[SWAPA↔B(Q)SWAPA↔B(ρA ⊗ ρB)] = Tr[SWAPA↔B(Q)ρB ⊗ ρA] , (26) 3This necessity is more of a technicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We use the purified distance smoothing so if we consider ρ ≈TD ε �ρ, then ρ ≈√ε �ρ whereas if we use trace distance directly there is a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This is a problem as in the converse we must convert to purified distance and √x : [0, 1] → [0, 1] but √ 2x : [0, 1] → [0, √ 2] which doesn’t work with smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' and the same argument for ρAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As swapping the or- dering of the spaces preserves positivity, we have the feasible set for Iε h(B : A)ρ is the same as Iε h(A : B)ρ up to a swapping of spaces and moreover (26) tells us the objective function values are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof of the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2) Recall from (7) that Dε s(ρAB||ρA ⊗ ρB) = sup{R ∈ R| Tr � ρAB{ρAB ≤ 2RρA ⊗ ρB} � ≤ ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note for any R, {ρAB ≤ 2RρA ⊗ ρB} ≡ Π is the projector onto the non-negative eigenspace of 2RρA ⊗ ρB − ρAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows SWAPA↔B(Π) is the projector onto the non-negative eigenspace of 2RρB ⊗ ρA − ρBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus, for all R ∈ R, Tr � ρAB{ρAB ≤ 2RρA ⊗ ρB} � = Tr � ρBA{ρBA ≤ 2RρB ⊗ ρA} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This implies the claim as the feasible R for fixed ε is the same for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' With these addressed, we state the one-shot soft- covering lemma and provide the part not proven in the cited paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ([26, Theorem 13 Achievability + A Little More]) Let ρXB ∈ D(X ⊗ B) and ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If δ ∈ (0, ε) and η ∈ ( 7 8ε, ε), then log � Mε(B : X)ρ � ≤ I↑,√ε−η max (B : X)ρ + log(ν) + g(ε)+ − log(1/ε − 1/8) + 3 log 3 + 7 , (27) where ν ≡ |spec(ρB)|, the number of distinct eigenvalues of ρB, ε = 2 + �ε − 2 √ 1 − �ε, and �ε = √ ε/8 − √ε − η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Similarly, log � Mε(B : X)ρ � ≤ I1−ε/8 h (B : X)ρ + log � ν · ν′� − 2 , (28) where ν′ = |spec(ρX ⊗ ρB)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We note our spectral terms for hypothesis testing mutual information differ from [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Our scaling is made explicit in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The proof for Iε h is effectively found in [26] except that we changed the factor of half of which one has to keep track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' How to address that is shown equivalently in proving the bound in terms of I↑↑,ε max, so we don’t show it explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To establish the bound in terms of I↑↑,ε max, the proof is still primarily found in [26], we just start from [26, Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='2] where we multiply both sides by 2: EC ����� 1 |C| ∑ x∈C ρx B − ρB ����� 1 ≤ 2 Tr � ρXB{PρB(ρXB) > cρX ⊗ ρB} � + 2 � |spec(ρB)|c |C| , (29) 15 where PρB is the pinching map on B according to ρB and c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then let δ ∈ (0, ε) and choose c ≡ exp � D1−(ε−δ)/2 s (PρB(ρXB)||ρX ⊗ ρB) + ζ} � for some small ζ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The reason to define c in this man- ner is because, by definition of ε−information spectrum divergence (7) we obtain: Tr � ρXB{PρB(ρXB) > cρX ⊗ ρB} � = Tr � PρB(ρXB){PρB(ρXB) > cρX ⊗ ρB} � =1 − Tr � PρB(ρXB){PρB(ρXB) ≤ cρX ⊗ ρB} � <1 − (1 − (ε − δ)/2) =(ε − δ)/2 where the first equality is because the projected space doesn’t change, and the strict inequality is using our definition of c with the definition of ε−information spectrum along with including small ζ and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Letting |C| = ⌈|spec(ρB)|c(2/δ)−2⌉ and plugging these bounds into (29) gets the target EC ��� 1 |C| ∑x∈C ρx B − ρB ��� 1 ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We remark so far the only change from the original proof is that we scaled ε, δ by two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To now establish bounds in terms of max mutual information, we will pick ε ∈ (0, 1/2) as explained at the start of the proof and we will let δ = ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We define ε′ ≡ 1 − ε/8, δ′ ≡ ε/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By our choice of |C|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' log � Mε(B : X)ρ � ≤ log |C| ≤ D1−(ε−δ)/2 s (PρB(ρXB)||ρX ⊗ ρB) + ζ + log |spec(ρB)| + 2 log(δ) = D1−(ε−δ)/2 s (PρB(ρBX)||ρB ⊗ ρX) + ζ + log(ν) + 2 log(δ) = Dε′−δ′ s (PρB(ρBX)||ρB ⊗ ρX) + ζ + log(ν) + 2 log(δ) ≤ D √ 1−ε′ max (ρBX||ρB ⊗ ρX) + log(ν) + 2 log(δ) − 3 log � δ′� + log � ε′� + 3 log(3) + ζ = I↑↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='√ ε/8 max (B : X)ρ + log(ν) − log(1/ε − 1/8) + 3 log 3 + 7 + ζ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' where the first inequality is by definition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' the second is our definition of |C|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' the first equality is by Item 2 of Proposition 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' the second equality is our definitions as 1 − ε/2 + ε/4 = 1 − ε/4 = ε′ − δ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' the third inequality is [33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 17] and where we have used PρB(ρXB) = PρX⊗ρB(ρXB) as ρXB is classical on the X register, the fifth is by definition and merging the logarithm terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As ζ > 0 was arbitrary, we let ζ tend to zero so that it goes away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lastly, we wish to upper bound I↑↑,√ ε/8 max by I↑,√ε−η max by applying Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This means we need √ε − η < √ ε/8, so η ∈ ( 7 8, 1) · ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then we need to solve 0 < �ε := √ ε/8 − √ε − η = ε + 2 √ ε for ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The equation a = x + 2√x holds for x = 2 + a − √ 1 + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus, ε = 2 + �ε − 2 √ 1 + �ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Plugging this value into Lemma 1 gets the max mutual information bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To get the hypothesis testing bound, it is effectively the same as in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' One starts from the second equality in the chain of inequalities above except ε′ ≡ 1 − 3ε 16 and δ′ ≡ ε/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Now we will bound the spectrum divergence term where we again use PρB(ρBX) = PρB⊗ρX(ρBX), Dε′−δ′ s (PρB⊗ρX(ρBX)||ρB ⊗ ρX) ≤Dε′ s (P||Q) − log � δ′� =Dε′+δ′−δ′ s (P||Q) − log � δ′� ≤Dε′+δ′ h (ρXB||ρX ⊗ ρB) + log � ν′� − 2 log � δ′� =I1−ε/8 h (B : X)ρ + log � ν′/ε2� + 8 , where the first inequality is [33, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 19] and P, Q are the Nussbaum-Szkoła distributions as discussed in [33] and the second inequality is [33, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 28] as the inequality always holds for θ(σ) replaced with ν′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The equalities just use definitions of ε′, δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We combine this with the bound we started from and this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We are now ready to prove achievability of distributed source simulation for quantum states from the above one-shot soft covering lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The idea is that if the state ρAC can be sufficiently approximated by a separable state �ρAC, then there is a QMC extension �ρA−X−C with recov- ery maps R, R which will allow us to source simulate using the random codebook rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note in particular this means that for separable states, one can source simulate to arbitrary non-zero error, but for an entangled state, there is a fundamental limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C) such that 2ET(A : C)ρ ̸= ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε1, ε2 ∈ (0, 1) such that 2ε1 + ε2 < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, Cε U,F(A : C)ρ ≤ Cε1 max(A : C)ρ + κ(ε2) , where the κ(x) := log(ν) + g(x) − log(1/x − 1/8) + 3 log 3 + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, there exist choices of ε1, ε2 such that this is finite whenever 2ET(A : C)ρ < ε, so it holds for all separable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Furthermore, if ρAC ∈ SepD(A : C), and η ∈ (7/8ε, ε), { �Cε F(A : C)ρ, �Cε F(A : C)ρ} ≤ min{ �C √ε−η max (A : C)ρ + κ(ε), �C1−ε/8 h (A : C)ρ + log � ν · ν′� − 2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Define 2ET(A : C)ρ = δE < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε1 ∈ (δE/2, ε/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε2 ∈ (0, ε − 2ε1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By our choice of ε1, it follows that there is at least one separable state contained in Bε1(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Denote an arbitrary choice �ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By Lemma 3, we can think of �ρAC as the marginal of the CQ state ρA−X−C = ∑x∈X p(x) |x⟩⟨x| ⊗ ρA x ⊗ ρC x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By definition, 16 Mε2(AC : X)�ρA−X−C is the minimal size such that the ex- pectation over random codebooks is a ε−covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It fol- lows for this size of codebook, there must exist a a code- book C′ with size Mε2(AC : X)ρ that is a ε2−covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Fix this codebook C′ and let | �X| := Mε2(AC : X)�ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note that using the Markov chain extension, we have that the ensemble is of the form {�ρA x ⊗ �ρC x }x∈C′ with each element equiprobable and by Lemma 9 it satisfies ∥ 1 |C′| ∑x∈C′ �ρA x ⊗ �ρC x − �ρAC∥1 ≤ ε2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Using our definition of ε1, ε2 along with the fact that (4) tells us purified distance upper bounds trace distance, we have ����� 1 |C′| ∑ x∈C′ �ρA x ⊗ �ρC x − ρAC ����� 1 ≤ 2ε1 + ε2 < ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (30) We will now build the strategy using this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let the distributed source produce χ|π �X �X′ and distribute the �X register to Alice and �X′ to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Now note that the recovery maps’ actions are of the form R : |x⟩⟨x| �→ �ρx C ⊗ |x⟩⟨x|, R : |x⟩⟨x| �→ �ρx A ⊗ |x⟩⟨x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, upon receiving their copies of |x⟩⟨x|, they may apply their recovery maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Ignoring their local copies of |x⟩⟨x|, the joint state is then 1 |C′| ∑ x∈C′ �ρA x ⊗ �ρC x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows from (30) that this generated state is a distributed source simulation to error ε and thus log � Mε2(AC : X)�ρA−X−C � = H0( �X)χ is an upper bound on Cε U,F(A : C)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As the choice of Markov chain exten- sion we picked was arbitrary, we can infimize over the choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, as we established in Lemma 8, the dimension of X may be bounded, and so this becomes a minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, we could minimize over a Bε1(ρ) as every state σ contained satisfies ∥σ − ρ∥1 ≤ 2ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, we have Cε U,F(A : C)ρ ≤ min �ρ∈Bε1(ρ) min A−X−C I √ε2−η max (AC : X)�ρ + κ = min �ρ∈Bε1(ρ) �Cε2−η max (A : C)�ρ + κ(ε2) , where we used Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To get the second upper bound in the lemma, we note the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First, Cmax(A : C) = �C0 max(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Second, �Cε max monotonically decreases as smoothing parameter ε increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, Cε1 max(A : C)ρ = min �ρ∈Bε1(ρ) �C0 max(A : C)�ρ ≥ min �ρ∈Bε1(ρ) �C √ε2−η max (A : C)�ρ , which establishes this second upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For the moreover statements, note we could do the same argument on Markov chain extensions of ρAC so long as ρAC is separable by Lemma 3, and then we only have a single smoothing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' CONVERSES FOR CORRELATION OF FORMATIONS With the achievability established from the one-shot cover lemma, we stress why the one-shot converse for random coding is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The converse for the minimal random codebook size for ε−covering gets the minimal size to achieve an ε−covering with respect to expectation over random codebooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' What we are interested in is the minimal size for any codebook to achieve distributed source simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As these are distinct settings, we turn our attention to establishing a converse to the one relevant for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This will follow from the DPI of the measures that induce the common informations and the relationship between Cmax and H0 for perfectly correlated classical states χ, which we now establish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For any distribution p ∈ P(X ), let χ|p XX′ = ∑x p(x) |x⟩⟨x| ⊗ |x⟩⟨x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then Cmax(X : X′)χ = H0(X)χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First note that the optimal Markov chain is pXX′ �X = ∑x p(x) |x⟩⟨x|⊗3 whose recovery maps are merely copying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' While intuitive, one also may make this rigorous in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The seed will have to be ∑y p′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By symmetry, the recovery maps for both parties will be conditional distributions {q(x|y)}y∈Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' These conditional distributions will in fact have to be deterministic as otherwise the X and X′ spaces won’t be perfectly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus ∑y p′(y) can be partitioned into sets Yxi := {y ∈ Y : q(xi|y) = 1} and then we can apply a coarse graining map that takes Φ that takes Yxi → xi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note Φ(pXX′Y) = pXX′ �X and as it is a coarse grain- ing map, by data-processing, Imax(XX′ : �X)pXX′ �X = Imax(XX′ : �X)Φ(pXX′Y) ≤ Imax(XX′ : Y)pXX′Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus, as we want to minimize Imax, this establishes we have the optimal Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' With this established, by Corollary 2, we have Imax(XX′ : �X)pXX′ �X = log � ∑ x p(x) exp(Imax(px XX′||pX)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (31) Noting px XX′ = |x⟩⟨x|⊗2, Imax(px XX′||pX) = min q∈P(X ′) � λ : |x⟩⟨x|⊗2 ≤ 2λpX ⊗ ∑ x′′ q(x′′) ��x′�� x′�� � = min � λ : |x⟩⟨x|⊗2 ≤ 2λp(x′) ��x′�� x′�� ⊗ |x⟩⟨x| ∀x′� = min � λ : 1 ≤ 2λp(x) � ⇒ λ = p(x)−1 , where in the first line we have just used the definition and Proposition 17, in the second we have used it is clear that qX = |x⟩⟨x| will decrease λ and then we are dealing with diagonal operators so the bound must hold entry- wise, the third is because the L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' only has support 17 on |x⟩⟨x|⊗2 and this completes the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, plugging this into (31), we Imax(XX′ : �X)pXX′ �X = log � ∑ x p(x) exp(log(1/p(x))) � = log � ∑ x 1 � = log(|X |) = Hmax(X)χ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We now present the one-shot converses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We note the square root in the correlation measure is due to purified distance smoothing rather than something fundamental per se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We begin with establishing the converse for cor- relation of formation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' distributed source simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We again stress that the results hold for the entire range of smoothing parameters so long as the state is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C) such that 2ET(ρ) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' C √ε max(A : C)ρ ≤ Cε F(A : C)ρ ≤ Cε U,F(A : C)ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In particular, the bound holds for any ρAC ∈ SepD(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let Cε F(A : C)ρ = n < ∞ where n is finite because ET(ρ) < ε so there is �ρAC ∈ Sep(A : C) such that �ρAC ≈TD ε ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This means, P(�ρAC, ρ) ≤ √ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, by Lemmas 3 and 4, this means there exists �ρA− �X−C such that ∥�ρAC − ρAC∥1 ≤ ε and | �X| = n is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Define X |p with respect to the distribution defining ρX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The recovery maps are preparation channels so that R �X→ �XA Define R �X→A ≡ TrX ◦R and likewise for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows �ρA− �X−C = (R� X′→C ◦ R �X→A)(χ|p X �XX′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then we have C √ε max(A : C)ρ = min �ρ∈B √ε(ρ) min �ρA− �X−C Imax(AC : �X)�ρ ≤Imax(AC : �X)�ρ =Imax(AC : �X)(R◦R)(χ) ≤Imax(XX′ : �X)χ = H0(X)χ = n = Cε F(A : C)ρ , where the first equality is by definition, the first in- equality is by our choice of �ρ being feasible, the second equality is by the equivalence established previously, the second inequality is by data processing, and the last steps are by Proposition 10 and our assumption respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We now present the converse for the entanglement- assisted correlation of formation and the private corre- lation of formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ SepD(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then �C √ε max(A : C) ≤ �Cε F(A : C) ≤ �Cε F(A : C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, for δ ∈ (0, 1 − ε), �C1−ε−δ h (A : C) + 3 log(δ) − 3 log 3 − log(1 − ε) ≤ �Cε F(A : C) ≤ �Cε F(A : C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As �Cε F ≤ �Cε F, one can focus on �Cε F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The proof for �Cε F is effectively the same as the previous except we need ρA−X−C ∈ SepD(A : C) for the value to be finite and then we consider a preparation channels of the form Φ : x �→ σx AC where ∥σA �XC − ρA− �X−C∥1 ≤ ε and | �X| = n, but as we consider DPI for Imax this does not conflict with Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Finally, to get the hypothesis testing bound, one defines √ 1 − ε = ε and applies [33, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Finally, we may combine the results into our one- shot bound for distributed source simulation and its entanglement-assisted equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' These were reported in the summary of results, but for simplicity we re-state them here with proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C) with 2ET(A : C)ρ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε1, ε2 ∈ (0, 1) such that 2ε1 + ε2 < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, C √ε max(A : C)ρ ≤ Cε F ≤ Cε U,F ≤ Cε1 max(A : C)ρ + κ , where κ is defined in Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As 2ET(A : C)ρ < ε was the constraint in both Lemmas 10 and 11, we’ve satisfied the conditions to apply them both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ SepD(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If δ ∈ (0, ε) and η ∈ ( 7 8ε, ε), then �C √ε max(A : C)ρ ≤{ �Cε F(A : C)ρ, �Cε F(A : C)ρ} ≤ �C √ε−η max (A : C)ρ + κ , or if δ′ ∈ (0, 1 − ε), �C1−ε−δ′ h (A : C)ρ + γ ≤ { �Cε F(A : C)ρ, �Cε F(A : C)ρ} ≤ �C1−ε/8 h (A : C)ρ + log � ν · ν′� − 2 , where γ ≡ 3 log(δ′) − 3 log 3 − log(1 − ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The conditions stated mean we satisfy the condi- tions of Lemmas 10 and Lemma 12 at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' RECOVERING ASYMPTOTIC RESULTS VIA A WEAK AEP Theorems 13 and 14 provide one-shot rates of dis- tributed source simulation and its entanglement-assisted counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A natural, and perhaps necessary, ques- tion would be whether we can in fact recover Wyner’s asymptotic result, and Hayashi’s extension, from our one-shot bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' There are a few reasons for this ques- tion being so important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First, while we have established 18 one-shot achievable and converse bounds, it is not a pri- ori obvious these bounds will asymptotically converge properly, though it would be surprising if they did not given previous work on the smooth entropy framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Second, we have actually established bounds for an operational task more general than Wyner’s setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That is, we have established upper and lower bounds for distributed source simulation where the randomness is not uniform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Cε F, as well as when it is uniform, Cε U,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It would therefore be interesting to determine whether this setting has the same asymptotic rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Finally, as was discussed in determining the correlation measures themselves, there seem to be further nuances in how these correlation measures work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Indeed, as noted in the introduction, I↑,ε max inherits a strong AEP from its chain rule decomposition into smooth min- and max-entropies and their AEPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, for this to hold, every register must be n−fold, but as we optimize over an extension, we lose this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, any asymptotic be- haviour we can prove is new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In this section we establish weak AEPs for our max mutual information induced correlation measures, as we summarize in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ρAC ∈ D(A ⊗ C), the Wyner Common information is defined as C(A : C)ρ := min A−X−C I(AC : X) , where the minimization is over QMC extensions of ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ SepD(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, lim ε→0 lim n→∞ � 1 nCε max(An : Cn)ρ⊗n � = C(A : C)ρ lim ε→0 lim n→∞ � 1 n �Cε max(An : Cn)ρ⊗n � = C(A : C)ρ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' These follow from Lemmas 17,18, 20, and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' An immediate corollary of these is that, assuming the error is required to go to zero and the state is separable, the rate of distributed source simulation with or without uniform randomness, and the entanglement-assisted and private distributed source simulation all are given by the common information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρ ∈ SepD(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The rates of dis- tributed source simulation with or without uniform ran- domness, and the entanglement-assisted and private dis- tributed source simulation all are given by the common information: C(A : C)ρ = lim ε→0 lim n→∞ � 1 nCε F(An : Cn)ρ⊗n � = lim ε→0 lim n→∞ � 1 nCε U,F(An : Cn)ρ⊗n � = lim ε→0 lim n→∞ � 1 n �Cε F(An : Cn)ρ⊗n � = lim ε→0 lim n→∞ � 1 n �Cε F(An : Cn)ρ⊗n � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This follows from applying Theorem 15 to The- orems 13 and 14 by considering ρ⊗n AB, taking the limit as n → ∞ and then the limit as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note this is because the smoothing parameters in the one-shot bounds will go to zero as well as ε → 0, and because the spectra of an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' state scales polynomially in n and log(O(poly(n)))/n → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We note two points in particular about this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First, this generalizes Wyner’s result and Hayashi’s ex- tension as it shows it does not matter if the seeded ran- domness was restricted to being uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This is in some sense intuitive as one would expect that asymptotically one would only need the conditional states ρ xn 1 A ⊗ ρ xn 1 C where xn 1 is typical, and the typical set is approximately equiprobable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Indeed, this is the intuition that allows us to establish achievability for the AEP for Cε max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Second, the above result shows, at least in the vanishing error scenario, there is no advantage to using entanglement nor disadvantage to leaking, or rather broadcasting, the X register to everyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The rest of this section proves the AEPs given in Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Specifically, we first establish achievability for each AEP, though for conciseness the actual proof of achievability for Cε max is provided in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In both cases the achievability holds for all ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We then prove a weak converse for each measure’s AEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That is, our proof methods for the converse require the limit ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We end the section with a discussion on what it would take to establish a strong converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Achievability for Weak AEPs We begin with the achievability for the alternative smooth max hypothesis testing common informations �Cε max, �Cε h as the nuance with the smooth max common information is more easily seen in contrast to this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The proof for the alternative common informations in effect follows directly from well-known second-order expansions on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' states [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, lim n→∞ � 1 n �Cε max(A : C)ρ⊗n AC � ≤ C(A : C)ρAC Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We prove the �Cε max version and then state why the other quantity is effectively the same proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If ρAC is entangled the bound is trivial so we assume it is separable and has a Markov chain extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let τA−X−C be the minimizer of C(A : C) �Cε max(An : Cn)ρ⊗n = min ˆρ∈QMC(ρ⊗n) min �ρ∈Bε( ˆρ) I↑ max(AnCn : �X)�ρ ≤ min �τ∈Bε(τ⊗n) I↑ max(AnCn : Xn)�τ ≤ min �τ∈Bε(τ⊗n) Dmax(�τ⊗n||τ⊗n AC ⊗ τ⊗n X ) 19 =Dε max(τ⊗n||τ⊗n AC ⊗ τ⊗n B ) =D(τABC||τAC ⊗ τB) − O(√ n) + O(log(n)) where the first inequality is choosing τ⊗n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' the second is because Imax minimizes over σBn and we have set it to τ⊗n B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' the next equality is by definition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' and then we have taken the second order expansion [33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Now dividing by n and taking the limit, lim n→∞ � 1 n �Cε max(An : Cn)ρ⊗n � ≤ lim n→∞ � 1 n Dε max(τ⊗n ABC||τ⊗n AC ⊗ τ⊗n B ) � =D(τABC||τAC ⊗ τB) = I(AC : B)τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof for �Cε max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The proof for �Cε h is effectively the same by choosing the n−fold copy of τA−X−C for the minimization over quantum Markov chain extensions and then using the second order expan- sion for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' states for Dε h given in [33, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We now note why the proof strategy given above won’t work for Cε max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Recall Cε max(A : C)ρ := min �ρ∈Bε(ρ) min A−X−C I↑ max(AC : X)�ρA−X−C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' What was crucial in the above proof was that Imax itself was smoothed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, as Iε max monotonically decreases as ε increases, we cannot smooth the Imax in the above equation as we want an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, if we let �ρ be the optimizer, then we lose smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As such, it seems we actually need to appeal to (strong conditional) typicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The proof is tedious with little intuition so we present the result here and sketch the proof for the intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The actual proof is presented in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, lim n→∞ � 1 nCε max(A : C)ρ⊗n AC � ≤ C(A : C)ρAC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof Sketch, See Appendix for Full Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If ρAC ̸∈ SepD(A : C), C(A : C)ρ = +∞ and it is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore we can focus on ρAC ∈ SepD(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Consider ρ⊗n A−X−C where ρA−X−C is the minimizer of C(A : C)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We now want to use typicality to achieve I(AC : X)ρA−X−C up to some error which we can take the limit of to make vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To do this, we construct a state τn AnXnCn which is the strongly typical sequences of ρ⊗n Xn on the Xn space and the strong conditionally typical ρ|xn 1 An , ρ|xn 1 Cn states on the other spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' One may use the chain rule [27] Imax(AC : X)ρ = HR(AC)ρ − Hmin(AC|X)ρ , where HR is defined in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By using properties of strong typicality, this decomposition allows one to establish Cε max(An : Cn)ρ⊗n ≤ nI(AC : X)ρ + nO(δ) + log(1 − ε) , where δ ∈ (0, 1) is a parameter of strong typicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By dividing by n and taking the limits δ → 0, n → ∞, we establish the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Weak Converse for AEPs Having established achievability, all that is left is to establish is the (weak) converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Before doing so, we ex- plain why it does not trivially follow from known prop- erties of Imax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In effect a weak converse for Iε max(A : B) would intuitively follow from the fact Imax(A : B)ρ ≥ I(A : B)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, as noted in the background, one may use chain rules to decompose Iε max(A : B) into smooth conditional min- and max-entropies (17) at which point it inherits a strong AEP from the strong AEP for these measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, all of these results do not apply because the max common information involves a non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' auxiliary random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Instead, we will need to find a way to get bounds which are independent of the random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To do so, we in part follow the original converse of Wyner’s result [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To present these proofs, we will need the following definition and a well- known lemma that is a direct corollary of (a form of) strong subadditivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρ ∈ D(A ⊗ C) The smoothed common information is Cε(A : C)ρ := min �ρ∈Bε(ρ) min A−X−C I(AC : X)�ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' H(An 1|B) ≤ ∑i=1 H(Ai|B), with saturation if and only if there exists a labeling of [n] such that ρAn i B is a Ai − B − An i+1 Markov chain for all i ∈ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In other words, with saturation only if Ai can be generated from B for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ρ ∈ D(A ⊗ C), lim ε→0 lim n→∞ � 1 nCε max(An : Cn)ρ⊗n AC � ≥ C(A : C)ρAC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let σAnCn ∈ Bε(ρ⊗n AC) be the minimizer of Cε max(An : Cn)ρ⊗n AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then Cε max(An : Cn)ρ⊗n AC =Cmax(An : Cn)σ ≥ C(An : Cn)σ = I(AnCn : X)σ , where we used that Imax(A : B) ≥ I(A : B) and that σAnXCn is a An − X − Cn Markov chain by definition of common information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Now we decompose the final right hand side of this chain of inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' I(AnCn : X)σ = [H(AnCn) − H(AnCn|X)] = � H(AnCn)σ − n ∑ i=1 H(AiCi|X)σ � = � H(AnCn)σ − n ∑ i=1 H(AiCi)σ 20 + n ∑ i=1 H(AiCi)σ − n ∑ i=1 H(AiCi|X)σ � =H(AnCn)σ − n ∑ i=1 H(AiCi)σ + n ∑ i=1 I(AiCi : X)σ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' where the first equality is a well-known chain rule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' the second equality is by Lemma 19 because we saturate as a An − X − Cn Markov chain is also a Ai − X − Ci Markov chain for all i ∈ [n] as you could trace off the marginals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' and the final identity is again using the same chain rule as the first equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, dividing this by n we have 1 nCε max(An : Cn)ρ⊗n AC = 1 n H(AnCn)σ − 1 n n ∑ i=1 H(AiCi)σ + 1 n n ∑ i=1 I(AiCi : X)σ (32) We now aim to introduce Cε(A : C)ρ into the above bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Consider τA−X−C ≡ σAk−X−Ck where k := argmink∈[n]I(AiCi : X)σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This is a Markov chain for the reason explained above and σAkCk ∈ Bε(ρAC) as purified distance only decreases under partial trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows that τ is a feasible point for Cε(A : C), so we have Cε(A : C) ≤ I(AkCk : X)σ ≤ 1 n n ∑ i=1 I(AiCi : X)σ , where the second inequality is the minimizer lower bounds the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Combining this with (32), 1 nCε max(An : Cn)ρ⊗n AC ≥Cε(A : C)ρ + 1 n H(AnCn)σ − 1 n n ∑ i=1 H(AiCi)σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note by assumption σAnCn ∈ Bε(ρ⊗n AC), and σAiCi ∈ Bε(ρAC) for all i ∈ [n] as explained earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As purified distance upper bounds trace distance, we may use the Fannes-Audenaert inequality to conclude H(AnCn)σ ≥ nH(AC)ρ + εn log(|AC|) + h2(ε) H(AiCi)σ ≥ H(AC)ρ − ε log(|AC|) − h2(ε) , where we have used that the von Neumann entropy is additive over tensor products in the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Plugging these in and cancelling the von Neumann entropy terms, we obtain 1 nCε max(An : Cn)ρ⊗n AC ≥ Cε(A : C)ρ + � 1 + 1 n � h2(ε) + 2ε log(|AC|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then if one lets n → ∞ and then ε → 0 on both sides, one obtains the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note that this held for all density matrices because if Bε(ρ) contains a separable state, a Markov chain exists and so both Cε max, Cε are finite and otherwise, both are infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As mentioned, effectively the same proof establishes a weak converse for �Cε max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ρ ∈ D(A ⊗ C), lim ε→0 lim n→∞ � 1 n �Cε max(An : Cn)ρ⊗n AC � ≥ C(A : C)ρAC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The proof is similar so we only note the major dif- ferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First, σAnXCn that optimizes �Cε max(An : Cn) is not necessarily a Markov chain, though it is classical on the auxiliary register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This follows as the smoothing is done on the choice of Markov chain along with Proposition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Nonetheless, using Lemma 19, we establish 1 n I(AnCn : B) ≥ 1 n H(AnCn)σ − 1 n n ∑ i=1 H(AiCi)σ + 1 n n ∑ i=1 I(AiCi : X)σ , (33) where the inequality is because we no longer have that σ is a Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For the second step, there is more to change as now we don’t know if σAiXCi is ever a Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let �ρAn−B−Cn be a (in case it is not unique) Markov chain that corresponds to the optimizer σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First, this means �ρAnCn = ρ⊗n AC, so �ρAiCi = ρAC for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It also means �ρAiBCi is a Ai − B − Ci Markov chain for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows τABIC = 1 n ∑i∈[n] |i⟩⟨i|I ⊗ �ρAiBCi is a A − IB − C Markov chain as conditioned on I, the remaining state is a Markov chain conditioned on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, τAC = 1 n ∑i∈[n] TrB �ρAiBCi = 1 n ∑i∈[n] ρAC = ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus, τ is a Markov chain extension of ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, we have C(A : C)ρ ≤I(AC : XI)τ =I(AC : I)τ + I(AC : X|I)τ =I(AC : I)τ + 1 n n ∑ i=1 I(AiCi : X)�ρ =H(AC)τ − H(AC|I)τ + 1 n n ∑ i=1 I(AiCi : X)�ρ =H(AC)ρ − 1 n n ∑ i=1 H(AiCi)�ρ + 1 n n ∑ i=1 I(AiCi : X)�ρ , which implies − 1 n n ∑ i=1 H(AiCi)�ρ ≥C(A : C)ρ − H(AC)ρ − 1 n n ∑ i=1 I(AiCi : X)�ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' where the issue is everything is in terms of σ and �ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, we can use the Alicki-Fannes-Winter (AFW) inequalities in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As �ρ ≈ε σ, we have �ρAiBCi ≈ε σAiBCi for all i ∈ [n] and likewise if we trace 21 off B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore using the AFW inequalities (for both unconditional entropy and mutual information [30]), − 1 n n ∑ i=1 H(AiCi)σ ≥ C(A : C)ρ − H(AC)ρ − 1 n n ∑ i=1 I(AiCi : X)σ − 4ε log |AC| − h2(ε) − 2(ε + 1) log(ε + 1) + ε log(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then we can plug this into (33) and use Fannes- Audenaert inequality on H(AnCn)σ in the same fashion as the previous proof to obtain 1 n �Cε max(An : Cn)ρ⊗n AC ≥C(A : C)ρ − 3ε log(|AC|) − 2(ε + 1) log(ε + 1) + ε log(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Taking the limit n → ∞ followed by letting ε → 0 completes the proof where we use 0 log(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In principle one could establish bounds in terms of �C1−ε−δ h , however this would either require prov- ing a new chain rule or simply use converting I1−ε−δ h to Iε max, neither of which would provide particular insight for our purposes, so we do not do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In Regards to a Strong Converse for the AEP We end this section with some remarks on what it would take to establish a strong converse for the AEP which would not depend on the smoothing parame- ter ε ∈ (0, 1), which we believe to be an interesting general problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First note that how the converse is proven currently, one would have to guarantee the opti- mizer Cε max(An : Cn)ρ⊗n AC was the optimizer of C(A : C)ρAC, which seems difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Any relaxation of either Cε max or C, such as to C → Cε will then require a continuity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, this certainly does not imply there is no strong AEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note at the start of the proof we immediately relax from Imax(A : C) to I(A : C) to make it upper bound Cε, which presumably is adding looseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, [6] established a strong converse in the classical setting for distributed source simulation, which would at least suggest there should be a strong converse for the AEP when restricted to classical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For this reason it is worthwhile to discuss what it would take to establish a strong converse for the AEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First, for the smooth min- and max-entropy, the strong AEP is proven via duality [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However the duality of mutual informations requires the inverse of a quantum state and therefore isn’t physical in the same fashion [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As such, it seems we cannot establish an AEP in this manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To the best of the authors’ knowledge this is the only known way to prove a strong AEP for smooth min- and max-entropy, so we cannot borrow a strategy from there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Before discussing other strategies, we reduce the prob- lem to one pertaining to min-entropy rather than max- mutual information for conceptual clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As mentioned in the introduction, using a straightforward general- ization of known chain rules for I↑,ε max, one can estab- lish a strong AEP for I↑,ε max from the strong AEPs for conditional min- and max-entropies (see the appendix and Proposition 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' One can modify these chain rules to establish chain rules for SMCI that show the issue could be converted to a question regarding conditional min-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That is, one can show the following (see appendix for proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A strong AEP for smooth max common information holds if for all ε ∈ (0, 1), there exists ε ∈ (ε, 1) such that 0 < ε + δ(ε) < 1 where δ(ε) := 2 � ε(1 − ε) so that lim n→∞ � 1 n max �ρ∈Bε+δ(ε)(ρ⊗n) max An−X−Cn Hmin(AnCn|X) � ≤ max A−X−C H(AC|X)ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Immediately we can see this sufficient condition is a converse in the sense that we are looking for the regularized smoothed quantity to be upper bounded by the conditional von Neumman entropy term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, it is distinct from the smooth min-entropy strong converse as we take the extension on the purified state and have a non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' random auxiliary variable to deal with still.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, it is unclear how to switch the smoothing from before taking the extension to after, even if we restrict to separable states where both resulting sets are non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='4 However, it does not seem moving the smoothing through is sufficient to obtain a strong converse AEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We propose this is fundamentally because it is in some sense fundamentally a parallel reduction, though even in the fully classical case where it can be forced to be sequential, it won’t trivially follow from current results that include quantum side-information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To show this, we present these problems in terms of what would be sufficient for a strong converse for the alternative SCMI, �Cε max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By the same chain rule argument as before, its strong converse would be implied by the inequality lim n→∞ � 1 n max An−X−Cn Hε min(AnCn|X) � ≤?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' max A−X−C H(AC|X) being true, which makes the smoothing simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Next, one can convert the smooth min-entropy to a smooth max-entropy in exchange for a correction term that goes away in the regularization, as is done in establishing the strong converse for the min-entropy AEP [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus one is interested in establishing something of the form lim n→∞ � 1 n max An−X−Cn Hε max(AnCn|X) � ≤?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' max A−X−C H(AC|X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 4If this switching of optimizations could be done, the open problem would become a Markov chain extension AEP for partially smoothed min-entropy [45], or partially smoothed mutual information if one did not use the chain rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 22 This looks rather similar to the max-entropy version of the entropy accumulation theorem (EAT) or its gener- alization [46], [47], which says that a sufficiently well- behaved (as-if-sequential) process that outputs entropy- generating registers per round converges to the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We now show why this does not work in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note we consider An − X − Cn, which technically is a parallel process where maps only act on X once to generate An,Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, this structure implies Ai − Ai−1 1 XCi−1 1 − Ci for all i ∈ [n] [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This means our process can be forced to look sequential in such a way that it satisfies the constraints of both versions of the EAT [46], [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, in altering the problem in this man- ner, the issue is that the effective maps will need access to Ai−1 1 Ci−1 1 as side-information for generating the next round as well as being the previous entropy generating registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This means one needs to have copies of these registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It is not obvious that this can be done in the separable state case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='5 This may be viewed as the general difficulty of applying the EAT to this parallel setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We do note that we could restrict to a classical Markov chain problem, Xn − Y − Zn, where registers can always be copied so we can properly define EAT maps Gi per round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, in this setting and by modifying the notation of some of the side-information registers for clarity, the generalised EAT [47, Appendix A] will be of the form Hε max(XnZn|YE)ρ ≤ ∑ i max |ω⟩ H(XiZi|Xi−1 1 Zi−1 i YEi)(Gi⊗idEi )(ω) , where ω is any purification of an input to Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The problem then is the LHS conditions on quan- tum side-information we don’t want to consider and by strong subadditivity of smooth max-entropy, Hε max(XnZn|Y �E) ≤ Hε max(XnZn|Y), so this would require modifying the max-entropy version of the EAT to not include the quantum side-information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To summarize, even for the alternative common in- formation, it appears that a strong converse for the fully separable case would require an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' reduction that takes into account the Markov chain structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This does not align with the EAT nor a traditional deFinetti theorem where a purification is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To the best of our knowledge, none of the problems for establishing a strong converse that we have noted are addressed in the smooth entropy framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We do however note there are certainly other ways of establishing strong converses in the face of auxiliary variables even in the quantum setting, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 5Technically, this is because while we know An − X − Cn is an exten- sion of ρ⊗n AC, it is not obvious from the structure of the Petz recovery map [30] nor the general structure of the recovery map for Markov chains specifically [48] that the recovery map RCi−1 1 XAi−1 1 →Ci 1XAi−1 1 won’t entangle some of the quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' EXTENDING TO MANY RECEIVERS We have now established the general framework of one-shot distributed source simulation and its relation smooth max common information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We now explain that it is straightforward to generalize beyond simulating a bipartite distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In the classical case this was addressed by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In that work if one goes to the appendix where they prove it, they just point out it is in effect the same as proof as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Indeed, this observation lifts to our setting with one nuance: it is not clear how to argue the minimizer for multipartite systems should be classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This is because the proof of Proposition 17 uses the decomposition of a quantum Markov chain and it is not clear how to generalize this to more systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Regardless, this is in effect irrelevant with regards to the operational task at hand, so we focus on the classical seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We begin with notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' A − X − C is very natural as it shows the two registers splaying out from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We can’t do this for more than two registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As such, if we imagine we want ρA1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=', ρAm each generated from seed X independently, then we will write this as X Am 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' With this notation established we can define the following operational tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAm 1 ∈ D(Am 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The one- shot correlation formation is defined as Cε F( : Am 1 : ) = min � H0(X)�ρ : �ρX Am 1 : �ρ ∼ε ρAm 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (34) Moreover, the one-shot uniform correlation of formation, which is the one-shot common information, is defined as Cε U,F(A : B) = min � H0(X)�ρ : �ρπ Am 1 : �ρ ∼ε ρAm 1 � , (35) where we remind the reader π means the register is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We can define the entanglement-assisted cases in a similar fashion, so we omit them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then we can define the extended smoothed correlation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Cmax( : Am 1 : )ρAm 1 := min �ρ∈Bε(ρAm 1 ) min �ρX Am 1 Imax(Am 1 : X)�ρAmX �Cmax( : Am 1 : )ρAm 1 := min ρX Am 1 min �ρ∈Bε(ρAm 1 X) Imax(Am 1 : X)�ρAmX We can establish the one-shot converse the same way as before using data-processing by extending the χ|p to more parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We can establish achievability using the one-shot random covering as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Similarly, the weak AEPs for these will follow the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The achievability of �Cε max( : Am 1 : ) can still be proven using the AEP for Iε max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The achievability of Cmax( : Am 1 : ) can be proven using strong typicality in the 23 same fashion as before since we can still decompose the optimizer of C( : Am 1 : ) as ∑x∈X p(x) |x⟩⟨x| � i∈[n] ρAi x and construct a state using strong conditional typicality from that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The converses are established the same as before with the replacement An 1Cn 1 �→ An 1,1An 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An m,1 where the first subscript here denotes the party label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus we have extended all our results to multiple parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' a) Monotonicity in Number of Parties: In [5] the au- thors note, albeit in the classical setting, that given ρAm 1 and ρAk 1 ≡ TrAm k (ρAm 1 ), C( : Am 1 : ) ≥ C( : Ak 1 : ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' that common information can only decrease as you decrease the number of parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' They suggest (1) this is surprising and (2) no such property is known to hold for mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We wish to briefly address these points in case they provide conceptual clarity for the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First, the authors suggest this monotonicity is surpris- ing because “if the information is common it ought to be non-increasing when more random variables are in- cluded.” This suggests the authors view common infor- mation as measuring the intersection of the randomness of the states (random variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, the common information is measuring the randomness needed to produce each random variable independently, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' to generate a common variable, and this is like measuring the union of the randomness of the states in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Whether or not one agrees this is what “common” should denote, if one views it in this fashion, it is clear that it must increase when you add more random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Second, that the common information has such a monotonic property is an immediate corollary of the mutual information having the same property, which in the quantum information community is called the data- processing inequality (not to be confused with the data- processing inequality for a Markov chain as is common in classical information theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We show this in the following generic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Given ρAm 1 and ρAk 1 ≡ TrAm k (ρAm 1 ), C( : Am 1 : ) ≥ C( : Ak 1 : ) where C is the Wyner common information defined using any mutual information I satisfying data-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρ⋆ Am 1 X be the minimizer of C( : Am 1 : ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, C( : Am 1 : ) =I(Am 1 : X)ρ⋆ ≥I(Ak 1 : X)TrAm k (ρ⋆) ≥ min ρX Ak 1 I(Ak 1 : X)ρ =C( : Ak 1 : )ρ , where the first inequality is the data-processing in- equality and the second is using minimizing TrAm k (ρ⋆) is one seeding option for ρAk 1 and so we can further minimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ENTANGLED STATE SOURCE SIMULATION We have now established the limits of distributed source simulation in the one-shot and asymptotic setting in terms of smooth entropic quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' These results however have only applied to separable states, and so it is worthwhile to ask what can be said about entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It is known that one cannot convert entangled states with zero communication and no auxilliary re- source [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, it is also known that there exists a sufficiently large (pure) entangled state that can be used to generate any (pure) entangled state up to small error with zero communication [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Specifically, what the authors show is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ([29]) For any ε > 0 and target bipartite pure state |ϕ⟩AB with Schmidt rank m, the catalyst state |µ(n)⟩A′B′ = 1 √Hn ∑n j=1 1√ j |j⟩A′ |j⟩B′ is such that for n > m1/ε there exist unitaries UAA′, WBB′ so that F(U ⊗ W(|µ(n)⟩A′B′ |0⟩A |0⟩B), |µ(n)⟩A′B′ ⊗ |ϕ⟩AB) ≥ 1 − ε , where F(·, ·) is the fidelity and Hn is the Harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This means that Alice and Bob, when given the proper seed state |µ(n)⟩, can prepare the target state |ϕ⟩AB using local operations and zero communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In this sense it seems the natural extension of distributed source simulation to the quantum setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, there are technical distinctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Specifically, while in both cases the seed state is (at least approximately) preserved, in embezzling, the remaining seed state is (approximately) decoupled from the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This has the further advantage of allowing the seed state to be used in further protocols in exchange for further degrading the total approximation, which we note none of the correlation of formations could guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' These similarities and distinctions are summarized in Fig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' With this established, we define the embezzleable entanglement of simulation, which measures the amount of entanglement (with respect to a specific choice of measure) necessary to source simulate a state in a dis- tributed manner and is named in such so as to avoid any confusion with entanglement of formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To define this, we will need the definition of entanglement rank [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For all r ≥ 1, the set of all operators R ∈ Pos(A ⊗ B) for which there exists a finite alphabet X and collection of linear operators {Mx}x∈X ⊂ L(A, B) such that rank(Mx) ≤ r for all x ∈ X and R = ∑ x∈X vec(Mx) vec(Mx)∗ , where vec : L(A, B) → A ⊗ B defined via vec(|i⟩ ⟨j|) = |j⟩ |i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We say a density matrix ρAB ∈ D(A ⊗ B) has entanglement rank r′ if it is contained Entr′(A : B) but not Entr′−1(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For notational simplicity, we define 24 pX Copy ΦX→A A C ΨX′→C ≈ε ρAC X (a) Distributed Source Simulation σ �A �C A′ C′ Φ �A→AA′ Φ �C→CC′ A C A′ C′ ρAC ≈ε ⊗ σA′C′ (b) Embezzling Source Simulation σA′C′ A′ C′ ΦA′→AA′ ΦC′→CC′ A C A′ C′ ρAC ≈ε (c) Entangled Source Simulation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 7: Comparison between distributed source simula- tion and the entangled state versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Grey lines rep- resent allowed correlations of either classical or quan- tum mechanical nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (a) Distributed source simulation where the X register is possibly strongly correlated to A and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (b) Embezzling source simulation where the auxiliary state is required to be output approximately decoupled from the simulated state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (c) Entangled source simulation where the input is an arbitrary quantum state and an appropriate marginal of the output must achieve the target state to tolerable error ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' EntA : B : D(A ⊗ B) → N≥1 as the function that takes a density matrix and returns its entanglement rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The subscript is because the partitioning is relevant as will be shown in a following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note that the set of separable operators is equivalent to the set Ent1(A : B) and if A ∼= B ∼= Cd, then all positive semidefinite operators are contained within Entd(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That is to say, the entanglement rank measures ‘how entangled’ an operator is and may be viewed as a mixed state extension of Schmidt rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C) and ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The ε−embezzleable entanglement of simulation is the logarithm of the minimal entanglement rank of a bi- partite state such that under local operations it may be converted to ρ up to ε− error in purified distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Formally, Cε EE,S(A : C)ρ := log min{r ≥ 1 : ∃σ ∈ Entr(A′ : C′) : (Φ ⊗ Ψ)(σ) ≈F ε σ ⊗ ρ} , where Φ ∈ C(A′, A′ ⊗ A) and Ψ ∈ C(C′, C′ ⊗ C) and σ ≈F ε ρ means F(σ, ρ) ≥ 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It is worthwhile to relate this back to the correlation of formation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Much like Cε F, it measures the logarithm of the ‘dimension’ of the resource, in this case entanglement rank, but does not care about the uniformity of the resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Second, one way of viewing distributed source simulation is that the channels are the recovery maps R : X → X ⊗ C, R : X → A ⊗ X, and the definitions of Φ, Ψ and the error condition mirror this as the channels (approximately) preserve the input resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note that while it only measures the entanglement rank, its demand on the ancillary state being (approx- imately) unchanged and uncorrelated means that it is not clear how one would make use of a classical ancillary state, at least when ε is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This is because if ρ is built conditionally on σ, it will not be uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For this reason it seems to properly capture the notion of embezzling as being the strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' With this definition introduced, we establish achiev- ability bounds and then argue that these bounds should be approximately tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAB ∈ D(AB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then SR(AR : B), SR(A : BR) is the same for all purifications |ψ⟩ABR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By isometric equivalence of purifications, |ψ⟩R′AB = (VR→R′ ⊗ 1) |ψ⟩RAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let |ψ⟩RAB = ∑i∈[m] √pi |φi⟩RA ⊗ |ϕi⟩B be its Schmidt decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then |ψ⟩R′AB = ∑i∈[m] √piVR→R′ |φi⟩RA ⊗ |ϕi⟩B is its Schmidt decomposition as an isometry maps pure states to pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' An identical argument holds for the other partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The following lemma shows it is necessary to take the minimization in the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For pure state |ψ⟩RAB, in general SR(AR : B) ̸= SR(A : BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, the exists |ψ⟩ABR such that SR(A : BR) − SR(AR : B) = dB − 1, the maximum possible difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We present an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAB = |φ⟩⟨φ|A ⊗ πB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then |ψ⟩ABR = |φ⟩A ⊗ |Φ+⟩BB′ ⊗ |φ⟩A′ where R ≡ A′B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows SR(A : BR) = 1 as it is product across this partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' On the other hand, |ψ⟩ABR = |φ⟩A ⊗ � �d−1/2 B ∑ i∈[dB] |i⟩B |i⟩B′ � � ⊗ |φ⟩A′ =d−1/2 B ∑ i∈[dB] |i⟩B ⊗ |φ⟩A ⊗ (|φ⟩ ⊗ |i⟩)R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' as {|φ⟩⊗2 ⊗ |i⟩}i form the Schmidt vectors for the AR space, SR(A : BR) = dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 25 Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ε ∈ (0, 1), one can construct ρAB to accuracy ε via embezzling using |µ(n)⟩ for n > m1/ε where m = min{SR(AR : B), SR(A : BR)} , Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Embezzling is a function of the Schmidt rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By the previous lemma, we only need to consider purifi- cation |ψ⟩ABR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To have a notion of locality, either Alice or Bob must embezzle in the purifying space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By the previous proposition, in general there is a difference in Schmidt rank depending on who purifies the state, so we take the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A : C) and ε ∈ (0, 1), then Cε EE,S(A : C)ρ ≤ 1 ε log(min{EntAR:C(ψ), EntA:CR(ψ)}) , where ψACR is an arbitrary purification of ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This follows the definition of Cε EE,S and the previ- ous proposition where we have taken the logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' There are two questions: the first would be if this strategy, when no classical side-information is allowed, is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Roughly speaking, it is in the sense that [29] showed that if one allowed LOCC and a state dependent catalyst that the error is bounded below by Ω(1/ log(n)), but the universal embezzling strategy scales as O(1/ log(n)) where n is the Schmidt rank of the seed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Of course this scaling requires the error demanded be small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' if the error is sufficiently large, it may be feasible to use less entanglement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' this is developed further by authors of this work in a separate paper [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also stress that Cε EE,S seems to be captured effectively by embezzling as we already argued why, in general, a classical auxiliary state could not be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The second question would be if this strategy has any notion of compressibility in the sense that it requires less resources for many copies of an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This is not so: we show this strategy scales in the number of copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ε ∈ (0, 1), one can construct ρ⊗k AB to accuracy ε via embezzling using |µ(n)⟩ for n > m1/ε where m = k · min{SR(AR : B), SR(A : BR)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Consider a purification of ρ⊗k AB, |ψ⟩AkBkR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Con- sider a purification of ρAB, |φ⟩ABR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows |φ⟩⊗k is a purification of ρ⊗k AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By the isometric equivalence of purifications, there is an isometry or reversed isometry taking R′ to Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As this is a local map, it can’t change the Schmidt rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus, SR(AkR′ : Bk) = SR(AkRk : Bk) and likewise for the other partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Finally, SR(AkRk : Bk) = k · SR(AR : B), and likewise for the other partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' a) Entanglement of Simulation without Decoupling: In the previous strategy, as already noted, the inclusion of a classical register is effectively not feasible because we require the catalyst to be decoupled from the output state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, distributed source simulation does not require this decoupling condition as the X seed register will be correlated with the A and C registers (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows one may argue the setting of Cε EE,S is too re- strictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The question then becomes what is the natural setup to correspond with distributed source simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The most general setting would be local operations and shared entanglement (LOSE) with no constraints, that is, the input can be any state and a marginal of the output is the target state up to some tolerated error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This aligns with the notation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, this unconstrained LOSE setting, if we mea- sure the needed entanglement in terms of entanglement rank, has a simple characterization as we briefly prove following a few definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1) and ρAC ∈ D(A ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The entanglement of simulation is given by Cε E,S := log min{EntA:C(σ) : TrA′C′(Φ ⊗ Ψ)(σ) ≈F ε ρ} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' where Φ ∈ C( �A, A′ ⊗ A), Ψ ∈ C( �C, C′ ⊗ C) and ρ ≈F ε means F(ρ, σ) ≥ 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAB ∈ D(A ⊗ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For r ≥ 1, the fidelity of entanglement rank is defined as EF,r(A : B)ρ := sup σAB∈EntrD(A:B) F(ρ, σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' From this, we define the ε−approximate entanglement rank as Entε A:B(ρ) := min{r ≥ 1 : EntF,r′(A : B)ρ ≥ 1 − ε} , which measures the smallest entanglement rank for a state σ to be ε-close to ρ under fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ε ∈ [0, 1], ρAC ∈ D(A ⊗ C), Cε E,S = log(Entε A:C(ρ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We prove this is a lower bound and then note it can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It is known that local operations can only decrease entanglement rank [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note that the throwing out of the A′, C′ spaces are also local operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, the input state σ must have an entanglement rank that is lower bounded by Entε A:C(ρ) or else it will violate the error restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, the state τAC that minimizes Entε A:C(ρ) can be forwarded and the local operations be trivial, so this is also achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Taking the logarithm completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We note part of the simplicity of this setting is our choice of measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [51] analyzed the quantum correlation complexity, which is the same setting but measuring the logarithm of the rank of the seed state rather than its entanglement rank and the analysis in this case is more arduous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, given the triviality of the 26 unconstrained case under our choice of measure, we choose to restrict the input state to aribtrary classical correlation and embezzling states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Definition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1) and ρAC ∈ D(A ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The entanglement of simulation restricted to embezzling states is Cε E|Emb,S(A : C)ρ := log min{r ≥ 1 : TrA′C′ ◦(Φ ⊗ Ψ)(σ(r)) ≈F ε ρ} , where σ(r) �A �C := |µ(r)⟩ ⊗ σXAXC, �A ∼= A′ ⊗ XA, �C ∼= C′ ⊗ XC, Φ ∈ C( �A, A′ ⊗ A), Ψ ∈ C( �C, C′ ⊗ C), and σ ≈F ε ρ means F(ρ, σ) ≥ 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We remark that one advantage of choosing the embez- zling state is that we have a notion of a consistent resource, which mirrors that the uniform correlation of formation has a consistent resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='6 The difference between embezzleable entanglement of simulation, Cε EE,S, and the entanglement of simulation restricted to embezzling states, Cε E|E,S is of course that we allow for arbitrary classical assistance in the latter which as addressed we cannot in general do with the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This in particular allows us to distribute a flag state in the latter setting, which results in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Theorem 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρ ∈ D(A ⊗ C) and ε ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then log(Entε A:C(ρ)) ≤ Cε SREE,S(A : C)ρ ≤ 1 ε log(EntA:C(ρ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Moreover, for sufficiently small ε ∈ (0, 1) the upper bound is nearly optimal for pure state |ψ⟩ that is not local unitarily equivalent to |µ(r′)⟩ for any r′, and the lower bound is tight when ε = 0 and |ψ⟩ is local unitarily equivalent to some |µ(r′)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' These two points imply these bounds are approximately tight although they don’t match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We first prove the achievability and converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then we provide examples where these are (nearly) tight bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (Achievability) The upper bound is trivial when ε = 0, so we assume ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Consider the decomposi- tion of ρAC = ∑x∈X p(x) |ψx⟩⟨ψx| such that the max Schmidt rank of {|ψx⟩}x∈X is minimized over all pos- sible decompositions of ρAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let m be this max Schmidt rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let |µ(r)⟩ such that r > m1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then let σ(r) = |µ(r)⟩⟨µ(r)|A′C′ ⊗ ∑x p(x) |x⟩⟨x|XA ⊗ |x⟩⟨x|XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let Alice receive A′XA and Charlie receive C′XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Conditioned on x, Alice and Charlie embezzle in |ψx⟩ using µ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For notational simplicity, we describe this in terms of conditional isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Call the set of local isometries conditioned on x ∈ X that do this {Ux∈X } and {Vx}x∈X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' These isometries include the local unitaries U∗ x,W∗ x from Theorem 22 which we have pushed into the isometries 6One might note that maximally entangled states would be the natural equivalent of uniform shared randomness, but [21] proves this won’t work for LOSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' using the isometric invariance of fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Call the total maps that implement this Φ and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Already tracing out the classical registers and the purification registers, F(|µ(r)⟩⟨µ(r)| ⊗ ρAC, TrA′C′(Φ ⊗ Ψ)(σ)) ≥ ∑ x∈X p(x)F(|µ(n)⟩ ⊗ |ψx⟩ , (Ux ⊗ Vx) |µ(n)⟩ |0⟩ |0⟩) ≥ ∑ x∈X p(x)(1 − ε) =1 − ε , where the first inequality is going to the isometric picture and using the joint concavity of fidelity and we’ve used the fidelity guarantee of Theorem 22 in the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Taking a logarithm of r due to Definition 24 completes the achievability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (Converse) Let r′ := Entε A:C(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By definition, the strategy must be a product map Φ ⊗ Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows it is a separable map [22, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='17], and thus it cannot increase the entanglement rank [22, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus, the entanglement rank of the seed state is at least this m′ value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (Near Tightness of Upper Bound) Consider the target state ρAB is a pure φAB such that it is not local unitarily equiv- alent to any member of {|µ(r)⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' There is no advantage to the shared randomness setting as there is an optimal pair of local maps from |µ(r′)⟩ to φ for any r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus, we have reduced the problem to embezzling, which is near optimal as shown in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (Tightness of Lower Bound) Consider the target state ρAB is local unitarily equivalent to |µ(r′)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then as F(ρ, σ) = 1 if and only if ρ = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus if ε = 0, one must forward |µ(r′)⟩ instead of any smaller embezzling state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' While the above result is approximately tight in certain settings, we remark the lower bound is clearly loose in general as the inability to convert one pure state to another with no communication is not only a function of the Schmidt rank, but the Schmidt coefficients [21], [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For example, in [21], the authors show that the amount of communication necessary for pure state transmission depends on the ‘entanglement spread,’ which is effec- tively a function of the maximum Schmidt coefficient and the number of Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' CONCLUSION AND OPEN PROBLEMS In this paper we have considered the task of dis- tributed source simulation in the one-shot setting for fully quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We introduced various one-shot operational quantities related to distributed source sim- ulation and similar tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We then introduced one-shot correlation measures and established that these correla- tion measures bound these operational quantities and thus characterize one-shot distributed source simulation and its related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In particular, this established a one-shot version of Wyner’s common information result [2] in the smooth entropy framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In doing so, we 27 generalized the support lemma to preparation channels and showed nuances about smoothing measures when an auxiliary random variable is involved— ideas which likely will have further use in one-shot quantum net- work theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' One particular technical point of interest is that we found it is important to not be smoothing the auxiliary variable and this led to inducing the measure via Dmax but one could not do the same using Dε h as it automatically smooths all argmuents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We then proceeded to recover asymptotic results from the one-shot results by establishing asymptotic equipar- tition properties for our one-shot correlation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In doing so, we perhaps intuitively extended Wyner’s original result [2], and Hayashi’s extension to separable states [16], by showing that asymptotically there is no advantage in non-uniform shared randomness in the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also showed that asymptotically the variations on the one-shot distributed source simulation we had considered, private entanglement-assisted and private distributed source simulation, converge are given by the same rate, at least in the case where we demand asymptotically vanishing error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This is to say many variations of the tasks that are clearly in general different in the one shot setting become equivalent asymptotically in the vanishing error setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' An open question would be whether these results generalize to not requiring vanishing error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Motivated by Yu and Tan’s recent result strong converse in the classical setting [6], [7], it would be our expectation they would.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To remain in the smooth entropy framework, this would require establishing a strong converse for our asymptotic equipartition prop- erty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We showed one option would be to establish a new property for conditional min entropy (Proposition 11) that likely would require new tools as we explained in that section why current methods do not seem sat- isfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Alternatively, one could establish the strong converse in some other fashion outside of the smooth entropy calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' An open problem related to the establishment of a strong converse would be the establishment of second- order asymptotics, though this may be difficult with the rate being a function of the auxiliary random vari- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Relatedly, we note that our one-shot upper and lower bounds (Theorem 13) are not both linear in the smoothing parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If one were to establish second- order asymptotics in a similar fashion as is standard in the smooth entropy framework [33], this would need to be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We note that the Renes-Renner one-shot bounds for compression with quantum side-information [52] have similar smoothing problems as our Theorem 13 and this was resolved by Tomamichel and Hayashi using they hypothesis testing entropy [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' One hope would be to do the same by establishing a converse for one-shot distributed source simulation for only separable states in terms of �Cε h and then finding a method for a second- order expansion of that quantity, though it’s not clear why that would be easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' After establishing the general framework of one-shot distributed source simulation for bipartite states we con- sidered two variations: more parties and an entangled state variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In the multipartite setting we explained while all the proofs extend in a straightforward manner and made some clarifications with regards to comments in previous work [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In the quantum setting, we dis- cussed the ability to use embezzling as a strategy for the equivalent of distributed source simulation as con- version of entangled states to arbitrary error with zero communication is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We considered both the setting where the seed state cannot use any classical correlation and when it can and gave tight upper and lower bounds that in general do not match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ACKNOWLEDGMENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' would like to thank Vincent Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Tan for pro- viding a draft of Yu’s and his monograph on common information [19] prior to its publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This work was supported in part by the National Science Foundation under Grant 2112890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' acknowledges the support of an Illinois Distinguished Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Shannon, “A mathematical theory of communication,” Bell System Technical Journal, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 379–423, 1948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Wyner, “The common information of two dependent random variables,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 163–179, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' G´acs and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' K¨orner, “Common information is far less than mutual information,” Problems of Control and Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 149–162, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [4] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Witsenhausen, “On sequences of pairs of dependent random variables,” SIAM Journal on Applied Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 28, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 100–113, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [5] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Liu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Xu, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Chen, “The common information of n dependent random variables,” in 2010 48th Annual Allerton Confer- ence on Communication, Control, and Computing (Allerton), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 836– 843, IEEE, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [6] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Yu and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Tan, “Wyner’s common information under renyi divergence measures,” in 2018 IEEE International Symposium on Information Theory (ISIT), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 156–160, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Yu and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Tan, “Corrections to “wyner’s common infor- mation under r´enyi divergence measures” [may 18 3616-3632],” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2599– 2608, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Winter, “Secret, public and quantum correlation cost of triples of random variables,” in Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' International Symposium on Information Theory, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ISIT 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2270–2274, IEEE, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [9] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Chitambar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Hsieh, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Winter, “The private and pub- lic correlation cost of three random variables with collaboration,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 62, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2034– 2043, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Chitambar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Fortescue, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Hsieh, “Quantum versus classical advantages in secret key distillation (and their links to quantum entanglement),” arXiv preprint arXiv:1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='07344, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Chitambar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Fortescue, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Hsieh, “Classical analog to entanglement reversibility,” Physical review letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 115, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 090501, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Chitambar and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Hsieh, “Round complexity in the local transformations of quantum and classical states,” Nature commu- nications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1–7, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Cuff, “Communication requirements for generating correlated random variables,” in 2008 IEEE International Symposium on Infor- mation Theory, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1393–1397, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [14] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Kumar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Li, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' El Gamal, “Exact common informa- tion,” in 2014 IEEE International Symposium on Information Theory, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 161–165, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 28 [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Yu and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Tan, “On exact and ∞-R´enyi common infor- mations,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 3366–3406, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Hayashi, “General nonasymptotic and asymptotic formulas in channel resolvability and identification capacity and their appli- cation to the wiretap channel,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1562–1575, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [17] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Cuff, “Distributed channel synthesis,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 7071–7096, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Yu and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Tan, “R´enyi resolvability and its applications to the wiretap channel,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 65, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1862–1897, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Yu and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Tan, “Common information, noise stability, and their extensions,” Foundations and Trends® in Communications and Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 107–389, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Bell, “On the einstein podolsky rosen paradox,” Physics Physique Fizika, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 195, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [21] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Hayden and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Winter, “Communication cost of entanglement transformations,” Physical Review A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 67, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 012326, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Watrous, The Theory of Quantum Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Cambridge Uni- versity Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Hayashi, Quantum Information: An Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Springer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Ahlswede and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Winter, “Strong converse for identification via quantum channels,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 48, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 569–579, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [25] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Cheng and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Gao, “Error exponent and strong converse for quantum soft covering,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [26] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Shen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Gao, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Cheng, “Optimal second-order rates for quantum soft covering and privacy amplification,” arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='11590, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Berta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Christandl, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Renner, “The quantum reverse shannon theorem based on one-shot information theory,” Com- munications in Mathematical Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 306, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 579–615, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [28] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Ciganovi´c, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Beaudry, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Renner, “Smooth max- information as one-shot generalization for mutual information,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 60, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1573– 1581, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [29] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' van Dam and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Hayden, “Universal entanglement transforma- tions without communication,” Physical Review A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 67, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 060302, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Wilde, “From classical to quantum shannon theory,” arXiv preprint arXiv:1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='1445, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Tomamichel, Quantum information processing with finite re- sources: mathematical foundations, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [32] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Han, Information-Spectrum Methods in Information Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Springer, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Tomamichel and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Hayashi, “A hierarchy of information quantities for finite block length analysis of quantum tasks,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 7693–7710, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [34] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Renner and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Wolf, “Smooth r´enyi entropy and applications,” in International Symposium onInformation Theory, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ISIT 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=', p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 233, IEEE, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Berta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Christandl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Colbeck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Renes, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Renner, “The uncertainty principle in the presence of quantum memory,” Nature Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 659–662, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' McKinlay and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Tomamichel, “Decomposition rules for quan- tum r´enyi mutual information with an application to information exclusion relations,” Journal of Mathematical Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 61, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 072202, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [37] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Khatri and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Wilde, “Principles of quantum communica- tion theory: A modern approach,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [38] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Physical Review Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 80, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2245, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [39] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Chitambar and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Gour, “Quantum resource theories,” Reviews of Modern Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 91, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 025001, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [40] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Hayden, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Jozsa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Petz, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Winter, “Structure of states which satisfy strong subadditivity of quantum entropy with equality,” Communications in mathematical physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 246, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 359–374, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' El Gamal and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Kim, Network Information Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Cam- bridge University Press, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [42] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Csisz´ar and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' K¨orner, Information theory: coding theorems for discrete memoryless systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Cambridge University Press, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [43] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Gharibian, “Strong np-hardness of the quantum separability problem,” 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Hayashi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Tomamichel, “Correlation detection and an operational interpretation of the r´enyi mutual information,” Journal of Mathematical Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 57, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 102201, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [45] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Anshu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Berta, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Jain, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Tomamichel, “Partially smoothed information measures,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 5022–5036, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [46] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Dupuis, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Fawzi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Renner, “Entropy accumulation,” Communications in Mathematical Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 379, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 867– 913, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [47] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Metger, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Fawzi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Sutter, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Renner, “Generalised entropy accumulation,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [48] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Sutter, “Approximate quantum markov chains,” in Approximate Quantum Markov Chains, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 75–100, Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [49] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Cheng, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Datta, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Rouz´e, “Strong converse bounds in quantum network information theory,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 67, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 2269–2292, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [50] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' George and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Chitambar, “Revisiting pure state transforma- tions with zero communication.” In Preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [51] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Jain, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Shi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Wei, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Zhang, “Correlation/communication complexity of generating bipartite states,” 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [52] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Renes and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Renner, “One-shot classical data compression with quantum side information and the distillation of common randomness or secret keys,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1985–1991, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' [53] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' M¨uller-Lennert, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Dupuis, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Szehr, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Fehr, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Tomamichel, “On quantum r´enyi entropies: A new generalization and some properties,” Journal of Mathematical Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 122203, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' APPENDIX MUTUAL INFORMATION LEMMAS In this section of the appendix, we establish various properties of mutual information measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Remember there are three versions of (smoothed) max mutual infor- mation (Definition 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In each result we specify which mutual information we mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Many of these results are relatively straightforward variations of proofs from [53] and/or [31, Chapter 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The first establishes that when smoothing a quantum state with classical registers, you can restrict to optimiz- ers that are classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAXBY ∈ Pos(AXBY) be classical on X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For ε ∈ [0, � Tr(ρ)), the smoothing ball Bε(ρ) may be restricted to QCQC states and the optimal τ, τ will be classical on the same registers when optimizing over Iε,x(AX : BY)ρ for all x ∈ {↓, ↑, ↑↑} where I is any mutual information defined on any R´enyi divergence D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By definition, Iε,x(AX : BY)ρ = min �ρ∈Bε(ρ) min τ∈SAX |�ρ ,τ∈SBY |�ρ D(�ρ||τ ⊗ τ) , where the sets S will also depend which x ∈ {↓, ↑ , ↑↑} we are using.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Next, by data-processing, D((PX ⊗ PY)(�ρ)||PX(τ) ⊗ PY(τ)) ≤ D(�ρ||τ ⊗ τ), where PX, PY are the pinching maps onto the computational bases for the registers X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As ρAXBY is classical on X and Y, by data-processing of purified distance, we may restrict minimizing Bε(ρ) to states that are classical on X as ρ is invariant under pinching on X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Furthermore then SAX |�ρ may be restricted to being classical on X and same idea for the other set with respect to Y as the optimizing 29 choice of �ρ, τ, τ will be classical on those spaces as we showed via DPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Next, we will need to establish that we can write terms proportional to max mutual information in terms of an expectation on the classical register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This will be broken up into multiple steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Each step has a conditional entropic equivalent which can be found in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρABX ∈ D≤(ABX) such that it is classical on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That is, ρABX = ∑x p(x) |x⟩⟨x| ⊗ ρx AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let α′ := α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then for α ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' 1) ∪ (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ∞) we have the following I↑↑ α (A : BX) = 1 α′ log � ∑ x px exp �(α′)Dα(ρx AB||ρA ⊗ ρx B) � � I↑ α(A : BX)ρ = α −α′ log � ∑ x px exp �α′ α I↑ α(ρx AB||ρA) �� I↓ α(A : BX) = min q∈P(X ) τA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='{τx B}x 1 −α′ log � ∑ x pα xq(x)−α′ exp �(α′)Dα(ρx AB||τA ⊗ ρx B) � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' where Iα is any Petz or Sandwiched R´enyi mutual information over the specified ranges and for the middle quantity we define I↑ α(ρAB||σA) := min τB Dα(ρAB||σA ⊗ τB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The proof is largely identical to that of [31, Propo- sition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='1], but we provide it for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We begin from the fact that for any CQ states ρXA, σXA, it holds Dα(ρXA||σXA) = 1 α − 1 log � ∑ x pα xq1−α y exp((α − 1)Dα(ρx A||σx A)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As I↑↑ α (A : BX) = Dmax(ρABX||ρA ⊗ ρBX), we have its simplification by direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For I↓ α(A : BX) we have established the minimizers τBX can be restricted to being classical in Proposition 17, thus Iα(A : BX)ρ = minτA,τBX Dmax(ρ||τ ⊗ τBX) so again a direct calculation can be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lastly we establish the remaining case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First recall I↑ α(A : BX) = minτBX Dα(ρ||ρA ⊗ τBX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows from above then that we have I↑ α(A : BX)ρ = 1 α′ min τBX log � ∑ x pα xq−α′ x exp �(α′)Dα(ρx AB||ρA ⊗ τx B) � � = 1 −α′ min q∈P(X ) log � ∑ x pα xq−α′ x exp((α − 1)Iα(ρx AB||ρA)) � , where the first is by definition and the second is by defining I↑ α(ρAB||σA) := minτB Dα(ρAB||σA ⊗ τB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The reason this definition suffices in the second step is be- cause minimizing over τBX = ∑x qx |x⟩⟨x| ⊗ τx B is equiv- alent to minimizing the set {τx B}x and distribution qx independently, so for each x we can move the choice of minimizing τx B in front of the relative entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Next, we define rx := px exp � α−1 α I↑ α(ρx AB||ρA) � for every x ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This means rα xp−α x = exp � (α − 1)I↑ α(A : B)ρx AB � for each x ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus we can plug this substitution back in to obtain: I↑ α(A : BX)ρ = 1 α − 1 min q∈P(X) log � ∑ x pα xq1−α x rα xp−α x � = 1 α − 1 min q∈P(X) log � ∑ x q1−α x rα x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The last optimization problem is a straightforward min- imization over a simplex and thus can be solved using KKT conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' One could skip over this, but just to be complete, it is provided at the end of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It finds the optimizer is qx = rx/(∑x rx) for all x, so we will plug this in to get the answer: I↑ α(A : BX)ρ = 1 α − 1 log � ∑ x q1−α x rα x � = 1 α − 1 log � �∑ x rα x � ∑ x′ rx′/rx �α−1� � = 1 α − 1 log � �∑ x rx � ∑ x′ rx′ �α−1� � , where the second equality is by definition of qx and the third is because rα xr1−α x = rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Continuing onwards, = α α − 1 log � � � ∑ x rx �1/α � (∑ x′ rx′) �1−1/α� � = α α − 1 log � ∑ x rx � , where the first line we have multiplied and divided by α and then pulled the 1/α factor into the logarithm and distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Plugging in the definition of rx completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (Solving the KKT Criteria) We can move the log out so we are optimizing min � ∑ x q1−α x rα x : ∑ x qx − 1 = 0 , −qx ≤ 0 ∀x ∈ X � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Denote the objective function f (q), the equality con- straint h(q), and the inequality constraints gi(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note ∂ ∂qx f (q) = (1 − α)(rx/qx)α, ∂ ∂qx h(q) = 1, and 30 ∂ ∂qx gi(q) = −δx,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus the Lagrangian constraint is (1 − α) ∑x(rx/qx)αex + λ ∑x ex − ∑x µxex = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Since this is effectively entry-wise, this means (α − 1)(rx/qx)α + µx = λ for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note if there is any x such that qx = 0, then this would make λ = ∞, but to be primal feasible there must exists qx ∈ (0, 1] which would make λ also finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This is a contradiction, therefore we can conclude qx > 0 for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If qx > 0 for all x, then complementary slackness requires µx = 0 for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This simplifies the Lagrange constraint so that by moving things around we conclude λ = (α − 1)(rx/qx)α ⇒ qx = rx(α − 1)1/α/λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then by primal feasibility condition ∑ x qx = 1 ⇒ λ = (α − 1)1/α ∑ x rx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Finally, plugging this value of λ into qx, we have qx = rx/(∑x rx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note that the objective function is linear as α is fixed as are the constraints so we have linear constraint qualifications which tells us this is indeed a minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note what the above shows us is that only I↑ α can be expressed as a mixture of the same information measure over the conditional states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This gives us the following nice corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρABX ∈ D≤(ABX) such that it is classical on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then I↑ max(A : BX)ρ = log � ∑ x px exp � I↑ max(ρx AB||ρA) �� , where I↑ max(ρx AB||ρA) := min τB∈D(B) Dmax(ρx AB||ρA ⊗ τB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By the previous proposition, we know this to hold for α ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Since I↑ max := limα→∞ I↑ α, we just need to take the limit on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This just means we need to use the product law of limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By L’Hopital’s rule, α/(α − 1) and (α − 1)/α both go to one and as limα→∞ Dα = Dmax, I↑ α is Imax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also show explicitly that I↑↑ max and I↓ max won’t sat- isfy the property we need for applying the generalized support lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAX ∈ Pos(AX) be classical on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then we have I↑↑ max(A : X) = max x Dmax(ρx A||ρA) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We can just focus on ρAX ∈ D(AX) by nor- malization property of Dmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' So we can write ρAX = ∑x p(x) |x⟩⟨x| ⊗ ρx A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then we have I↑↑ max(A : X)ρ =Dmax(ρAX||ρA ⊗ ρX) = min{λ : ∑ x p(x) |x⟩⟨x| ⊗ ρx A ≤ exp(λ)∑ x (|x⟩⟨x| ⊗ ρA)} = min{λ : ρx A ≤ exp(λ)ρA} = max x Dmax(ρx A||ρA) , where the second equality is the definition of Dmax and expanding the states, Proposition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAX ∈ Pos(AX) be classical on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then I↓ max(A : X)ρAX = min q∈P(X ) τA∈D(A) log � p(x) q(x) ∥τ−1/2 A ρx Aτ−1/2 A ∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By definition, I↓ max(A : X)ρAX = min τA∈D(A) σX∈D(X) Dmax(ρAX||τA ⊗ σX) = log min q∈P(X ) τA∈D(A) max x p(x) q(x) ∥τ−1/2 A ρx Aτ−1/2 A ∥∞ , where we have used that we can restrict σX to clas- sical states by Proposition 17 and that Dmax(ρ||σ) = log ∥σ−1/2ρσ−1/2∥ to simplify in the second line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We also show that Iε h(A : BX) may be written as an expectation like D↑ max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We will use the following well- known lemma [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' For Dε h(ρ||σ), without loss of generality the optimizer 0 ≤ Λ⋆ ≤ 1 satisfies the constraint with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' That is, Tr[Λ⋆ρ] = 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρABX ∈ D(A ⊗ B ⊗ X) be classical on the X register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then Iε h(A : BX)ρ = − log � ∑ x p(x)Dε h(ρx AB||ρA ⊗ ρx B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By definition and using the preceding lemma, exp �−Iε h(A : BX)ρ � := inf 0≤Λ≤1{Tr[Λ(ρA ⊗ ρBX)] : Tr[ΛρABX] = 1 − ε} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Next, since ρA ⊗ ρBX and ρABX are both invariant under dephasing onto the X register, without loss of generality the optimizer must be of the form Λ = ∑x |x⟩⟨x| ⊗ Λx where 0 ≤ Λx ≤ 1AB for all x ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Also note that we can decompose ρABX = ∑x p(x) |x⟩⟨x| ⊗ ρx AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, exp �−Iε h(A : BX)ρ � = inf {Λx}x � ∑ x p(x) Tr[Λx ABρA ⊗ ρx B] : ∑ x p(x) Tr[Λx ABρx AB] = 1 − ε} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Now if Tr � Λx ABρx AB � ̸= 1 − ε for any x ∈ X , the constraint will not be satisfied, so every Tr � Λxρx AB � = 1 − ε for each 31 x ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Denote Λx,⋆ := argmin(Dε h(ρx AB||ρA ⊗ ρx B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If the constraint is satisfied but there is an x such that Λx ̸= Λx,⋆, then, as Iε h is a minimization, we could improve the objective function by replacing Λx with Λx,⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, the optimal is choosing Λx = Λx,⋆ for all x ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By definition of the Λx,⋆’s, this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Smoothed Max Common Information Lemmas We first establish the alternative smoothed max com- mon information is always optimized by a classical auxiliary random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' min �ρ∈QMC(ρ) Iε max(AC : B)�ρ = min A−X−C Iε max(AC : X)�ρ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This follows by the same argument as Lemma 5, which we briefly explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let (ρA−B−C, �ρABC, σB) be the optimizers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' �Cε max(A : C)ρ = Dmax(�ρABC||�ρAC ⊗ σB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let E : B → BX be as defined in Lemma 5 with respect to ρA−B−C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then by data-processing of Dmax, �Cε max(A : C)ρ ≥ I↑ max(AC : X)(TrB ◦E)(�ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' However, as explained in Lemma 5, (TrB ◦E)(ρA−B−C) is a Markov Chain extension of ρAC, ρA−X−C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As purified distance decreases under quantum channels, (TrB ◦E)(�ρ) ∈ Bε(ρA−X−C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Thus this was a minimizer to begin with, which shows we can restrict to a classical Markov chain extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Here we establish that Cε max is always optimized by a normalized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The proof is effectively identical to that of [28, Lemma 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We make use of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ([28, Lemma 21]) Let ρAB ∈ D(A ⊗ B) and ε ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If �ρ ∈ Bε(ρ), then �ρ/ Tr(�ρ) ∈ Bε(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C) and ε ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then there exists normalized quantum Markov chain state �ρA−X−C such that �ρAC ∈ Bε(ρ) such that Cε max(A : C)ρ = I↑ max(AC : X)�ρA−X−C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let �ρAC ∈ Bε(ρ) with QMC extension �ρA−X−C be the optimizer of Cε max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then there exists σX ∈ D(X) such that exp � Cε max(A : C)ρ ��ρAC ⊗ σX ≥ �ρA−X−C ⇒ exp � Cε max(A : C)ρ � �ρAC Tr(�ρAC) ⊗ σX ≥ �ρA−X−C Tr(�ρAC) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows Cε max(A : C)ρ ≥ Dmax � �ρAC Tr(�ρAC) ⊗ σX|| �ρA−X−C Tr(�ρAC) � , but as this state is contained in the optimization by Lemma 26 and Cε max minimizes over states, this renor- malized state would be the optimizer so long as it were a QMC extension of �ρAC/ Tr(�ρAC), which we will now show it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By definition of QMC extensions on subnormalized states we know Tr(�ρAC) = Tr(�ρX) so �ρA−X−C/ Tr(�ρAC) = (R ◦ R)(�ρX)/ Tr(�ρX) ∈ D(A ⊗ X ⊗ C) and is a QMC extension of �ρAC by the recoverability characterization of QMCs in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ACHIEVABILITY PROOF FOR SMOOTH MAX COMMON INFORMATION AEP In the section we establish the achievability for the SMCI AEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To avoid restating many standard results about strong conditional typicality, we will refer to the relevant results in [30] when needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' If ρAC ̸∈ Sep(A : C), this is trivial as C(A : C) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We therefore assume ρAC ∈ Sep(A : C) for the rest of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' At a high level, the proof is as follows: we bound Cε max in terms of one shot entropies evaluated on any MC extension of the smoothed initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We then construct a MC extension for the n−fold case that for sufficiently large n guarantees its marginal can be arbitrarily close to ρ⊗n AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Lastly, we bound the one-shot entropies of this specific MC extension using strong conditional typicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Taking the appropriate limits then completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We begin by bounding Cε max(A : C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Cε max(A : C)ρ = min �ρ∈Bε(ρ) min A−X−C Imax(AC : X)�ρA−X−C ≤ min �ρ∈Bε(ρ) min A−X−C � HR(AC)�ρ − Hmin(AC|X)�ρ � (36) Where the inequality follows from [27, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='11] and HR(A)ρ := − log sup{γ ∈ R : γΠρA ≤ ρA} , (37) where ΠρA is the projector onto the support of ρA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' In particular, this means exp �−HR(A)ρ � = λmin(ρA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Next we construct a Markov chain distribution whose marginal is contained in Bε(ρ⊗n AC) for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρA−X−C be any Markov chain extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows it is of the form ∑x∈X p(x) |x⟩⟨x| ⊗ ρx A ⊗ ρx C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then we consider τn AnXnCn := Pr � xn ∈ Tδ Xn �−1 ∑ xn∈Tδ xn p(xn) |xn⟩⟨xn| ⊗ τxn An ⊗ τxn Cn , where Tδ Xn is the strongly typical set for ρX, τxn An := Tr � ρxn AnΠδ An|xn �−1 Πδ An|xnρxn AnΠδ An|xn where Πδ An|xn is the projector onto the strong conditionally typical subspace, and similarly for τxn Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This is a Markov chain by its algebraic structure, so we just need to verify its purified distance can be made arbitrarily small as n grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We will instead just use trace norm as purified distance goes to zero as trace norm does by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='���τn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='�A �X �C − ρ⊗n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='A−X−C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='����� Pr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='xn ∈ Tδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='�−1 ∑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='xn∈Tδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='p(xn) |xn⟩⟨xn| ⊗ τxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ τxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='− ∑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='xn∈X n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='p(xn) |xn⟩⟨xn| ⊗ ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='����� ∑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='xn∈Tδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Pr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='xn ∈ Tδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='p(xn) − p(xn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='|xn⟩⟨xn| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='⊗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='τxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ τxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn − ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='+ Pr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='xn ̸∈ Tδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='= ∑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='xn∈Tδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Pr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='xn ∈ Tδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='p(xn) − p(xn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ���τxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ τxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn − ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='1 + Pr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='xn ̸∈ Tδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='where everything so far is just expanding definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We now bound the trace norm for a single xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To simplify notation, let Π := Πδ An|xn ⊗ Πδ Cn|xn and Π⊥ := (IAnCn − Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='���τxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ τxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn − ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='���τxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ τxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn − Πρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='CnΠ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn − Πρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='CnΠ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='���τxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ τxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn − Πρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='CnΠ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='���ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn − Πρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An ⊗ ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='CnΠ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='AnΠδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An|xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='CnΠδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn|xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='1 − Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='AnΠδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='An|xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='ρxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='CnΠδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='Cn|xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='2ε′ − ε′2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='(1 − ε′)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='+ 2ε′ − ε′2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='≤2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='ε′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='(1 − ε′)2 + 2ε′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='≤4ε′ + 2ε′ = 6ε′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='where the first inequality is the triangle inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' the following equality is using the definition of τxn An ⊗ τxn Cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' the second inequality is for sufficiently large n using the ‘unit probability’ property of conditionally typical state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' and we have assumed that ε′ ≤ 1/2 so that ε′/(1 − ε′)2 ≤ 2ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Now we plug this back into our equation to get ���τn �A �X �C − ρ⊗n A−X−C ��� 1 =6ε′ · ∑ xn∈Tδ Xn (Pr � xn ∈ Tδ Xn �−1 p(xn) − p(xn)) + Pr � xn ̸∈ Tδ Xn � ≤6ε′ · � Pr � xn ∈ Tδ Xn �−1 Pr � xn ∈ Tδ Xn � − Pr � xn ∈ Tδ Xn �� + Pr � xn ̸∈ Tδ Xn � ≤6ε′ · (1 − (1 − ε′)) + ε′ = 6ε′2 + ε′ ≤ 4ε′ , where we again use that n must be sufficiently large and in the final inequality we have used that we already assumed ε′ ≤ 1/2, so 6ε′2 ≤ 3ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows for sufficiently large n, for any ε ∈ (0, 1), you can pick a ε′ small enough that τn ∈ Bε(ρ⊗n A−X−C) as purified distance is upper bounded by � 2∥τn − ρ⊗n∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, for sufficiently large n, using (36), Cε max(An : Cn)ρ ≤ HR(AC)τn − Hmin(AC|X)τn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (38) Now we just need to bound these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We start with the Hmin term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First, by properties of strong typicality for a classical system [30, Section 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='2], the strongly typical sequences satisfy p(xn) ≤ 2−n(H(X)−cδ) and |Tδ Xn| ≤ 2n(H(X)+cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Second, the min-entropy of a conditional state ρxn An is given by Hmin(An)ρxn An = − log ∥ρxn An∥∞ and recall ∥ · ∥∞ = λmax(·) for a positive semidefinite opera- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' So we want to bound this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note, using properties of strong conditional quantum typicality [30, Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='4], τxn An = Tr � ρxn AnΠδ An|xn �−1 Πδ An|xnρxn AnΠδ An|xn ≤(1 − ε)−12−n(H(A|X)−δ′′)Πδ An|xn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It follows ∥τxn An∥∞ ≤ (1 − ε)−12−n(H(A|X)−δ′′) , where we have used Πδ An|xn is a projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' and by an identical argument one can bound ∥τCn |xn∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Combining these points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' we have − Hmin(AC|X) = log � � ∑ x∈Tδ Xn p(xn) exp � −Hmin(ρAC |xn ) � � � = log � ∑ x∈Tδ Xn p(xn) exp � −Hmin(ρA |xn) � exp � −Hmin(ρC |xn) �� = log � � ∑ x∈Tδ Xn p(xn)∥ρA |xn∥∞∥ρC |xn∥∞ � � ≤ log � 2−n(H(X)−cδ)2n(H(X)+cδ)� + log � (1 − ε)−12−n(H(A|X)−δ′′)� + log � (1 − ε)−12−n(H(C|X)−δ′′)� = − n [H(A|X) + H(C|X)] + 2(cδ + δ′′) − 2 log(1 − ε) = − nH(AC|X) + 2n(cδ + δ′′) − 2 log(1 − ε) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (39) 33 where the first equality is from is the expansion of min-entropy conditioned a classical register [31] and the second equality is additivity of min-entropy over tensor products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Next we bound HR(AC)τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' First note that τn AnCn =∑ xn �p(xn)τAn |xn ⊗ τCn |xn ≥(1 − ε)−3 ∑ xn p(xn)2−n(H(A|X)+δ′′)2−n(H(C|X)+δ′′) Πδ An|xn ⊗ Πδ Cn|xn ≥(1 − ε)−32−n(H(X)+δ)2−nH(AC|X)2−2nδ′′ ∑ xn∈Tδ Xn Πδ An|xn ⊗ Πδ Cn|xn =(1 − ε)−32−nH(AC)2−n(2δ′′−δ) ∑ xn∈Tδ Xn Πδ An|xn ⊗ Πδ Cn|xn ≥(1 − ε)−32−nH(AC)2−n(2δ′′−δ)Πδ An|xn ⊗ Πδ Cn|xn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' where we have just used strong conditional typicality properties again and at the end we have just picked an arbitrary xn and its conditional state which is dominated by itself and thus the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It is an immediate conse- quence that λmin(τn AnCn) ≥ (1 − ε)−32−nH(AC)2−n(2δ′′−δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' (40) Note by definition, (37), exp(−HR(AC)) = λmin(ρAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, using (40) HR(AC) ≤ − log � (1 − ε)−32−nH(AC)2−n(2δ′′−δ)� =nH(AC)ρ + n(2δ′′ − δ) + 3 log(1 − ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We can plug this bound into the RHS of (38) along with (39) to obtain Cε max(An : Cn)ρ ≤n � H(AC)ρ − H(AC|X)ρ � + n � 4δ′′ + (2c − 1)δ) � + 1 log(1 − ε) =nI(AC : X)ρ + n � 4δ′′ + (2c − 1)δ) � + 1 log(1 − ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note this holds for any choice of Markov Chain exten- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, picking ρ⋆ as the Markov Chain that optimizes C(A : C)ρ and dividing by n, we have 1 nCε max(An : Cn) ≤C(A : C) + � 4δ′′ + (2c − 1)δ) � + 1 n log(1 − ε) , so letting n → ∞ and δ → 0 completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' CHAIN RULES FOR SMOOTH MAX COMMON INFORMATION In this section we establish lemmas for how to reduce the asymptotic behaviour of SMCI to properties of the conditional min-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To do so, we first show how to generalize results of [27], [28] to get a strong AEP for I↑,ε max, and then just note how to modify these proofs to establish chain rules for SMCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We begin by proving the lower bounds in detail as this requires the most alteration from the previous proof [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' We first refine the basic trick for a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1) and ρ ∈ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then there exists 0 ≤ Π ≤ 1 such that [Π, ρ] = 0, P(ρ, ΠρΠ) ≤ 2 � ε(1 − ε) and Hε min(A)ρ ≤ Hmin(A)ΠρΠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By [35, Lemma 18], there exists a Π as speci- fied such that Tr �(1 − Π2)ρ � ≤ 2ε satisfying the min- entropy bound given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note 2ε ≥ Tr �(1 − Π2)ρ � = 1 − Tr � Π2ρ � which gives us Tr � Π2ρ � ≥ 1 − 2ε, so Tr � Π2ρ �2 ≥ (1 − 2ε)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Finally this means � 1 − Tr(Π2ρ)2 ≤ � 1 − (1 − 2ε)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By [27, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='7], we have P(ρ, ΠρΠ) ≤ Tr(ρ)−1/2 � Tr(ρ)2 − Tr(Π2ρ)2 = � 1 − Tr(Π2ρ)2 ≤ � 1 − (1 − 2ε)2 = 2 � ε(1 − ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Now we can use this refinement to extend the result of [28] to the full parameter range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ([28, Lemma 6, Extended to Full Param- eter Range]) Let ρAB ∈ D(A ⊗ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let 0 < ε < ε < 1 such that such that ε + δ(ε) ∈ (0, 1) where δ(ε) := 2 � ε(1 − ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then it holds I↑,ε max(A : B)ρ ≥ Hε−ε min(A)ρ − Hε+δ(ε) min (A|B)ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The proof is largely identical to the original except that we have refined certain steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' By re-arranging [27, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='13] and maximizing over the smoothing ball, we have Hε+δ((ε) min (A|B)ρ ≥ max �ρ∈Bε+δ(ε)(ρ) � Hmin(A)�ρ − Imax(A : B)�ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, Hε+δ((ε) min (A|B)ρ ≥ max �ρ∈Bε+δ(ε)(ρ) � Hmin(A)�ρ − Imax(A : B)�ρ � ≥ max ω∈Bε(ρ) � max Π [Hmin(A)ΠωΠ − Imax(A : B)ΠωΠ] � , where the new maximization is over 0 ≤ ΠA ≤ 1A such that ΠωΠ ≈δ(ε) ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This is a lower bound because we restricted smoothing to Bε, so ω ≈ε ρ which using purified distance is a metric implies ΠωΠ ≈ε+δ(ε) ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ω⋆ ∈ Bε(ρ) ∩ D(A ⊗ B) be the optimizer of Iε max(A : B)ρ which is normalized without loss of generality [28, Lemma 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, Hε+δ(ε) min (A|B)ρ ≥ max Π [Hmin(A)Πω⋆Π − Imax(A : B)Πω⋆Π] ≥ max Π [Hmin(A)Πω⋆Π] − Imax(A : B)ω⋆ , 34 where the first we have chosen ω⋆ rather than maxi- mizing and the second we have used that Dmax actually satisfies data-processing for CPTNI maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then as ω⋆ is normalized and we range over Π such that Πω⋆Π ≈δ(ε) ω⋆, we know by Proposition 24 we can bound the state Hε+δ(ε) min (A|B)ρ ≥ Hε min(A)ω⋆ − Imax(A : B)ω⋆ ≥ Hε−ε min(A)ρ − I↑,ε max(A : B)ρ , where the first line is using ω⋆ was the optimizer for I↑,ε max(A : B)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The second line is because if �ρ ∈ Bε−ε(ρ), as ω ≈ε ρ, we can conclude �ρ ≈ε ω and thus is included in the previous line’s optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As smooth min-entropy is maximized, this suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Re-ordering the terms completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' As a corollary, we have the following lower bound on SMCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let 0 < ε < ε < 1 such that ε + δ(ε) where δ(ε) := 2 � ε(1 − ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then it holds, Cε max(A : C)ρ ≥Hε−ε min(AC)ρ − max �ρ∈Bε+δ(ε)(ρ) max A−X−C Hmin(AC|X) , Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The proof is effectively the same as the previous proposition except one must keep track of the min- imiziation over Markov chain extensions along with the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' One will need to minimize over the Markov chain extension under the projection Π and maintain the Markov chain property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' consider a min(ΠωΠ)A−B−C Imax(AC : B)ΠωΠ term where we still de- mand ΠωΠ ≈δ(ε) ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note if the feasible set is empty, then this term is infinite and so, as we subtract it, in this setting the lower bound trivially holds, so the proof will go in both when such a Π exists or does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The upper bound chain rule for Iε max is already suffi- cient, so we just state it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' ([27, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content='12]) Let ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, I↑,ε max(A : B)ρ ≤ Hε2/48 max (A)ρ − Hε2/48 min (A|B)ρ − 2 log � ε2/24 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1) and ρAC ∈ D(A ⊗ C), then Cε max(A : C)ρ ≤ Hε2/48 max (AC)ρ − max A−X−C Hε2/48 min (AC|X)ρ − 2 log � ε2/24 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The proof is the same as the previous proposi- tion except you push the minimization over Markov extensions through and note the minus sign flips the minimization into a maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' It is clear that the chain rules for I↑,ε max can then be used to establish a strong AEP for I↑,ε max as we quickly show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1) and ρ ∈ D(A ⊗ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then, lim n→∞ � 1 n Iε max(An : Bn)ρ⊗n � = I(A : C)ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Fix ε, δ(ε) that satisfy Proposition 25 to get Hε−ε min(A⊗n)ρ⊗n − Hε+δ(ε) min (An|Bn)ρ⊗n ≤ I↑,ε max(An : Bn)ρ⊗n ≤ Hε2/48 max (An)ρ⊗n − Hε2/48 min (An|Bn)ρ⊗n − 2 log � ε2/24 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Dividing by n and taking the limit as n → ∞, using the AEP for smooth min and max-entropies [31], H(A)ρ − H(A|B)ρ ≤ lim n→∞ � I↑,ε max(An : Bn)ρ⊗n � ≤H(A)ρ − H(A|B)ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Noting that H(A) − H(A|B) = I(A : B) by standard chain rules completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' This general proof method then gets us the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Proposition 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ρAC ∈ D(A ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Let ε ∈ (0, 1), δ ∈ (0, ε), ε ∈ (ε, 1) such that 0 < ε + δ((ε)) < 1 where δ(ε) := 2 � ε(1 − ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Then H(AC)− lim n→∞ � 1 n max �ρ∈Bε+δ(ε)(ρ⊗n) max An−X−Cn Hmin(AnCn|X) � ≤ lim n→∞ � 1 nCε max(An : Cn)ρ⊗n � ≤H(AC) − lim n→∞ � max An−X−Cn 1 n Hε−δ min(AnCn|X)ρ � , Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' The proof method is the same as the previous proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Note that we don’t particularly care about the upper bound as our achievability result held for all ε ∈ (0, 1) (Lemma 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' Therefore, our interest is in if the lower bound in the previous proposition can upper bound the common information in the regularized limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} +page_content=' To get Proposition 11, note that by a standard chain rule, C(A : C)ρ = min A−X−C I(AC : X)ρ =H(AC) − max A−X−C H(AC|X)ρ , so to acquire the asymptotic bound, it would suffice that H(AC) − max A−X−C H(AC|X)ρ ≤H(AC) − lim n→∞ � 1 n max �ρ∈Bε+δ(ε)(ρ⊗n) max An−X−Cn Hmin(AnCn|X) � , which by cancelling and multiplying by negative one gets us the term in Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQfFAlo/content/2301.04301v1.pdf'} diff --git a/XdE2T4oBgHgl3EQfugjd/content/tmp_files/2301.04082v1.pdf.txt b/XdE2T4oBgHgl3EQfugjd/content/tmp_files/2301.04082v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f2d443435316781632be1225fe281b77f39cde3 --- /dev/null +++ b/XdE2T4oBgHgl3EQfugjd/content/tmp_files/2301.04082v1.pdf.txt @@ -0,0 +1,962 @@ +arXiv:2301.04082v1 [quant-ph] 10 Jan 2023 +A novel way of calculating scattering integrals +Alfredo Takashi Suzuki +Department of Physics, La Sierra University +4500 Riverwalk Pkw., Riverside, CA 92505 +Timothy Suzuki +Department of Physics and Astronomy, Michigan State University +567 Wilson Rd., East Lansing, MI 48824 +Abstract +The technique coined as NDIM - Negative Dimensional Integration Method by their discoverers, +relies on a three-pronged basis: Gaussian integration, series expansion and analytic continuation. +The technique has been successfully applied to the calculation of covariant and non covariant +Feynman integrals in a generic dimensional regularization space, i.e., D-dimensional space-time for +D including the negative domain values. Since the dimensionality is general, we can use specifically +for one-dimensional integrals. In this work we show how this technique can be applied to tackle +certain improper integrals and give an example of a particular improper integral that appears +in quantum mechanical scattering process. Traditionally, improper integrals are ascribed certain +values through the limiting approach or as is known, by the Cauchy principal value via residues +concept technique. Here we use the NDIM approach to do the calculations and show it works fine +for the improper integrals. This novel approach we believe is more straightforward and does not +require to handle poles, residues, or difficult closed contours as in the traditional approach. +PACS numbers: +Keywords: integration techniques +1 + +I. +INTRODUCTION +NDIM (Negative Dimensional Integration Method) [1] was born within the context of +perturbative quantum field theories where Feynman diagrams that give a pictorial view of +physical processes are calculated. When the perturbative series entails loop calculations, +these often lead to diverging Feynman integrals, for which a regularization procedure is +called for. +Among several regularization procedures, dimensional regularization [2] is a +popular choice, where a four-dimensional integration measure is analytically extended to +a D-dimensional one. In essence, what NDIM does is to allow this analytic continuation +to span even into the negative dimensionality region. With the help of a D-dimensional +Gaussian integration and series expansion, an integrand with poles are transformed into +polynomials that are integrated in negative dimensions. After the polynomial integration in +negative dimensions is carried out, analytic continue the result back into positive dimensions. +[3]. +This D-dimensional NDIM procedure can be recast into a one-dimensional integration +measure without difficulty [4, 5]. In our previous works, we considered integrands with pure +imaginary poles, so the calculation entails no closed circuit with real poles standing on the +integration path. In the present work we consider improper integrals with real poles, and +show that even for this case NDIM can be applied successfully. +II. +IMPROPER INTEGRALS +Let us consider the following improper integral: +I = ++∞ +� +−∞ +dx +(x2 − a2) = ++∞ +� +−∞ +dx +(x − a) (x + a) ; +a ∈ R. +(1) +Since the integrand is not well-defined (diverges) at x = ±a, we may try to ascribe a +value to this integral by the limiting procedure as follows: +I = +1 +2a + + + lim +ǫ→0+; +R→∞ + + +a−ǫ +� +−R +dx +(x − a) + ++R +� +a+ǫ +dx +(x − a) + + ++ lim +ǫ′→0+; +R→∞ + + +−a−ǫ′ +� +−R +dx +(x + a) + ++R +� +−a+ǫ′ +dx +(x + a) + + + + + . +(2) +2 + +We can ascribe a principal value (PV) to it in the sense of Cauchy by using the same +infinitesimal, that is, ǫ = ǫ′ → 0+, in the limiting analysis +IPV = +1 +2a + + + lim +ǫ→0+; +R→∞ + + +a−ǫ +� +−R +dx +(x − a) + ++R +� +a+ǫ +dx +(x − a) + + ++ lim +ǫ→0+; +R→∞ + + +−a−ǫ +� +−R +dx +(x + a) + ++R +� +−a+ǫ +dx +(x + a) + + + + + = 0. +(3) +We could also use the closed contour integration analytically continuing the integration +variable into complex variables, x → z, +IPV = +� +dz +(z2 − a2) = +� +dz +(z − a) (z + a) ; +z ∈ C; a ∈ R. +(4) +Here, because we have two poles on the real axis, x = ±a, we need to distort the contour +around them by a semi-circle of infinitesimal radius (r → 0+) in order to avoid the poles. We +can do this in several manners, closing the contour either on the upper or lower hemispheres: +1. Excluding both poles; +2. Including both poles; +3. Excluding the pole x = −a and including the pole x = a; +4. Including the pole x = −a and excluding the pole x = a. +All theses choices are totally equivalent, so, if we choose, for instance, the following +contour, excluding both poles from the closed upper hemisphere circuit: +Rez +Imz +−R ++R +CR +CR +C− +r +C+ +r +⋆ +−a +⋆a +Figure 1 +Cauchy residue theorem yields this closed loop integration as, with f(z) ≡ (z2 − a2)−1 = +[(z − a)(z + a)]−1 +IPV = +� +dz +(z2 − a2) = 2iπ +� +Residuesf(z) = 0, +(5) +since the chosen closed contour does not contain any poles. +3 + +The contour integration, on the other hand, can be broken down into +IPV = +� +f(z)dz += +� +CR +f(z)dz + +−a−r +� +−R +f(z)dz + +� +C− +r +f(z)dz ++ +a−r +� +−a+r +f(z)dz + +� +C+ +r +f(z)dz + ++R +� +a+r +f(z)dz +(6) +Let us look at in detail these integrals. First, let us consider +CR ≡ +� +CR +f(z)dz = +� +CR +dz +(z − a)(z + a). +(7) +For this integral, along the upper hemisphere counterclockwise semi-circle, we have +z = Reiθ , +0 ≤ θ ≤ π; +dz = iReiθdθ. +(8) +Therefore, +CR = +� π +0 +iReiθdθ +(Reiθ − a)(Reiθ + a). +(9) +For the relevant limit R → ∞, the integrand’s leading term is +lim +R→∞ CR ≈ +i +R +� π +0 +e−iθdθ → 0. +(10) +Next, let us examine +C− +r ≡ +� +C− +r +f(z)dz = +� +C− +r +dz +(z − a)(z + a). +(11) +For this integral, along the upper hemisphere clockwise semi-circle around x = −a, we +have +z = −a + reiϕ , +π ≤ ϕ ≤ 0; +dz = ireiϕdϕ. +(12) +Therefore, +C− +r += +� 0 +π +ireiϕdϕ +(−2a + reiϕ)(reiϕ) += i +� 0 +π +dϕ +(−2a + reiϕ). +(13) +4 + +Here, for the relevant limit r → 0+, the integrand’s leading term is +lim +r→0+ C− +r +≈ − i +2a +� 0 +π +dϕ → iπ +2a. +(14) +Now, for the +C+ +r ≡ +� +C+ +r +f(z)dz = +� +C+ +r +dz +(z − a)(z + a). +(15) +For this integral, along the upper hemisphere clockwise semi-circle around x = a, we have +z = a + reiϕ , +π ≤ ϕ ≤ 0; +dz = ireiϕdϕ. +(16) +Therefore, +C+ +r += +� 0 +π +ireiϕdϕ +(reiϕ)(2a + reiϕ) += i +� 0 +π +dϕ +(2a + reiϕ). +(17) +Here, for the relevant limit r → 0+, the integrand’s leading term is +lim +r→0+ C+ +r +≈ +i +2a +� 0 +π +dϕ → −iπ +2a . +(18) +Finally, plugging the results for CR, C− +r and C+ +r into equation (6) with the relevant limits +taken, R → ∞; r → 0+, we have +IPV = +� +f(z)dz += 0 + +−a +� +−∞ +f(x)dx + iπ +2a + +a +� +−a +f(x)dx − iπ +2a + ++∞ +� +a +f(x)dx += ++∞ +� +−∞ +f(x)dx ≡ ++∞ +� +−∞ +dx +(x2 − a2). +(19) +Comparing with equation (5), we get, in accordance with equation (3) ++∞ +� +−∞ +dx +(x2 − a2) = 0. +(20) +Since the integral in equation (1) is an improper one, we can in principle ascribe other +values to it, other than the PV value just calculated. It is possible to consider the original +integral, equation (1), with a ∈ C, by letting the real pole be dislocated by an infinitesimal +imaginary part, i.e., either a+ = a + iε or a− = a − iε. +5 + +A. +Improper integral for a+ shift +Let us consider now the following integral, +I+ = lim +ε→0+ I+(ε), +(21) +where +I+(ε) ≡ ++∞ +� +−∞ +dz +(z2 − a2 ++) ; +a+ ∈ C += ++∞ +� +−∞ +dz +(z − a − iε) (z + a + iε) , +a ∈ R. +(22) +For this, we consider the following closed circuit, +Rez +Imz +−R ++R +CR +CR +⋆ +−a −iε +⋆ +a +iε +Figure 2 +Then, we have +I+(ε) ≡ +� +dz +(z2 − a2 ++) = +� +dz +(z − a − iε) (z + a + iε) += 2iπ +� +Residuesf+(z) , +f+(z) ≡ +1 +(z − a − iε) (z + a + iε) += 2iπ +� +lim +z→a+iε(z − a − iε)f+(z) +� += +iπ +a + iε. +(23) +So, remembering that the large counterclockwise contour does not contribute, i.e., CR → 0 +for R → ∞, we have +I+ = lim +ε→0+ +iπ +a + iε = iπ +a +(24) +6 + +B. +Improper integral for a− shift +For this integral, we have +I− = lim +ε→0+ I−(ε), +(25) +where +I−(ε) ≡ ++∞ +� +−∞ +dz +(z2 − a2 +−) ; +a− ∈ C += ++∞ +� +−∞ +dz +(z − a + iε) (z + a − iε) , +a ∈ R. +(26) +The same previous closed circuit now encloses the pole in the second quadrant, +Rez +Imz +−R ++R +CR +CR +⋆ +−a +iε +⋆ +a −iε +Figure 3 +Then, we have +I−(ε) ≡ +� +dz +(z2 − a2 +−) = +� +dz +(z − a + iε) (z + a − iε) += 2iπ +� +Residuesf−(z) , +f−(z) ≡ +1 +(z − a + iε) (z + a − iε) += 2iπ +� +lim +z→−a+iε(z + a − iε)f−(z) +� += +iπ +−a + iε. +(27) +So, finally, +I− = lim +ε→0+ +iπ +−a + iε = −iπ +a . +(28) +We note that, as the PV is also defined by +IPV = 1 +2 lim +ε→0+ {I+(ε) + I−(ε)} = 0. +(29) +7 + +III. +NDIM AND THE IMPROPER INTEGRALS +NDIM has already been applied for some definite integrals [4, 5]. Here we are going +to employ the NDIM to evaluate equation (1). To implement the NDIM technique for the +sought integral, let us introduce the generating functional Gaussian integral that is pertinent +to our calculation: +Ga = ++∞ +� +−∞ +dx e−λ(x2−a2). +(30) +The Gaussian integration can be performed without difficulty, yielding +Ga = eλa2 ++∞ +� +−∞ +e−λx2 += eλa2 +�π +λ. +(31) +Next, expanding the exponential function in the result above in power series we get +Ga = π1/2 +∞ +� +n=0 +a2n +n! λn−1/2. +(32) +On the other hand, expanding in power series the original Gaussian integral (30), we have +Ga = ++∞ +� +−∞ +dx +∞ +� +k=0 +(−1)k λk +k! (x2 − a2)k. +(33) +As the above integral has a polynomial integrand equivalent to the desired integral with +poles, we have the negative dimensional integration characterized by this procedure, i.e, +introducing the notation +INDIM(k) = ++∞ +� +−∞ +ˆdx +� +x2 − a2�k , +(34) +with ˆdx to remind that we are in the negative dimension measure continuation. Then, the +later (33) can be written as +Ga = +∞ +� +k=0 +(−1)k λk +k! INDIM(k). +(35) +8 + +Comparing the two series expansion for Ga, term by term, we have that n = k + 1/2 and +INDIM(k) can be obtained as +INDIM(k) = (−1)−kk!π1/2 +a2k+1 +(k + 1/2)! += (−1)−kπ1/2a2k+1 +Γ(1 + k) +Γ(1 + k + 1/2) += (−1)−kπ1/2a2k+1 +1 +(1 + k)1/2 +, +(36) +where in the last line above we have introduced the Pochhammer’s symbol +(α)β ≡ Γ(α + β) +Γ(α) +. +(37) +Since we want (34) analytically continued (AC) to allow for negative values of k (in +positive dimensional measure) we proceed by using in (36) the following Pochhammer’s +identity +(1 − α)β = (−1)β +1 +(α)−β +. +(38) +Then, +IAC +NDIM(k) = (−1)−k−1/2π1/2a2k+1(−k)−1/2, +(39) +In (34), we want k = −1 to reproduce the original integral in positive dimensions, so we +are interested in +IAC +NDIM(k = −1) = (−1)1/2π1/2a−1(1)−1/2 += iπ +a = I+. +(40) +Since the original integral is, of course invariant under the symmetry a → −a, it follows +that we also have +IAC +NDIM(k = −1) = (−1)1/2π1/2(−a)−1(1)−1/2 += −iπ +a = I−. +(41) +And once again, we can calculate the PV value, +IPV = 1 +2 lim +ε→0+ {I+(ε) + I−(ε)} = 0. +(42) +9 + +IV. +NDIM IN QUANTUM MECHANICS SCATTERING CALCULATION +In quantum mechanics scattering problems, there is an improper integral of the following +form to be calculated: +S(σ) = ++∞ +� +−∞ +dx x sin x +(x2 − σ2), +σ ∈ R +(43) +In order to perform this integration using NDIM, we first need to express it in terms of +integrands that would be fitting for applying the technique. So the ‘road preparation’ for it +is done by considering the integral +E(a, σ) = ++∞ +� +−∞ +dx +eiax +(x2 − σ2) , +a ∈ R. +(44) += +∞ +� +m=0 +(ia)m +m! Im(σ), +where we have defined +Im(σ) = ++∞ +� +−∞ +dx +xm +(x2 − σ2). +(45) +We introduce then the relevant generating functional Gaussian integration as follows +G(α, β) = ++∞ +� +−∞ +dxeαx−β(x2−σ2) +(46) += e +α2 +4β +βσ2�π +β += +�π +β +∞ +� +k,l=0 +(βσ2)k +k! +α2l +4lβll!. +(47) +On the other hand, series expansion of integrand in (46) gives +G(α, β) = +∞ +� +r,s=0 +(−1)sαrβs +r!s! ++∞ +� +−∞ +dx xr � +x2 − σ2�s . +(48) +Comparing (47) and (48) we have that +∞ +� +r,s=0 +(−1)sαrβs +r!s! ++∞ +� +−∞ +dx xr � +x2 − σ2�s = π1/2 +∞ +� +k,l=0 +σ2k +4l +βk−l−1/2 +k! +α2l +l! . +(49) +10 + +The term by term equality entails +r = 2l +s = k − l − 1/2. +(50) +Then, we have that k = s + r/2 + 1/2 and l = r/2, leading to ++∞ +� +−∞ +dx xr � +x2 − σ2�s = (−1)−sr!s!π1/2β2s+r+1 +4r/2 +1 +(s + r/2 + 1/2)! +1 +(r/2)! += (−1)−sπ1/2σr+2s+1 +4r/2 +(1 + r/2)r/2 +(1 + s)r/2+1/2 +. +(51) +Note that the integral +INDIM(r, s) ≡ ++∞ +� +−∞ +dx xr � +x2 − σ2�s , +(52) +will be equivalent to (45) when r = m and s = −1. Note also that henceforth we are going +to be omitting the reminder sign over ˆdx for the NDIM integration measure. Since we need +s to assume negative values in INDIM(r, s), we analytic continue the Pochhammer’s symbol +on the righthandside of (51) that contains s, i.e., +1 +(1 + s)r/2+1/2 +AC +−→ (−s)−r/2−1/2 +(−1)r/2+1/2 . +(53) +Then, +IAC +NDIM(r, s) = (−1)−s−r/2−1/2π1/2σr+2s+1 +4r/2 +(1 + r/2)r/2(−s)−r/2−1/2. +(54) +Comparing (45) with (52) we see that we need to have r = m and s = −1 in (52) to get +the relevant result for (45). So +IAC +NDIM(m, −1) = (−1)1/2−m/2π1/2σm−1 +4m/2 (1 + m/2)m/2(1)−m/2−1/2 += iπ1/2σm +σ +1 +(−4)m/2 +Γ(1 + m)Γ(1/2 − m/2) +Γ(1 + m/2) +. +(55) +Now, using the gamma function relation +Γ(1/2 − m/2) = π1/2(−4)m/2Γ(1 + m/2) +Γ(1 + m) . +(56) +Then +IAC +NDIM(m, −1) ≡ Im(σ) = iπσm +σ +. +(57) +11 + +Substituting this last result into (44) we get +E(a, σ) = +∞ +� +m=0 +(ia)m +m! +iπσm +σ += iπ +σ +∞ +� +m=0 +(iaσ)m +m! +. +(58) +Thus, finally: ++∞ +� +−∞ +dx +eiax +(x2 − σ2) = iπ +σ eiaσ. +(59) +Next, we need to evaluate +Ex(a, σ) = ++∞ +� +−∞ +dx +x eiax +(x2 − σ2). +(60) +We do not need to recalculate from scratch; we use the identity +x eiax = −i d +daeiax, +(61) +in the result (59). Doing this, we get +Ex(a, σ) ≡ ++∞ +� +−∞ +dx +x eiax +(x2 − σ2) = iπeiaσ. +(62) +From the result (62) we have that ++∞ +� +−∞ +dxx cos(ax) +(x2 − σ2) = 0 +(63) ++∞ +� +−∞ +dx x sin(ax) +(x2 − σ2) = π eiaσ. +(64) +Our original quantum mechanics integral S(σ) is our result (64) for a = 1, so +E+ +x (1, σ) ≡ S(σ) += ++∞ +� +−∞ +dx x sin(x) +(x2 − σ2) = π eiσ. +(65) +The above result for quantum mechanical scattering problem corresponds to an outgoing +wave. +This result is equivalent to letting σ → σ + iε in the Cauchy contour technique +12 + +calculation. Noting that the integral on the left handside is invariant under σ ↔ −σ, we +have the result for quantum mechanical scattering problem corresponding to an incoming +wave. +E− +x (1, −σ) ≡ S(−σ) += ++∞ +� +−∞ +dx x sin(x) +(x2 − σ2) = π e−iσ. +(66) +This corresponds to letting σ → σ − iε in the Cauchy residue calculation. And the PV +result for the integral is, as before, +SPV(σ) = 1 +2 +� +E+ +x (1, σ) + E− +x (1, −σ) +� += 1 +2 {S(σ) + S(−σ)} += π +�eiσ + e−iσ +2 +� += π cos σ. +(67) +Since the original integral S(±σ) is an improper integral, we can see that the value we +ascribe to it will depend on which type of physical boundary conditions we want: incoming +or outgoing waves or even the average between those boundary conditions (PV). +V. +CONCLUSIONS +We have shown that using the NDIM technique, it was possible to evaluate some improper +integrals and more specifically, the one integral that is relevant for quantum mechanical +scattering problems. We have shown how to do the calculations in the NDIM technique and +shown that the results it provides are equivalent to the results obtained using the Cauchy +residue procedure, in which poles in the real axis is given an infinitesimal shift into the +complex plane, either σ + iε or σ − iε. In the NDIM procedure, each pole residue gives +a distinct answer [4]. +In our present calculation for the quantum mechanical scattering +problem, there are two poles, and so two residues; and the PV result can be calculated by +the simple average between the two residue calculations. +[1] I.G.Halliday and R.M.Ricotta, Phys. Lett. B 193 (1987) 241-246. +[2] G.’t Hooft and M.I.G.Veltman, +Nucl. Phys. B +44 (1972) 189-213; +C.G.Bollini and +J.J.Giambiagi, Il Nuovo Cim. B 12 (1972) 20-26; J.F.Ashmore, Lett. Nuovo Cim. 4 (1972) +289-290. +13 + +[3] A. T. Suzuki, A. G. M. Schmidt, and R. Bentin, Nucl. Phys. B537, 549 (1999); A. T. Suzuki +and A. G. M. Schmidt, Prog. Theor. Phys. 103, 1011 (2000); Eur. Phys. J. C 12, 361 (2000); +A. T. Suzuki and A. G. M. Schmidt, Phys. Lett. B494, 332 (2000), A.T.Suzuki and T.Suzuki, +Phys. Rev. D 106, 085007 (2022). +[4] A.T. Suzuki, Acta Phys. Polonica B 37 (2006) 2767-2779. +[5] A.T.Suzuki, arXiv:math-ph/0407032v1 (2004); A.T.Suzuki, arXiv:0806.3216v1 (2008) +[6] D. M. Capper, J. J. Dulwich, and M. J. Litvak, Nucl. Phys. B 241(2) (1984) 463. +[7] P.A.M.Dirac, Rev.Mod.Phys. 21 (1949) 392. +[8] Table of Integrals, Series and Products, I.S.Gradshteyn and I.M.Ryzhik, 8th Edition, Edited +by Daniel Zwillinger, Academic Press, Oxford, UK (2015). +14 + diff --git a/XdE2T4oBgHgl3EQfugjd/content/tmp_files/load_file.txt b/XdE2T4oBgHgl3EQfugjd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3624b333a06da523aa2d584768c04f1056477472 --- /dev/null +++ b/XdE2T4oBgHgl3EQfugjd/content/tmp_files/load_file.txt @@ -0,0 +1,246 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf,len=245 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='04082v1 [quant-ph] 10 Jan 2023 A novel way of calculating scattering integrals Alfredo Takashi Suzuki Department of Physics, La Sierra University 4500 Riverwalk Pkw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=', Riverside, CA 92505 Timothy Suzuki Department of Physics and Astronomy, Michigan State University 567 Wilson Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=', East Lansing, MI 48824 Abstract The technique coined as NDIM - Negative Dimensional Integration Method by their discoverers, relies on a three-pronged basis: Gaussian integration, series expansion and analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' The technique has been successfully applied to the calculation of covariant and non covariant Feynman integrals in a generic dimensional regularization space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=', D-dimensional space-time for D including the negative domain values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Since the dimensionality is general, we can use specifically for one-dimensional integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' In this work we show how this technique can be applied to tackle certain improper integrals and give an example of a particular improper integral that appears in quantum mechanical scattering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Traditionally, improper integrals are ascribed certain values through the limiting approach or as is known, by the Cauchy principal value via residues concept technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Here we use the NDIM approach to do the calculations and show it works fine for the improper integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' This novel approach we believe is more straightforward and does not require to handle poles, residues, or difficult closed contours as in the traditional approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' PACS numbers: Keywords: integration techniques 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' INTRODUCTION NDIM (Negative Dimensional Integration Method) [1] was born within the context of perturbative quantum field theories where Feynman diagrams that give a pictorial view of physical processes are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' When the perturbative series entails loop calculations, these often lead to diverging Feynman integrals, for which a regularization procedure is called for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Among several regularization procedures, dimensional regularization [2] is a popular choice, where a four-dimensional integration measure is analytically extended to a D-dimensional one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' In essence, what NDIM does is to allow this analytic continuation to span even into the negative dimensionality region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' With the help of a D-dimensional Gaussian integration and series expansion, an integrand with poles are transformed into polynomials that are integrated in negative dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' After the polynomial integration in negative dimensions is carried out, analytic continue the result back into positive dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' This D-dimensional NDIM procedure can be recast into a one-dimensional integration measure without difficulty [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' In our previous works, we considered integrands with pure imaginary poles, so the calculation entails no closed circuit with real poles standing on the integration path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' In the present work we consider improper integrals with real poles, and show that even for this case NDIM can be applied successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' IMPROPER INTEGRALS Let us consider the following improper integral: I = +∞ � −∞ dx (x2 − a2) = +∞ � −∞ dx (x − a) (x + a) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (1) Since the integrand is not well-defined (diverges) at x = ±a, we may try to ascribe a value to this integral by the limiting procedure as follows: I = 1 2a \uf8f1 \uf8f2 \uf8f3 lim ǫ→0+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' R→∞ \uf8ee \uf8f0 a−ǫ � −R dx (x − a) + +R � a+ǫ dx (x − a) \uf8f9 \uf8fb + lim ǫ′→0+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' R→∞ \uf8ee \uf8f0 −a−ǫ′ � −R dx (x + a) + +R � −a+ǫ′ dx (x + a) \uf8f9 \uf8fb \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (2) 2 We can ascribe a principal value (PV) to it in the sense of Cauchy by using the same infinitesimal, that is, ǫ = ǫ′ → 0+, in the limiting analysis IPV = 1 2a \uf8f1 \uf8f2 \uf8f3 lim ǫ→0+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' R→∞ \uf8ee \uf8f0 a−ǫ � −R dx (x − a) + +R � a+ǫ dx (x − a) \uf8f9 \uf8fb + lim ǫ→0+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' R→∞ \uf8ee \uf8f0 −a−ǫ � −R dx (x + a) + +R � −a+ǫ dx (x + a) \uf8f9 \uf8fb \uf8fc \uf8fd \uf8fe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (3) We could also use the closed contour integration analytically continuing the integration variable into complex variables, x → z, IPV = � dz (z2 − a2) = � dz (z − a) (z + a) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' z ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (4) Here, because we have two poles on the real axis, x = ±a, we need to distort the contour around them by a semi-circle of infinitesimal radius (r → 0+) in order to avoid the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' We can do this in several manners, closing the contour either on the upper or lower hemispheres: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Excluding both poles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Including both poles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Excluding the pole x = −a and including the pole x = a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Including the pole x = −a and excluding the pole x = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' All theses choices are totally equivalent, so, if we choose, for instance, the following contour, excluding both poles from the closed upper hemisphere circuit: Rez Imz −R +R CR CR C− r C+ r ⋆ −a ⋆a Figure 1 Cauchy residue theorem yields this closed loop integration as, with f(z) ≡ (z2 − a2)−1 = [(z − a)(z + a)]−1 IPV = � dz (z2 − a2) = 2iπ � Residuesf(z) = 0, (5) since the chosen closed contour does not contain any poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' 3 The contour integration, on the other hand, can be broken down into IPV = � f(z)dz = � CR f(z)dz + −a−r � −R f(z)dz + � C− r f(z)dz + a−r � −a+r f(z)dz + � C+ r f(z)dz + +R � a+r f(z)dz (6) Let us look at in detail these integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' First, let us consider CR ≡ � CR f(z)dz = � CR dz (z − a)(z + a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (7) For this integral, along the upper hemisphere counterclockwise semi-circle, we have z = Reiθ , 0 ≤ θ ≤ π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' dz = iReiθdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (8) Therefore, CR = � π 0 iReiθdθ (Reiθ − a)(Reiθ + a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (9) For the relevant limit R → ∞, the integrand’s leading term is lim R→∞ CR ≈ i R � π 0 e−iθdθ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (10) Next, let us examine C− r ≡ � C− r f(z)dz = � C− r dz (z − a)(z + a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (11) For this integral, along the upper hemisphere clockwise semi-circle around x = −a, we have z = −a + reiϕ , π ≤ ϕ ≤ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' dz = ireiϕdϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (12) Therefore, C− r = � 0 π ireiϕdϕ (−2a + reiϕ)(reiϕ) = i � 0 π dϕ (−2a + reiϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (13) 4 Here, for the relevant limit r → 0+, the integrand’s leading term is lim r→0+ C− r ≈ − i 2a � 0 π dϕ → iπ 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (14) Now, for the C+ r ≡ � C+ r f(z)dz = � C+ r dz (z − a)(z + a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (15) For this integral, along the upper hemisphere clockwise semi-circle around x = a, we have z = a + reiϕ , π ≤ ϕ ≤ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' dz = ireiϕdϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (16) Therefore, C+ r = � 0 π ireiϕdϕ (reiϕ)(2a + reiϕ) = i � 0 π dϕ (2a + reiϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (17) Here, for the relevant limit r → 0+, the integrand’s leading term is lim r→0+ C+ r ≈ i 2a � 0 π dϕ → −iπ 2a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (18) Finally, plugging the results for CR, C− r and C+ r into equation (6) with the relevant limits taken, R → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' r → 0+, we have IPV = � f(z)dz = 0 + −a � −∞ f(x)dx + iπ 2a + a � −a f(x)dx − iπ 2a + +∞ � a f(x)dx = +∞ � −∞ f(x)dx ≡ +∞ � −∞ dx (x2 − a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (19) Comparing with equation (5), we get, in accordance with equation (3) +∞ � −∞ dx (x2 − a2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (20) Since the integral in equation (1) is an improper one, we can in principle ascribe other values to it, other than the PV value just calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' It is possible to consider the original integral, equation (1), with a ∈ C, by letting the real pole be dislocated by an infinitesimal imaginary part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=', either a+ = a + iε or a− = a − iε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Improper integral for a+ shift Let us consider now the following integral, I+ = lim ε→0+ I+(ε), (21) where I+(ε) ≡ +∞ � −∞ dz (z2 − a2 +) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' a+ ∈ C = +∞ � −∞ dz (z − a − iε) (z + a + iε) , a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (22) For this, we consider the following closed circuit, Rez Imz −R +R CR CR ⋆ −a −iε ⋆ a +iε Figure 2 Then, we have I+(ε) ≡ � dz (z2 − a2 +) = � dz (z − a − iε) (z + a + iε) = 2iπ � Residuesf+(z) , f+(z) ≡ 1 (z − a − iε) (z + a + iε) = 2iπ � lim z→a+iε(z − a − iε)f+(z) � = iπ a + iε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (23) So, remembering that the large counterclockwise contour does not contribute, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=', CR → 0 for R → ∞, we have I+ = lim ε→0+ iπ a + iε = iπ a (24) 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Improper integral for a− shift For this integral, we have I− = lim ε→0+ I−(ε), (25) where I−(ε) ≡ +∞ � −∞ dz (z2 − a2 −) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' a− ∈ C = +∞ � −∞ dz (z − a + iε) (z + a − iε) , a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (26) The same previous closed circuit now encloses the pole in the second quadrant, Rez Imz −R +R CR CR ⋆ −a +iε ⋆ a −iε Figure 3 Then, we have I−(ε) ≡ � dz (z2 − a2 −) = � dz (z − a + iε) (z + a − iε) = 2iπ � Residuesf−(z) , f−(z) ≡ 1 (z − a + iε) (z + a − iε) = 2iπ � lim z→−a+iε(z + a − iε)f−(z) � = iπ −a + iε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (27) So, finally, I− = lim ε→0+ iπ −a + iε = −iπ a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (28) We note that, as the PV is also defined by IPV = 1 2 lim ε→0+ {I+(ε) + I−(ε)} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (29) 7 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' NDIM AND THE IMPROPER INTEGRALS NDIM has already been applied for some definite integrals [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Here we are going to employ the NDIM to evaluate equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' To implement the NDIM technique for the sought integral, let us introduce the generating functional Gaussian integral that is pertinent to our calculation: Ga = +∞ � −∞ dx e−λ(x2−a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (30) The Gaussian integration can be performed without difficulty, yielding Ga = eλa2 +∞ � −∞ e−λx2 = eλa2 �π λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (31) Next, expanding the exponential function in the result above in power series we get Ga = π1/2 ∞ � n=0 a2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' λn−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (32) On the other hand, expanding in power series the original Gaussian integral (30), we have Ga = +∞ � −∞ dx ∞ � k=0 (−1)k λk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (x2 − a2)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (33) As the above integral has a polynomial integrand equivalent to the desired integral with poles, we have the negative dimensional integration characterized by this procedure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='e, introducing the notation INDIM(k) = +∞ � −∞ ˆdx � x2 − a2�k , (34) with ˆdx to remind that we are in the negative dimension measure continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Then, the later (33) can be written as Ga = ∞ � k=0 (−1)k λk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' INDIM(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (35) 8 Comparing the two series expansion for Ga, term by term, we have that n = k + 1/2 and INDIM(k) can be obtained as INDIM(k) = (−1)−kk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='π1/2 a2k+1 (k + 1/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' = (−1)−kπ1/2a2k+1 Γ(1 + k) Γ(1 + k + 1/2) = (−1)−kπ1/2a2k+1 1 (1 + k)1/2 , (36) where in the last line above we have introduced the Pochhammer’s symbol (α)β ≡ Γ(α + β) Γ(α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (37) Since we want (34) analytically continued (AC) to allow for negative values of k (in positive dimensional measure) we proceed by using in (36) the following Pochhammer’s identity (1 − α)β = (−1)β 1 (α)−β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (38) Then, IAC NDIM(k) = (−1)−k−1/2π1/2a2k+1(−k)−1/2, (39) In (34), we want k = −1 to reproduce the original integral in positive dimensions, so we are interested in IAC NDIM(k = −1) = (−1)1/2π1/2a−1(1)−1/2 = iπ a = I+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (40) Since the original integral is, of course invariant under the symmetry a → −a, it follows that we also have IAC NDIM(k = −1) = (−1)1/2π1/2(−a)−1(1)−1/2 = −iπ a = I−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (41) And once again, we can calculate the PV value, IPV = 1 2 lim ε→0+ {I+(ε) + I−(ε)} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (42) 9 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' NDIM IN QUANTUM MECHANICS SCATTERING CALCULATION In quantum mechanics scattering problems, there is an improper integral of the following form to be calculated: S(σ) = +∞ � −∞ dx x sin x (x2 − σ2), σ ∈ R (43) In order to perform this integration using NDIM, we first need to express it in terms of integrands that would be fitting for applying the technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' So the ‘road preparation’ for it is done by considering the integral E(a, σ) = +∞ � −∞ dx eiax (x2 − σ2) , a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (44) = ∞ � m=0 (ia)m m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Im(σ), where we have defined Im(σ) = +∞ � −∞ dx xm (x2 − σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (45) We introduce then the relevant generating functional Gaussian integration as follows G(α, β) = +∞ � −∞ dxeαx−β(x2−σ2) (46) = e α2 4β +βσ2�π β = �π β ∞ � k,l=0 (βσ2)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' α2l 4lβll!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='. (47) On the other hand, series expansion of integrand in (46) gives G(α, β) = ∞ � r,s=0 (−1)sαrβs r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' +∞ � −∞ dx xr � x2 − σ2�s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (48) Comparing (47) and (48) we have that ∞ � r,s=0 (−1)sαrβs r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' +∞ � −∞ dx xr � x2 − σ2�s = π1/2 ∞ � k,l=0 σ2k 4l βk−l−1/2 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' α2l l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (49) 10 The term by term equality entails r = 2l s = k − l − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (50) Then, we have that k = s + r/2 + 1/2 and l = r/2, leading to +∞ � −∞ dx xr � x2 − σ2�s = (−1)−sr!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='π1/2β2s+r+1 4r/2 1 (s + r/2 + 1/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' 1 (r/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' = (−1)−sπ1/2σr+2s+1 4r/2 (1 + r/2)r/2 (1 + s)r/2+1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (51) Note that the integral INDIM(r, s) ≡ +∞ � −∞ dx xr � x2 − σ2�s , (52) will be equivalent to (45) when r = m and s = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Note also that henceforth we are going to be omitting the reminder sign over ˆdx for the NDIM integration measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Since we need s to assume negative values in INDIM(r, s), we analytic continue the Pochhammer’s symbol on the righthandside of (51) that contains s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=', 1 (1 + s)r/2+1/2 AC −→ (−s)−r/2−1/2 (−1)r/2+1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (53) Then, IAC NDIM(r, s) = (−1)−s−r/2−1/2π1/2σr+2s+1 4r/2 (1 + r/2)r/2(−s)−r/2−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (54) Comparing (45) with (52) we see that we need to have r = m and s = −1 in (52) to get the relevant result for (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' So IAC NDIM(m, −1) = (−1)1/2−m/2π1/2σm−1 4m/2 (1 + m/2)m/2(1)−m/2−1/2 = iπ1/2σm σ 1 (−4)m/2 Γ(1 + m)Γ(1/2 − m/2) Γ(1 + m/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (55) Now, using the gamma function relation Γ(1/2 − m/2) = π1/2(−4)m/2Γ(1 + m/2) Γ(1 + m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (56) Then IAC NDIM(m, −1) ≡ Im(σ) = iπσm σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (57) 11 Substituting this last result into (44) we get E(a, σ) = ∞ � m=0 (ia)m m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' iπσm σ = iπ σ ∞ � m=0 (iaσ)m m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (58) Thus, finally: +∞ � −∞ dx eiax (x2 − σ2) = iπ σ eiaσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (59) Next, we need to evaluate Ex(a, σ) = +∞ � −∞ dx x eiax (x2 − σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (60) We do not need to recalculate from scratch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' we use the identity x eiax = −i d daeiax, (61) in the result (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Doing this, we get Ex(a, σ) ≡ +∞ � −∞ dx x eiax (x2 − σ2) = iπeiaσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (62) From the result (62) we have that +∞ � −∞ dxx cos(ax) (x2 − σ2) = 0 (63) +∞ � −∞ dx x sin(ax) (x2 − σ2) = π eiaσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (64) Our original quantum mechanics integral S(σ) is our result (64) for a = 1, so E+ x (1, σ) ≡ S(σ) = +∞ � −∞ dx x sin(x) (x2 − σ2) = π eiσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (65) The above result for quantum mechanical scattering problem corresponds to an outgoing wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' This result is equivalent to letting σ → σ + iε in the Cauchy contour technique 12 calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Noting that the integral on the left handside is invariant under σ ↔ −σ, we have the result for quantum mechanical scattering problem corresponding to an incoming wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' E− x (1, −σ) ≡ S(−σ) = +∞ � −∞ dx x sin(x) (x2 − σ2) = π e−iσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (66) This corresponds to letting σ → σ − iε in the Cauchy residue calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' And the PV result for the integral is, as before, SPV(σ) = 1 2 � E+ x (1, σ) + E− x (1, −σ) � = 1 2 {S(σ) + S(−σ)} = π �eiσ + e−iσ 2 � = π cos σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' (67) Since the original integral S(±σ) is an improper integral, we can see that the value we ascribe to it will depend on which type of physical boundary conditions we want: incoming or outgoing waves or even the average between those boundary conditions (PV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' CONCLUSIONS We have shown that using the NDIM technique, it was possible to evaluate some improper integrals and more specifically, the one integral that is relevant for quantum mechanical scattering problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' We have shown how to do the calculations in the NDIM technique and shown that the results it provides are equivalent to the results obtained using the Cauchy residue procedure, in which poles in the real axis is given an infinitesimal shift into the complex plane, either σ + iε or σ − iε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' In the NDIM procedure, each pole residue gives a distinct answer [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' In our present calculation for the quantum mechanical scattering problem, there are two poles, and so two residues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' and the PV result can be calculated by the simple average between the two residue calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Halliday and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Ricotta, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' B 193 (1987) 241-246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' [2] G.’t Hooft and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Veltman, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' B 44 (1972) 189-213;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Bollini and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Giambiagi, Il Nuovo Cim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' B 12 (1972) 20-26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Ashmore, Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Nuovo Cim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' 4 (1972) 289-290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' 13 [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Suzuki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Schmidt, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Bentin, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' B537, 549 (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Suzuki and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Schmidt, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' 103, 1011 (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' C 12, 361 (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Suzuki and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Schmidt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' B494, 332 (2000), A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Suzuki and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Suzuki, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' D 106, 085007 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Suzuki, Acta Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Polonica B 37 (2006) 2767-2779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Suzuki, arXiv:math-ph/0407032v1 (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Suzuki, arXiv:0806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='3216v1 (2008) [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Capper, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Dulwich, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Litvak, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' B 241(2) (1984) 463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Dirac, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' 21 (1949) 392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' [8] Table of Integrals, Series and Products, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Gradshteyn and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content='Ryzhik, 8th Edition, Edited by Daniel Zwillinger, Academic Press, Oxford, UK (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfugjd/content/2301.04082v1.pdf'} diff --git a/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf b/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf new file mode 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Myers1 +1Mani L. Bhaumik Institute for Theoretical Physics +Department of Physics & Astronomy, University of California, Los Angeles, CA 90095, USA +Abstract +We undertake a general study of the boundary (or edge) modes that arise in gauge +and gravitational theories defined on a space with boundary, either asymptotic or at finite +distance, focusing on efficient techniques for computing the corresponding boundary action. +Such actions capture all the dynamics of the system that are implied by its asymptotic +symmetry group, such as correlation functions of the corresponding conserved currents. +Working in the covariant phase space formalism, we develop a collection of approaches for +isolating the boundary modes and their dynamics, and illustrate with various examples, +notably AdS3 gravity (with and without a gravitational Chern-Simons terms) subject to +assorted boundary conditions. +arXiv:2301.02964v1 [hep-th] 8 Jan 2023 + +Contents +1 +Introduction +2 +2 +Review of Boundary Actions +6 +2.1 +U(1) CS theory on M = D × R . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.2 +Review of covariant phase space formalism . . . . . . . . . . . . . . . . . . . +8 +2.3 +Where boundary actions come from . . . . . . . . . . . . . . . . . . . . . . . +13 +2.4 +Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +3 +Computing Gauge Orbit Actions +17 +3.1 +The momentum method +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +3.2 +The transfer field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +3.3 +The transfer field method +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +3.4 +Computing Ω from the Noether charges . . . . . . . . . . . . . . . . . . . . . +24 +3.5 +Examples +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +3.6 +Comparison of techniques +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +37 +4 +Application to Topologically Massive Gravity +38 +4.1 +Einstein-Hilbert contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +4.2 +CS contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +4.3 +Total charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +42 +4.4 +Lower spin gravity formulation . . . . . . . . . . . . . . . . . . . . . . . . . . +42 +4.5 +Boundary action for warped AdS3 . . . . . . . . . . . . . . . . . . . . . . . . +44 +5 +Emergent Boundary Modes +46 +5.1 +Background solution +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +47 +5.2 +Boundary photons +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +48 +5.3 +Boundary gravitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +50 +A Forms Conventions +52 +B Identically Closed Forms +54 +1 + +B.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +55 +B.2 General algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +57 +B.3 Examples +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +60 +C Diffeomorphism Charges for Generally non-Covariant Lagrangians +62 +D Non-Abelian CS +62 +E Relation to Schwazrzian Action for JT gravity +65 +1 +Introduction +The purpose of this paper is to systematically understand the mechanism by which gauge +theories1 defined on spaces with boundaries (either at finite distance or asymptotic) are +found to host local degrees confined to the boundary. These boundary degrees of freedom +are governed by a boundary action and we aim to develop general and efficient methods +for calculating it. +The simplest and most familiar example is provided by the Chern- +Simons/WZW correspondence [1,2]. +The existence of boundary modes is tied to the crucial distinction between those gauge +symmetries that act nontrivially at the boundary versus those which are suitably localized +away from the boundary. The former take the system from one point in phase space (or +from one quantum state) to another, while the latter do not. We refer to these as large and +small gauge transformations respectively. For example, in the case of General Relativity in +asymptotically flat spacetime two black hole configurations, one at rest and one in uniform +motion, are related by a large coordinate transformation. Large gauge transformations are +generated by nontrivial conserved charges, and the large versus small distinction implies that +such charges can be expressed as boundary integrals. This fact has as its most elementary +incarnation the flux integral expression for charge in electrodynamics, and finds its most +general expression in the covariant phase space formalism [3]. +Of interest to us in this work are cases in which the large gauge transformations are +associated to physical degrees of freedom localized at the boundary. In a canonical formu- +lation, physical degrees of freedom are nonzero modes of the symplectic form. In a gauge +theory the symplectic form breaks up into bulk and boundary pieces. Boundary degrees of +freedom are zero modes of the bulk part of the symplectic form, but not of the boundary +1We use the term “gauge symmetry” to denote any local symmetry, including Yang-Mills and +diffeomorphisms. +2 + +part. The boundary modes are furthermore governed by a boundary action. Particularly +interesting is the case in which the group of large gauge transformations, i.e the asymptotic +symmetry group, is infinite dimensional, in which case there is a boundary field theory. One +motivation for our work is that these boundary field theory degrees of freedom are every bit +as physical as any other and so should be understood. Although we do not discuss it here, +boundary modes also play an import role in attempts to formulate entanglement entropy in +gauge theory and gravity; a few references include [4–7]. +In the case of Chern-Simons (CS) theory there is a simple well known procedure for +obtaining the WZW boundary action [2]. In CS theory the Gauss law constraint is the +statement that the spatial components of the field strength vanish. The general solution +to this constraint is obtained by writing the gauge fields as a gauge transformation of a +given flat connection. Inserting this back into the action the CS Lagrangian becomes a total +derivative and the resulting boundary term is the WZW action.2 +This strategy is not necessarily so easy to carry out in other theories, in particular if +there is no easy way to find the general solution of the constraints. A main objective of +this paper is to develop more widely applicable methods for deducing boundary actions, +illustrated by explicit examples. To this end we primarily work in a covariant phase space +framework [8,9,3], where the main actors are the symplectic form and the boundary charges. +We take as examples three-dimensional gravity, possibly supplemented with a gravitational +CS term, subject to various boundary conditions. As is well known [10,11], pure 3D gravity +with AdS3 boundary conditions admits a CS formulation to which the basic CS/WZW +procedure above can be applied (with some modifications due to the change of boundary +conditions), and one obtains a theory of boundary gravitons [12,13]. The metric formulation +version of this procedure was worked out in [14,15], and extended to the case of a finite cutoff +boundary. Here we illustrate our more general methods by considering non-AdS boundary +conditions, including those of warped AdS3 [16] supported by the gravitational CS term [17]; +note that in general the addition of the gravitational CS term breaks the correspondence +with the usual SL(2, R) × SL(2, R) CS formulation. +We now briefly summarize our various approaches to deriving boundary actions. In the +canonical approach followed here, the boundary action is built out of a canonical 1-form Θ, +whose exterior variation yields the symplectic form Ω, and the boundary Hamiltonian. These +objects are defined on a phase space which consists of a particular gauge orbit, obtained by +acting on a chosen background solution with all possible boundary condition preserving gauge +transformations. The basic point is that on a given orbit the symplectic form as well as the +charges generating the large gauge transformations (i.e. asymptotic symmetries) all localize +2To get a fully explicit two-dimensional boundary action one needs to choose an explicit parametrization +of the gauge group elements, as we review in Appendix D. +3 + +to the boundary. If the boundary charges QV associated to large gauge transformations V can +be computed, the symplectic form may in principle be found by solving the equation iV Ω = +−δQV . Solving this can be quite laborious, and one still needs to compute the potential Θ +for Ω. For the class of examples that arise in AdS3 gravity with various boundary conditions, +it is possible to bypass all this and pass directly from the conserved (angular)momentum +and Hamiltonian to the boundary action, as was employed in the example of cutoff AdS3 +gravity in [15] leading to a boundary Nambu-Goto action, whose origin was clarified in [18]. +As this approach may not always be possible, we also develop more general methods +for computing Ω. These are based on identifying a phase space 1-form valued vector field +W, which we refer to as the transfer field, which obeys the relation δφ = iWδφ when δφ +is restricted to a single gauge orbit. +Given knowledge of W and of the charges Q we +show how one can use these to read off the boundary symplectic form. Furthermore, we +demonstrate another technique which is somewhat less efficient for computing Ω, but has the +advantage of allowing one to sometimes obtain expressions for the boundary contributions to +Ω independent of the chosen boundary conditions. We describe when this can be done, most +notably for diffeomorphisms. As an example, the Einstein-Hilbert action in any dimension +with any cosmological constant always produces the contribution (3.66) to the boundary +symplectic form, independent of the boundary conditions. +The main example we use to illustrate our general methods is warped AdS3 [19,20,16]. +This is a well-studied solution of topologically massive gravity (TMG) [17]. The warped +asymptotics make it less obvious a priori on what surface the boundary action should be +thought of as living. We will work out the boundary action in detail, and also show how the +same results may be obtained via “lower spin gravity” [21] which is a CS formulation that +can be used to describe a subsector of the full TMG phase space. +These boundary actions are important inasmuch as the the boundary modes are part of +the dynamical degrees of freedom of the theory. For example the boundary photons and +gravitons arising in the CS and AdS3 gravity theories contribute to the thermal partition +function [22]. However, at first sight the physical relevance of these modes may seem elusive, +given that they are generated by performing gauge transformations. This point is clarified +by coupling another system to the theory containing the boundary modes. We give a simple +example of this in which a boundary scalar couples to a CS theory defined on a spatial disk, +showing how correlators of the boundary scalar are modified by the coupling to the boundary +photons. +Another situation occurs when the boundary is not the true “end” of the spacetime, +but rather an interface marking a transition between two different regions with distinct +asymptotics. It is interesting to ask whether and how the modes that “would have been +4 + +there” had the interface been an actual boundary manifest themselves in the full system. +One can think of this as a version of the setup described in the previous paragraph, where +one side of the transition region now functions as the additional system. This situation +arises very naturally in gravity. For example, one can have a solution with a near horizon +AdS3 region, which by itself supports boundary modes, embedded inside an asymptotically +AdSD>3 solution. The latter solution has a finite dimensional asymptotic symmetry group, so +one may wonder whether the near horizon boundary modes are detectable at the asymptotic +boundary. We answer this question in the affirmative, showing how the pure gauge modes +in the near horizon are promoted to non-pure gauge modes in the full spacetime. +It is worthwhile to clarify our usage of certain terminology in what follows. In partic- +ular, when we refer to a gauge transformation, depending on context we may or may not +distinguish whether we mean small or large gauge transformations, and likewise for diffeo- +morphisms. When this matters, which is often, we will distinguish the two. Recall that small +gauge transformations/diffeomorphisms describe redundancies, and in a canonical framework +are zero modes of the symplectic form. Large gauge transformations/diffeomorphisms instead +move us between distinct points in phase space, and are nonzero modes of the symplectic +form. Finally, we occasionally use the term “local symmetry”, which is meant to encompasses +both gauge and diffeomorphism symmetry, whether small or large. The usage should always +be clear from the context. +The rest of this paper is organized as follows. In Section 2, after quickly reviewing the +standard approach to U(1) CS theory on a disk and the physical relevance of boundary +modes, we go on to a general discussion of the origin and identification of boundary modes +within the framework of the covariant phase space. In Section 3 we develop specific methods +for computing boundary actions, illustrated through particular examples. In Section 4 we +consider the case of warped AdS3 asymptotics in topologically massive gravity, which is a +useful and nontrivial example to illustrate various issues. We obtain the boundary action in +both the metric formulation and in the so-called lower spin gravity formulations. Section 5 +discusses how boundary modes can appear in the IR. A series of appendices lay out some +conventions, review an important theorem regarding identically closed forms, review the +proper method for handling non-diffeomorphism invariant actions in the Wald formalism, +review non-Abelian CS theory and apply our methods to it, and explain the connection of +our 3D gravity results to 2D JT gravity. +5 + +2 +Review of Boundary Actions +In this section we discuss general aspects of boundary modes, their origin in terms of large +gauge transformations, and the construction of an action that describes them. +2.1 +U(1) CS theory on M = D × R +To get oriented, we first quickly review the simple and classic example of Chern-Simons +theory on a spatial disk and the corresponding boundary gauge modes, following the original +Lagrangian approach [2]. This approach is based on solving the Gauss law constraint and +substituting back into the action, yielding a total derivative. While this method works well +here, it is not so easy to adapt to other examples such as gravity in the metric description. +For this reason we go on to develop more flexible methods based on a covariant phase space +analysis. +The action for for Abelian CS theory on a spatial disk cross time is +S = k +� +M +A ∧ dA + Sbndy += −k +� +M +d3x(Ar∂tAφ − Aφ∂tAr + 2AtFφr) + S′ +bndy , +(2.1) +where we integrated by parts and absorbed the boundary term into S′ +bndy. +We choose +boundary conditions δ(At − Aφ)|∂M = 0. +Specializing to the case (At − Aφ)|∂M = 0, a good variational principle is achieved by +taking +Sbndy = 0, +S′ +bdy = −k +� +∂M +dtdφA2 +φ . +(2.2) +At is a Lagrange multiplier enforcing the constraint Fφr = 0, which is solved by writing +Aφ = ∂φα , +Ar = ∂rα , +(2.3) +with α(r, t, φ + 2π) = α(r, t, φ). Plugging this back into the action, the bulk terms become +a total derivative, and we arrive at the chiral boson action +S = k +� +∂M +dtdφ(∂φα∂tα − ∂φα∂φα) . +(2.4) +6 + +The basic equal time Poisson-Dirac bracket is +{α(φ), ∂φα(φ′)} = 1 +2kδ(φ − φ′) . +(2.5) +The Hamiltonian and charges generating infinitesimal gauge transformations by λ are +Ht = k +� +dφ(∂φα)2, +H[λ] = 2k +� 2π +0 +λ∂φαdφ . +(2.6) +The theory describes a U(1) current J = k∂φα whose Fourier modes obey a U(1) current +algebra. The current-current-correlator is +GJJ(w) = ⟨J(w)J(0)⟩ = − +k +4 sin2 � w +2 +� , +w = φ + t, +w = φ − t . +(2.7) +2.1.1 +Physical relevance of boundary modes +Inasmuch as the preceding analysis shows that boundary modes can carry nonzero energy +and momentum, they are established as being nontrivial physical states. Nonetheless, their +“pure gauge” character leads one to wonder, at least upon first hearing, whether they might +be ignorable in some sense, for example by decoupling from the rest of the physical system +in which they are embedded. However, it is not hard to show that the boundary modes do +have measurable consequences on other observables. +To expose these effects we can think of coupling the theory in the disk region to some +external system comprised of charged matter that couples to the CS gauge field at the +boundary of the disk. In the simplest incarnation we can take the system to live on the +boundary of the disk, and to be completely explicit we consider a charged scalar field example, +S = k +� +M +A ∧ dA + +� +∂M +d2x(DµΦ)∗DµΦ . +(2.8) +The covariant derivative is taken to correspond to a gauging of the scalar shift symmetry, +DµΦ = ∂µΦ − iqAµ. This is convenient, since the associated current, Jµ = i(∂µΦ − ∂µΦ∗) +decomposes into dimension (1, 0) and (0, 1) operators, which is not the case for the current +associated to phase rotations of the scalar. Repeating the previous steps we arrive at the +action +S = +� +d2x [−2k∂φα∂wα + 4∂wΦ∗∂wΦ + 2iq∂φα(∂wΦ − ∂wΦ∗)] +(2.9) +with ∂w = 1 +2(∂φ + ∂t), ∂w = 1 +2(∂φ − ∂t). The coupling of α to Φ has no effect on the energy +7 + +spectrum of the theory, as follows from the fact that the coupling can be removed by a +redefinition of Φ. There is a nontrivial effect on scalar correlators, in particular on the two- +point function of the current Jw. This effect reflects the fluctuating phase acquired by a +charged particle on traveling between the two operator insertion points on the boundary. +Treating q as a perturbation, we can readily sum up contributing diagrams by performing +Wick contractions, resulting in +G(q) +JJ(p) = ⟨J−(p)J−(−p)⟩q += +∞ +� +m=0 +(−4q2)m[GJJ(p)]m+1[Gαα(p)]m += +GJJ(p) +1 + 4q2GJJ(p)Gαα(p) , +(2.10) +where the q = 0 correlators are +GJJ(p) = ⟨Jw(p)Jw(−p)⟩q ∼ pw +pw +Gαα(p) = ⟨∂φα(p)∂φλ(−p)⟩q ∼ pφ +kpw +. +(2.11) +At q = 0 the correlator behaves as pw/pw corresponding to 1/ sin2( w +2 )2 in position space. As +q→∞ the correlator tends to zero, with leading behavior +k +q2 +pw +pφ ; being polynomial in pt, this +vanishes for unequal times. +The point to be emphasized here is that the boundary modes leave a detectable imprint +on the scalar correlators, and so are clearly “real.” +2.2 +Review of covariant phase space formalism +We begin with a brief review of the covariant phase space formalism, which will also serve to +establish notation for the remainder of this paper. Along the way, we make comments about +precisely where boundary conditions enter the formalism, as these will be useful to keep in +mind later. For a more detailed review, see for example [8,3,23,24]. +2.2.1 +Action and covariant phase space +We consider a theory defined on a D = d + 1 dimensional spacetime M which admits +a foliation by codimension-1 slices which we will generally denote by Σ. The, potentially +asymptotic, boundary structure of M can then be decomposed into ∂M = Σ+∪Γ∪Σ− where +Σ± are the slices in the asymptotically far future and past and Γ is formed by unioning the +8 + +boundaries of all the slices. On this spacetime we consider a theory with fields φ whose +dynamics are described by an action +S[φ] = +� +M +L + +� +∂M +ℓ +(2.12) +where L is the D-form Lagrangian and ℓ is some allowed d-form boundary contribution3, +which we assume to be local functionals of the fields. +Throughout we will use δ to denote the exterior variational derivative, which we refer to +as the variation. Particular infinitesimal transformations of the fields will be thought of as +vector fields V on field space. The action of a vector V on a field φ will then be denoted by +the contraction4 iV δφ. By integrating by parts, the variation of the Lagrangian may always +be written +δL = E ∧ δφ + dθ +(2.13) +for some θ. +Setting E = 0 will be our equations of motion. +We note that θ is always +ambiguous up to addition of a d-closed form which we will return to shortly5. +Using this identity, the variation of the action is given by +δS = +� +M +E ∧ δφ + +� +Σ+−Σ− +(θ + δℓ) + +� +Γ +(θ + δℓ). +(2.14) +In order to have a good variational principle we require that the on-shell variation have no +support on Γ which then requires +(θ + δℓ)|Γ = dB +(2.15) +for some B. Of course, this B can always be absorbed into a redefinition of θ. However, we +note that for a generic theory, the LHS above will not automatically take the form of a total +derivative. Instead, there will be terms which only vanish upon the imposition of boundary +conditions. This means that the B here generally depends on the boundary conditions we +3Though we use the same symbol ℓ for the boundary contribution over all of ∂M, there need be no +relation between ℓ on Γ and ℓ on Σ±. Shifts in this ℓ on either always produce shifts in the canonical 1-form +by something δ exact and so do not change the symplectic structure. +4Though here we prefer the contraction on variation notation to denote infinitesimal transformation, the +reader may find it helpful to recall that the following are equivalent: iV δF[φ] = LV F[φ] = V (F[φ]) where +LV denotes the Lie derivative on field space and V (F[φ]) is the action of the vector field V on the function +F on field space. +5In the literature, e.g. [3], it is often mentioned that θ is also ambiguous up to addition of a δ-closed +form. While this is true, any such shift in θ is equivalent to shifting ℓ. So we take the perspective that θ has +no δ ambiguity, but ℓ remains to be chosen. +9 + +choose for our theory, and the existence of B may impose conditions on what we choose for +ℓ6. As a simple example, starting from the action (2.1) we find +θ = −kA ∧ δA, +B = 0 +(2.16) +with chiral boundary conditions. +It will be useful later to keep explicit which objects depend on the boundary conditions +and which do not. So while it is possible to absorb B into a redefinition of θ, we will refrain +from doing so in order to avoid reference to boundary conditions when writing θ. +It’s also useful to observe that having a good variational principle is equivalent to slice +independence of the symplectic form. The potential for the symplectic form is always found +by extracting what remains of the action’s variation from the initial and final time slices; we +write A to denote this. Here we have +δS = +� +Σ+−Σ− +A = +� +Σ+−Σ− +(θ + δℓ − dB) +(2.17) +after the imposition of (2.15). This means we should choose our symplectic form to be7 +Ω = +� +Σ +ω = +� +Σ +δA = +� +Σ +δ(θ − dB). +(2.18) +Though the ultimate argument for using this object as our symplectic form will be that it +produces the desired Poisson brackets, we can see an immediate benefit by taking a second +variation of (2.17), which implies that this Ω is independent of the slice we choose. +It is, however, useful to observe that we can show more directly that the slice indepen- +dence of Ω is precisely equivalent to the demand (2.15), and hence the demand for a good +variational principle. To see this we take a second variation of (2.13) to find +−δ(E ∧ δφ) = dδθ = dω +(2.19) +so the symplectic current is closed on-shell. Hence Ω(Σ) is independent of the slice Σ if +and only if the pullback of ω to Γ vanishes. +This demand can be rewritten as ω|Γ = +δ(θ|Γ − dB) = δ[(θ + δℓ)|Γ − dB] so the condition (2.15), obtained from demanding a good +variational principle, is equivalent8 to the slice independence of the symplectic form. +6A standard example of this would be the need to include the Gibbons-Hawking-York term in the +Einstein-Hilbert action with Dirichlet boundary conditions, though in that case we may choose B = 0 +depending on our gauge fixing, see e.g. [24] for details. +7Throughout this work we ignore the complications that come from the possibility of non-trivial phase +space topology, including the possibility of a symplectic form with non-trivial De Rahm cohomology. +8Strictly speaking, the slice independence of the symplectic form only implies (2.15) up to a δ-closed +10 + +As a final comment about this definition for the symplectic form, we should discuss the +distinction between the phase space and prephase space. We will generally take prephase +space to consist of all configurations of the fields which obey the boundary conditions and +the equations of motion. On this space the quantity (2.18) will generally be degenerate and +hence cannot be a proper symplectic form9. For this reason, it’s often referred to as the +presymplectic form. +To form the actual phase space, we need to perform a symplectic quotient and mod out +the null directions of the symplectic form. This is a matter of viewing the prephase space +as a bundle whose fibers are the null directions and whose base space is our true phase +space. The mathematical details were laid out in [9], but in practice the result is that Ω is a +non-degenerate 2-form on the base space and so working on the true phase space is a matter +is ignoring those variables whose variation lies along the pure gauge directions10. +2.2.2 +Symmetries and charges +With the covariant phase space framework now in place, it will be important for us to review +how symmetries enter the picture. A vector field V on phase space is typically defined to be +a symmetry if its action on the Lagrangian is a total derivative: +iV δL = dkV +(2.20) +for some kV . Contracting V onto (2.13) it now follows that +dJV ≡ d(iV θ − kV ) = E ∧ iV δφ. +(2.21) +So JV = iV θ − kV is the conserved Noether current associated to V . +Though any vector field V satisfying (2.20) admits a conserved Noether current, con- +structing the Noether charge is not always as a simple as integrating the current over a time +slice. There may be non-trivial boundary contributions to the true Noether charge HV in +form, but locally on phase space such a form can be written as exact and absorbed into a redefinition of ℓ. +9This is closely related to the lack of deterministic evolution on prephase space; specifying the fields +and some number of their derivatives on a Cauchy slice may not uniquely determine the same data on a +later time slice. The most common way to exhibit this non-uniqueness is by specifying a configuration and +applying to it a gauge transformation which differs from the identity only at times later than the first slice. +We would thus have two solutions to the equations of motion whose data agree on one slice but disagree on +another. +10If we are being strict, this is the statement that, at least locally, a section of the bundle is diffeomorphic +to the base space. +11 + +order for it to satisfy +iV Ω = −δHV . +(2.22) +For general symmetries, one must directly evaluate the contraction on the symplectic +form, but for gauge symmetries we may find the charges by another, sometimes more efficient, +method. +This was pointed out in [25] for the special case of diffeomorphism charges in +diffeomorphism invariant theories, but with the Theorem 1 of Appendix B it’s simple to +generalize this calculation to any gauge transformation, as we review now11. +We suppose Vλ generates a gauge transformation with gauge parameter λ, defined such +that V0 = 0 so λ = 0 is the identity transformation. Taking an additional variation of (2.21) +it follows that, on-shell, +d(LVλθ − δkλ) = 0 +(2.23) +where we have abbreviated kVλ = kλ. Thus we have a form closed for all free functions λ +and theorem 1 tells us that there must exist a phase space 1-form Πλ, constructed locally +from the fields and λ, such that +LVλθ − δkλ = dΠλ. +(2.24) +With this, it now follows from (2.18) that +δJλ = −iVλω + d(Πλ − iVλδB). +(2.25) +Thus if there exists a function Cλ such that +Πλ − iVλδB = δCλ, +(2.26) +(2.25) implies +iVλΩ = −δ +� +Σ +(Jλ − dCλ). +(2.27) +We note that the existence of Cλ is not guaranteed and will typically depend on the boundary +conditions chosen for the theory12. The expression (2.27) now identifies the correct boundary +11Some additional simplifications that can help in computations are possible in the special case of +diffeomorphisms even when the theory is not diffeomorphism invariant, as when gravitational Chern-Simons +terms are included in the action. This still makes use of Theorem 1 and was pointed out in [26]. We review +it in Appendix C. +12The insufficiency of (2.20) alone to ensure the existence of a charge satisfying (2.22) is pointed out +12 + +Noether charge as being the integral of Jλ with some additional boundary contributions. +Since Jλ is closed for all λ, and is linear in λ, we can go further and compute a local +functional, referred to as the Noether-Wald charge, Qλ such that Jλ = dQλ. With this the +Noether charge may be written +H[λ] = +� +∂Σ +(Qλ − Cλ) +(2.28) +which has support only on the boundary of our Cauchy slice. Furthermore, we note that the +only place the boundary conditions enter into this expression is through Cλ, as Jλ and Qλ +depend only on the Lagrangian of the theory. +Since the charges (2.28) have support only on the boundary, it follows immediately that +any gauge transform whose parameters λ have compact support away from any boundaries +must produce vanishing Noether charge. The vector fields generating these transformations +are thus identified from (2.27) as null directions of the presymplectic form which need to be +modded out in the symplectic quotient. The non-zero Noether charges generate the large +gauge transformations of the theory and are evidently localized to the boundaries of the +spacetime. In the case these boundaries are asymptotic, the large gauge transformations are +said to be asymptotic symmetries. +2.3 +Where boundary actions come from +Using the machinery of the covariant phase space we can understand the, rather weak, +sufficient conditions for producing boundary modes and gain some insight into the conditions +under which the action for the theory is supported exclusively on the boundary. To transform +this question into one which is easier to work with we first recall the phase space action. +Writing the symplectic form for the theory as Ω = δΘ, the phase space action is given by13 +S[γ] = +� +γ +Θ − +� +γ +Htdt +(2.29) +many placed in the literature. In e.g. [3,23], integrability conditions are required as we see here and in [24] +an auxiliary condition, there eq. (4.16), is required. +13Of course, Θ is not unique. Different Θ correspond to holding different data fixed in the initial and final +configurations. For a point particle, Θ = pδx is the correct potential for varying with the initial and final +position of the particle held fixed. +13 + +where γ is some path through phase space parametrized by t. To orient ourselves it’s useful +to recall that for point particles Θ = pδx so +S = +� +pδx − +� +Htdt = +� +(p ˙x − Ht)dt +(2.30) +where the dot denotes the derivative along the path. +If time translation is a local symmetry, as it is in a diffeomorphism invariant theory, +then by Theorem 1 the Hamiltonian generating time translation Ht is supported on the +boundary. While this gives a boundary contribution to the action, we would still be left +asking about when we receive boundary contributions from Θ and about what happens +when time translation is not a local symmetry. We can obtain a better characterization of +boundary contributions to the action which will help answer both of these questions by first +introducing some notation. +Consider a generic theory with fields φ and some gauge group. We denote the action +of a gauge group element with gauge parameters14 α on φ by Tα[φ]. For example, if φ is +a complex scalar field of charge q, Tα[φ] = eiqαφ. If φ is a U(1) connection we would have +Tα[φ] = φ + dα. +Now, we are always free to consider the field redefinition to gauge orbit variables where +we pick a class of gauge-fixed configurations φ so a general configuration is in the gauge orbit +of some φ: φ = Tα[φ]. This separates the prephase space into gauge directions, parametrized +by the α, and non-gauge directions, parametrized by the gauge inequivalent φ. +In these variables if we write15 Ht = +� +Σ Jt, Jt[φ, α] is a local function of α. Since Jt is +closed for all functions α, Theorem 1 tells us that we can construct a local potential Qt[φ, α] +such that +Jt[φ, α] = Jt[φ, 0] + dQt[φ, α]. +(2.31) +That is to say the current separates into a component we would find if we had immediately +fixed the gauge, and a boundary contribution which contains all the effects of the gauge +modes α, though it’s notably also free to depend upon boundary excitations in the gauge +fixed φ directions. We note that Jt[φ, 0] = 0 when time translation is a gauge symmetry. +A similar argument works on the symplectic form. There are only three types of com- +ponents that can appear in the symplectic form when we go to gauge orbit variables. This +14For later convenience we assume that α = 0 is the identity transformation and that these α are free +functions on spacetime as might be obtained by exponentiating a Lie algebra about the identity. +15In this section only we assume for convenience that Jt has been defined to include all necessary boundary +contributions already so Ht is the full Noether charge satisfying (2.22). +14 + +fixes the generic form of the symplectic form to be +ω = ωφφ[φ, α]δφ ∧ δφ + ωφα[φ, α]δφ ∧ δα + ωαα[φ, α]δα ∧ δα +(2.32) +where the coefficients should generally be understood to contract on any indices we have +suppressed in δφ and δα, and may also contain derivatives that operate on the field variations. +Since the δα are completely free function in the bulk and dω = 0 on-shell by (2.19), we +may again apply theorem 1 but now using δα instead of α as our free functions. We thus +conclude16 +ω = ωφφ[φ, α]δφ ∧ δφ + d˜ωb[φ, α] +(2.33) +where we have put a tilde on ˜ωb because it will not be the complete contribution to the +boundary symplectic form. Importantly, ˜ωb is a local functional of the fields. +We can go further by invoking theorem 1, this time on the α dependence of ωφφ[φ, α]δφ∧ +δφ. The result is evaluating ωφφ at α = 0 and an additional, locally constructed contribution +to the boundary symplectic form: +ω = ωφφ[φ, 0]δφ ∧ δφ + dωb[φ, α]. +(2.34) +Like with H, the symplectic form separates into a bulk component that we would have found +by gauge fixing from the very beginning and a boundary term which encapsulates the entire +effect of the gauge symmetry. +We can choose both of these terms to be δ closed on phase space separately17. It then +follows that we can find separate potentials and the symplectic potential current A breaks +up into two terms, +A[φ, α] = AM[φ] + dA∂M[φ, α]. +(2.35) +Our arguments about the Hamiltonian and symplectic form together imply that the +action must also break up into two terms, one supported on the bulk obtained from gauge +16Less formally, one can understand that there ultimately cannot be any δα components appearing in the +bulk or Ω would have non-zero contractions on small gauge transformations. +17To see this, suppose to the contrary that the variation of the bulk term in (2.34) is not δ closed but +varies to something spacetime exact, δ(ωφφ[φ, 0]δφ ∧ δφ) = dωM, which then cancels against part of dδωb. +This ωM cannot contain any dependence on α or δα, so if a cancellation occurs it’s sufficient to consider +α = δα = 0. But our use of theorem 1 implies ωb = 0 in this case, so no cancellation can occur, requiring +dωM = 0, and the terms in (2.34) are separately closed. +15 + +fixing, and one supported only on the boundary18: +S = +� +dt +� +Σ +� +iV AM[φ] − Jt[φ, 0] +� ++ +� +dt +� +∂Σ +� +iV A∂M[φ, α] − Qt[φ, α] +� +(2.36) +where V generates the path the action is evaluated on. So all gauge theories will reduce to a +bulk component which is gauge fixed, and a boundary term which contains all the dynamics +of the boundary gauge modes. Importantly, this boundary action may contain interactions +not only among the boundary fields α, but also with the boundary values of the bulk gauge +fixed fields φ. +2.4 +Discussion +The result (2.36) establishes boundary contributions to the action as a generic feature of +gauge theory independent of whether we are able to solve constraints and directly reduce +the action to the boundary as in 3d CS theory. But the nature of this boundary action varies +from case to case, the main controlling factor being the structure of the asymptotic symmetry +group. For example, Einstein gravity in AdSD>3 with standard asymptotically AdS boundary +conditions has a finite dimensional asymptotic symmetry given by the SO(D−1, 2) isometry +group of the global AdS vacuum. Thus instead of a boundary field theory we have a boundary +quantum mechanics, with one quantum mechanical degree of freedom corresponding to each +generator. +To illustrate this we consider the even simpler case of pure Maxwell theory in asymptot- +ically flat space. The phase space variables are ( ⃗A, ⃗E) subject to the Gauss law constraint +⃗∇ · ⃗E = 0, and we impose that these vanish at spatial infinity. The symplectic form is +Ω ∼ +� +Σ δ ⃗E · ∧δ ⃗A. This symplectic form together with the boundary conditions require that +the gauge parameters α(⃗x) become spatially constant at infinity. Writing ⃗A = ⃗A + ⃗∇α, we +have +Ω ∼ +� +∂Σ +⃗n · δ ⃗E ∧ δα ∼ δQ ∧ δα +(2.37) +where Q is the total electric charge. As the Hamiltonian has no dependence on α, the bound- +ary action is simply Sα = +� +dtQ ˙α, whose equation of motion is simply charge conservation. +This boundary contribution is actually familiar in another guise. In particular, consider the +Euclidean theory with periodic imaginary time, t ∼= t + β. Since Q is constant on-shell we +18One should be careful to take the orientation of ∂Σ in these integrals to be the one induced by writing +Vol(M) = τ ∧ n ∧ Vol(∂Σ) where τ is the normal form to the slices Σ and n is the (outward) normal form +to Γ. This is a convention consistent with Stokes theorem, see [24] for more details. +16 + +have Sα = Q +� +dt ˙α ≡ βµQ, which identifies the boundary mode α as being proportional to +the chemical potential. +Turning now to the case of an infinite dimensional asymptotic symmetry group, here we +do expect to get a boundary field theory. This boundary field theory may either be a free +theory, as in U(1) CS theory, or interacting, as in non-Abelian CS theory or 3D gravity. In +general, it’s clear that the existence of interactions is closely tied to the bulk theory having +a non-Abelian asymptotic symmetry group. +We may also ask more practically how to compute (2.36). +In principle, once we go +to gauge orbit variables all stages of the calculation here are algorithmic as reviewed in +Appendix B. But to use the gauge orbit variables we would first need to classify the gauge- +inequivalent solutions. Since we are working in a canonical formulation it would be sufficient +to find all gauge-inequivalent initial data, but that would still mean solving the constraints +of the theory explicitly. +Instead, one could could imagine taking φ to be some perhaps incomplete class of initial +data solving the constraints and consider the dynamics of this subspace of the full phase +space. For example by considering a special subspace such as a moduli space of black hole +solutions. +The logical extreme of these ideas would be to take φ to be a single solution, so we are +looking at the gauge orbits about a background configuration. In this case δφ = 0 and +AM = 0 so the entire symplectic form lives on the boundary. This simplification allows more +efficient computational methods than the gauge orbit variable strategy described above. +Three such methods are described in the next section. +3 +Computing Gauge Orbit Actions +As discussed in Section 2.4, we may have reason to consider only certain sectors of a theory. +The particular case of interest here is where we consider a single gauge fixed configuration +φ and orbits around it. This means δφ = 0 and AM = 0. The remainder of the bulk term in +the action (2.36) then integrates to some constant and we are left with only the boundary +contribution to the action and more efficient methods of calculation are available to us. This +section is concerned with describing three such methods. +The first is less generically applicable, but very efficient when it applies as it directly +computes A∂M instead of the boundary symplectic form ωb. The other two methods are +concerned with computing ωb and are based on the existence of a particular 1-form valued +vector field on phase space. +17 + +It’s important to note that as δφ = 0 suggests, these methods treat φ as a background +field so any components A∂M might have had in the δφ directions will not be captured by +these computations. +3.1 +The momentum method +For all the 3D gravity examples considered in this work there exists a very efficient method +for deducing the boundary action [15]. We describe it here in general terms. +Supposing that expressions for boundary charges Hξ are known, to write down the +boundary action we require knowledge of the boundary symplectic potential Θ, defined via +Ω = δΘ. A direct approach to obtaining it is to first extract Ω via the relation iVξΩ = −δHξ, +and then compute Θ by solving δΘ = Ω. We now discuss how, under suitable assumptions, we +can write down the solution for Θ directly, bypassing the laborious procedure just mentioned. +This method was used in the case of cutoff AdS3 gravity [15]. +We consider a boundary with a single spatial dimension with coordinate x, which may +live either on the circle or line. The boundary field theory variables are written as (Φ, Ψ), +where Ψ could stand for a collection of fields. On the circle or line, these fields are taken +to obey periodic boundary conditions or vanish at infinity respectively. Under phase space +vector fields Vξ, which we can think of as reparametrizations of x, the fields transform as +δξΦ = iVξδΦ = ξ + Φ′ξ +δξΨ = iVξδΨ = Ψ′ξ +(3.1) +where ′ = ∂x. The inhomogeneous term in δξΦ corresponds to Φ being the field associated +with x-reparametrizations. Now suppose that the charge corresponding to constant ξ (we +take ξ = 1 and denote this charge by P) is constrained to take the form +P = +� +dx +� +κΦΦΦ′2 + κΦΨΦ′Ψ′ + κΨΨΨ′2 + P ′ +ΦΦ′ + P ′ +ΨΨ′� +(3.2) +where the κ’s are constant and the functions (PΦ, PΨ) are local functions of (Φ′, Φ′′, . . . ; Ψ′, Ψ′′, . . .), +i.e. do not depend on undifferentiated fields. The basic result is that up to a δ-exact term, +the unique Θ which solves +iVξΩ = −δP , +ξ = 1 +(3.3) +18 + +is given by +Θ = +� +dx [κΦΦΦ′δΦ + κΦΨΦ′δΨ + κΨΨΨ′δΨ + P ′ +ΦδΦ + P ′ +ΨδΨ] . +(3.4) +To prove this, we first note that it is straightforward to verify that (3.3) is satisfied, so only +the question of uniqueness remains. To analyze uniqueness we consider a correction to the +symplectic form, +∆Ω = +� +dx [δX′ +Φ ∧ δΦ + δX′ +Ψ ∧ δΨ] , +(3.5) +where the functions (XΦ, XΨ) are constrained to obey the same general properties of (PΦ, PΨ). +We compute +iVξ∆Ω = δ +� +dx [X′ +ΦΦ′ + X′ +ΨΨ′] += −δ +� +dx [X′′ +ΦΦ + X′′ +ΨΨ] +(3.6) +We need this to vanish in order not to spoil (3.3). Vanishing of the integrand in the second +line of (3.6) requires X′′ +Φ = X′′ +Ψ = 0, since the two terms cannot cancel each other under the +assumed form of (XΦ, XΨ). However, this implies that (X′ +Φ, X′ +Ψ) are constants, which implies +that δX′ +Φ = δX′ +Ψ = 0, so that ∆Ω = 0. The remaining possibility is that the integrand in +(3.6) is a total derivative. This requires X′ +Φ = δF +δΦ and X′ +Ψ = δF +δΨ for some F. However, this +leads to ∆Ω = +� +dxδ2F = 0, so we again get no contribution. +The upshot is that just from consideration of P we are led to a unique result for Ω and +an explicit result for its potential Θ. It follows, assuming that our underlying theory is +consistent, that this Ω will solve iVξΩ = −δHξ for all large gauge transformations Vξ, as can +of course be checked in specific examples. +We conclude this section with comments on the assumed structure (3.2), which follows +from a particular gauge invariance. To make the discussion concrete we consider the bound- +ary theory in the context of pure gravity in AdS3. Setting κ = 0 corresponds to the orbit +built on a pure AdS3 background. This background is invariant under isometries that act +as translations, boosts, and dilatations on the boundary coordinates. Taking the functions +(Φ, Ψ) to be general linear functions of x corresponds to acting on the background by one of +these isometries; since this does nothing to the background such functions are pure gauge, +and so all charges must vanish for such functions. This implies that only derivatives of such +functions can appear in P, and that each term must contain at least one second derivative. +This leads to (3.2), possibly after integrating by parts. The same comments apply to ∆Ω. +19 + +Turning on κ, which corresponds to a nonzero mass AdS3 background, breaks some of the +isometries, allowing the quadratic terms like κΦΦΦ′2 to appear. +In fact, a more general +expression like κΦΦΦ′n for some n > 2 might also be anticipated, but only the n = 2 case +will arise in our examples. This is important, since the logic leading to (3.4) no longer holds +for the n > 2 case. In fact, given the simple form of the κ dependent part of that charges +will arise, it’s easy to write down the corresponding contribution to the boundary action by +inspection. +3.2 +The transfer field +Consider a variation of our fields in the gauge orbit variables φ = Tα[φ], as in Section 2.3. +Since we are taking φ to be fixed, the variation can only change α. But since φ is constructed +by applying a gauge group element to the configuration φ, this variation must be equivalent +to the action of some Lie algebra element on φ. +The situation is essentially identical to the story which comes up when studying sponta- +neous symmetry breaking, in particular the broken part of the symmetry group’s non-linear +action on Nambu-Goldstone bosons when studying spontaneous symmetry breaking, see for +example [27, 28]. The idea is that δα defines a Lie algebra element at the origin of the +gauge group while φ has been transported to g. So to perform the variation of φ we need to +transport δα to the tangent space at g. Schematically, we can think of this as inserting the +identity in the action on φ to write +δφ = (δTg)[φ] = (δTgT −1 +g )[φ] +(3.7) +so the variation of φ is equivalent to the action of this Lie algebra element valued as a 1-form +on phase space. +Since Lie algebra elements are the generators of group transformations, there must exist +some W which implements the action of this Lie algebra element. +Because the algebra +element is valued as a 1-form on phase space, W must be a (1, 1) tensor on phase space, +which we think of as a 1-form valued vector field and refer to as the transfer field. This gives +W the very special defining property +δφ = iWδφ. +(3.8) +While the property (3.8) is ultimately the formal property we will exploit to compute the +boundary symplectic form, we will also need to explicitly compute the transfer field. So it +will be useful to see how the rather abstract discussion above can be realized in some simple +20 + +examples. +The simplest example is of a U(1) connection A = A + dα. Here we have explicitly +δA = dδα = iVδαδA +(3.9) +so W = Vδα implements the gauge transformation with gauge parameter δα, as might be +expected for an Abelian group where translating δα is trivial. +The next simplest case is where A is a connection for some non-Abelian group. Now the +gauge orbit is A = g−1Ag + g−1dg. Here g = g(α) is defined by the exponential map. For +simplicity we could take g = exp(α) with α valued in the Lie algebra. We may now write +δA = [A, g−1δg] + d(g−1δg) = iVg−1δgδA +(3.10) +so W = Vg−1δg generates a gauge transformation with 1-form valued gauge parameter g−1δg. +We also note that this examples gives a concrete realization of the schematic manipulation +(3.7). +As a final example of W for this section, we consider diffeomorphisms. Take the diffeo- +morphism to be yα = f α(x) and consider for simplicity the diffeomorphism orbit of a scalar +field φ(x) = φ(f(x)). Taking the variation of this scalar field, +δφ = δf α(x)∂φ(y) +∂yα +��� +y=f(x) += δf α(x)∂xµ +∂yα +��� +y=f(x) +∂φ(f(x)) +∂xµ += Lξφ +(3.11) +so the variation is equivalent to the action of the Lie derivative with respect to the 1-form +valued (spacetime) vector field19 +ξµ = ∂(f −1)µ +∂yα +��� +f(x)δf α(x). +(3.12) +Hence W = Vξ. This fact was also noted in [24] where it was used to circumvent some +technical difficulties in JT gravity. +With these examples in mind, there is one additional property of W which will be +important going forward. Denote by w the Lie algebra element whose action W generates. +19In practical calculations, it is much simpler to move the Jacobian factor to the left and compute the +variation of f by a diffeomorphism, then defining ξ by inverting the Jacobian. This is the approach taken in +the examples of Sections 3.5.2 and 3.5.3. +21 + +In our examples we have w = δα for the U(1) gauge group, w = g−1δg for the non-Abelian +gauge group, and for diffeomorphisms we have (3.12). +Importantly, this w is some functional of δα. Equally as important, this functional need +not be invertible as can be seen in the non-Abelian example20. This will be important going +forward because to apply Theorem 1 we need free functions. In our setup δα are free, but w +may or may not be, depending on the details of the gauge group. For example, the relation +between δf and ξ in (3.12) remains invertible despite diffeomorphisms being non-Abelian. +While the relation between w and δα is not invertible, it is injective, and hence is invertible +on the image of all δα. This means δw may always be expressed in terms of only w again. +We can see this in our examples. The U(1) case is uninteresting because δw = 0, but in the +non-Abelian case we find +δw = δ(g−1δg) = −(g−1δg) ∧ (g−1δg) = −w ∧ w, +(3.13) +so the general invertibility of w(δα) is unimportant to this rewriting of δw, as claimed. +The example of diffeomorphisms is slightly more complicated, but nonetheless can be +worked out to find +δξµ = ξν ∧ ∇νξµ, +(3.14) +assuming we are working with a torsionless connection so the affine part of the covariant +derivative does not contribute to this expression. +3.3 +The transfer field method +Now that we have an understanding of the transfer field W, we can give a technique for +computing the boundary symplectic form. From the definition (2.18), it would clearly be +sufficient to show that the bulk δθ term is a total derivative. We may use the transfer field +to write +θ = iWθ +(3.15) +since in every term of θ, the 1-form factor can be replaced as in (3.8). +20Invertibility fails for the same reason A = g−1dg is not invertible for dα: if A is not flat no such dα +exists. +22 + +Since W generates the action of the Lie algebra element w, it follows that +θ = iWθ = Jw + kw +(3.16) +where Jw is the Noether current (2.21) associated to the gauge transformation generated by +w and kw is defined by (2.20). +But as discussed in Section 2.2, Jw = dQw and hence +ω = δθ − dδB = δkw + dδ(Qw − B). +(3.17) +So to find ωb it is sufficient to find a potential for δkw by solving δkw = d˜kw. +Since by (2.19) we have dω = 0 on-shell, we must also have dδkw = 021. Now δkw depends +on the free functions22 δα so we may apply theorem 1 to conclude the existence of a locally +constructed ˜kw. +It should be commented that while δα, and the fact that δα = 0 corresponds to the +identity, ensures the existence of ˜kw, it would be more computationally efficient to use w +as the free function whenever possible. +This can be done when the relation between w +and δα is invertible, as discussed in the previous section. The invertibility in the case of +diffeomorphisms will be leveraged in Section 3.5.2. +Hence we have found +ωb = ˜kw + δ(Qw − B) +(3.18) +which has the advantage of cleanly separating the boundary condition dependence of ωb into +the δB term since ˜kw and Qw depend only on the Lagrangian. +This technique is straightforward to apply to U(1) CS theory on D × R with chiral +boundary conditions. We recall for this theory (2.16), note kλ = kλdA, and consider a +solution A to the equations of motion, so dA = 0. The gauge orbit around this solution is +then A = A + dα. As discussed in the previous section we have w = δα. +Since kλ = 0 on-shell in this theory, we have ˜kw = 0. Then following (3.18), the boundary +21We could also argue more directly that δdkw = δiW δL = δ2L = 0 since iW δL = δL by (3.8). +22In general, the imposition of boundary conditions will restrict what functions α we allow ourselves to +consider near the boundary. However, the only place the boundary conditions enter in these arguments is +through B, and in particular the closure of δkw requires no reference to boundary conditions. So we should +imagine performing these manipulations before imposing any boundary conditions which would then restrict +the allowed δα. +23 + +symplectic form must be +ωb = δQw = kδA ∧ δα +(3.19) +where we have used B = 0. This integrates to +Ω = k +� 2π +0 +dφ ∂φδα ∧ δα +(3.20) +and produces the usual Kac-Moody algebra (2.5) expected for U(1) CS theory on the disk. +Using the Hamiltonian (2.6) and the phase space action (2.36) we reproduce the expected +boundary action (2.4). +3.4 +Computing Ω from the Noether charges +In the previous section we were able to leverage the transfer field to write (3.15). It’s simple +to see that this generalizes to any 1-form on the gauge orbit, not just θ. With some additional +though we can go beyond 1-forms and obtain a similar result for any p-form on the orbit. +To see how this works, consider iW(δφa ∧ δφb) where we have restored the indices a and +b labeling our fields. Computing the contraction we find +iW(δφa ∧ δφb) = (iWδφa) ∧ δφb + δφa ∧ (iWδφb) += 2δφa ∧ δφb +(3.21) +where in the second line we have used the defining relation (3.8) for W. It’s important to +keep track of the sign on the second term because W is 1-form valued and must be commuted +past the first factor. +With this sign understood, the generalization to a form of any degree is immediate: we +simply march through the factors in the wedge product to perform the contraction and then +use the definition of W to rewrite the contraction back in terms of the original form. Hence, +for any p-form Z on phase space we have the generalized identity +iWZ = pZ. +(3.22) +We can leverage this to very efficiently compute the boundary symplectic form ωb on the +gauge orbit directly from the Noether charges. The key is to observe that the contraction +24 + +on an arbitrary 2-form may be rewritten into the form +iW(δφa ∧ δφb) = − +� +δφb ∧ (iWδφa) − δφa ∧ (iWδφb) +� +. +(3.23) +That is, if we make sure to commute the contracted part to the right of the expression, we +obtain the negative of the expression we would have found if W was not valued as a 1-form. +It follows that if we compute iWΩ and make sure to commute all the contractions to the +right, we must obtain +iWΩ = +(δH[ ˜w])| ˜w=w +(3.24) +since we have already argued that W generates the action of a Lie algebra element. Note +that on the right hand side we compute the variation of H before evaluating at the gauge +parameter w. This is because w enters through the contraction on the right and so clearly +cannot have a variation applied to it. +Combining this with our general observation (3.22) it follows that the symplectic form +on the gauge orbit is given by +Ω = 1 +2(δH[ ˜w])| ˜w=w +(3.25) +where the expression on the right should be understood as placing all factors of w to the right +of any variations, and the variation taken treating ˜w as an arbitrary, but not 1-form valued, +Lie algebra element. One must be cautious not to interpret (3.25) as meaning Θ = 1 +2H[w]. +The order in which the variation and the evaluation at ˜w = w occur are important for this +result, as will be seen our examples in Section 3.5. +We may again revisit our U(1) CS example to demonstrate this approach. Indeed, looking +at (3.19) we can already see the basic structure of (3.25) since B = 0 with our chiral boundary +conditions. But going through the details to make sure the coefficients come out correct, +particularly since (3.19) uses the Noether current and not the complete Noether charge, +we start with the charges (2.6). +Then (3.25) tells us to vary the charge, treating ˜w as +any large gauge parameter in the theory, which generally may mean it’s state-independent +or some state-dependent function of some other parameters determining the large gauge +transformations. Here there are no such complications as ˜w is state-independent and we find +1 +2δH[ ˜w] = k +� 2π +0 +˜wδAφdφ. +(3.26) +The final step in computing (3.25) is to evaluate this charge at ˜w = w ≡ δα. But when we +25 + +do so, we must make sure that we first move all ˜w factors to the right of δAφ. Doing so, +Ω = k +� 2π +0 +dφδAφ ∧ δα = k +� 2π +0 +dφ∂φδα ∧ δα +(3.27) +which again matches the expected Kac-Moody expression (2.4). +3.5 +Examples +Here we apply our methods to three examples. First we consider U(1) CS theory in D = 5. +This theory, with our chosen boundary conditions, is only marginally more complex than +the example of U(1) CS theory in D = 3 that we have been considering. It is also simple +enough that we are able to implement the manipulations in Section 2.3 to find (2.34) with +non-gauge directions included. +In Sections 3.5.2 and 3.5.3 we consider Einstein-Hilbert gravity in D = 3 with two +different sets of boundary conditions. +First we consider standard asymptotically AdS3 +boundary conditions and then the boundary conditions described in [29]. The latter are +designed to produce a theory similar to what we will see in topologically massive gravity, +considered in Section 4. +Finally, we mention that the example of SU(2) CS theory in D = 3 is considered in +Appendix D. There we also show that the results of our methods match the standard results +expected from the CS/WZW correspondence. +3.5.1 +U(1) CS in D = 5 +As a first example which isn’t quite as trivial as U(1) CS in D = 3 we consider U(1) CS in +D = 5. Since we work near the boundary the details of the bulk geometry are irrelevant. +We fix the boundary geometry, though the details of this geometry will not be paramount +to most of our manipulations, to be R × S1 × S2. On the boundary we choose coordinates +xi, i = 1, 2, 3, 4 with x1 the coordinate on the R, which we think of as time, x2 the angular +coordinate on S1, and x3, x4 some coordinates on the S2. To make contact with the D = 3 +CS theory, it will sometimes be useful to write t ≡ x1 and φ ≡ x2. +For this example we choose our boundary conditions to fix A3 and A4 to be arbitrary +functions of x3 and x4 on the boundary while we fix A1 = A2 to be an arbitrary function +of x1 and x2. This will allow us to draw parallels to U(1) CS in D = 3 on the cylinder in +the final result. It is useful to note that with these boundary conditions the only non-zero +boundary components of F are F12, which depends dynamically on A1 (and hence A2), and +26 + +F34 which is a fixed function of only x3 and x4. +The Lagrangian for this theory is +L = A ∧ F ∧ F +(3.28) +which varies to produce +δL = 3F ∧ F ∧ δA − 2d(F ∧ A ∧ δA). +(3.29) +Thus the equations of motion, F ∧F = 0, do not imply that all solutions are flat connections. +Considering the condition (2.15), we see that our chosen boundary conditions allow us to +choose ℓ = B = 0. +Since B = 0, the full symplectic current for the theory is given by +ω = −2δ(F ∧ A) ∧ δA. +(3.30) +Now, this theory is sufficiently simple that we are able to explicitly carry out the gauge-orbit +described in Section 2.3. We suppose A is some solution to the equations of motion and we +consider the gauge orbits around this configuration, A = A+dα. It’s straightforward to find +ω = −2δ(F ∧ A) ∧ δA − 2d[δ(αF) ∧ dδα + δ(αF) ∧ δA − δ(F ∧ A) ∧ δα] +(3.31) +so we realize the expected split into the gauge-fixed part and the boundary part depending +on the large gauge transformations parametrizing the orbit. Here we can also see that the +boundary component of the symplectic form supports components which mix variations +in the gauge-fixed configuration with variations along the orbit. Indeed, there is even a +component which involves no variations of α. +But here we are interested only in the components along the orbit directions, so we take +δA = 0 to find +ωb = −2Fδα ∧ dδα. +(3.32) +Integrating this over the boundary and using our boundary conditions the full symplectic +form on the orbit is given by +Ω = −2 +� +∂Σ +F 34δα ∧ ∂2δαdx2 ∧ dx3 ∧ dx4 += 2 +�� +S2 F +� � 2π +0 +∂φδα ∧ δαdφ. +(3.33) +27 + +Because we have chosen our boundary conditions to factor the S2 from the R×S1, we obtain +the same Kac-Moody symplectic form as would be expected from the U(1) CS theory on the +cylinder, but now with an effective level set by our boundary conditions via the magnetic +flux through the S2. +Using that the Hamiltonian generating time evolution is +Ht = 2 +� +∂Σ +F 34A2 +2dx2 ∧ dx3 ∧ dx4 += 2 +�� +S2 F +� � 2π +0 +(Aφ + ∂φα)2dφ. +(3.34) +Constructing the phase space action via (2.36) we thus find +S = 2 +�� +S2 F +� � � +∂φα +� +∂tα − ∂φα − 2Aφ +� +− A +2 +φ +� +dtdφ. +(3.35) +Since we have restricted ourselves to the gauge orbit, α is our only dynamical variable in +this action and, in particular, this means the final A +2 +φ term can be dropped as an additive +constant to the action. In this example it’s possible to observe explicitly from (3.31) that, +had we not restricted to the gauge orbit of some fixed A, the Aφ terms here would represent +an explicit coupling between the bulk, A, and boundary, α, degrees of freedom. +This result can, of course, also be obtained by the techniques introduced in the previous +sections. Taking first the approach of Section 3.3 we note that +iVλδL = dλ ∧ F ∧ F = d(λF ∧ F) +(3.36) +so kλ vanishes on-shell and will make no contribution. For the other ingredient, we calculate +Jλ = iVλθ − kλ = 2d(λA ∧ F). +(3.37) +Thus, since B = 0 with our chosen boundary conditions, we find from (3.19) +ωb = 2δ(A ∧ F) ∧ δα = 2dδα ∧ F ∧ δα +(3.38) +since w = δα for the U(1) orbit, matching (3.32). From this point, imposing the boundary +conditions to find the true symplectic form (3.33), and hence the action (3.35), is unchanged. +If we instead took the route of Section 3.4 we would need to compute the full Noether +28 + +charges +H[λ] = 4 +�� +S2 F +� � +S1 λA +(3.39) +which requires that we use the boundary conditions to obtain. For example, that we fix A3 +and A4 completely on the boundary tells us that λ = λ(x1, x2). The formulation (3.25) now +tells us to write +Ω = 2 +�� +S2 F +� � +δA ∧ δα. +(3.40) +Writing this explicitly in coordinates and with δA = δdα this clearly reproduces (3.33) and +the rest of the boundary action story follows. +3.5.2 +Asymptotically AdS3 Einstein-Hilbert gravity +As a second example, we can derive the Alekseev-Shatashvili symplectic form [30] for asymp- +totically AdS3 Einstein-Hilbert gravity. +All three approaches can be worked out in this +example. We start by recalling some facts this theory. +The action and canonical 1-form for this theory are given by +L = +1 +16πG +√−g(R + 2)d3x, +θ = +1 +16πG +√−g(∇νδgλν − gµν∇λδgµν)(d2x)λ. +(3.41) +Where expressions are more conventionally expressed in terms of the Brown-Henneaux +central charge we write c = 3/2G. We will also need later that +kξ = iξL = − 1 +4πG +√−gξµ(d2x)µ +(3.42) +where we have used the equations of motion to simplify things. +In Fefferman-Graham coordinates the asymptotically AdS3 solutions to the equations of +motion are given by the Ba˜nados metrics [31] +ds2 = dρ2 +4ρ2 + 1 +ρ(dw + ρL(w)dw)(dw + ρL(w)dw) +(3.43) +where w = φ + t and w = φ − t are convenient coordinates on the boundary and ρ > 0 is +the radial coordinate such that the boundary is located at ρ = 0. Note that w is not the +complex conjugate of w as we are working in Lorentzian signature. The boundary stress +29 + +tensor associated to these metrics is given by +Tww = − 1 +4GL, +Tww = − 1 +4GL, +Tww = 0. +(3.44) +The asymptotic vector fields which preserve the boundary conditions are +ξw = ϵ(w) − 1 +2∂2 +wϵ(w)ρ + O(ρ2) +(3.45) +ξw = ϵ(w) − 1 +2∂2 +wϵ(w)ρ + O(ρ2) +(3.46) +ξρ = (∂wϵ(w) + ∂wϵ(w)) ρ + O(ρ0), +(3.47) +where ϵ(w) and ϵ(w) are free functions labeling the vector field. While these vector fields do +not change the conformal metric on the boundary, they do change the components subleading +in ρ, and hence the stress tensor by +iVξδTww = 2Twwϵ′ + T ′ +wwϵ + c +12ϵ′′′ +(3.48) +iVξδTww = 2Twwϵ′ + T ′ +wwϵ + c +12ϵ′′′. +(3.49) +The charges associated with the diffeomorphisms are then easily constructed by +H[ξ] = − 1 +2π +� 2π +0 +(Twwϵ − Twwϵ)dφ. +(3.50) +Going to the gauge orbit means integrating up the variations (3.48) under an infinitesimal +diffeomorphism to a finite one. If the finite diffeomorphism on the boundary is given by +w′ = f(w), +w′ = f(w) +(3.51) +then we have +Tww = c +12 +�κ +2f ′2 + {f, w} +� +, +Tww = c +12 +�κ +2f +′2 + {f, w} +� +(3.52) +where the Schwarzian derivative is +{f, w} = f ′′′ +f ′ − 3 +2 +f ′′2 +f ′2 . +(3.53) +In this orbit, the diffeomorphism f(w) = w and f(w) = w evidently produces constant +Tww, Tww. The κ and κ parametrize what this constant value is, and thereby parametrize +the diffeomorphism inequivalent metrics (3.43). The case κ = κ = 1 produces global AdS. +30 + +At ρ = 0, (3.45) defines a diffeomorphism on the boundary which acts on the f and f as +iVξδf = f ′ϵ, +iVξδf = f +′ϵ. +(3.54) +This can be obtained by composing two diffeomorphisms and we may also verify that +this transformation rule is compatible with (3.48) and (3.52) together. Since we have the +momentum written down now as +P = H[∂φ] = − c +24π +� 2π +0 +dφ +��κ +2f ′ − 1 +2 +� 1 +f ′ +�′′� +f ′ − +� +κ +2f +′ − 1 +2 +� +1 +f +′ +�′′� +f +′ +� +(3.55) +where we have used that {f, w} = − 1 +2 +� +1 +f′ +�′′ +f ′ up to addition of total derivatives, we may +observe that it takes the form (3.2) required for the momentum method. +It therefore follows that the canonical 1-form is given by +Θ = − c +24π +� 2π +0 +dφ +��κ +2f ′ − 1 +2 +� 1 +f ′ +�′′� +δf − +� +κ +2f +′ − 1 +2 +� +1 +f +′ +�′′� +δf +� +. +(3.56) +The variational of this potential can indeed be massaged into the more standard form +Ω = − c +48π +� 2π +0 +��δf ′ ∧ δf ′′ +f ′2 +− κδf ∧ δf ′ +� +− +� +δf +′ ∧ δf +′′ +f +′2 +− κδf ∧ δf +′ +�� +dφ. +(3.57) +of the Alekseev-Shatashvili symplectic form. +Using the canonical 1-form (3.56) in (2.29) together with the Hamiltonian H[∂t] produces +the Alekseev-Shatashvili action +S = − c +24π +� +d2x +� +κf ′∂wf − +� 1 +f ′ +�′′ +∂wf − κf +′∂wf − +� +1 +f +′ +�′′ +∂wf +� +. +(3.58) +We may also obtain these results by the technique in Section 3.4. Since we already know +the charges, the only ingredient still needed in (3.25) is the 1-form valued gauge parameter +w. Since we are working with diffeomorphisms this means we need to work out (3.12). In +this case things are simplified somewhat because we evidently only require the vector field +ξ evaluated at ρ = 0 since the subleading components of (3.45) are not needed to compute +the charge H. +Now, (3.54) gives the variations of f and f under an arbitrary asymptotic diffeomorphism, +31 + +the 1-form valued vector field with parameters ϵδ and ϵδ must satisfy +� +δf +δf +� += +� +f ′ +0 +0 +f +′ +� � +ϵδ +ϵδ +� +. +(3.59) +This is evidently (3.12) evaluated on the boundary and with the Jacobian factor moved to +the other side of the equality. +It now follows that (3.25) gives +Ω = − 1 +4π +� 2π +0 +(δTww ∧ ϵδ − δTww ∧ ϵδ)dφ += − c +48π +� 2π +0 +� +δ +�κ +2f ′2 + {f, w} +� +∧ δf +f ′ − δ +�κ +2f +′2 + {f, w} +� +∧ δf +f +′ +� +dφ. +(3.60) +Integrating by parts this expression can can be shown to equal (3.57). +We can also obtain this result by computing (3.18). For this computation we will need the +potential Qξ for the Noether current Jξ and the potential ˜kξ for δkξ. Much of this discussion +is independent of the choice Λ = −1 and D = 3, so we will leave these undetermined until +they are actually needed to simplify our expressions. The Noether-Wald charge is well known +as the so-called Komar term [32,23], +Qξ = − +1 +16πG +√−g∇µξν(dD−2x)µν. +(3.61) +The computation of ˜kξ is straightforward from (3.42) if we use (3.14). With this we find +16πGδkξ = − 4Λ +D − 2 +� +δξµ√−g + 1 +2 +√−ggαβδgαβ ∧ ξµ +� +(dD−1x)µ += +4Λ +D − 2 +√−g∇λ(ξλ ∧ ξµ)(dD−1x)µ. +(3.62) +From this we identify +˜kξ = − +1 +16πG +2Λ +D − 2 +√−gξµ ∧ ξν(dD−2x)µν. +(3.63) +The boundary symplectic form is therefore given by (3.18) to find +ωb = δQξ + ˜kξ += − +1 +16πG +√−g +� +∇λξλ ∧ ∇µξν + δ(∇µξν) + +2Λ +D − 2ξµ ∧ ξν +� +(dD−2x)µν. +(3.64) +32 + +Computing +δ(∇µξν)(dD−2x)µν = − +� +∇λξµ ∧ ∇λξν − 1 +2R +µν +αβ +ξα ∧ ξβ +� +(dD−2x)µν +(3.65) +the boundary symplectic form becomes +ωb = − +1 +16πG +√−g +� +∇λξλ ∧ ∇µξν − ∇λξµ ∧ ∇λξν − 1 +2R +µν +αβ +ξα ∧ ξβ + +2Λ +D − 2 +� +(dD−2x)µν. +(3.66) +If we now specialize to D = 3 where +R +µν +αβ += δµ +αRν +β − δν +αRµ +β − δµ +βRν +α + δν +βRµ +α − 1 +2R(δµ +αδν +β − δν +αδµ +β) +(3.67) +and use the equations of motion to write +Rµν = 2Λgµν, +R = 6Λ, +(3.68) +(3.66) can be simplified to +ωb = − c +24π +√−g +� +∇λξλ ∧ ∇µξν − ∇λξµ ∧ ∇λξν + Λξµ ∧ ξν� +(d1x)µν. +(3.69) +It’s straightforward to evaluate (3.69) on the metric (3.43) on the computer. We restrict +the vector field ξ to take the form (3.45), where now ϵ and ϵ should be understood as the +1-form valued ϵδ and ϵδ we used earlier and introduced in (3.59). Doing so we find the +φ = 1 +2(w + w) component of (3.69) to be +ωb|∂Σ = +c +24π +1 +ρ∂φ(ϵ ∧ ϵ)dφ ++ +c +48π +� +4Lϵ′ ∧ ϵ − 4Lϵ′ ∧ ϵ + ϵ′ ∧ ϵ′′ + ϵ′′ ∧ ϵ′ − ϵ′′′ ∧ ϵ + ϵ′′′ ∧ ϵ +� +dφ + O(ρ). (3.70) +Since we will integrate this over the circle to form the symplectic form we are free to drop total +derivatives. In particular this means the divergent term above may be dropped. Furthermore, +in the O(ρ0) term the two terms which mix ϵ and ϵ combine to a total derivative. So the +symplectic form factorizes into a holomorphic and antiholomorphic term as expected, +Ω = +c +48π +� 2π +0 +� +(4Lϵ′ − ϵ′′′) ∧ ϵ − (4Lϵ′ − ϵ′′′) ∧ ϵ +� +dφ. +(3.71) +33 + +But recalling (3.44) and (3.48) we have +δL ∧ ϵ = 1 +2(4Lϵ′ − ϵ′′′) ∧ ϵ +(3.72) +and similarly for L. Of course, we identify the RHS here as being precisely the factor which +appears in (3.71). Making the replacement we recover the first line of (3.60). +3.5.3 +New AdS boundary conditions +Here we again consider the Einstein-Hilbert Lagrangian (3.41) in D = 3, but this time we +consider the modified boundary conditions described in [29]. Many of the details in the +calculations here are similar to those which appear in topologically massive gravity, to be +discussed in Section 4, so the manipulations here will be useful as a warmup in addition to +a demonstration of the techniques in Section 3. We write an arbitrary Fefferman-Graham +gauge metric as +ds2 = dρ2 +4ρ2 + 1 +ρ +� +g(0) +ab + ρg(1) +ab + O(ρ2) +� +dxadxb +(3.73) +where again ρ > 0 is the radial coordinate with asymptotic boundary at ρ and xa = (t, φ) +are the coordinate on the boundary with φ ∼ φ + 2π. As before we will use the coordinates +w = t + φ and w = −t + φ for convenience23. +The boundary conditions considered in [29] can be phrased as fixing the components +g(0) +ww = 0, +∂wg(0) +ww = 0, +g(0) +ww = 1 +2, +g(0) +ww = 4G∆ +(3.74) +with ∆ a fixed constant. Importantly, one can check that, while it is slightly modified from +the Gibbons-Hawking-York boundary term, there indeed exists a choice of ℓ such that B = 0 +in (2.15). +The Einstein equations can be solved exactly for metrics obeying these boundary condi- +tions. The analog of the Ba˜nados metrics for this case are +ds2 =dρ2 +4ρ2 + 1 +ρdw(dw + ∂wPdw) + 4G[Ldw2 + ∆(dw + ∂wPdw)2] ++ (4G)2∆ρLdw(dw + ∂wPdw) +(3.75) +where P = P(w) and L = L(w) are free functions parametrizing the solutions. +23These coordinates relate to the coordinates (r, t+, t−) which appear in [29] by ρ = 1/r2 and t+ = w, +t− = −w. Later we will meet a pair of functions (P, L) which were (P, L) in [29]. +34 + +Furthermore, the asymptotic vector fields which preserve these boundary conditions are +given by +ξρ = ρϵ′ +ξw = ϵ + O(ρ2) +ξw = σ − 1 +2ρϵ′′ + O(ρ2) +(3.76) +where ϵ(w) and σ(w) are free functions parametrizing the diffeomorphism. +These diffeomorphisms act on the parameters P and L by +iVξδP = σ + P ′ϵ, +iVξδL = 2Lϵ′ + ϵL′ − 1 +8Gϵ′′′. +(3.77) +The charges associated to these diffeomorphisms can be computed to be +H[ϵ, σ] = +� 2π +0 +dφ +2π +�� +∆P ′2 − L +� +ϵ + ∆(1 + 2P ′)σ +� +. +(3.78) +From (3.77) it’s clear that the ϵ part of the transformation acts just like (3.48) on L while +the σ part of the transformations acts to shift P. This means we can integrate up (3.77) to +find +P = g(w), +L = − 1 +8G{f, w}. +(3.79) +This can be done explicitly by applying the finite diffeomorphism +ρ′ = f ′ρ + G∆f ′′2 +f ′ ρ3 + O(ρ4), +w′ = f + 2G∆f ′′ρ2 + O(ρ3), +w′ = w + g − 1 +2 +f ′′ +f ′ ρ + O(ρ3), +(3.80) +where f(w) and g(w) are free functions, to the background metric +ds2 = dρ′2 +4ρ′2 + 1 +ρ′dw′dw′ + 4G∆dw2. +(3.81) +35 + +Diffeomorphisms act on the orbit parameters by +iVξδf = f ′ϵ, +iVξδg = σ + g′ϵ, +(3.82) +which can be found by composing diffeomorphisms or deducing the transformation rule from +(3.77) and (3.79). Using (3.79) we may also write the momentum as +P = H[1, 1] = ∆ + +� 2π +0 +dφ +2π +� +∆(2 + g′)g′ + 1 +8G{f, w} +� += ∆ + +� 2π +0 +dφ +2π +� +∆g′g′ − 1 +2 +1 +8G +� 1 +f ′ +�′′ +f ′ +� +(3.83) +where in the second line we have used that {f, w} = − 1 +2 +� +1 +f′ +�′′ +f ′ up to the addition of total +derivatives. As was the case in (3.55), this momentum is of the form (3.2) for the momentum +method. The canonical 1-form is given by +Θ = +� 2π +0 +dφ +2π +� +∆g′δg − +1 +16G +� 1 +f ′ +�′′ +δf +� +. +(3.84) +From this and the Hamiltonian Ht = H[1, −1] we obtain the phase space action +S = +� dtdφ +2π +� +∆g′(˙g + g′) − +1 +16G +� 1 +f ′ +�′′ +( ˙f − f ′) +� +. +(3.85) +To instead apply our other techniques, (3.82) supplies the analog of (3.59), now taking +the form +� +δf +δg +� += +� +f ′ +0 +g′ +1 +� � +ϵδ +σδ +� +. +(3.86) +With all of these expressions for this theory we are well positioned to compute the +boundary symplectic form via the methods in Sections 3.3 and 3.4. To make things even +easier, we may note that (3.69) applies for any theory governed by the Einstein-Hilbert +Lagrangian, independent of the boundary conditions we impose, so long as B = 0, which +happens to be the case for this theory. +Performing the evaluation we find that again the divergent 1/ρ term is a total derivative +as it was for asymptotically Dirichlet boundary conditions while the finite term can be +36 + +massaged to yield +Ω = +� 2π +0 +dφ +4π +� +δ(L − ∆P ′2) ∧ ϵδ − 2∆δP ′ ∧ σδ +� +, +(3.87) +which one may check has canonical 1-form (3.84). Furthermore, comparing this expression +to the the charges (3.78) it’s immediately clear that the prescription (3.25) reproduces (3.87) +as well. Thus both the techniques of Sections 3.3 and 3.4 lead to the same phase space action +(3.85). +3.6 +Comparison of techniques +We have presented several techniques for computing actions governing boundary modes. It +is useful to discuss the computational advantages offered by each approach. +When applicable, the momentum method is the most efficient approach. As presented, +it applies to D = 3 diffeomorphism orbits built on bulk solutions that are translationally +invariant in the boundary directions. +The needed inputs are simply expressions for the +boundary Hamiltonian and momentum charges written in terms of the orbit variables. The +prescription involves using integration by parts to write the momentum in a canonical form, +after which the canonical 1-form Θ can be read off. Together with the expression for the +Hamiltonian, the boundary action follows. A great advantage here is that one gets Θ directly, +bypassing the need to solve δΘ = Ω. Also noteworthy is that it can be applied order by order +in perturbation theory, for example by expanding in powers of the boundary fluctuations. +This method proved very effective in [15]. +The other methods we discussed are less efficient but have a wider range of applicability. +Using the transfer vector to invert iV Ω = −δHV requires that we we have computed all of +the Noether charges, rather than just the momentum. Of course this has the advantage that +it is applicable in any dimension and with any type of gauge orbit. Additionally, while the +computation of the Noether charges will generally be sensitive to the boundary conditions +of the theory, once the charges are computed the actual computation (3.25) is immediate. +As seen in the examples (3.60) and (3.87), the resulting form for Ω may not make solving +δΘ = Ω immediate. +The transfer field method is likely the most computationally involved of the three ap- +proaches, but has the unique advantage of offering a clean separation between contributions +depending on the Lagrangian of the theory and the boundary conditions imposed. In the +case one wishes to study a Lagrangian under a variety of boundary conditions, this method +could become more efficient than the others which would require that we recompute the +37 + +Noether charges with each set of boundary conditions. For example, once (3.69) was known +for D = 3 Einstein-Hilbert theory, there was essentially no additional computation required +to apply it in the example in Section 3.5.3 beyond working out the new class of asymptotic +vector fields. +As a final point here, we note that while computing the Noether charges generally involves +solving a condition (2.26) to obtain the charges, each step in the transfer field method can +in principle be completed algorithmically. This opens the possibility that, given θ and kw for +the theory, the computation of both Qw and ˜kw can be automated on the computer, leaving +only the computation of B to obtain an expression for (3.18). +4 +Application to Topologically Massive Gravity +Topologically massive gravity [17] is described by the Einstein-Hilbert action supplemented +with a gravitational CS term, +STMG = +1 +16πGSEH − +l +96πGν SCS +(4.1) +with24 +SEH = +� +d3x√g +� +R + 2 +l2 +� ++ Sbndy +SCS = +� +Tr +� +Γ ∧ dΓ + 2 +3Γ ∧ Γ ∧ Γ +� +. +(4.2) +Here the connection 1-form is Γα +β = Γα +βµdxµ, where Γα +βµ are the usual Christoffel symbols +built out of the metric. We take ν > 1 and henceforth set l = 1. Our interest in applying +our general techniques to this theory is to illustrate various issues that are not present in +simpler examples, in particular the the non-diffeomorphism invariance of the action and the +existence of solutions with relatively exotic “warped” asymptotics. We find that these pose +no obstacle to implementing our general procedure to find the boundary action. +We will be particularly interested in solutions [16] that are the warped analog of the more +familiar Ba˜nados geometries [31], +ds2 = L2 +� +dr2 +r2 + u2 � +dt + ˆKdφ + (r + r−1 ˆL)dφ +�2 +− (r − r−1 ˆL)2dφ2 +� +, +φ ∼= φ + 2π (4.3) +24It is straightforward to check that the boundary action may be chosen such that B = 0 in (2.15). +38 + +where +L2 = +1 +ν2 + 3 , +u2 = +4ν2 +ν2 + 3 . +(4.4) +(4.3) is a solution to STMG for any functions ˆK = ˆK(φ) and ˆL = ˆL(φ). Setting ˆK = ˆL = 0 +gives the warped vacuum solution (the analog of global AdS3); constant values of ( ˆK, ˆL) +can be obtained by quotienting the vacuum solution (analogous to how one obtains BTZ); +( ˆK, ˆL) with nontrivial dependence on φ (analogous to the Ba˜nados geometries) are obtained +by applying asymptotic symmetry transformations to the solutions with constant values. +The solutions (4.3) live in a phase space defined by boundary conditions that are preserved +by the asymptotic coordinate transformations xµ→xµ + χµ which have the large r behavior +χφ = ξφ + 1 +2r2∂2 +φξφ + . . . +χt = ξt − 1 +r∂2 +φξφ + . . . +χr = −r∂φξφ + . . . +(4.5) +Here ξt,φ = ξt,φ(φ) are arbitrary periodic functions of φ. Under an infinitesimal diffeomor- +phism of the form (4.5) we have +δ ˆL = −1 +2∂3 +φξφ + 2 ˆL∂φξφ + ∂φ ˆLξφ +δ ˆK = ∂φξt + ∂φ( ˆKξφ) +(4.6) +Associated to any boundary preserving vector field ξ = ξµ∂µ is a charge H[ξ]. For example, +the charges associated with rigid time and angular translations, i.e energy and angular +momentum, are +M = 1 +uLH[∂t] = +1 +12πG +� 2π +0 +ˆKdφ +J = H[∂φ] = − 1 +6πGuL +� 2π +0 +�� +1 + 1 +u2 +� +ˆL − 1 +4 +ˆK2 +� +dφ +(4.7) +as will be derived below.25 +To derive the charges we can consider SEH and SCS independently of each other. That +is, the action S = C1SEH + C2SCS yields charges Q′ = C1Q′ +EH + C2Q′ +CS, so we can extract +Q′ +EH and Q′ +CS by setting one of C1,2 = 0. This is allowed because the procedure to obtain +Q is linear, and the current JEH is conserved in the SEH theory, and likewise for JCS. +25Original references for the derivation of the charges and asymptotic symmetry algebras include [33–37]. +39 + +4.1 +Einstein-Hilbert contribution +As in (3.61) we have +QEH +χ += −εαβ +φ∇αχβdφ +(4.8) +Evaluating this on (4.3) gives +QEH +χ += Lu(u2 − 1)r2ξφ + 2Lu(u2 − 1) ˆKrξφ + Lu3rξt ++ Lu +� +u2(2 ˆL + ˆK2) + 6 ˆL +� +ξφ + Lu3 ˆKξt + O(r−1) +(4.9) +The term proportional to r2 is constant on the phase space and so will be omitted in what +follows. The same goes for the last term on the first line. On the other hand, there is a +linearly diverging term in the first line which is not constant on phase space; this term will +cancel a similar term in the CS contribution, yielding a finite result as r→∞. +4.2 +CS contribution +The Lagrangian 3-form LCS = Tr(Γ ∧ dΓ + 2 +3Γ3) has variation +δLCS = Eγρδgγρ +√−gd3x + dθCS +(4.10) +with +Eγρ = −∇βRβρ +µνεγµν = −2Cγρ +θCS = Tr(δΓ ∧ Γ) − 2δgκρRρ +δdxκ ∧ dxδ , +(4.11) +where Cγρ is the Cotton tensor and the Ricci tensor Rµν is formed from the Riemann tensor +R = dΓ + Γ2 as Rµν = Rα +µαν. We also recall that in D = 3 the Riemann tensor can be +expressed in terms of the Ricci tensor; in terms of the Riemann two-form this amounts to +the identity +Rα +β = +� +δα +γ Rδβ − gβγRα +δ − 1 +2Rδα +γ gδβ +� +dxγ ∧ dxδ +(4.12) +Following the discussion in appendix C to compute the charges, the Noether current +corresponding to the bulk vector field ξ is +JCS +ξ += iVξθCS − iξLCS − Yξ . +(4.13) +40 + +where Yξ defined via +dYξ = δξLCS − LξLCS += Tr(dv ∧ dΓ) +(4.14) +with vα +β = ∂βξα. We take +Yξ = − Tr(dv ∧ Γ) . +(4.15) +We also have +iVξθCS = ∇β∇γξαdxβ ∧ Γγ +α + Tr(iξR ∧ Γ) − 2(∇κξρ + ∇ρξκ)Rρ +δdxκ ∧ dxδ +iξLCS = Tr(iξΓdΓ − Γ ∧ iξR) +(4.16) +Using these relations along with (4.12), some algebra leads to +JCS +ξ += dQCS +ξ +(4.17) +with +QCS +ξ += ∂γξαΓγ +α + ∇γξαΓγ +α + ξa� +− 4Raδ + Rgaδ +� +dxδ . +(4.18) +The full charge is +HCS +ξ += +� +∂Σ +� +QCS +ξ +− CCS +ξ +� +(4.19) +where CCS +ξ +is obtained by solving δCCS +ξ += iξθ, which can be written explicitly as +δCCS +ξ += (δΓα +κβΓβ +δα − δΓα +δβΓβ +κα)ξκdxδ − 2(δgκρRρ +δ − δgδρRρ +κ)ξκdxδ +(4.20) +To proceed further we need to specify the asymptotic form of the solutions of interest. Taking +the solutions to be of the form (4.3) it is straightforward to compute +QCS +ξ += +� +Lu(u2 − 1)r2ξφ + 2u(u2 − 1)L ˆKrξφ + u(u2 − 1)Lrξt ++ L +� +u +� +u2 − 2 +3 +� +ˆK2 + +� +2u3 − 10 +3 u + 8 +3u +� +ˆL +� +ξφ + Lu +� +u2 − 2 +3 +� +ˆKξt + O(r−1) +� +dφ +CCS +ξ += +� +− u +� +u2 − 2 +3 +� +L ˆKξt + O(r−1) +� +dφ +(4.21) +41 + +Again, we can ignore terms that are constant on phase space. +4.3 +Total charge +The total charge is +Hξ = +1 +16πG +� +∂Σ +� +Qξ − 1 +6ν (QCS +ξ +− CCS +ξ +) +� +. +(4.22) +Noting the cancellation between (non-constant) diverging term, we arrive at a finite result +in the large r limit, +Hξ = +1 +16πG +� +∂Σ +�4 +3uL ˆKξt − 8 +3uL +�� +1 + 1 +u2 +� ˆL − 1 +4 +ˆK2 +� +ξφ +� +dφ . +(4.23) +4.4 +Lower spin gravity formulation +While Einstein gravity in three dimensions can be recast as a CS theory (with gauge group +SL(2, R)×SL(2, R)), this no longer holds in the presence of a gravitational CS term because +the theory is no longer topological. However, the subsector of the theory described by the +solutions (4.3) can be described by a CS theory, namely one with gauge group SL(2, R) × +U(1), so-called “lower spin gravity” [21, 38]. The choice of gauge group is dictated by the +isometry group of the warped vacuum solution. In particular, there is a fairly simple relation +such that the charges and symplectic form of the two theories are mapped to each other. +Since the equations of CS theory are much easier to deal with than those of TMG, this +provides a simpler route to isolating the boundary degrees of freedom. The simplification +arises essentially because the bulk degrees of freedom have been omitted. +The action is +S = k +4π +� +Tr +� +A ∧ dA + 2 +3A ∧ A ∧ A +� ++ k +8π +� +A ∧ dA +(4.24) +Here A is an SL(2, R) connection and A is a U(1) connection. The SL(2, R) generators +obey [Lm, Ln] = (m − n)Lm+n and we use the two-dimensional representation, for which +Tr L1L−1 = −1. +a = (L1 − ˆLL−1)dφ + (ω1L0 + ω2L−1)dt +a = Kdφ + dt . +(4.25) +Here, as is standard, the boundary connections (a, a) are related to bulk connections (A, A) +42 + +by a gauge transformation that encodes the radial dependence, +A = b−1ab + b−1db , +A = bab−1 + bdb−1 , +(4.26) +where b = e− 1 +2 L0 ln r. Gauge transformations act as +δa = dϵ + [a, ϵ] , +δa = dϵ +(4.27) +with ϵ = ϵ1L1 + ϵ0L0 + ϵ−1L−1. The form of (4.25) is preserved by taking ϵ0 = −∂φϵ1 and +ϵ−1 = 1 +2∂2 +φϵ1 − ˆLϵ1, To connect to the metric description we trade (ϵ1, ϵ) for (ξφ, ξt) according +to +ϵ1 = ξφ , +ϵ = ξt + ˆKξφ , +(4.28) +with +ˆK = 4π +k K . +(4.29) +The gauge transformations then act as +δ ˆL = −1 +2∂3 +φξφ + 2 ˆL∂φξφ + ∂φ ˆLξφ +δ ˆK = ∂φξt + ∂φ( ˆKξφ) +(4.30) +reproducing (4.6). +We can now apply relations reviewed in Appendix D for CS theory. Letting V denote the +phase space vector field implementing an infinitesimal gauge transformation with parameters +(ϵ, ϵ) we have iV Ω = −δH[ϵ, ϵ] with +H[ϵ, ϵ] = k +2π +� +∂Σ +Tr(ϵa) + k +4π +� +∂Σ +ϵa += +� 2π +0 +dφ +�� k +2π +ˆL + k +8π +ˆK2 +� +ξφ + k +4π +ˆKξt +� += +� 2π +0 +dφ(Lξφ + Kξt) , +(4.31) +where we have defined L according to +ˆL = 2π +k (L − 2π +k K2) . +(4.32) +43 + +The second line of (4.31) agrees with (4.23) under the identification +k = uL +3G +� +1 + 1 +u2 +� +, +k = −uL +3G . +(4.33) +We conclude that there is a simple relation between the canonical structure of lower spin +gravity and the class of solutions (4.3) to TMG. +One can also build the metrics (4.3) out of the SL(2, R)×U(1) connection, in analogy with +the corresponding relation in ordinary 3D gravity. See [21] for details. Gauge transformations +in the CS theory will map to diffs in the metric formulation, as follows from the fact that +we have already mapped the transformations on the phase space variables ( ˆL, ˆK). +4.5 +Boundary action for warped AdS3 +To compute the boundary action our first task is to obtain expressions for the charges +evaluated on a given orbit. The finite diff functions are written as (Φ, T) with +Φ = φ + ξφ , +T = t + ξt . +(4.34) +The infinitesimal transformations are given in (4.30). The first expression is familiar and +exponentiates via the Schwarzian derivative and the second expression is easily handled to +yield +ˆL(φ) = (∂φΦ)2 ˆL0 − 1 +2{Φ(φ), φ} +ˆK(φ) = ∂φΦ ˆK0 + ∂φT(Φ) +(4.35) +where the constant values ( ˆL0, ˆK0) serve as parameters labelling the orbit under considera- +tion. Translating to unhatted variables, as appear in the charge expression Q = +� +dφ(Lξφ + +Kξt) we have +L(φ) = (∂φΦ)2L0 − k +4π{Φ, φ} + ∂φΦ∂φTK0 + k +8π(∂φT)2 +K(φ) = ∂φΦK0 + k +4π∂φT . +(4.36) +To extract the boundary symplectic potential Θ we use the momentum method. The +44 + +momentum, i.e. generator of φ translations, is +P = H[ξ = ∂φ] = +� 2π +0 +L(φ)dφ . +(4.37) +To apply the momentum method we are instructed to write P in the form +P = +� +(PΦΦ′ + PTT ′)dφ +(4.38) +which is achieved by taking26 +PΦ = L0Φ′ + k +8π +� 1 +Φ′ +�′′ +PT = K0Φ′ + k +8πT ′ . +(4.39) +The symplectic potential is then +Θ = +� +dφ +�� +L0Φ′ + k +8π +� 1 +Φ′ +�′′� +δΦ + +� +K0Φ′ + k +8πT ′ +� +δT +� +. +(4.40) +The Hamiltonian corresponding to time translations is +H = Q[ξ = ∂t] = +� +Kdφ = 2πK0 , +(4.41) +so the kinetic term in the action is obtained by making the replacements (δΦ, δT)→( ˙Φ, ˙T). +The fact that this H is a constant on phase space reflects the fact the solutions in this theory +are time independent. More interesting dynamics are obtained by choosing a Hamiltonian +that generates translations along the vector field ∂t+Ω∂φ. This is appropriate for computing +a partition function of the form Tr e−β(H+ΩP). Such partition functions are the appropriate +ones to consider for two reasons [39]. First, unitary representations of the warped current +algebra have H unbounded from below while P is positive definite, so Ω > 0 is required +for convergence. Second, the black hole solutions (4.3) (with constant ( ˆL, ˆK)) have horizon +26We should note that L(φ) has several terms which contain only first derivatives; as discussed in section +(3.1), since they are purely quadratic their contribution to Ω is easily found by directly solving iV Ω = −δQV +for such terms. +45 + +Killing vector field (defined to be the vector field that vanishes at the bifurcation surface) +ξH = ∂t + Ω∂φ += ∂t + +1 +2 +� +ˆL + ˆK +∂φ +(4.42) +Ω must be positive for a smooth horizon, as can be seen from the expression for the surface +gravity +κ = +� +ˆL +� +ˆL + ˆK/2 +. +(4.43) +As in all of our examples, the boundary action captures all of the information regarding +the warped spacetimes that is dictated by symmetry. For instance, it can be used to compute +correlators of the currents, as well one loop corrections to the partition function coming from +the boundary modes. +5 +Emergent Boundary Modes +As developed so far, boundary field theories arise due to the imposition of boundary con- +ditions that are preserved by some infinite dimensional group of transformations that act +nontrivially at the boundary. However, situations can arise in which the boundary in question +is not a sharp boundary in the sense of terminating the space on which the theory lives, but +rather an interface that connects the original region to an “outer region”. The question is +whether or not the boundary modes survive; that is, are they effectively transported to the +outer region? +This situation arises naturally in the AdS/CFT context. For example, one might have +an AdS3 solution which appears as the near horizon limit of some non-asymptotically AdS3 +solution M. Is the asymptotic symmetry group of the AdS3 region visible at the boundary of +the larger spacetime M? From the dual CFT point of view, the AdS3 region encodes physics +in the deep IR, so one is asking about how to these IR degrees of freedom are realized in +terms of the UV description. +We will flesh out how this works in a particular example. Five-dimensional Einstein- +Maxwell theory (with Λ < 0) admits an asymptotically AdS5 solution that has a near horizon +AdS3×R2 factor which we compactify to T 2 [40]. The T 2 is supported by a nonzero Maxwell +field strength. +We show how the boundary graviton phase space is visible at the AdS5 +boundary. Including also a CS term A∧F ∧F, the AdS3 region supports boundary photons, +46 + +and we again show how these appear in the full description. These results are consistent with +the low energy correlators computed in [41], which are sensitive to the emergent boundary +modes. +5.1 +Background solution +We first briefly summarize the relevant features of the solution; more details may be found +in [40]. The action is +S = − +1 +16πG5 +� +d5x√g +� +R + F MNFMN − 12 +� ++ +k +12πG5 +� +A ∧ F ∧ F +(5.1) +We consider a solution of the form +ds2 = +dr2 +L(r)2 + 2L(r)dx+dx− + e2V (r)dxidxi , +i = 1, 2 +F = bdx1 ∧ dx2 +(5.2) +The function L can be found in terms of V , but numerics are required to find V . We will +only need the asymptotics. As r→∞ we have AdS5 asymptotics, +L(r) = 2r + . . . , +e2V (r) = cV r + . . . +(5.3) +for some constant cV . +As r→0 we have AdS3 × T 2 asymptotics, +L(r) = 2br + . . . , +e2V (r) = 1 + . . . +(5.4) +Fluctuations of the metric and gauge field with polarizations and spacetime dependence +restricted to the AdS3 directions are governed by an effective 3d Einstein-CS theory, so if we +place a boundary in this region we will find the usual 2d boundary photons and gravitons. On +the other hand, the asymptotic symmetry group at the AdS5 boundary is finite dimensional, +which leads to the aforementioned question of how the near horizon boundary modes are +visible in terms of observables computed at the AdS5 boundary. +47 + +5.2 +Boundary photons +5.2.1 +Linearized solutions +We considered a linearized gauge field perturbation of the form +δA = a+(r, x+)dx+ . +(5.5) +It obeys +Le2V ∂a+ +∂r + −2kba+ = 0 +(5.6) +The solution which is smooth at the origin (assuming k > 0) is +a+(r, x+) = ϵ(x+)e +−2kb +� ∞ +r +dr′ +L(r′)e2V (r′) +(5.7) +This has asymptotics: +r→0 : +a+(r, x+) = ϵ(x+)rk + . . . +r→∞ : +a+(r, x+) = ϵ(x+) +� +1 − kb +cV r + . . . +� +. +(5.8) +The 1/r falloff term implies a nonzero boundary current J+ ∼ ϵ(x+). This solution is not +quite what we want however, since normalizable solutions, corresponding to vanishing source +in the dual CFT, should vanish in the large r limit. We can remedy this by performing a +gauge transformation. We first of all write +ϵ(x+) = −∂+λ(x+) +(5.9) +and then perform a gauge transformation with parameter Λ(r, x+) = f(r)λ(x+) where +lim +r→0 f(r) = 0 , +lim +r→∞ f(r) = 1 . +(5.10) +After the gauge transformation we have +a+ + ∂+Λ = ∂+λ(x+) +� +f(r) − e +−2kb +� ∞ +r +dr′ +L(r′)e2V (r′) +� +ar + ∂rΛ = λ(x+)∂rf(r) +a− + ∂−Λ = 0 +(5.11) +48 + +such that all components vanish at both small and large r, as desired. It is apparent that +these modes are not pure gauge, as they carry a nonzero field strength. +5.2.2 +Symplectic form +We now wish to compute the symplectic form restricted to the space of these linearized +solutions. In particular, we compute the full D = 4 + 1 symplectic form for these non-pure +gauge solutions. We will see that the result agrees with what we would get by considering +pure gauge modes living in the near horizon AdS3, with a boundary imposed in that region. +Since all polarizations and spacetime dependence of the fluctuations is confined to three +dimensions it is convenient to dimensionally reduce the action (5.1). Keeping the gauge field +terms, after integrating over the T 2 the action can be written as +S = +bV2 +4πG5 +� +M3 +L +(5.12) +where V2 is the coordinate volume of T 2 and +L = 1 +2ΦF ∧ ⋆F + kA ∧ F +(5.13) +where Φ = b−1e2V . We will consider S = +� +L and tack on the +bV2 +4πG5 prefactor at the end. +Proceeding as usual, we vary the action with respect to the gauge field and write the result +as in (2.13), δL = EAδA + dθ yielding the field equation +d[Φ ⋆ dA + 2kA] = 0 +(5.14) +and the symplectic form Ω = +� +Σ ω with +ω = δθ = −ΦδA ∧ ⋆dδA − kδA ∧ δA +(5.15) +The fluctuation (5.11) takes the form A = a + dΛ where a obeys +Φ ⋆ da + 2ka = 0 . +(5.16) +Using this we have +Ω = k +� +Σ +� +δa ∧ δa − dδΛ ∧ dδΛ +� +. +(5.17) +In the case of interest a = a+dx+ and so the first term vanishes, while the second terms is +49 + +an exact form. Restoring the prefactor, we then obtain the symplectic form +Ω = − kbV2 +4πG5 +� +∂Σ +δΛ ∧ dδΛ . +(5.18) +This symplectic form lives on the AdS5 boundary. However, and this is the main point, the +form of the result is precisely the same as would have been obtained by just considering the +pure CS action +kb +4πG5 +� +A∧F defined on the near horizon AdS3 region. It is furthermore easy +to verify that the boundary currents take the same form as well. This shows very explicitly +how the near horizon boundary photon degrees of freedom are effectively transported to the +AdS5 boundary. The mechanism for this is not entirely trivial; in particular, note that the +fluctuation modes we used in this analysis are not pure gauge in the near horizon region, yet +the symplectic form defined on them agrees with that of pure gauge modes in AdS3. +5.3 +Boundary gravitons +5.3.1 +Linearized solutions +We now consider a fluctuation around the metric in (5.2) by considering +ds2 = dr2 +L2 + 2Ldx+dx− + M(dx+)2 + e2V dxidxi +(5.19) +where we will work to linear order in M = M(r, x+). For r ≪ 1 background metric contains +the AdS3 factor ds2 +3 = +dr2 +4b2r2 + 4brdx+dx−. In this near horizon region, a boundary graviton +is obtained by applying a diff xµ→xµ + ξµ with +ξ = ϵ(x+)∂+ − ∂+ϵ(x+)r∂r − +1 +8b3r∂2 ++ϵ(x+)∂− . +(5.20) +This yields a fluctuation of the form +M = − 1 +2b2∂3 ++ϵ(x+) . +(5.21) +The fluctuation (5.21) is not a solution of the linearized Einstein equations outside the near +horizon region. Our task is to extend (5.21) to a solution in the full spacetime that respects +the AdS5 asymptotics. As shown in [41] such a solution is obtained by multiplying (5.21) by +−2bLc(r), where +Lc(r) = L(r) +� r +∞ +dr′ +L(r′)2e2V (r′) . +(5.22) +50 + +The asymptotics of this function are +r→0 +Lc +0(r) ∼ − 1 +2b +r→∞ +Lc +0(r) ∼ − +1 +4cV r . +(5.23) +So the desired fluctuation mode is +M = b−1Lc(r)∂3 ++ϵ(x+) . +(5.24) +This non-pure gauge solution carries a nonzero AdS5 boundary stress tensor (c = 3/2G3 is +the near horizon Brown-Henneaux central charge), +T++ = − cV c +96π∂3 ++ϵ(x+) +(5.25) +which is the same form as the contribution to the AdS3 boundary stress tensor coming from +the pure gauge mode (5.21). +5.3.2 +Symplectic form +The fluctuation modes found above are not yet in a form suitable for computing the sym- +plectic form. +Indeed, contracting two such perturbations with the general gravitational +symplectic form will give zero; this follows from symmetry considerations, as each perturba- +tion carries two lower + indices and there are no available upper + indices to soak these up. +The issue is that the perturbation is singular at r = 0 due to the breakdown of coordinates +there. We can fix this by performing a compensating diff that zeroes out the perturbation as +r→0 but acts trivially at large r. Let f(r) interpolate smoothly between −1 and 0 in going +from small to large r. We then act with a diff of the form (5.20) except with ϵ(x+) replaced +by f(r)ϵ(x+). We write the combined perturbation as +φ + φf . +(5.26) +By construction, it vanishes as r→0 and is equivalent to our original perturbation at large +r. The symplectic form contracted against two such perturbations is27 +Ω(φ1 + φ1f, φ2 + φ2f) . +(5.27) +27Here we are using the notation Ω(φ1, φ2) = iV2iV1Ω, where Vi denotes the phase space vector field +corresponding to the linearized perturbation φi. +51 + +Now, Ω(φ1, φ2) = 0 as already noted. Also Ω(φ1f, φ2 + φ2f) = 0. This follows since φ1f is a +pure diff mode, and Ω contracted against a pure diff mode localizes to the boundary of Σ, +but φ2 + φ2f vanishes as r→0 and φ1f vanishes as r→∞, so both boundary terms are zero. +All that survives is therefore Ω(φ1, φ2f). This again localizes to the boundary, but now +we get a nonzero boundary term at small r, Since φ1 = −φ1f at small r, this boundary +term is precisely the same expression as appears in the symplectic form of pure diff modes +in AdS3.28 So we once again find that the perturbations defined on the full spacetime carry +the same symplectic form as the pure gauge near horizon modes. +Acknowledgements +We thank Ruben Monten for useful discussions. P.K. is supported in part by the National +Science Foundation grant PHY-2209700. +A +Forms Conventions +For convenience we collect some conventions and useful expressions involving forms which +are used elsewhere in this work. For a general p-form, +ω = 1 +p!ωµ1···µpdxµ1 ∧ · · · ∧ dxµp, +dω = 1 +p!∂νωµ1···µpdxν ∧ dxµ1 ∧ · · · ∧ dxµp += 1 +p!∇νωµ1···µpdxν ∧ dxµ1 ∧ · · · dxµp +(A.1) +so long as the connection ∇ is torsionless. This also implies +1 +p!∂[νωµ1···µp] = +1 +(p + 1)!(dω)νµ1···µp +(A.2) +where the antisymmetrization is defined to include a division by the order of the symmetric +group29. The contraction is defined by +iξω = +1 +(p − 1)!ξνωνµ2···µpdxµ2 ∧ · · · ∧ dxµp, +(iξω)µ2···µp = ξνωνµ2···µp. +(A.3) +28This relative minus sign is cancelled by the minus sign that occurs when we switch from taking the +boundary to be inner versus outer. +29For example these conventions imply (dω)αβ = ∂αωβ − ∂βωα. +52 + +We often find it useful to define +(dD−px)µ1···µp = +1 +(D − p)!ϵµ1···µpν1···νD−pdxν1 ∧ · · · ∧ dxνD−p +(A.4) +and it is useful to note +iξ(dD−px)µ1···µp = ξν(dD−(p+1)x)µ1···µpν. +(A.5) +With these notations the Hodge star is defined by +⋆ω = +� +|g| +p!(D − p)!ωµ1···µpϵµ1···µpνp+1···νDdxνp+1 ∧ · · · ∧ dxνD = +� +|g| +p! ωµ1···µp(dD−px)µ1···µp (A.6) +where ϵµ1···µD is the totally antisymmetric numeric array with entries ±1. With this definition +⋆2 = sgn(g)(−1)p(D−p). +One may also show, assuming torsionless connection, that for p + 1 ̸= D +∇νXµ1···µD−(p+1)ν(dp+1x)µ1···µD−(p+1) = d +� +1 +D − (p + 1)Xµ1···µD−(p+1)ν(dpx)µ1···µD−(p+1)ν +� +. +(A.7) +Note there is no factorial in the coefficient. A useful special case of this is p = D − 2 (so dω +is a (D − 1)-form current) is +∇νXµν(dD−1x)ν = d +�1 +2Xµν(dD−2x)µν +� +. +(A.8) +For the special case p = D, if ω = ∇µjµ� +|g|dDx, then ω = dJ with J = ⋆j = +� +|g|jµ(dD−1x)µ. +Finally, suppose Σ is a codimension-1 (non-null) surface in M and let φ∗ be the pullback +to Σ, as would appear in an integral over Σ. Then +φ∗ �� +|g|(dD−1x)µ +� += σˆnµ Vol(Σ) +(A.9) +where ˆnµ is the unit normal to Σ and σ = ˆnµˆnµ depends on whether Σ is timelike or lightlike. +The orientation choice here is Vol(M) = ˆn ∧ Vol(Σ). +53 + +B +Identically Closed Forms +The Poincar´e lemma ensures that if J is any closed form on spacetime then, at least locally, +there exists a potential Q satisfying J = dQ. Though such conserved currents are common +in physics due to Noether’s theorem, the Poincar´e lemma is not typically useful because it +carries no guarantee that Q can be constructed locally from the fields in our theory. Indeed, +for a general current J, Q will indeed be non-local. The exception to this, of course, are the +Noether currents associated to gauge symmetries. These are currents Jξ which are closed for +every possible set of free functions ξ parametrizing the gauge transformation. This additional +ingredient, closure despite depending on an arbitrary function, is what we need to ensure the +potential Qξ can be constructed as a local functional of ξ and the other fields in the theory, +assuming Jξ was local to begin with. +This result has appeared in the physics literature in [42] and [43,44], the latter referring +to the older mathematical literature on the bivariational complex30 where this result is +established by finding a homotopy operator. While [42] is clear on the recursive algorithm +for constructing the potential Qξ, insights from the more mathematical literature on the +utility of higher Euler operators allow us solve the recursion explicitly in a natural way31. +As most of this work only relies on the existence of an algorithm to compute a local Qξ +and not on its details, we state the result as a theorem here32: +Theorem 1 As elsewhere, suppose M is a D-dimensional spacetime, φ are some collections +of fields over M, and ξ are some functions over M. Let J be a p-form over M (p < D) and +a local functional of φ and ξ, meaning at each point x ∈ M it depends on only φ and ξ, and +finitely many of their derivatives, at x. Suppose further that dJ = 0 for all free functions ξ +and that J = J0 when ξ = 0. Then there exists a local functional Q of φ and ξ such that +J = J0 + dQ. +For the interested reader, we have included a short but pedagogical review of how the +algorithm can be derived, by way of a simple example, in section B.1. The statement of the +algorithmic procedure can be found in section B.2, along with a small notational dictionary +to some other locations in the literature. +30See [45,46] for textbook treatments on this perspective. +31This is essentially deriving the homotopy operator, though we will only show that it constructs local +potentials and not that it obeys the stronger property of defining a homotopy operator. +32The original statement in [42] is slightly more general as the ξ are allowed to be sections of an arbitrary +bundle rather than functions, but this is all we will require and allows us to include the statement about +ξ = 0. +54 + +B.1 +Motivation +The idea underlying this algorithm, in this presentation, relies on a simple observation which +can be demonstrated by considering the special case where J is a 2-form which depends only +linearly on the functions ξ up to the first derivative. It is also conceptually simpler to start +with a potential for J, say ˜Q, rather than J itself33. Writing ˜Q as an expansion in the +derivative order of ξ we have +˜Q = +� +˜Qkαξk + ˜Qµ +kα∂µξk� +dxα += +� +ξkEk( ˜Qα) + ∂µ[ξkEµ +k ( ˜Qα)] +� +dxα +(B.1) +where in the second line we have integrated by parts until no derivatives act directly on the +ξk. We will refer to the coefficients Eµ1···µr +k +( ˜Qα) as the Euler coefficients34. Here we would +have +Ek( ˜Qα) = ˜Qkα − ∂µ ˜Qµ +kα, +Eµ +k ( ˜Qα) = ˜Qµ +kα. +(B.2) +Since the ξk are free functions, the coefficients in the derivative expansion are unique. Since +there are only finitely many derivatives present we can always convert between the derivative +and Euler expansions by integration by parts identities, so the Euler expansion must also be +unique35. +Of course, this is merely a rewriting of ˜Q, but it’s a rewriting with the advantage that +when we take the differential of ˜Q we find +d ˜Q = 1 +1! +� +0 ++ +∂ρ[ξkEk( ˜Qα)] ++ +∂ρ∂µ[ξkEµ +k ( ˜Qα)] +� +dxρ ∧ dxα += 1 +2! +� +ξkEk(d ˜Qρα) ++ +∂µ[ξkEµ +k (d ˜Qρα)] ++ +∂µ∂ν[ξkEµν +k (d ˜Qρα)] +� +dxρ ∧ dxα +(B.3) +where in the second line we have written the generic Euler expansion for d ˜Q = J. But +33This ˜Q will be related to the potential Q we construct in the end, but in general will be different. +The actual algorithm makes no reference to this ˜Q, so it’s only purpose here is as a useful intermediary to +motivate the key ideas. +34This language is inherited from the mathematical literature where the Eµ1···µr +k +would be called the higher +Euler operators as they are generalizations of the Euler-Lagrange operator which produces the equations of +motion from a Lagrangian. +35This is the step which requires the “finite derivative order” part of locality. In [42] it was that the +recursion started with the highest derivative order. +55 + +because the Euler expansion is unique, the aligned terms must match so +1 +2!Ek(Jρα) = 0, +1 +2!Eµ +k (Jρα) = δµ +[ρEk( ˜Qα]), +1 +2!Eµν +k (Jρα) = δ(µ +[ρ Eν) +k ( ˜Qα]). +(B.4) +Throughout this section, antisymmetrization will not include the k index as it’s generically +of a different type. If we were able to invert these equations for the Euler coefficients of ˜Q, +we would have determined ˜Q in terms of J. +Unfortunately, we cannot invert these equations, but we don’t need to because J doesn’t +have a unique potential. So really since our goal is to construct a potential for J, we only +need to reconstruct ˜Q up to a closed form. To do this we want to eliminate the Kronecker +deltas, and we may do so by taking a trace between µ and ρ in (B.4). We find +1 +2!Eµ +k (Jµα) = 1 +2! +� +DEk( ˜Qα) − Ek( ˜Qα) +� +, +1 +2!Eµν +k (Jµα) = +1 +2!2! +� +DEν +k( ˜Qα) − Eν +k( ˜Qα) + Eν +k( ˜Qα) − δν +αEµ +k ( ˜Qµ) +� +. +(B.5) +The overall factors come from the (anti-)symmetrization and we see that we can classify the +types of terms which appear. If either µ or ρ (or both) appear on the delta, we obtain a +term proportional to an Euler coefficient of ˜Q. If both indices do not appear on the delta, +then we obtain a term which still contains a delta. In the case both µ and ρ appear on the +delta we find the coefficient to be the dimension. If only µ is on the delta we obtain a minus +sign36, if only ρ is on the delta then we obtain a plus sign. +We may rewrite (B.5) in the form +Ek( ˜Qα) = +1 +D − 1Eµ +k (Jµα). +Eν +k( ˜Qα) = 2 +DEµν +k (Jµα) + 1 +Dδν +αEµ +k ( ˜Qµ). +(B.6) +The second equation does not specify the Euler coefficient37 of ˜Q in terms of only J, but +observe that if we try to build ˜Q back from these expressions we find +˜Q = +� +1 +D − 1ξkEµ(Jµα) + ∂ν +� +ξk +� 2 +DEµν +k (Jµα) + 2δν +αEµ +k ( ˜Qµ) +��� +dxα. +(B.7) +36That this observation generalizes to higher forms and more derivatives of ξ is not completely obvious, +but is nonetheless true. The proof is essentially an exercise in combinatorics. +37In fact, the inability to invert these equations is related to the fact that we are really defining a homotopy +operator. If the homotopy operator is denoted h, then ˜Q = hd ˜Q + dh ˜Q = hJ + dh ˜Q. The “delta terms” in +our organization of the contraction collects into the second term here as we argue shortly. See e.g. [45] for a +proof that these terms organize into the particular form dh ˜Q. These extra terms are not important to our +particular application. +56 + +So the term preventing us from completely determining the Euler coefficients of ˜Q in terms +of those of J has collected itself into an exact form! Furthermore, we can see that this +will be a generic conclusion because all of the upper spacetime indices (besides µ which is +already contracted) have to contract on derivatives when we construct the form from the +Euler coefficients. In the “delta type” terms, at least one of these derivatives will therefore +contract on the delta and hence on a form index, making the term a total differential. +This means we can always write ˜Q = Q+dα where α is constructed from all of the “delta +type” terms. Since Q, which is thus constructed entirely from the Euler coefficients of J, is +related to ˜Q by an exact form, both are equally good potentials for J and so there is no loss +in taking Q over ˜Q. +The generalization to higher form degrees and arbitrary derivative orders in ξ is now +mostly a matter of making sure we get the coefficients on the various terms in the contraction +correct. This is not completely trivial, but a straightforward combinatorics problem, the +main difficulty resting with the result mentioned in Footnote 36. +B.2 +General algorithm +Suppose that J is an identically closed p-form (p < D) depending on a set of free functions +ξk and finitely many derivatives thereof. We give the algorithm in three steps. First we +state it for the special case where J depends only linearly on ξ and its derivatives. We then +point out the simple generalization to other derivative operators. Finally, we state how the +algorithm for the linear case also gives the solution to the non-linear case. +Since J is linear in ξ and its derivatives, it can always be written in the form +J = +� +Jkα1···αpξk + Jµ +kα1···αp∂µξk + · · · +� +dxα1 ∧ · · · ∧ dxαp += +� +ξkEk(Jα1···αp) + ∂µ[ξkEµ +k (Jα1···αp)] + · · · +� +dxα1 ∧ · · · ∧ dxαp +(B.8) +where in the second line we have integrated by parts to remove all derivatives from ξ. The +Euler coefficients, Eµ1···µr +k +(Jα1···µp), are all the data we need to construct a local potential for +J. Note that the upper indices of the Euler coefficients are assumed to be symmetrized. +With this, the potential is given by +Q = +1 +(p − 1)! +� +r=0 +r + 1 +D − p + r + 1∂µ1 · · · ∂µr[ξkEνµ1···µr +k +(Jνα2···αp)]dxα2 ∧ · · · dxαp +(B.9) +where the upper bound on the sum over r depends on the derivative order J as it’s a sum over +all of J’s Euler coefficients. We also take the convention where r = 0 indicates no derivatives +57 + +should be taken. The factor of (p−1)! can be absorbed into a contraction against the vector +∂ν to write +Q = +� +r=0 +r + 1 +D − p + r + 1∂µ1 · · · ∂µr[ξkEνµ1···µr +k +(i∂νJ)]. +(B.10) +This formula appears elsewhere in the literature using slightly different notation. +In +[44,47] a multi-index notation is used (multi-index notation is also employed in [45,46], but +the conventions are slightly different which makes coefficients look different) wherein (µ) is +a tuple of indices, |µ| is its length, and ((µ)ν) is the concatenation of the tuple (µ) with the +additional index ν. If we further denote Eµ1···µr +k += +δ +δξk +(µ) and i∂νJ = +∂J +∂dxν then (B.10) may be +written compactly as +Q = +|µ| + 1 +D − p + |µ| + 1∂(µ) +� +ξk +δ +δξk +(ν(µ)) +∂J +∂dxν +� +(B.11) +where a summation over the length of the tuple (µ) is to be understood. For example, this +is equation (A36) in [47]. +The generalization from partial derivatives to covariant derivatives of arbitrary type is +described in [42]. In the proof of (B.9) the only property of the derivatives used is that they +obey the product rule and that the upper indices of the Euler coefficients are completely +symmetric. Symmetry of the indices follows trivially for partial derivatives, but for covariant +derivatives we can always work with the symmetrized derivatives at the cost of introducing +field strengths. So if we assume that all derivatives have first been symmetrized, we can +write38 +J = +� +ξkEk(Jα1···αp) + ∇µ[ξkEµ +k (Jα1···αp)] + · · · +� +dxα1 ∧ · · · ∧ dxαp +(B.12) +with potential +Q = +1 +(p − 1)! +� +r=0 +r + 1 +D − p + r + 1∇µ1 · · · ∇µr[ξkE(νµ1···µr) +k +(Jνα2···αp)]dxα2 ∧ · · · ∧ dxαp. (B.13) +For the generalization to non-linear dependence of J on ξ we consider an arbitrary 1- +parameter path ξ(λ) through ξ-space. Denote by Jλ the evaluation of J on ξ(λ). This Jλ is +38Though we use the same symbol for the Euler coefficient, the Euler coefficients computed using +symmetrized covariant derivatives and using partial derivatives will, of course, not generally be the same. +58 + +still identically closed and so it’s λ derivative is as well: +d ˙Jλ = 0 +(B.14) +where we have denoted the λ derivative by a dot. +Now ˙Jλ is an identically closed form which depends only linearly on ˙ξ, which is arbitrary +and independent of ξ(λ). Hence we may apply the (B.13) for the linear case to ˙J using ˙ξ as +our free function to find +Qλ = +� +r=0 +r + 1 +D − p + r + 1∇µ1 · · · ∇µr[ ˙ξkE(νµ1···µr) +k +(i∂νJ)] +(B.15) +which satisfies ˙Jλ = dQλ. The Euler coefficients now, of course, must be understood to have +been computed from integrating by parts on ˙ξk. +Taking the λ integral we find +J1 = J0 + d +� 1 +0 +Qλdλ. +(B.16) +If we choose the flow ξ(λ) such that ξ(0) = 0 and ξ = ξ(1), then theorem 1 is an immediate +consequence. +A useful choice of path is the contracting flow39 ξ(λ) = λξ. In this case the potential for +J is +Q = +� 1 +0 +dλ +� +r=0 +r + 1 +D − p + r + 1∇µ1 · · · ∇µr[ξkE(νµ1···µr) +k +(i∂νJ(λξ))] +(B.17) +where it should be understood that the Euler coefficients used here should be those of ˙Jλ. +Making the notational changes described above (B.11), we see that this expression is (A.9) +in [44]. +39Much like the Poincar´e contracting homotopy, this choice is not invariant under redefinitions of the ξ. +Furthermore, there may be cases where this path is not suitable. We refer the reader to the discussion in [42] +for some considerations in this direction. The contracting flow is always possible when the ξ are functions +on spacetime. +59 + +B.3 +Examples +As a simple first example we may derive the Komar term (3.61). Using (3.41) and (3.42) we +find +Jξ = +√−g +16πG (∇ν(∇µξν + ∇νξµ) − 2∇µ∇νξν) (ddx)µ − iξL. +(B.18) +The potential is simple to compute in this case without using the heavy machinery introduced +above. We need only write +∇µ∇νξν = −Rµ +νξν + ∇ν∇µξν +(B.19) +so +Jξ = +�√−g +8πG Rµ +νξν(ddx)µ − iξL +� ++ +√−g +16πG∇ν(∇νξµ − ∇µξν)(ddx)µ. +(B.20) +The first pair of terms, linear in ξ with no derivatives thereof, vanish on the equations of +motion. The latter pair of terms can be identified as yielding the Komar term via (A.8). +To demonstrate the heavy machinery, note first that we only need the linear case (B.10) +and that only Euler derivatives with at least one upper index (meaning at least one derivative +in the Euler expansion of Jξ) contribute. So any terms linear in ξ with no derivatives, after +putting Jξ in Euler form, will not contribute. Indeed, we can see in the example (B.4) that +these terms must vanish when Jξ is closed40. In particular, this means the iξL term will not +contribute to Qξ, which we also found in (3.67). +Since only the iVξθ terms in Jξ will contribute. +In the remaining terms we need to +symmetrize the covariant derivatives so the Euler coefficients will have symmetrized upper +derivatives. +Here the derivatives are all 2nd order, so the anti-symmetrization will pro- +duce Riemann curvatures with no additional derivatives on ξ. These terms will again not +contribute. The result is now +Jξ = +√−g +16πG∇(α∇β) +�� +gβµδα +k + gαβδµ +k − 2gαµδβ +k +� +ξk� +(ddx)µ + (· · · )kξk. +(B.21) +The relevant Euler coefficients are then +Eα +k (Jξ) = 0, +Eαβ +k (Jξ) = 1 +2 +� +2gαβδµ +k − gαµδβ +k − gβµδα +k +� +(ddx)µ. +(B.22) +40This is actually a familiar statement. Note that the zeroth Euler coefficient is precisely the Euler- +Lagrange operator acting on Jξ. This is then the standard statement that the Euler-Lagrange equations +annihilate total derivatives. +60 + +The reconstruction (B.10) now produces +Qξ = +√−g +16πG +� +0 + +1 + 1 +D − (D − 1) + 1 + 1∇α +� +ξkEαβ +k (i∂βJξ) +�� += +√−g +16πG +2 +3 +1 +2∇α +� +ξk � +gαβδµ +k − gαµδβ +k − gβµδα +k +� +(dd−1x)µβ +� +(B.23) +which is again the Komar term, though this algorithm is obviously not unnecessary for this +example. +Another example, this time non-linear in ξ, would be the calculation (3.62). This example +is again simple enough that we can simply obtain the result (3.63) without applying the +general algorithm. To do this we need only (3.14) to find +16πGδkξ = +4Λ +D − 2 +√−g [ξν ∧ ∇νξµ + ∇νξν ∧ ξµ] (ddx)µ +(B.24) +which is (3.63) upon collecting the total derivative and using (A.8). +To find this result using the general algorithm, we first take a derivative of δkξ to obtain +16πGδ ˙kξ = +4Λ +D − 2 +√−g∇ν +� +˙ξν ∧ ξµ + ξν ∧ ˙ξµ� +(ddx)µ += +4Λ +D − 2 +√−g∇ν +� +˙ξk ∧ (δν +kξµ − δµ +kξν) +� +(ddx)µ. +(B.25) +From this we extract the Euler coefficient +Eν +k(δ ˙kξ) = +1 +16πG +4Λ +D − 2 +√−g ∧ (δν +kξµ − δµ +kξν) (ddx)µ +(B.26) +where we have included a dangling wedge to remind ourselves that when we perform the +potential reconstruction the ˙ξ appears to the left of this coefficient. +Since δkξ = 0 when ξ = 0, we have (B.16) with J0 = 0 if we use the contracting flow +(B.17). The reconstruction (B.17) then yields +16πG˜kξ = +� 1 +0 +dλ +0 + 1 +D − (D − 1) + 0 + 1 +4Λ +D − 2 +√−gξk ∧ λ(δν +kξµ − δµ +kξν)(dd−1x)µν += − 2Λ +D − 2 +√−gξµ ∧ ξν(dd−1x)µν, +(B.27) +as we have already shown should be the case. +61 + +C +Diffeomorphism Charges for Generally non-Covariant +Lagrangians +In section 2.2.2, we reviewed a general procedure for computing the Noether charge associated +to an arbitrary gauge symmetry. In the special case where the gauge transformation is a +diffeomorphism, additional simplifications are possible [26]. +The transformation of the Lagrangian under a diffeomorphism ξ can generally be sepa- +rated into covariant and non-covariant pieces as +kξ = iξL + Yξ. +(C.1) +We may do the same for the action of the diffeomorphism on θ: +LVξθ = Lξθ + ˜Πξ. +(C.2) +With these separations together, (2.24) becomes +dΠξ = diξθ + (˜Πξ − δYξ). +(C.3) +That is, the covariant part of the transformations collect automatically into a total derivative, +so the task of computing Πξ is reduced to just finding a potential Σξ satisfying +dΣξ = ˜Πξ − δYξ. +(C.4) +The computation of Cξ in (2.26) is now +δCξ = iξθ + Σξ − iVξδB. +(C.5) +These considerations only alter the computations we need to find Cξ and do not change +the identification of the charge in (2.27) as (2.28). +D +Non-Abelian CS +Here we consider a non-Abelian CS theory in D = 3. This is an important example and +stress test of our techniques because of the CS/WZW correspondence which, at first glance, +might seem to be in tension with our methods. In particular, recall the well known fact that +one must supply an explicit parametrization in order to reduce a WZW model to an explicit +62 + +boundary theory, and in particular the theory cannot be reduced explicitly to the boundary +in terms of the gauge element g. +On the other hand, the charges for CS theory can be explicitly constructed on the +boundary from g and its derivatives. The construction (3.25) would then seem to suggest +that Ω can also be constructed explicitly from g, and we will see that this is indeed the case. +In order for our construction (2.29) of the boundary action to be consistent with this known +property of WZW models, it must be the case that while Ω can be explicitly constructed +from g, the canonical 1-form Θ cannot. This is a non-trivial requirement and we will see +that it is indeed the case. +So then to define the theory we take the background manifold to be M = R × D where +the spatial slices are disks. The Lagrangian and some useful quantities are +L = tr +� +A ∧ dA + 2 +3A3 +� +, +E = 2F, +θ = − tr(A ∧ δA), +kλ = tr(λdA). +(D.1) +We choose the chiral boundary conditions At = Aφ so there is no need for a boundary +contribution to the action, B = ℓ = 0, in light of (2.15). +The Noether current associated to gauge transformations is +Jλ = tr [d(λA) − 2λF] = d tr(λA) +(D.2) +so Qλ = tr(λA). Thus one may check that the full Noether charges are +H[λ] = 2 +� 2π +0 +dφ tr(λA), +Ht = +� 2π +0 +dφ tr(A2 +φ). +(D.3) +Before attempting to apply our techniques to this theory, we should use the WZW +correspondence to see that the boundary action produced by standard techniques should +be. Since we will need an explicit parametrization, and hence gauge group, we wil choose +for simplicity SU(2) with the parametrization +g = eiα1σ1eiα2σ2eiα3σ3, +z = 2 sin(2α2) +(D.4) +where σk are the Pauli matrices and z turns out to be a convenient definition. +By solving the Lagrange multiplier constraint, the action produced by the Lagrangian +(D.1) becomes the WZW model +S = +� +M +tr +�� +g−1dg +�3� +− +� +tr +� +g−1∂tgg−1∂φg +� +dt ∧ dφ + +� +∂M +tr +� +A2 +φ +� +dt ∧ dφ. +(D.5) +63 + +Using tr(g−1dg)3 = d(zdα1 ∧ dα3) the boundary action may be written +S = −2 +� +∂M +dtdφ [(α′ +1 − zα′ +3)( ˙α1 − α′ +1) + α′ +2( ˙α2 − α′ +2) + α′ +3( ˙α3 − α′ +3)] +(D.6) +where the minus sign comes from identifying the boundary orientation to be dφ ∧ dt, as +commented on in footnote 18. +With this result in mind, since we already have the full Noether charges in hand, the +simplest approach is to compute the symplectic form via (3.25). This produces +Ω = +� 2π +0 +tr +� +δAφ ∧ g−1δg +� +dφ = +� 2π +0 +tr +� +δ(g−1∂φg) ∧ g−1δg +� +dφ. +(D.7) +As promised at the beginning of this section, due to the g−1 factor, this cannot be written +as δ of something directly in terms of g. Instead we must go to the explicit parametrization +(D.4) in order to find the canonical 1-form. +But this is a straightforward calculation to perform, the result being +Ω = −2δ +� 2π +0 +[α′ +1δα1 + α′ +2δα2 + (α′ +3 − zα′ +1)δα3] dφ. +(D.8) +Together with +Ht = −2 +� 2π +0 +� +α′2 +1 + α′2 +2 + (α′ +3 − zα′ +1)α′ +3 +� +dφ +(D.9) +we find the phase space action +S = −2 +� +dt +� +∂Σ +dφ [α′ +1( ˙α1 − α′ +1) + α′ +2( ˙α2 − α′ +2) + (α′ +3 − zα′ +1)( ˙α3 − α′ +3)] . +(D.10) +We could also find this result by the methods in section 3.3. Since we already have Qλ +from (D.2), we only need to compute the potential for δkw with w = g−1δg. In the case +of diffeomorphisms it’s simple to use (3.12) as our free function with respect to which the +potential ˜kw can be constructed. Here, however, w = g−1δg is not a completely free function +and we must use δα as our free function instead. This unfortunately means that we cannot +write down a potential ˜kw without specifying the gauge group. +Using the parametrization (D.4) we find +˜kw = δz ∧ δα3dα1 + δα3 ∧ δα1dz + δα1 ∧ δzdα3. +(D.11) +64 + +Furthermore we may compute +δQw = δz ∧ (dα3δα1 + dα1δα3) + z(dδα3 ∧ δα1 + dδα1 ∧ δα3) ++ 2(δα1 ∧ dδα1 + δα2 ∧ dδα2 + δα3 ∧ dδα3). +(D.12) +Adding these forms we find +δQw + ˜kw = −2δ [dα1δα1 + dα2δα2 + (dα3 − zdα2)δα2] + d(zδα2 ∧ δα1). +(D.13) +Integrating this over the φ circle the total derivative term does not contribute and we +evidently reproduce (D.8), from which the phase space action (D.10) follows. +E +Relation to Schwazrzian Action for JT gravity +From the perspective of the current paper, it is most illuminating to view JT gravity and its +Schwarzian action description [48] as a special case of our approach to 3D gravity. The 3D +origin of the JT/Schwarzian is discussed in [49]. +In 3D we have the Euclidean action +S3 = − +1 +16πG +� +d3x√g3(R3 + 2) − +1 +8πG +� +∂M +d2x +� +h3(K3 − 1) +(E.1) +The 3D class of metrics we consider are +ds2 +3 = dz2 +z2 + +� 1 +z2 + z2 +4 LL +� +dwdw − 1 +2Ldw2 − 1 +2Ldw2 +(E.2) +with w = φ + it, w = φ − it, (L, L) are functions of (φ, t) that take the form +L = {F(φ, t), φ} + κ +2F ′2 +L = {F(φ, t), φ} + κ +2F +′2 +(E.3) +where (κ, κ) are constants and ′ = ∂φ. +At fixed t, the functions (F, F) are elements of +diff(S1). In general, the metrics (E.2) are off-shell; to be on-shell the functions (F, F) must +be, respectively, holomorphic and anti-holomorphic in w. +It is rather tricky to obtain the off-shell action governing (E.2) by direct substitution into +(E.1), in part because the coordinates are not globally smooth. Instead, we apply phase space +methods, viewing (F, F) at fixed time as points on phase space and building an action out of +65 + +the gravitational symplectic form and Hamiltonian on this phase space. This procedure [14] +leads to the Alekseev-Shatashvili action [30], +SAS = − +1 +16πG +� +d2x +� +κF ′∂ ¯wF − +� 1 +F ′ +�′′ +∂ ¯wF + ¯κ ¯F ′∂w ¯F − +� 1 +¯F ′ +�′′ +∂w ¯F +� +. +(E.4) +We now turn to the 2D story. One approach is to KK reduce the 3D action by considering +metrics +ds2 +3 = ds2 +2 + Φ2dt2 , +t ∼= t + 2π +(E.5) +taking the metric components to be t independent. Using +� +M3 +d3x√g3(R3 − Λ) = +� +M2 +d2x√g2Φ(R2 − Λ) − 2 +� +∂M2 +dx +√ +hnµ∂µΦ +(E.6) +where the metric on the 2d boundary is h and nµ is the outward pointing unit normal to the +boundary, along with +√g3K3 = +√ +hΦK2 + +√ +hnν∂µΦ +(E.7) +we find that the 3D action (E.2) reduces to the JT action +S3 = − +1 +16πG2 +� +d2x +√ +hΦ(R2 + 2) − +1 +8πG2 +� +dφ +√ +hΦ(K2 − 1) +(E.8) +with G2 = G/2π. +Coming back to our 3D picture, to be in accordance with (E.5) we should take F = F(φ) +and F = F(φ), along with κ = κ. We are thus considering 3D metrics (generically off-shell) +of the form +ds2 +3 = dz2 +z2 + 1 +z2 +� +1 − Lz2 +2 +�2 +dφ2 + 1 +z2 +� +1 + Lz2 +2 +�2 +dt2 +(E.9) +with L(φ) = {F(φ), φ}+ κ +2F ′2. For this restricted class of off-shell metrics it is straightforward +to evaluate the 3D action S3 on the form (E.9). In particular, we have R2 = −2 so only the +boundary term survives. Cutting off the space at z = zc and taking Φ|z=zc = z−1 +c , we find +the action as zc→0, +S3 = − +1 +8πG2 +� +L(φ)dφ + constant +(E.10) +66 + +which is the Schwarzian action (supplemented with the κ term).41 Alternatively, we can apply +the reduction to the AS action (E.4). After integrating over t and doing some integration +by parts, we find that (E.4) is equal to (E.10). +References +[1] E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 +(1989) 351–399. +[2] S. Elitzur, G. W. Moore, A. Schwimmer, and N. Seiberg, Remarks on the Canonical +Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108–134. +[3] R. M. Wald and A. Zoupas, A General definition of ’conserved quantities’ in +general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027, +arXiv:gr-qc/9911095. +[4] W. Donnelly and A. C. Wall, Entanglement entropy of electromagnetic edge modes, +Phys. Rev. Lett. 114 no. 11, (2015) 111603, arXiv:1412.1895 [hep-th]. +[5] S. Ghosh, R. M. Soni, and S. P. Trivedi, On The Entanglement Entropy For Gauge +Theories, JHEP 09 (2015) 069, arXiv:1501.02593 [hep-th]. +[6] W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 +(2016) 102, arXiv:1601.04744 [hep-th]. +[7] A. Blommaert, T. G. Mertens, and H. Verschelde, Edge dynamics from the path integral +— Maxwell and Yang-Mills, JHEP 11 (2018) 080, arXiv:1804.07585 [hep-th]. +[8] C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical +theories, Three hundred years of gravitation (1986) 676–684. +[9] J. Lee and R. M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) +725–743. +[10] A. Achucarro and P. K. Townsend, A Chern-Simons Action for Three-Dimensional +anti-De Sitter Supergravity Theories, Phys. Lett. B 180 (1986) 89. +[11] E. Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 +(1988) 46. +41The original approach to obtaining this action involved considering cutouts of AdS2 with wiggly +boundary. +Here the boundary is at fixed coordinate location z = zc, and wiggles are instead encoded +in fluctuations of the metric via the function L(φ). +67 + +[12] O. Coussaert, M. Henneaux, and P. van Driel, The Asymptotic dynamics of three- +dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. +12 (1995) 2961–2966, arXiv:gr-qc/9506019. +[13] J. Cotler and K. Jensen, A theory of reparameterizations for AdS3 gravity, JHEP 02 +(2019) 079, arXiv:1808.03263 [hep-th]. +[14] P. Kraus, R. Monten, and R. M. Myers, 3D Gravity in a Box, SciPost Phys. 11 (2021) +070, arXiv:2103.13398 [hep-th]. +[15] S. Ebert, E. Hijano, P. Kraus, R. Monten, and R. M. Myers, Field Theory of Interacting +Boundary Gravitons, SciPost Phys. 13 no. 2, (2022) 038, arXiv:2201.01780 [hep-th]. +[16] D. Anninos, W. Li, M. Padi, W. Song, and A. Strominger, Warped AdS(3) Black Holes, +JHEP 03 (2009) 130, arXiv:0807.3040 [hep-th]. +[17] S. Deser, R. Jackiw, and S. Templeton, Topologically Massive Gauge Theories, Annals +Phys. 140 (1982) 372–411. [Erratum: Annals Phys. 185, 406 (1988)]. +[18] P. Kraus, R. Monten, and K. Roumpedakis, Refining the cutoff 3d gravity/TT +correspondence, JHEP 10 (2022) 094, arXiv:2206.00674 [hep-th]. +[19] Y. Nutku, Exact solutions of topologically massive gravity with a cosmological constant, +Class. Quant. Grav. 10 (1993) 2657–2661. +[20] M. G¨urses, Perfect Fluid Sources in 2+1 Dimensions, Class. Quant. Grav. 11 no. 10, +(1994) 2585. +[21] D. M. Hofman and B. Rollier, Warped Conformal Field Theory as Lower Spin Gravity, +Nucl. Phys. B 897 (2015) 1–38, arXiv:1411.0672 [hep-th]. +[22] A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, +JHEP 02 (2010) 029, arXiv:0712.0155 [hep-th]. +[23] G. +Comp`ere +and +A. +Fiorucci, +Advanced +Lectures +on +General +Relativity, +arXiv:1801.07064 [hep-th]. +[24] D. Harlow and J.-Q. Wu, Covariant phase space with boundaries, JHEP 10 (2020) 146, +arXiv:1906.08616 [hep-th]. +[25] R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 no. 8, (1993) +R3427–R3431, arXiv:gr-qc/9307038. +68 + +[26] Y. Tachikawa, Black hole entropy in the presence of Chern-Simons terms, Class. Quant. +Grav. 24 (2007) 737–744, arXiv:hep-th/0611141. +[27] S. R. Coleman, J. Wess, and B. Zumino, Structure of phenomenological Lagrangians. +1., Phys. Rev. 177 (1969) 2239–2247. +[28] C. G. Callan, Jr., S. R. Coleman, J. Wess, and B. Zumino, Structure of phenomenological +Lagrangians. 2., Phys. Rev. 177 (1969) 2247–2250. +[29] G. Comp`ere, W. Song, and A. Strominger, New Boundary Conditions for AdS3, JHEP +05 (2013) 152, arXiv:1303.2662 [hep-th]. +[30] A. Alekseev and S. L. Shatashvili, Path Integral Quantization of the Coadjoint Orbits +of the Virasoro Group and 2D Gravity, Nucl. Phys. B 323 (1989) 719–733. +[31] M. Ba˜nados, Three-dimensional quantum geometry and black holes, AIP Conf. Proc. 484 +no. 1, (1999) 147–169, arXiv:hep-th/9901148. +[32] V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical +black hole entropy, Phys. Rev. D 50 (1994) 846–864, arXiv:gr-qc/9403028. +[33] C. Nazaroglu, Y. Nutku, and B. Tekin, Covariant Symplectic Structure and Con- +served Charges of Topologically Massive Gravity, Phys. Rev. D 83 (2011) 124039, +arXiv:1104.3404 [hep-th]. +[34] A. Bouchareb and G. Clement, Black hole mass and angular momentum in topologically +massive gravity, Class. Quant. Grav. 24 (2007) 5581–5594, arXiv:0706.0263 [gr-qc]. +[35] G. Compere and S. Detournay, Semi-classical central charge in topologically massive +gravity, Class. Quant. Grav. 26 (2009) 012001, arXiv:0808.1911 [hep-th]. [Erratum: +Class.Quant.Grav. 26, 139801 (2009)]. +[36] G. Compere and S. Detournay, Boundary conditions for spacelike and timelike warped +AdS3 spaces in topologically massive gravity, JHEP 08 (2009) 092, arXiv:0906.1243 +[hep-th]. +[37] M. Blagojevic and B. Cvetkovic, Asymptotic structure of topologically massive gravity +in spacelike stretched AdS sector, JHEP 09 (2009) 006, arXiv:0907.0950 [gr-qc]. +[38] T. Azeyanagi, S. Detournay, and M. Riegler, Warped Black Holes in Lower-Spin Gravity, +Phys. Rev. D 99 no. 2, (2019) 026013, arXiv:1801.07263 [hep-th]. +[39] S. Detournay, T. Hartman, and D. M. Hofman, Warped Conformal Field Theory, Phys. +Rev. D 86 (2012) 124018, arXiv:1210.0539 [hep-th]. +69 + +[40] E. D’Hoker and P. Kraus, Magnetic Brane Solutions in AdS, JHEP 10 (2009) 088, +arXiv:0908.3875 [hep-th]. +[41] E. D’Hoker, P. Kraus, and A. Shah, RG Flow of Magnetic Brane Correlators, JHEP 04 +(2011) 039, arXiv:1012.5072 [hep-th]. +[42] R. M. Wald, On identically closed forms locally constructed from a field, Journal of Math- +ematical Physics 31 no. 10, (1990) 2378–2384, https://doi.org/10.1063/1.528839. +https://doi.org/10.1063/1.528839. +[43] G. Barnich, F. Brandt, and M. Henneaux, Local BRST cohomology in gauge theories, +Phys. Rept. 338 (2000) 439–569, arXiv:hep-th/0002245. +[44] G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation +laws and central charges, Nucl. Phys. B 633 (2002) 3–82, arXiv:hep-th/0111246. +[45] P. J. Olver, Applications of Lie Groups to Differential Equations, . Graduate Texts in +Mathematics. Springer New York, NY, 1986. +[46] I. M. Anderson, The Variational Bicomplex, . Utah State University, 1989. +[47] G. Compere, Symmetries and conservation laws in Lagrangian gauge theories with +applications to the mechanics of black holes and to gravity in three dimensions, . PhD +thesis, Brussels U., 2007. arXiv:0708.3153 [hep-th]. +[48] J. Maldacena, D. Stanford, and Z. Yang, Conformal symmetry and its breaking +in two dimensional Nearly Anti-de-Sitter space, PTEP 2016 no. 12, (2016) 12C104, +arXiv:1606.01857 [hep-th]. +[49] T. +G. +Mertens, +The +Schwarzian +theory +— +origins, +JHEP +05 +(2018) +036, +arXiv:1801.09605 [hep-th]. +70 + diff --git a/atE1T4oBgHgl3EQfKgMk/content/tmp_files/load_file.txt b/atE1T4oBgHgl3EQfKgMk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..12031403d9888aa052a24f211f1474d31e8430d6 --- /dev/null +++ b/atE1T4oBgHgl3EQfKgMk/content/tmp_files/load_file.txt @@ -0,0 +1,2432 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf,len=2431 +page_content='Systematics of Boundary Actions in Gauge Theory and Gravity Seolhwa Kim1, Per Kraus1, Richard M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Myers1 1Mani L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Bhaumik Institute for Theoretical Physics Department of Physics & Astronomy, University of California, Los Angeles, CA 90095, USA Abstract We undertake a general study of the boundary (or edge) modes that arise in gauge and gravitational theories defined on a space with boundary, either asymptotic or at finite distance, focusing on efficient techniques for computing the corresponding boundary action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Such actions capture all the dynamics of the system that are implied by its asymptotic symmetry group, such as correlation functions of the corresponding conserved currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Working in the covariant phase space formalism, we develop a collection of approaches for isolating the boundary modes and their dynamics, and illustrate with various examples, notably AdS3 gravity (with and without a gravitational Chern-Simons terms) subject to assorted boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='02964v1 [hep-th] 8 Jan 2023 Contents 1 Introduction 2 2 Review of Boundary Actions 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1 U(1) CS theory on M = D × R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 57 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 Examples .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 60 C Diffeomorphism Charges for Generally non-Covariant Lagrangians 62 D Non-Abelian CS 62 E Relation to Schwazrzian Action for JT gravity 65 1 Introduction The purpose of this paper is to systematically understand the mechanism by which gauge theories1 defined on spaces with boundaries (either at finite distance or asymptotic) are found to host local degrees confined to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' These boundary degrees of freedom are governed by a boundary action and we aim to develop general and efficient methods for calculating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The simplest and most familiar example is provided by the Chern- Simons/WZW correspondence [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The existence of boundary modes is tied to the crucial distinction between those gauge symmetries that act nontrivially at the boundary versus those which are suitably localized away from the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The former take the system from one point in phase space (or from one quantum state) to another, while the latter do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We refer to these as large and small gauge transformations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For example, in the case of General Relativity in asymptotically flat spacetime two black hole configurations, one at rest and one in uniform motion, are related by a large coordinate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Large gauge transformations are generated by nontrivial conserved charges, and the large versus small distinction implies that such charges can be expressed as boundary integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This fact has as its most elementary incarnation the flux integral expression for charge in electrodynamics, and finds its most general expression in the covariant phase space formalism [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Of interest to us in this work are cases in which the large gauge transformations are associated to physical degrees of freedom localized at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In a canonical formu- lation, physical degrees of freedom are nonzero modes of the symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In a gauge theory the symplectic form breaks up into bulk and boundary pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Boundary degrees of freedom are zero modes of the bulk part of the symplectic form, but not of the boundary 1We use the term “gauge symmetry” to denote any local symmetry, including Yang-Mills and diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 2 part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The boundary modes are furthermore governed by a boundary action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Particularly interesting is the case in which the group of large gauge transformations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='e the asymptotic symmetry group, is infinite dimensional, in which case there is a boundary field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' One motivation for our work is that these boundary field theory degrees of freedom are every bit as physical as any other and so should be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Although we do not discuss it here, boundary modes also play an import role in attempts to formulate entanglement entropy in gauge theory and gravity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' a few references include [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In the case of Chern-Simons (CS) theory there is a simple well known procedure for obtaining the WZW boundary action [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In CS theory the Gauss law constraint is the statement that the spatial components of the field strength vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The general solution to this constraint is obtained by writing the gauge fields as a gauge transformation of a given flat connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Inserting this back into the action the CS Lagrangian becomes a total derivative and the resulting boundary term is the WZW action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2 This strategy is not necessarily so easy to carry out in other theories, in particular if there is no easy way to find the general solution of the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' A main objective of this paper is to develop more widely applicable methods for deducing boundary actions, illustrated by explicit examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To this end we primarily work in a covariant phase space framework [8,9,3], where the main actors are the symplectic form and the boundary charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We take as examples three-dimensional gravity, possibly supplemented with a gravitational CS term, subject to various boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As is well known [10,11], pure 3D gravity with AdS3 boundary conditions admits a CS formulation to which the basic CS/WZW procedure above can be applied (with some modifications due to the change of boundary conditions), and one obtains a theory of boundary gravitons [12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The metric formulation version of this procedure was worked out in [14,15], and extended to the case of a finite cutoff boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Here we illustrate our more general methods by considering non-AdS boundary conditions, including those of warped AdS3 [16] supported by the gravitational CS term [17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' note that in general the addition of the gravitational CS term breaks the correspondence with the usual SL(2, R) × SL(2, R) CS formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We now briefly summarize our various approaches to deriving boundary actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In the canonical approach followed here, the boundary action is built out of a canonical 1-form Θ, whose exterior variation yields the symplectic form Ω, and the boundary Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' These objects are defined on a phase space which consists of a particular gauge orbit, obtained by acting on a chosen background solution with all possible boundary condition preserving gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The basic point is that on a given orbit the symplectic form as well as the charges generating the large gauge transformations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' asymptotic symmetries) all localize 2To get a fully explicit two-dimensional boundary action one needs to choose an explicit parametrization of the gauge group elements, as we review in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 3 to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' If the boundary charges QV associated to large gauge transformations V can be computed, the symplectic form may in principle be found by solving the equation iV Ω = −δQV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Solving this can be quite laborious, and one still needs to compute the potential Θ for Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For the class of examples that arise in AdS3 gravity with various boundary conditions, it is possible to bypass all this and pass directly from the conserved (angular)momentum and Hamiltonian to the boundary action, as was employed in the example of cutoff AdS3 gravity in [15] leading to a boundary Nambu-Goto action, whose origin was clarified in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As this approach may not always be possible, we also develop more general methods for computing Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' These are based on identifying a phase space 1-form valued vector field W, which we refer to as the transfer field, which obeys the relation δφ = iWδφ when δφ is restricted to a single gauge orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Given knowledge of W and of the charges Q we show how one can use these to read off the boundary symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Furthermore, we demonstrate another technique which is somewhat less efficient for computing Ω, but has the advantage of allowing one to sometimes obtain expressions for the boundary contributions to Ω independent of the chosen boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We describe when this can be done, most notably for diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As an example, the Einstein-Hilbert action in any dimension with any cosmological constant always produces the contribution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='66) to the boundary symplectic form, independent of the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The main example we use to illustrate our general methods is warped AdS3 [19,20,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is a well-studied solution of topologically massive gravity (TMG) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The warped asymptotics make it less obvious a priori on what surface the boundary action should be thought of as living.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We will work out the boundary action in detail, and also show how the same results may be obtained via “lower spin gravity” [21] which is a CS formulation that can be used to describe a subsector of the full TMG phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' These boundary actions are important inasmuch as the the boundary modes are part of the dynamical degrees of freedom of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For example the boundary photons and gravitons arising in the CS and AdS3 gravity theories contribute to the thermal partition function [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' However, at first sight the physical relevance of these modes may seem elusive, given that they are generated by performing gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This point is clarified by coupling another system to the theory containing the boundary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We give a simple example of this in which a boundary scalar couples to a CS theory defined on a spatial disk, showing how correlators of the boundary scalar are modified by the coupling to the boundary photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Another situation occurs when the boundary is not the true “end” of the spacetime, but rather an interface marking a transition between two different regions with distinct asymptotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It is interesting to ask whether and how the modes that “would have been 4 there” had the interface been an actual boundary manifest themselves in the full system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' One can think of this as a version of the setup described in the previous paragraph, where one side of the transition region now functions as the additional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This situation arises very naturally in gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For example, one can have a solution with a near horizon AdS3 region, which by itself supports boundary modes, embedded inside an asymptotically AdSD>3 solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The latter solution has a finite dimensional asymptotic symmetry group, so one may wonder whether the near horizon boundary modes are detectable at the asymptotic boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We answer this question in the affirmative, showing how the pure gauge modes in the near horizon are promoted to non-pure gauge modes in the full spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It is worthwhile to clarify our usage of certain terminology in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In partic- ular, when we refer to a gauge transformation, depending on context we may or may not distinguish whether we mean small or large gauge transformations, and likewise for diffeo- morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' When this matters, which is often, we will distinguish the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Recall that small gauge transformations/diffeomorphisms describe redundancies, and in a canonical framework are zero modes of the symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Large gauge transformations/diffeomorphisms instead move us between distinct points in phase space, and are nonzero modes of the symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Finally, we occasionally use the term “local symmetry”, which is meant to encompasses both gauge and diffeomorphism symmetry, whether small or large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The usage should always be clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In Section 2, after quickly reviewing the standard approach to U(1) CS theory on a disk and the physical relevance of boundary modes, we go on to a general discussion of the origin and identification of boundary modes within the framework of the covariant phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In Section 3 we develop specific methods for computing boundary actions, illustrated through particular examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In Section 4 we consider the case of warped AdS3 asymptotics in topologically massive gravity, which is a useful and nontrivial example to illustrate various issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We obtain the boundary action in both the metric formulation and in the so-called lower spin gravity formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Section 5 discusses how boundary modes can appear in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' A series of appendices lay out some conventions, review an important theorem regarding identically closed forms, review the proper method for handling non-diffeomorphism invariant actions in the Wald formalism, review non-Abelian CS theory and apply our methods to it, and explain the connection of our 3D gravity results to 2D JT gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 5 2 Review of Boundary Actions In this section we discuss general aspects of boundary modes, their origin in terms of large gauge transformations, and the construction of an action that describes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1 U(1) CS theory on M = D × R To get oriented, we first quickly review the simple and classic example of Chern-Simons theory on a spatial disk and the corresponding boundary gauge modes, following the original Lagrangian approach [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This approach is based on solving the Gauss law constraint and substituting back into the action, yielding a total derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' While this method works well here, it is not so easy to adapt to other examples such as gravity in the metric description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For this reason we go on to develop more flexible methods based on a covariant phase space analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The action for for Abelian CS theory on a spatial disk cross time is S = k � M A ∧ dA + Sbndy = −k � M d3x(Ar∂tAφ − Aφ∂tAr + 2AtFφr) + S′ bndy , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1) where we integrated by parts and absorbed the boundary term into S′ bndy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We choose boundary conditions δ(At − Aφ)|∂M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Specializing to the case (At − Aφ)|∂M = 0, a good variational principle is achieved by taking Sbndy = 0, S′ bdy = −k � ∂M dtdφA2 φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) At is a Lagrange multiplier enforcing the constraint Fφr = 0, which is solved by writing Aφ = ∂φα , Ar = ∂rα , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) with α(r, t, φ + 2π) = α(r, t, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Plugging this back into the action, the bulk terms become a total derivative, and we arrive at the chiral boson action S = k � ∂M dtdφ(∂φα∂tα − ∂φα∂φα) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) 6 The basic equal time Poisson-Dirac bracket is {α(φ), ∂φα(φ′)} = 1 2kδ(φ − φ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) The Hamiltonian and charges generating infinitesimal gauge transformations by λ are Ht = k � dφ(∂φα)2, H[λ] = 2k � 2π 0 λ∂φαdφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6) The theory describes a U(1) current J = k∂φα whose Fourier modes obey a U(1) current algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The current-current-correlator is GJJ(w) = ⟨J(w)J(0)⟩ = − k 4 sin2 � w 2 � , w = φ + t, w = φ − t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='7) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1 Physical relevance of boundary modes Inasmuch as the preceding analysis shows that boundary modes can carry nonzero energy and momentum, they are established as being nontrivial physical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Nonetheless, their “pure gauge” character leads one to wonder, at least upon first hearing, whether they might be ignorable in some sense, for example by decoupling from the rest of the physical system in which they are embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' However, it is not hard to show that the boundary modes do have measurable consequences on other observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To expose these effects we can think of coupling the theory in the disk region to some external system comprised of charged matter that couples to the CS gauge field at the boundary of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In the simplest incarnation we can take the system to live on the boundary of the disk, and to be completely explicit we consider a charged scalar field example, S = k � M A ∧ dA + � ∂M d2x(DµΦ)∗DµΦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8) The covariant derivative is taken to correspond to a gauging of the scalar shift symmetry, DµΦ = ∂µΦ − iqAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is convenient, since the associated current, Jµ = i(∂µΦ − ∂µΦ∗) decomposes into dimension (1, 0) and (0, 1) operators, which is not the case for the current associated to phase rotations of the scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Repeating the previous steps we arrive at the action S = � d2x [−2k∂φα∂wα + 4∂wΦ∗∂wΦ + 2iq∂φα(∂wΦ − ∂wΦ∗)] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='9) with ∂w = 1 2(∂φ + ∂t), ∂w = 1 2(∂φ − ∂t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The coupling of α to Φ has no effect on the energy 7 spectrum of the theory, as follows from the fact that the coupling can be removed by a redefinition of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' There is a nontrivial effect on scalar correlators, in particular on the two- point function of the current Jw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This effect reflects the fluctuating phase acquired by a charged particle on traveling between the two operator insertion points on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Treating q as a perturbation, we can readily sum up contributing diagrams by performing Wick contractions, resulting in G(q) JJ(p) = ⟨J−(p)J−(−p)⟩q = ∞ � m=0 (−4q2)m[GJJ(p)]m+1[Gαα(p)]m = GJJ(p) 1 + 4q2GJJ(p)Gαα(p) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='10) where the q = 0 correlators are GJJ(p) = ⟨Jw(p)Jw(−p)⟩q ∼ pw pw Gαα(p) = ⟨∂φα(p)∂φλ(−p)⟩q ∼ pφ kpw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='11) At q = 0 the correlator behaves as pw/pw corresponding to 1/ sin2( w 2 )2 in position space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As q→∞ the correlator tends to zero, with leading behavior k q2 pw pφ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' being polynomial in pt, this vanishes for unequal times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The point to be emphasized here is that the boundary modes leave a detectable imprint on the scalar correlators, and so are clearly “real.” 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2 Review of covariant phase space formalism We begin with a brief review of the covariant phase space formalism, which will also serve to establish notation for the remainder of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Along the way, we make comments about precisely where boundary conditions enter the formalism, as these will be useful to keep in mind later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For a more detailed review, see for example [8,3,23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1 Action and covariant phase space We consider a theory defined on a D = d + 1 dimensional spacetime M which admits a foliation by codimension-1 slices which we will generally denote by Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The, potentially asymptotic, boundary structure of M can then be decomposed into ∂M = Σ+∪Γ∪Σ− where Σ± are the slices in the asymptotically far future and past and Γ is formed by unioning the 8 boundaries of all the slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' On this spacetime we consider a theory with fields φ whose dynamics are described by an action S[φ] = � M L + � ∂M ℓ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='12) where L is the D-form Lagrangian and ℓ is some allowed d-form boundary contribution3, which we assume to be local functionals of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Throughout we will use δ to denote the exterior variational derivative, which we refer to as the variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Particular infinitesimal transformations of the fields will be thought of as vector fields V on field space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The action of a vector V on a field φ will then be denoted by the contraction4 iV δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' By integrating by parts, the variation of the Lagrangian may always be written δL = E ∧ δφ + dθ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='13) for some θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Setting E = 0 will be our equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We note that θ is always ambiguous up to addition of a d-closed form which we will return to shortly5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Using this identity, the variation of the action is given by δS = � M E ∧ δφ + � Σ+−Σ− (θ + δℓ) + � Γ (θ + δℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='14) In order to have a good variational principle we require that the on-shell variation have no support on Γ which then requires (θ + δℓ)|Γ = dB (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15) for some B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Of course, this B can always be absorbed into a redefinition of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' However, we note that for a generic theory, the LHS above will not automatically take the form of a total derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Instead, there will be terms which only vanish upon the imposition of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This means that the B here generally depends on the boundary conditions we 3Though we use the same symbol ℓ for the boundary contribution over all of ∂M, there need be no relation between ℓ on Γ and ℓ on Σ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Shifts in this ℓ on either always produce shifts in the canonical 1-form by something δ exact and so do not change the symplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 4Though here we prefer the contraction on variation notation to denote infinitesimal transformation, the reader may find it helpful to recall that the following are equivalent: iV δF[φ] = LV F[φ] = V (F[φ]) where LV denotes the Lie derivative on field space and V (F[φ]) is the action of the vector field V on the function F on field space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 5In the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [3], it is often mentioned that θ is also ambiguous up to addition of a δ-closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' While this is true, any such shift in θ is equivalent to shifting ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' So we take the perspective that θ has no δ ambiguity, but ℓ remains to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 9 choose for our theory, and the existence of B may impose conditions on what we choose for ℓ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As a simple example, starting from the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1) we find θ = −kA ∧ δA, B = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='16) with chiral boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It will be useful later to keep explicit which objects depend on the boundary conditions and which do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' So while it is possible to absorb B into a redefinition of θ, we will refrain from doing so in order to avoid reference to boundary conditions when writing θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It’s also useful to observe that having a good variational principle is equivalent to slice independence of the symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The potential for the symplectic form is always found by extracting what remains of the action’s variation from the initial and final time slices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' we write A to denote this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Here we have δS = � Σ+−Σ− A = � Σ+−Σ− (θ + δℓ − dB) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='17) after the imposition of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This means we should choose our symplectic form to be7 Ω = � Σ ω = � Σ δA = � Σ δ(θ − dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='18) Though the ultimate argument for using this object as our symplectic form will be that it produces the desired Poisson brackets, we can see an immediate benefit by taking a second variation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='17), which implies that this Ω is independent of the slice we choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It is, however, useful to observe that we can show more directly that the slice indepen- dence of Ω is precisely equivalent to the demand (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15), and hence the demand for a good variational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To see this we take a second variation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='13) to find −δ(E ∧ δφ) = dδθ = dω (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='19) so the symplectic current is closed on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Hence Ω(Σ) is independent of the slice Σ if and only if the pullback of ω to Γ vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This demand can be rewritten as ω|Γ = δ(θ|Γ − dB) = δ[(θ + δℓ)|Γ − dB] so the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15), obtained from demanding a good variational principle, is equivalent8 to the slice independence of the symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 6A standard example of this would be the need to include the Gibbons-Hawking-York term in the Einstein-Hilbert action with Dirichlet boundary conditions, though in that case we may choose B = 0 depending on our gauge fixing, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [24] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 7Throughout this work we ignore the complications that come from the possibility of non-trivial phase space topology, including the possibility of a symplectic form with non-trivial De Rahm cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 8Strictly speaking, the slice independence of the symplectic form only implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15) up to a δ-closed 10 As a final comment about this definition for the symplectic form, we should discuss the distinction between the phase space and prephase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We will generally take prephase space to consist of all configurations of the fields which obey the boundary conditions and the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' On this space the quantity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='18) will generally be degenerate and hence cannot be a proper symplectic form9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For this reason, it’s often referred to as the presymplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To form the actual phase space, we need to perform a symplectic quotient and mod out the null directions of the symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is a matter of viewing the prephase space as a bundle whose fibers are the null directions and whose base space is our true phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The mathematical details were laid out in [9], but in practice the result is that Ω is a non-degenerate 2-form on the base space and so working on the true phase space is a matter is ignoring those variables whose variation lies along the pure gauge directions10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2 Symmetries and charges With the covariant phase space framework now in place, it will be important for us to review how symmetries enter the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' A vector field V on phase space is typically defined to be a symmetry if its action on the Lagrangian is a total derivative: iV δL = dkV (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='20) for some kV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Contracting V onto (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='13) it now follows that dJV ≡ d(iV θ − kV ) = E ∧ iV δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='21) So JV = iV θ − kV is the conserved Noether current associated to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Though any vector field V satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='20) admits a conserved Noether current, con- structing the Noether charge is not always as a simple as integrating the current over a time slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' There may be non-trivial boundary contributions to the true Noether charge HV in form, but locally on phase space such a form can be written as exact and absorbed into a redefinition of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 9This is closely related to the lack of deterministic evolution on prephase space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' specifying the fields and some number of their derivatives on a Cauchy slice may not uniquely determine the same data on a later time slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The most common way to exhibit this non-uniqueness is by specifying a configuration and applying to it a gauge transformation which differs from the identity only at times later than the first slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We would thus have two solutions to the equations of motion whose data agree on one slice but disagree on another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 10If we are being strict, this is the statement that, at least locally, a section of the bundle is diffeomorphic to the base space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 11 order for it to satisfy iV Ω = −δHV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='22) For general symmetries, one must directly evaluate the contraction on the symplectic form, but for gauge symmetries we may find the charges by another, sometimes more efficient, method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This was pointed out in [25] for the special case of diffeomorphism charges in diffeomorphism invariant theories, but with the Theorem 1 of Appendix B it’s simple to generalize this calculation to any gauge transformation, as we review now11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We suppose Vλ generates a gauge transformation with gauge parameter λ, defined such that V0 = 0 so λ = 0 is the identity transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Taking an additional variation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='21) it follows that, on-shell, d(LVλθ − δkλ) = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='23) where we have abbreviated kVλ = kλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Thus we have a form closed for all free functions λ and theorem 1 tells us that there must exist a phase space 1-form Πλ, constructed locally from the fields and λ, such that LVλθ − δkλ = dΠλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='24) With this, it now follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='18) that δJλ = −iVλω + d(Πλ − iVλδB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) Thus if there exists a function Cλ such that Πλ − iVλδB = δCλ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='26) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) implies iVλΩ = −δ � Σ (Jλ − dCλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='27) We note that the existence of Cλ is not guaranteed and will typically depend on the boundary conditions chosen for the theory12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='27) now identifies the correct boundary 11Some additional simplifications that can help in computations are possible in the special case of diffeomorphisms even when the theory is not diffeomorphism invariant, as when gravitational Chern-Simons terms are included in the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This still makes use of Theorem 1 and was pointed out in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We review it in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 12The insufficiency of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='20) alone to ensure the existence of a charge satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='22) is pointed out 12 Noether charge as being the integral of Jλ with some additional boundary contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since Jλ is closed for all λ, and is linear in λ, we can go further and compute a local functional, referred to as the Noether-Wald charge, Qλ such that Jλ = dQλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' With this the Noether charge may be written H[λ] = � ∂Σ (Qλ − Cλ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='28) which has support only on the boundary of our Cauchy slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Furthermore, we note that the only place the boundary conditions enter into this expression is through Cλ, as Jλ and Qλ depend only on the Lagrangian of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since the charges (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='28) have support only on the boundary, it follows immediately that any gauge transform whose parameters λ have compact support away from any boundaries must produce vanishing Noether charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The vector fields generating these transformations are thus identified from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='27) as null directions of the presymplectic form which need to be modded out in the symplectic quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The non-zero Noether charges generate the large gauge transformations of the theory and are evidently localized to the boundaries of the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In the case these boundaries are asymptotic, the large gauge transformations are said to be asymptotic symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 Where boundary actions come from Using the machinery of the covariant phase space we can understand the, rather weak, sufficient conditions for producing boundary modes and gain some insight into the conditions under which the action for the theory is supported exclusively on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To transform this question into one which is easier to work with we first recall the phase space action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Writing the symplectic form for the theory as Ω = δΘ, the phase space action is given by13 S[γ] = � γ Θ − � γ Htdt (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='29) many placed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [3,23], integrability conditions are required as we see here and in [24] an auxiliary condition, there eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='16), is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 13Of course, Θ is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Different Θ correspond to holding different data fixed in the initial and final configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For a point particle, Θ = pδx is the correct potential for varying with the initial and final position of the particle held fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 13 where γ is some path through phase space parametrized by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To orient ourselves it’s useful to recall that for point particles Θ = pδx so S = � pδx − � Htdt = � (p ˙x − Ht)dt (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='30) where the dot denotes the derivative along the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' If time translation is a local symmetry, as it is in a diffeomorphism invariant theory, then by Theorem 1 the Hamiltonian generating time translation Ht is supported on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' While this gives a boundary contribution to the action, we would still be left asking about when we receive boundary contributions from Θ and about what happens when time translation is not a local symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We can obtain a better characterization of boundary contributions to the action which will help answer both of these questions by first introducing some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Consider a generic theory with fields φ and some gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We denote the action of a gauge group element with gauge parameters14 α on φ by Tα[φ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For example, if φ is a complex scalar field of charge q, Tα[φ] = eiqαφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' If φ is a U(1) connection we would have Tα[φ] = φ + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Now, we are always free to consider the field redefinition to gauge orbit variables where we pick a class of gauge-fixed configurations φ so a general configuration is in the gauge orbit of some φ: φ = Tα[φ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This separates the prephase space into gauge directions, parametrized by the α, and non-gauge directions, parametrized by the gauge inequivalent φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In these variables if we write15 Ht = � Σ Jt, Jt[φ, α] is a local function of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since Jt is closed for all functions α, Theorem 1 tells us that we can construct a local potential Qt[φ, α] such that Jt[φ, α] = Jt[φ, 0] + dQt[φ, α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='31) That is to say the current separates into a component we would find if we had immediately fixed the gauge, and a boundary contribution which contains all the effects of the gauge modes α, though it’s notably also free to depend upon boundary excitations in the gauge fixed φ directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We note that Jt[φ, 0] = 0 when time translation is a gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' A similar argument works on the symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' There are only three types of com- ponents that can appear in the symplectic form when we go to gauge orbit variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This 14For later convenience we assume that α = 0 is the identity transformation and that these α are free functions on spacetime as might be obtained by exponentiating a Lie algebra about the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 15In this section only we assume for convenience that Jt has been defined to include all necessary boundary contributions already so Ht is the full Noether charge satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 14 fixes the generic form of the symplectic form to be ω = ωφφ[φ, α]δφ ∧ δφ + ωφα[φ, α]δφ ∧ δα + ωαα[φ, α]δα ∧ δα (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='32) where the coefficients should generally be understood to contract on any indices we have suppressed in δφ and δα, and may also contain derivatives that operate on the field variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since the δα are completely free function in the bulk and dω = 0 on-shell by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='19), we may again apply theorem 1 but now using δα instead of α as our free functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We thus conclude16 ω = ωφφ[φ, α]δφ ∧ δφ + d˜ωb[φ, α] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='33) where we have put a tilde on ˜ωb because it will not be the complete contribution to the boundary symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Importantly, ˜ωb is a local functional of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We can go further by invoking theorem 1, this time on the α dependence of ωφφ[φ, α]δφ∧ δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The result is evaluating ωφφ at α = 0 and an additional, locally constructed contribution to the boundary symplectic form: ω = ωφφ[φ, 0]δφ ∧ δφ + dωb[φ, α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='34) Like with H, the symplectic form separates into a bulk component that we would have found by gauge fixing from the very beginning and a boundary term which encapsulates the entire effect of the gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We can choose both of these terms to be δ closed on phase space separately17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It then follows that we can find separate potentials and the symplectic potential current A breaks up into two terms, A[φ, α] = AM[φ] + dA∂M[φ, α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='35) Our arguments about the Hamiltonian and symplectic form together imply that the action must also break up into two terms, one supported on the bulk obtained from gauge 16Less formally, one can understand that there ultimately cannot be any δα components appearing in the bulk or Ω would have non-zero contractions on small gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 17To see this, suppose to the contrary that the variation of the bulk term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='34) is not δ closed but varies to something spacetime exact, δ(ωφφ[φ, 0]δφ ∧ δφ) = dωM, which then cancels against part of dδωb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This ωM cannot contain any dependence on α or δα, so if a cancellation occurs it’s sufficient to consider α = δα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' But our use of theorem 1 implies ωb = 0 in this case, so no cancellation can occur, requiring dωM = 0, and the terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='34) are separately closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 15 fixing, and one supported only on the boundary18: S = � dt � Σ � iV AM[φ] − Jt[φ, 0] � + � dt � ∂Σ � iV A∂M[φ, α] − Qt[φ, α] � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='36) where V generates the path the action is evaluated on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' So all gauge theories will reduce to a bulk component which is gauge fixed, and a boundary term which contains all the dynamics of the boundary gauge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Importantly, this boundary action may contain interactions not only among the boundary fields α, but also with the boundary values of the bulk gauge fixed fields φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4 Discussion The result (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='36) establishes boundary contributions to the action as a generic feature of gauge theory independent of whether we are able to solve constraints and directly reduce the action to the boundary as in 3d CS theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' But the nature of this boundary action varies from case to case, the main controlling factor being the structure of the asymptotic symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For example, Einstein gravity in AdSD>3 with standard asymptotically AdS boundary conditions has a finite dimensional asymptotic symmetry given by the SO(D−1, 2) isometry group of the global AdS vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Thus instead of a boundary field theory we have a boundary quantum mechanics, with one quantum mechanical degree of freedom corresponding to each generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To illustrate this we consider the even simpler case of pure Maxwell theory in asymptot- ically flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The phase space variables are ( ⃗A, ⃗E) subject to the Gauss law constraint ⃗∇ · ⃗E = 0, and we impose that these vanish at spatial infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The symplectic form is Ω ∼ � Σ δ ⃗E · ∧δ ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This symplectic form together with the boundary conditions require that the gauge parameters α(⃗x) become spatially constant at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Writing ⃗A = ⃗A + ⃗∇α, we have Ω ∼ � ∂Σ ⃗n · δ ⃗E ∧ δα ∼ δQ ∧ δα (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='37) where Q is the total electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As the Hamiltonian has no dependence on α, the bound- ary action is simply Sα = � dtQ ˙α, whose equation of motion is simply charge conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This boundary contribution is actually familiar in another guise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In particular, consider the Euclidean theory with periodic imaginary time, t ∼= t + β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since Q is constant on-shell we 18One should be careful to take the orientation of ∂Σ in these integrals to be the one induced by writing Vol(M) = τ ∧ n ∧ Vol(∂Σ) where τ is the normal form to the slices Σ and n is the (outward) normal form to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is a convention consistent with Stokes theorem, see [24] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 16 have Sα = Q � dt ˙α ≡ βµQ, which identifies the boundary mode α as being proportional to the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Turning now to the case of an infinite dimensional asymptotic symmetry group, here we do expect to get a boundary field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This boundary field theory may either be a free theory, as in U(1) CS theory, or interacting, as in non-Abelian CS theory or 3D gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In general, it’s clear that the existence of interactions is closely tied to the bulk theory having a non-Abelian asymptotic symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We may also ask more practically how to compute (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In principle, once we go to gauge orbit variables all stages of the calculation here are algorithmic as reviewed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' But to use the gauge orbit variables we would first need to classify the gauge- inequivalent solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since we are working in a canonical formulation it would be sufficient to find all gauge-inequivalent initial data, but that would still mean solving the constraints of the theory explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Instead, one could could imagine taking φ to be some perhaps incomplete class of initial data solving the constraints and consider the dynamics of this subspace of the full phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For example by considering a special subspace such as a moduli space of black hole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The logical extreme of these ideas would be to take φ to be a single solution, so we are looking at the gauge orbits about a background configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In this case δφ = 0 and AM = 0 so the entire symplectic form lives on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This simplification allows more efficient computational methods than the gauge orbit variable strategy described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Three such methods are described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 3 Computing Gauge Orbit Actions As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4, we may have reason to consider only certain sectors of a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The particular case of interest here is where we consider a single gauge fixed configuration φ and orbits around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This means δφ = 0 and AM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The remainder of the bulk term in the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='36) then integrates to some constant and we are left with only the boundary contribution to the action and more efficient methods of calculation are available to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This section is concerned with describing three such methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The first is less generically applicable, but very efficient when it applies as it directly computes A∂M instead of the boundary symplectic form ωb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The other two methods are concerned with computing ωb and are based on the existence of a particular 1-form valued vector field on phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 17 It’s important to note that as δφ = 0 suggests, these methods treat φ as a background field so any components A∂M might have had in the δφ directions will not be captured by these computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1 The momentum method For all the 3D gravity examples considered in this work there exists a very efficient method for deducing the boundary action [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We describe it here in general terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Supposing that expressions for boundary charges Hξ are known, to write down the boundary action we require knowledge of the boundary symplectic potential Θ, defined via Ω = δΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' A direct approach to obtaining it is to first extract Ω via the relation iVξΩ = −δHξ, and then compute Θ by solving δΘ = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We now discuss how, under suitable assumptions, we can write down the solution for Θ directly, bypassing the laborious procedure just mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This method was used in the case of cutoff AdS3 gravity [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We consider a boundary with a single spatial dimension with coordinate x, which may live either on the circle or line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The boundary field theory variables are written as (Φ, Ψ), where Ψ could stand for a collection of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' On the circle or line, these fields are taken to obey periodic boundary conditions or vanish at infinity respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Under phase space vector fields Vξ, which we can think of as reparametrizations of x, the fields transform as δξΦ = iVξδΦ = ξ + Φ′ξ δξΨ = iVξδΨ = Ψ′ξ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1) where ′ = ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The inhomogeneous term in δξΦ corresponds to Φ being the field associated with x-reparametrizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Now suppose that the charge corresponding to constant ξ (we take ξ = 1 and denote this charge by P) is constrained to take the form P = � dx � κΦΦΦ′2 + κΦΨΦ′Ψ′ + κΨΨΨ′2 + P ′ ΦΦ′ + P ′ ΨΨ′� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) where the κ’s are constant and the functions (PΦ, PΨ) are local functions of (Φ′, Φ′′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Ψ′, Ψ′′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' do not depend on undifferentiated fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The basic result is that up to a δ-exact term, the unique Θ which solves iVξΩ = −δP , ξ = 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) 18 is given by Θ = � dx [κΦΦΦ′δΦ + κΦΨΦ′δΨ + κΨΨΨ′δΨ + P ′ ΦδΦ + P ′ ΨδΨ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) To prove this, we first note that it is straightforward to verify that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) is satisfied, so only the question of uniqueness remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To analyze uniqueness we consider a correction to the symplectic form, ∆Ω = � dx [δX′ Φ ∧ δΦ + δX′ Ψ ∧ δΨ] , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) where the functions (XΦ, XΨ) are constrained to obey the same general properties of (PΦ, PΨ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We compute iVξ∆Ω = δ � dx [X′ ΦΦ′ + X′ ΨΨ′] = −δ � dx [X′′ ΦΦ + X′′ ΨΨ] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6) We need this to vanish in order not to spoil (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Vanishing of the integrand in the second line of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6) requires X′′ Φ = X′′ Ψ = 0, since the two terms cannot cancel each other under the assumed form of (XΦ, XΨ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' However, this implies that (X′ Φ, X′ Ψ) are constants, which implies that δX′ Φ = δX′ Ψ = 0, so that ∆Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The remaining possibility is that the integrand in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6) is a total derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This requires X′ Φ = δF δΦ and X′ Ψ = δF δΨ for some F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' However, this leads to ∆Ω = � dxδ2F = 0, so we again get no contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The upshot is that just from consideration of P we are led to a unique result for Ω and an explicit result for its potential Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It follows, assuming that our underlying theory is consistent, that this Ω will solve iVξΩ = −δHξ for all large gauge transformations Vξ, as can of course be checked in specific examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We conclude this section with comments on the assumed structure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2), which follows from a particular gauge invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To make the discussion concrete we consider the bound- ary theory in the context of pure gravity in AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Setting κ = 0 corresponds to the orbit built on a pure AdS3 background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This background is invariant under isometries that act as translations, boosts, and dilatations on the boundary coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Taking the functions (Φ, Ψ) to be general linear functions of x corresponds to acting on the background by one of these isometries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' since this does nothing to the background such functions are pure gauge, and so all charges must vanish for such functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This implies that only derivatives of such functions can appear in P, and that each term must contain at least one second derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This leads to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2), possibly after integrating by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The same comments apply to ∆Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 19 Turning on κ, which corresponds to a nonzero mass AdS3 background, breaks some of the isometries, allowing the quadratic terms like κΦΦΦ′2 to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In fact, a more general expression like κΦΦΦ′n for some n > 2 might also be anticipated, but only the n = 2 case will arise in our examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is important, since the logic leading to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) no longer holds for the n > 2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In fact, given the simple form of the κ dependent part of that charges will arise, it’s easy to write down the corresponding contribution to the boundary action by inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2 The transfer field Consider a variation of our fields in the gauge orbit variables φ = Tα[φ], as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since we are taking φ to be fixed, the variation can only change α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' But since φ is constructed by applying a gauge group element to the configuration φ, this variation must be equivalent to the action of some Lie algebra element on φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The situation is essentially identical to the story which comes up when studying sponta- neous symmetry breaking, in particular the broken part of the symmetry group’s non-linear action on Nambu-Goldstone bosons when studying spontaneous symmetry breaking, see for example [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The idea is that δα defines a Lie algebra element at the origin of the gauge group while φ has been transported to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' So to perform the variation of φ we need to transport δα to the tangent space at g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Schematically, we can think of this as inserting the identity in the action on φ to write δφ = (δTg)[φ] = (δTgT −1 g )[φ] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='7) so the variation of φ is equivalent to the action of this Lie algebra element valued as a 1-form on phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since Lie algebra elements are the generators of group transformations, there must exist some W which implements the action of this Lie algebra element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Because the algebra element is valued as a 1-form on phase space, W must be a (1, 1) tensor on phase space, which we think of as a 1-form valued vector field and refer to as the transfer field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This gives W the very special defining property δφ = iWδφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8) While the property (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8) is ultimately the formal property we will exploit to compute the boundary symplectic form, we will also need to explicitly compute the transfer field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' So it will be useful to see how the rather abstract discussion above can be realized in some simple 20 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The simplest example is of a U(1) connection A = A + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Here we have explicitly δA = dδα = iVδαδA (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='9) so W = Vδα implements the gauge transformation with gauge parameter δα, as might be expected for an Abelian group where translating δα is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The next simplest case is where A is a connection for some non-Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Now the gauge orbit is A = g−1Ag + g−1dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Here g = g(α) is defined by the exponential map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For simplicity we could take g = exp(α) with α valued in the Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We may now write δA = [A, g−1δg] + d(g−1δg) = iVg−1δgδA (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='10) so W = Vg−1δg generates a gauge transformation with 1-form valued gauge parameter g−1δg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We also note that this examples gives a concrete realization of the schematic manipulation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As a final example of W for this section, we consider diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Take the diffeo- morphism to be yα = f α(x) and consider for simplicity the diffeomorphism orbit of a scalar field φ(x) = φ(f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Taking the variation of this scalar field, δφ = δf α(x)∂φ(y) ∂yα ��� y=f(x) = δf α(x)∂xµ ∂yα ��� y=f(x) ∂φ(f(x)) ∂xµ = Lξφ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='11) so the variation is equivalent to the action of the Lie derivative with respect to the 1-form valued (spacetime) vector field19 ξµ = ∂(f −1)µ ∂yα ��� f(x)δf α(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='12) Hence W = Vξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This fact was also noted in [24] where it was used to circumvent some technical difficulties in JT gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' With these examples in mind, there is one additional property of W which will be important going forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Denote by w the Lie algebra element whose action W generates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 19In practical calculations, it is much simpler to move the Jacobian factor to the left and compute the variation of f by a diffeomorphism, then defining ξ by inverting the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is the approach taken in the examples of Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 21 In our examples we have w = δα for the U(1) gauge group, w = g−1δg for the non-Abelian gauge group, and for diffeomorphisms we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Importantly, this w is some functional of δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Equally as important, this functional need not be invertible as can be seen in the non-Abelian example20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This will be important going forward because to apply Theorem 1 we need free functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In our setup δα are free, but w may or may not be, depending on the details of the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For example, the relation between δf and ξ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='12) remains invertible despite diffeomorphisms being non-Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' While the relation between w and δα is not invertible, it is injective, and hence is invertible on the image of all δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This means δw may always be expressed in terms of only w again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We can see this in our examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The U(1) case is uninteresting because δw = 0, but in the non-Abelian case we find δw = δ(g−1δg) = −(g−1δg) ∧ (g−1δg) = −w ∧ w, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='13) so the general invertibility of w(δα) is unimportant to this rewriting of δw, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The example of diffeomorphisms is slightly more complicated, but nonetheless can be worked out to find δξµ = ξν ∧ ∇νξµ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='14) assuming we are working with a torsionless connection so the affine part of the covariant derivative does not contribute to this expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 The transfer field method Now that we have an understanding of the transfer field W, we can give a technique for computing the boundary symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' From the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='18), it would clearly be sufficient to show that the bulk δθ term is a total derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We may use the transfer field to write θ = iWθ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15) since in every term of θ, the 1-form factor can be replaced as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 20Invertibility fails for the same reason A = g−1dg is not invertible for dα: if A is not flat no such dα exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 22 Since W generates the action of the Lie algebra element w, it follows that θ = iWθ = Jw + kw (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='16) where Jw is the Noether current (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='21) associated to the gauge transformation generated by w and kw is defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' But as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2, Jw = dQw and hence ω = δθ − dδB = δkw + dδ(Qw − B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='17) So to find ωb it is sufficient to find a potential for δkw by solving δkw = d˜kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='19) we have dω = 0 on-shell, we must also have dδkw = 021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Now δkw depends on the free functions22 δα so we may apply theorem 1 to conclude the existence of a locally constructed ˜kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It should be commented that while δα, and the fact that δα = 0 corresponds to the identity, ensures the existence of ˜kw, it would be more computationally efficient to use w as the free function whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This can be done when the relation between w and δα is invertible, as discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The invertibility in the case of diffeomorphisms will be leveraged in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Hence we have found ωb = ˜kw + δ(Qw − B) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='18) which has the advantage of cleanly separating the boundary condition dependence of ωb into the δB term since ˜kw and Qw depend only on the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This technique is straightforward to apply to U(1) CS theory on D × R with chiral boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We recall for this theory (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='16), note kλ = kλdA, and consider a solution A to the equations of motion, so dA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The gauge orbit around this solution is then A = A + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As discussed in the previous section we have w = δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since kλ = 0 on-shell in this theory, we have ˜kw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Then following (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='18), the boundary 21We could also argue more directly that δdkw = δiW δL = δ2L = 0 since iW δL = δL by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 22In general, the imposition of boundary conditions will restrict what functions α we allow ourselves to consider near the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' However, the only place the boundary conditions enter in these arguments is through B, and in particular the closure of δkw requires no reference to boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' So we should imagine performing these manipulations before imposing any boundary conditions which would then restrict the allowed δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 23 symplectic form must be ωb = δQw = kδA ∧ δα (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='19) where we have used B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This integrates to Ω = k � 2π 0 dφ ∂φδα ∧ δα (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='20) and produces the usual Kac-Moody algebra (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) expected for U(1) CS theory on the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Using the Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6) and the phase space action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='36) we reproduce the expected boundary action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4 Computing Ω from the Noether charges In the previous section we were able to leverage the transfer field to write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It’s simple to see that this generalizes to any 1-form on the gauge orbit, not just θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' With some additional though we can go beyond 1-forms and obtain a similar result for any p-form on the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To see how this works, consider iW(δφa ∧ δφb) where we have restored the indices a and b labeling our fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Computing the contraction we find iW(δφa ∧ δφb) = (iWδφa) ∧ δφb + δφa ∧ (iWδφb) = 2δφa ∧ δφb (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='21) where in the second line we have used the defining relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8) for W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It’s important to keep track of the sign on the second term because W is 1-form valued and must be commuted past the first factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' With this sign understood, the generalization to a form of any degree is immediate: we simply march through the factors in the wedge product to perform the contraction and then use the definition of W to rewrite the contraction back in terms of the original form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Hence, for any p-form Z on phase space we have the generalized identity iWZ = pZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='22) We can leverage this to very efficiently compute the boundary symplectic form ωb on the gauge orbit directly from the Noether charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The key is to observe that the contraction 24 on an arbitrary 2-form may be rewritten into the form iW(δφa ∧ δφb) = − � δφb ∧ (iWδφa) − δφa ∧ (iWδφb) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='23) That is, if we make sure to commute the contracted part to the right of the expression, we obtain the negative of the expression we would have found if W was not valued as a 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It follows that if we compute iWΩ and make sure to commute all the contractions to the right, we must obtain iWΩ = +(δH[ ˜w])| ˜w=w (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='24) since we have already argued that W generates the action of a Lie algebra element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Note that on the right hand side we compute the variation of H before evaluating at the gauge parameter w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is because w enters through the contraction on the right and so clearly cannot have a variation applied to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Combining this with our general observation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='22) it follows that the symplectic form on the gauge orbit is given by Ω = 1 2(δH[ ˜w])| ˜w=w (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) where the expression on the right should be understood as placing all factors of w to the right of any variations, and the variation taken treating ˜w as an arbitrary, but not 1-form valued, Lie algebra element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' One must be cautious not to interpret (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) as meaning Θ = 1 2H[w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The order in which the variation and the evaluation at ˜w = w occur are important for this result, as will be seen our examples in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We may again revisit our U(1) CS example to demonstrate this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Indeed, looking at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='19) we can already see the basic structure of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) since B = 0 with our chiral boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' But going through the details to make sure the coefficients come out correct, particularly since (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='19) uses the Noether current and not the complete Noether charge, we start with the charges (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) tells us to vary the charge, treating ˜w as any large gauge parameter in the theory, which generally may mean it’s state-independent or some state-dependent function of some other parameters determining the large gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Here there are no such complications as ˜w is state-independent and we find 1 2δH[ ˜w] = k � 2π 0 ˜wδAφdφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='26) The final step in computing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) is to evaluate this charge at ˜w = w ≡ δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' But when we 25 do so, we must make sure that we first move all ˜w factors to the right of δAφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Doing so, Ω = k � 2π 0 dφδAφ ∧ δα = k � 2π 0 dφ∂φδα ∧ δα (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='27) which again matches the expected Kac-Moody expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5 Examples Here we apply our methods to three examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' First we consider U(1) CS theory in D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This theory, with our chosen boundary conditions, is only marginally more complex than the example of U(1) CS theory in D = 3 that we have been considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It is also simple enough that we are able to implement the manipulations in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 to find (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='34) with non-gauge directions included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 we consider Einstein-Hilbert gravity in D = 3 with two different sets of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' First we consider standard asymptotically AdS3 boundary conditions and then the boundary conditions described in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The latter are designed to produce a theory similar to what we will see in topologically massive gravity, considered in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Finally, we mention that the example of SU(2) CS theory in D = 3 is considered in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' There we also show that the results of our methods match the standard results expected from the CS/WZW correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1 U(1) CS in D = 5 As a first example which isn’t quite as trivial as U(1) CS in D = 3 we consider U(1) CS in D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since we work near the boundary the details of the bulk geometry are irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We fix the boundary geometry, though the details of this geometry will not be paramount to most of our manipulations, to be R × S1 × S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' On the boundary we choose coordinates xi, i = 1, 2, 3, 4 with x1 the coordinate on the R, which we think of as time, x2 the angular coordinate on S1, and x3, x4 some coordinates on the S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To make contact with the D = 3 CS theory, it will sometimes be useful to write t ≡ x1 and φ ≡ x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For this example we choose our boundary conditions to fix A3 and A4 to be arbitrary functions of x3 and x4 on the boundary while we fix A1 = A2 to be an arbitrary function of x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This will allow us to draw parallels to U(1) CS in D = 3 on the cylinder in the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It is useful to note that with these boundary conditions the only non-zero boundary components of F are F12, which depends dynamically on A1 (and hence A2), and 26 F34 which is a fixed function of only x3 and x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The Lagrangian for this theory is L = A ∧ F ∧ F (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='28) which varies to produce δL = 3F ∧ F ∧ δA − 2d(F ∧ A ∧ δA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='29) Thus the equations of motion, F ∧F = 0, do not imply that all solutions are flat connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Considering the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15), we see that our chosen boundary conditions allow us to choose ℓ = B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since B = 0, the full symplectic current for the theory is given by ω = −2δ(F ∧ A) ∧ δA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='30) Now, this theory is sufficiently simple that we are able to explicitly carry out the gauge-orbit described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We suppose A is some solution to the equations of motion and we consider the gauge orbits around this configuration, A = A+dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It’s straightforward to find ω = −2δ(F ∧ A) ∧ δA − 2d[δ(αF) ∧ dδα + δ(αF) ∧ δA − δ(F ∧ A) ∧ δα] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='31) so we realize the expected split into the gauge-fixed part and the boundary part depending on the large gauge transformations parametrizing the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Here we can also see that the boundary component of the symplectic form supports components which mix variations in the gauge-fixed configuration with variations along the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Indeed, there is even a component which involves no variations of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' But here we are interested only in the components along the orbit directions, so we take δA = 0 to find ωb = −2Fδα ∧ dδα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='32) Integrating this over the boundary and using our boundary conditions the full symplectic form on the orbit is given by Ω = −2 � ∂Σ F 34δα ∧ ∂2δαdx2 ∧ dx3 ∧ dx4 = 2 �� S2 F � � 2π 0 ∂φδα ∧ δαdφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='33) 27 Because we have chosen our boundary conditions to factor the S2 from the R×S1, we obtain the same Kac-Moody symplectic form as would be expected from the U(1) CS theory on the cylinder, but now with an effective level set by our boundary conditions via the magnetic flux through the S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Using that the Hamiltonian generating time evolution is Ht = 2 � ∂Σ F 34A2 2dx2 ∧ dx3 ∧ dx4 = 2 �� S2 F � � 2π 0 (Aφ + ∂φα)2dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='34) Constructing the phase space action via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='36) we thus find S = 2 �� S2 F � � � ∂φα � ∂tα − ∂φα − 2Aφ � − A 2 φ � dtdφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='35) Since we have restricted ourselves to the gauge orbit, α is our only dynamical variable in this action and, in particular, this means the final A 2 φ term can be dropped as an additive constant to the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In this example it’s possible to observe explicitly from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='31) that, had we not restricted to the gauge orbit of some fixed A, the Aφ terms here would represent an explicit coupling between the bulk, A, and boundary, α, degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This result can, of course, also be obtained by the techniques introduced in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Taking first the approach of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 we note that iVλδL = dλ ∧ F ∧ F = d(λF ∧ F) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='36) so kλ vanishes on-shell and will make no contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For the other ingredient, we calculate Jλ = iVλθ − kλ = 2d(λA ∧ F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='37) Thus, since B = 0 with our chosen boundary conditions, we find from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='19) ωb = 2δ(A ∧ F) ∧ δα = 2dδα ∧ F ∧ δα (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='38) since w = δα for the U(1) orbit, matching (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' From this point, imposing the boundary conditions to find the true symplectic form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='33), and hence the action (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='35), is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' If we instead took the route of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4 we would need to compute the full Noether 28 charges H[λ] = 4 �� S2 F � � S1 λA (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='39) which requires that we use the boundary conditions to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For example, that we fix A3 and A4 completely on the boundary tells us that λ = λ(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The formulation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) now tells us to write Ω = 2 �� S2 F � � δA ∧ δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='40) Writing this explicitly in coordinates and with δA = δdα this clearly reproduces (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='33) and the rest of the boundary action story follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2 Asymptotically AdS3 Einstein-Hilbert gravity As a second example, we can derive the Alekseev-Shatashvili symplectic form [30] for asymp- totically AdS3 Einstein-Hilbert gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' All three approaches can be worked out in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We start by recalling some facts this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The action and canonical 1-form for this theory are given by L = 1 16πG √−g(R + 2)d3x, θ = 1 16πG √−g(∇νδgλν − gµν∇λδgµν)(d2x)λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='41) Where expressions are more conventionally expressed in terms of the Brown-Henneaux central charge we write c = 3/2G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We will also need later that kξ = iξL = − 1 4πG √−gξµ(d2x)µ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='42) where we have used the equations of motion to simplify things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In Fefferman-Graham coordinates the asymptotically AdS3 solutions to the equations of motion are given by the Ba˜nados metrics [31] ds2 = dρ2 4ρ2 + 1 ρ(dw + ρL(w)dw)(dw + ρL(w)dw) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='43) where w = φ + t and w = φ − t are convenient coordinates on the boundary and ρ > 0 is the radial coordinate such that the boundary is located at ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Note that w is not the complex conjugate of w as we are working in Lorentzian signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The boundary stress 29 tensor associated to these metrics is given by Tww = − 1 4GL, Tww = − 1 4GL, Tww = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='44) The asymptotic vector fields which preserve the boundary conditions are ξw = ϵ(w) − 1 2∂2 wϵ(w)ρ + O(ρ2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='45) ξw = ϵ(w) − 1 2∂2 wϵ(w)ρ + O(ρ2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='46) ξρ = (∂wϵ(w) + ∂wϵ(w)) ρ + O(ρ0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='47) where ϵ(w) and ϵ(w) are free functions labeling the vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' While these vector fields do not change the conformal metric on the boundary, they do change the components subleading in ρ, and hence the stress tensor by iVξδTww = 2Twwϵ′ + T ′ wwϵ + c 12ϵ′′′ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='48) iVξδTww = 2Twwϵ′ + T ′ wwϵ + c 12ϵ′′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='49) The charges associated with the diffeomorphisms are then easily constructed by H[ξ] = − 1 2π � 2π 0 (Twwϵ − Twwϵ)dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='50) Going to the gauge orbit means integrating up the variations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='48) under an infinitesimal diffeomorphism to a finite one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' If the finite diffeomorphism on the boundary is given by w′ = f(w), w′ = f(w) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='51) then we have Tww = c 12 �κ 2f ′2 + {f, w} � , Tww = c 12 �κ 2f ′2 + {f, w} � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='52) where the Schwarzian derivative is {f, w} = f ′′′ f ′ − 3 2 f ′′2 f ′2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='53) In this orbit, the diffeomorphism f(w) = w and f(w) = w evidently produces constant Tww, Tww.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The κ and κ parametrize what this constant value is, and thereby parametrize the diffeomorphism inequivalent metrics (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The case κ = κ = 1 produces global AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 30 At ρ = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='45) defines a diffeomorphism on the boundary which acts on the f and f as iVξδf = f ′ϵ, iVξδf = f ′ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='54) This can be obtained by composing two diffeomorphisms and we may also verify that this transformation rule is compatible with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='48) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='52) together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since we have the momentum written down now as P = H[∂φ] = − c 24π � 2π 0 dφ ��κ 2f ′ − 1 2 � 1 f ′ �′′� f ′ − � κ 2f ′ − 1 2 � 1 f ′ �′′� f ′ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='55) where we have used that {f, w} = − 1 2 � 1 f′ �′′ f ′ up to addition of total derivatives, we may observe that it takes the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) required for the momentum method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It therefore follows that the canonical 1-form is given by Θ = − c 24π � 2π 0 dφ ��κ 2f ′ − 1 2 � 1 f ′ �′′� δf − � κ 2f ′ − 1 2 � 1 f ′ �′′� δf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='56) The variational of this potential can indeed be massaged into the more standard form Ω = − c 48π � 2π 0 ��δf ′ ∧ δf ′′ f ′2 − κδf ∧ δf ′ � − � δf ′ ∧ δf ′′ f ′2 − κδf ∧ δf ′ �� dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='57) of the Alekseev-Shatashvili symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Using the canonical 1-form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='56) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='29) together with the Hamiltonian H[∂t] produces the Alekseev-Shatashvili action S = − c 24π � d2x � κf ′∂wf − � 1 f ′ �′′ ∂wf − κf ′∂wf − � 1 f ′ �′′ ∂wf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='58) We may also obtain these results by the technique in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since we already know the charges, the only ingredient still needed in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) is the 1-form valued gauge parameter w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since we are working with diffeomorphisms this means we need to work out (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In this case things are simplified somewhat because we evidently only require the vector field ξ evaluated at ρ = 0 since the subleading components of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='45) are not needed to compute the charge H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Now, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='54) gives the variations of f and f under an arbitrary asymptotic diffeomorphism, 31 the 1-form valued vector field with parameters ϵδ and ϵδ must satisfy � δf δf � = � f ′ 0 0 f ′ � � ϵδ ϵδ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='59) This is evidently (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='12) evaluated on the boundary and with the Jacobian factor moved to the other side of the equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It now follows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) gives Ω = − 1 4π � 2π 0 (δTww ∧ ϵδ − δTww ∧ ϵδ)dφ = − c 48π � 2π 0 � δ �κ 2f ′2 + {f, w} � ∧ δf f ′ − δ �κ 2f ′2 + {f, w} � ∧ δf f ′ � dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='60) Integrating by parts this expression can can be shown to equal (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We can also obtain this result by computing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For this computation we will need the potential Qξ for the Noether current Jξ and the potential ˜kξ for δkξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Much of this discussion is independent of the choice Λ = −1 and D = 3, so we will leave these undetermined until they are actually needed to simplify our expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The Noether-Wald charge is well known as the so-called Komar term [32,23], Qξ = − 1 16πG √−g∇µξν(dD−2x)µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='61) The computation of ˜kξ is straightforward from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='42) if we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' With this we find 16πGδkξ = − 4Λ D − 2 � δξµ√−g + 1 2 √−ggαβδgαβ ∧ ξµ � (dD−1x)µ = 4Λ D − 2 √−g∇λ(ξλ ∧ ξµ)(dD−1x)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='62) From this we identify ˜kξ = − 1 16πG 2Λ D − 2 √−gξµ ∧ ξν(dD−2x)µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='63) The boundary symplectic form is therefore given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='18) to find ωb = δQξ + ˜kξ = − 1 16πG √−g � ∇λξλ ∧ ∇µξν + δ(∇µξν) + 2Λ D − 2ξµ ∧ ξν � (dD−2x)µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='64) 32 Computing δ(∇µξν)(dD−2x)µν = − � ∇λξµ ∧ ∇λξν − 1 2R µν αβ ξα ∧ ξβ � (dD−2x)µν (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='65) the boundary symplectic form becomes ωb = − 1 16πG √−g � ∇λξλ ∧ ∇µξν − ∇λξµ ∧ ∇λξν − 1 2R µν αβ ξα ∧ ξβ + 2Λ D − 2 � (dD−2x)µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='66) If we now specialize to D = 3 where R µν αβ = δµ αRν β − δν αRµ β − δµ βRν α + δν βRµ α − 1 2R(δµ αδν β − δν αδµ β) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='67) and use the equations of motion to write Rµν = 2Λgµν, R = 6Λ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='68) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='66) can be simplified to ωb = − c 24π √−g � ∇λξλ ∧ ∇µξν − ∇λξµ ∧ ∇λξν + Λξµ ∧ ξν� (d1x)µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='69) It’s straightforward to evaluate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='69) on the metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='43) on the computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We restrict the vector field ξ to take the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='45), where now ϵ and ϵ should be understood as the 1-form valued ϵδ and ϵδ we used earlier and introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Doing so we find the φ = 1 2(w + w) component of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='69) to be ωb|∂Σ = c 24π 1 ρ∂φ(ϵ ∧ ϵ)dφ + c 48π � 4Lϵ′ ∧ ϵ − 4Lϵ′ ∧ ϵ + ϵ′ ∧ ϵ′′ + ϵ′′ ∧ ϵ′ − ϵ′′′ ∧ ϵ + ϵ′′′ ∧ ϵ � dφ + O(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='70) Since we will integrate this over the circle to form the symplectic form we are free to drop total derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In particular this means the divergent term above may be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Furthermore, in the O(ρ0) term the two terms which mix ϵ and ϵ combine to a total derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' So the symplectic form factorizes into a holomorphic and antiholomorphic term as expected, Ω = c 48π � 2π 0 � (4Lϵ′ − ϵ′′′) ∧ ϵ − (4Lϵ′ − ϵ′′′) ∧ ϵ � dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='71) 33 But recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='44) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='48) we have δL ∧ ϵ = 1 2(4Lϵ′ − ϵ′′′) ∧ ϵ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='72) and similarly for L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Of course, we identify the RHS here as being precisely the factor which appears in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Making the replacement we recover the first line of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 New AdS boundary conditions Here we again consider the Einstein-Hilbert Lagrangian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='41) in D = 3, but this time we consider the modified boundary conditions described in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Many of the details in the calculations here are similar to those which appear in topologically massive gravity, to be discussed in Section 4, so the manipulations here will be useful as a warmup in addition to a demonstration of the techniques in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We write an arbitrary Fefferman-Graham gauge metric as ds2 = dρ2 4ρ2 + 1 ρ � g(0) ab + ρg(1) ab + O(ρ2) � dxadxb (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='73) where again ρ > 0 is the radial coordinate with asymptotic boundary at ρ and xa = (t, φ) are the coordinate on the boundary with φ ∼ φ + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As before we will use the coordinates w = t + φ and w = −t + φ for convenience23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The boundary conditions considered in [29] can be phrased as fixing the components g(0) ww = 0, ∂wg(0) ww = 0, g(0) ww = 1 2, g(0) ww = 4G∆ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='74) with ∆ a fixed constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Importantly, one can check that, while it is slightly modified from the Gibbons-Hawking-York boundary term, there indeed exists a choice of ℓ such that B = 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The Einstein equations can be solved exactly for metrics obeying these boundary condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The analog of the Ba˜nados metrics for this case are ds2 =dρ2 4ρ2 + 1 ρdw(dw + ∂wPdw) + 4G[Ldw2 + ∆(dw + ∂wPdw)2] + (4G)2∆ρLdw(dw + ∂wPdw) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='75) where P = P(w) and L = L(w) are free functions parametrizing the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 23These coordinates relate to the coordinates (r, t+, t−) which appear in [29] by ρ = 1/r2 and t+ = w, t− = −w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Later we will meet a pair of functions (P, L) which were (P, L) in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 34 Furthermore, the asymptotic vector fields which preserve these boundary conditions are given by ξρ = ρϵ′ ξw = ϵ + O(ρ2) ξw = σ − 1 2ρϵ′′ + O(ρ2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='76) where ϵ(w) and σ(w) are free functions parametrizing the diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' These diffeomorphisms act on the parameters P and L by iVξδP = σ + P ′ϵ, iVξδL = 2Lϵ′ + ϵL′ − 1 8Gϵ′′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='77) The charges associated to these diffeomorphisms can be computed to be H[ϵ, σ] = � 2π 0 dφ 2π �� ∆P ′2 − L � ϵ + ∆(1 + 2P ′)σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='78) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='77) it’s clear that the ϵ part of the transformation acts just like (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='48) on L while the σ part of the transformations acts to shift P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This means we can integrate up (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='77) to find P = g(w), L = − 1 8G{f, w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='79) This can be done explicitly by applying the finite diffeomorphism ρ′ = f ′ρ + G∆f ′′2 f ′ ρ3 + O(ρ4), w′ = f + 2G∆f ′′ρ2 + O(ρ3), w′ = w + g − 1 2 f ′′ f ′ ρ + O(ρ3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='80) where f(w) and g(w) are free functions, to the background metric ds2 = dρ′2 4ρ′2 + 1 ρ′dw′dw′ + 4G∆dw2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='81) 35 Diffeomorphisms act on the orbit parameters by iVξδf = f ′ϵ, iVξδg = σ + g′ϵ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='82) which can be found by composing diffeomorphisms or deducing the transformation rule from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='77) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='79) we may also write the momentum as P = H[1, 1] = ∆ + � 2π 0 dφ 2π � ∆(2 + g′)g′ + 1 8G{f, w} � = ∆ + � 2π 0 dφ 2π � ∆g′g′ − 1 2 1 8G � 1 f ′ �′′ f ′ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='83) where in the second line we have used that {f, w} = − 1 2 � 1 f′ �′′ f ′ up to the addition of total derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As was the case in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='55), this momentum is of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) for the momentum method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The canonical 1-form is given by Θ = � 2π 0 dφ 2π � ∆g′δg − 1 16G � 1 f ′ �′′ δf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='84) From this and the Hamiltonian Ht = H[1, −1] we obtain the phase space action S = � dtdφ 2π � ∆g′(˙g + g′) − 1 16G � 1 f ′ �′′ ( ˙f − f ′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='85) To instead apply our other techniques, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='82) supplies the analog of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='59), now taking the form � δf δg � = � f ′ 0 g′ 1 � � ϵδ σδ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='86) With all of these expressions for this theory we are well positioned to compute the boundary symplectic form via the methods in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To make things even easier, we may note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='69) applies for any theory governed by the Einstein-Hilbert Lagrangian, independent of the boundary conditions we impose, so long as B = 0, which happens to be the case for this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Performing the evaluation we find that again the divergent 1/ρ term is a total derivative as it was for asymptotically Dirichlet boundary conditions while the finite term can be 36 massaged to yield Ω = � 2π 0 dφ 4π � δ(L − ∆P ′2) ∧ ϵδ − 2∆δP ′ ∧ σδ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='87) which one may check has canonical 1-form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Furthermore, comparing this expression to the the charges (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='78) it’s immediately clear that the prescription (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) reproduces (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='87) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Thus both the techniques of Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4 lead to the same phase space action (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6 Comparison of techniques We have presented several techniques for computing actions governing boundary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It is useful to discuss the computational advantages offered by each approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' When applicable, the momentum method is the most efficient approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As presented, it applies to D = 3 diffeomorphism orbits built on bulk solutions that are translationally invariant in the boundary directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The needed inputs are simply expressions for the boundary Hamiltonian and momentum charges written in terms of the orbit variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The prescription involves using integration by parts to write the momentum in a canonical form, after which the canonical 1-form Θ can be read off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Together with the expression for the Hamiltonian, the boundary action follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' A great advantage here is that one gets Θ directly, bypassing the need to solve δΘ = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Also noteworthy is that it can be applied order by order in perturbation theory, for example by expanding in powers of the boundary fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This method proved very effective in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The other methods we discussed are less efficient but have a wider range of applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Using the transfer vector to invert iV Ω = −δHV requires that we we have computed all of the Noether charges, rather than just the momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Of course this has the advantage that it is applicable in any dimension and with any type of gauge orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Additionally, while the computation of the Noether charges will generally be sensitive to the boundary conditions of the theory, once the charges are computed the actual computation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As seen in the examples (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='60) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='87), the resulting form for Ω may not make solving δΘ = Ω immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The transfer field method is likely the most computationally involved of the three ap- proaches, but has the unique advantage of offering a clean separation between contributions depending on the Lagrangian of the theory and the boundary conditions imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In the case one wishes to study a Lagrangian under a variety of boundary conditions, this method could become more efficient than the others which would require that we recompute the 37 Noether charges with each set of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For example, once (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='69) was known for D = 3 Einstein-Hilbert theory, there was essentially no additional computation required to apply it in the example in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 beyond working out the new class of asymptotic vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As a final point here, we note that while computing the Noether charges generally involves solving a condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='26) to obtain the charges, each step in the transfer field method can in principle be completed algorithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This opens the possibility that, given θ and kw for the theory, the computation of both Qw and ˜kw can be automated on the computer, leaving only the computation of B to obtain an expression for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 4 Application to Topologically Massive Gravity Topologically massive gravity [17] is described by the Einstein-Hilbert action supplemented with a gravitational CS term, STMG = 1 16πGSEH − l 96πGν SCS (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1) with24 SEH = � d3x√g � R + 2 l2 � + Sbndy SCS = � Tr � Γ ∧ dΓ + 2 3Γ ∧ Γ ∧ Γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) Here the connection 1-form is Γα β = Γα βµdxµ, where Γα βµ are the usual Christoffel symbols built out of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We take ν > 1 and henceforth set l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Our interest in applying our general techniques to this theory is to illustrate various issues that are not present in simpler examples, in particular the the non-diffeomorphism invariance of the action and the existence of solutions with relatively exotic “warped” asymptotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We find that these pose no obstacle to implementing our general procedure to find the boundary action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We will be particularly interested in solutions [16] that are the warped analog of the more familiar Ba˜nados geometries [31], ds2 = L2 � dr2 r2 + u2 � dt + ˆKdφ + (r + r−1 ˆL)dφ �2 − (r − r−1 ˆL)2dφ2 � , φ ∼= φ + 2π (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) 24It is straightforward to check that the boundary action may be chosen such that B = 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 38 where L2 = 1 ν2 + 3 , u2 = 4ν2 ν2 + 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) is a solution to STMG for any functions ˆK = ˆK(φ) and ˆL = ˆL(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Setting ˆK = ˆL = 0 gives the warped vacuum solution (the analog of global AdS3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' constant values of ( ˆK, ˆL) can be obtained by quotienting the vacuum solution (analogous to how one obtains BTZ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' ( ˆK, ˆL) with nontrivial dependence on φ (analogous to the Ba˜nados geometries) are obtained by applying asymptotic symmetry transformations to the solutions with constant values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The solutions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) live in a phase space defined by boundary conditions that are preserved by the asymptotic coordinate transformations xµ→xµ + χµ which have the large r behavior χφ = ξφ + 1 2r2∂2 φξφ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' χt = ξt − 1 r∂2 φξφ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' χr = −r∂φξφ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) Here ξt,φ = ξt,φ(φ) are arbitrary periodic functions of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Under an infinitesimal diffeomor- phism of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) we have δ ˆL = −1 2∂3 φξφ + 2 ˆL∂φξφ + ∂φ ˆLξφ δ ˆK = ∂φξt + ∂φ( ˆKξφ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6) Associated to any boundary preserving vector field ξ = ξµ∂µ is a charge H[ξ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For example, the charges associated with rigid time and angular translations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='e energy and angular momentum, are M = 1 uLH[∂t] = 1 12πG � 2π 0 ˆKdφ J = H[∂φ] = − 1 6πGuL � 2π 0 �� 1 + 1 u2 � ˆL − 1 4 ˆK2 � dφ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='7) as will be derived below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25 To derive the charges we can consider SEH and SCS independently of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' That is, the action S = C1SEH + C2SCS yields charges Q′ = C1Q′ EH + C2Q′ CS, so we can extract Q′ EH and Q′ CS by setting one of C1,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is allowed because the procedure to obtain Q is linear, and the current JEH is conserved in the SEH theory, and likewise for JCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 25Original references for the derivation of the charges and asymptotic symmetry algebras include [33–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1 Einstein-Hilbert contribution As in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='61) we have QEH χ = −εαβ φ∇αχβdφ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8) Evaluating this on (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) gives QEH χ = Lu(u2 − 1)r2ξφ + 2Lu(u2 − 1) ˆKrξφ + Lu3rξt + Lu � u2(2 ˆL + ˆK2) + 6 ˆL � ξφ + Lu3 ˆKξt + O(r−1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='9) The term proportional to r2 is constant on the phase space and so will be omitted in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The same goes for the last term on the first line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' On the other hand, there is a linearly diverging term in the first line which is not constant on phase space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' this term will cancel a similar term in the CS contribution, yielding a finite result as r→∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2 CS contribution The Lagrangian 3-form LCS = Tr(Γ ∧ dΓ + 2 3Γ3) has variation δLCS = Eγρδgγρ √−gd3x + dθCS (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='10) with Eγρ = −∇βRβρ µνεγµν = −2Cγρ θCS = Tr(δΓ ∧ Γ) − 2δgκρRρ δdxκ ∧ dxδ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='11) where Cγρ is the Cotton tensor and the Ricci tensor Rµν is formed from the Riemann tensor R = dΓ + Γ2 as Rµν = Rα µαν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We also recall that in D = 3 the Riemann tensor can be expressed in terms of the Ricci tensor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' in terms of the Riemann two-form this amounts to the identity Rα β = � δα γ Rδβ − gβγRα δ − 1 2Rδα γ gδβ � dxγ ∧ dxδ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='12) Following the discussion in appendix C to compute the charges, the Noether current corresponding to the bulk vector field ξ is JCS ξ = iVξθCS − iξLCS − Yξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='13) 40 where Yξ defined via dYξ = δξLCS − LξLCS = Tr(dv ∧ dΓ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='14) with vα β = ∂βξα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We take Yξ = − Tr(dv ∧ Γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15) We also have iVξθCS = ∇β∇γξαdxβ ∧ Γγ α + Tr(iξR ∧ Γ) − 2(∇κξρ + ∇ρξκ)Rρ δdxκ ∧ dxδ iξLCS = Tr(iξΓdΓ − Γ ∧ iξR) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='16) Using these relations along with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='12), some algebra leads to JCS ξ = dQCS ξ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='17) with QCS ξ = ∂γξαΓγ α + ∇γξαΓγ α + ξa� − 4Raδ + Rgaδ � dxδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='18) The full charge is HCS ξ = � ∂Σ � QCS ξ − CCS ξ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='19) where CCS ξ is obtained by solving δCCS ξ = iξθ, which can be written explicitly as δCCS ξ = (δΓα κβΓβ δα − δΓα δβΓβ κα)ξκdxδ − 2(δgκρRρ δ − δgδρRρ κ)ξκdxδ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='20) To proceed further we need to specify the asymptotic form of the solutions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Taking the solutions to be of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) it is straightforward to compute QCS ξ = � Lu(u2 − 1)r2ξφ + 2u(u2 − 1)L ˆKrξφ + u(u2 − 1)Lrξt + L � u � u2 − 2 3 � ˆK2 + � 2u3 − 10 3 u + 8 3u � ˆL � ξφ + Lu � u2 − 2 3 � ˆKξt + O(r−1) � dφ CCS ξ = � − u � u2 − 2 3 � L ˆKξt + O(r−1) � dφ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='21) 41 Again, we can ignore terms that are constant on phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 Total charge The total charge is Hξ = 1 16πG � ∂Σ � Qξ − 1 6ν (QCS ξ − CCS ξ ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='22) Noting the cancellation between (non-constant) diverging term, we arrive at a finite result in the large r limit, Hξ = 1 16πG � ∂Σ �4 3uL ˆKξt − 8 3uL �� 1 + 1 u2 � ˆL − 1 4 ˆK2 � ξφ � dφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='23) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4 Lower spin gravity formulation While Einstein gravity in three dimensions can be recast as a CS theory (with gauge group SL(2, R)×SL(2, R)), this no longer holds in the presence of a gravitational CS term because the theory is no longer topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' However, the subsector of the theory described by the solutions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) can be described by a CS theory, namely one with gauge group SL(2, R) × U(1), so-called “lower spin gravity” [21, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The choice of gauge group is dictated by the isometry group of the warped vacuum solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In particular, there is a fairly simple relation such that the charges and symplectic form of the two theories are mapped to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since the equations of CS theory are much easier to deal with than those of TMG, this provides a simpler route to isolating the boundary degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The simplification arises essentially because the bulk degrees of freedom have been omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The action is S = k 4π � Tr � A ∧ dA + 2 3A ∧ A ∧ A � + k 8π � A ∧ dA (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='24) Here A is an SL(2, R) connection and A is a U(1) connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The SL(2, R) generators obey [Lm, Ln] = (m − n)Lm+n and we use the two-dimensional representation, for which Tr L1L−1 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' a = (L1 − ˆLL−1)dφ + (ω1L0 + ω2L−1)dt a = Kdφ + dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) Here, as is standard, the boundary connections (a, a) are related to bulk connections (A, A) 42 by a gauge transformation that encodes the radial dependence, A = b−1ab + b−1db , A = bab−1 + bdb−1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='26) where b = e− 1 2 L0 ln r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Gauge transformations act as δa = dϵ + [a, ϵ] , δa = dϵ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='27) with ϵ = ϵ1L1 + ϵ0L0 + ϵ−1L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The form of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) is preserved by taking ϵ0 = −∂φϵ1 and ϵ−1 = 1 2∂2 φϵ1 − ˆLϵ1, To connect to the metric description we trade (ϵ1, ϵ) for (ξφ, ξt) according to ϵ1 = ξφ , ϵ = ξt + ˆKξφ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='28) with ˆK = 4π k K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='29) The gauge transformations then act as δ ˆL = −1 2∂3 φξφ + 2 ˆL∂φξφ + ∂φ ˆLξφ δ ˆK = ∂φξt + ∂φ( ˆKξφ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='30) reproducing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We can now apply relations reviewed in Appendix D for CS theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Letting V denote the phase space vector field implementing an infinitesimal gauge transformation with parameters (ϵ, ϵ) we have iV Ω = −δH[ϵ, ϵ] with H[ϵ, ϵ] = k 2π � ∂Σ Tr(ϵa) + k 4π � ∂Σ ϵa = � 2π 0 dφ �� k 2π ˆL + k 8π ˆK2 � ξφ + k 4π ˆKξt � = � 2π 0 dφ(Lξφ + Kξt) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='31) where we have defined L according to ˆL = 2π k (L − 2π k K2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='32) 43 The second line of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='31) agrees with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='23) under the identification k = uL 3G � 1 + 1 u2 � , k = −uL 3G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='33) We conclude that there is a simple relation between the canonical structure of lower spin gravity and the class of solutions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) to TMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' One can also build the metrics (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) out of the SL(2, R)×U(1) connection, in analogy with the corresponding relation in ordinary 3D gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' See [21] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Gauge transformations in the CS theory will map to diffs in the metric formulation, as follows from the fact that we have already mapped the transformations on the phase space variables ( ˆL, ˆK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5 Boundary action for warped AdS3 To compute the boundary action our first task is to obtain expressions for the charges evaluated on a given orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The finite diff functions are written as (Φ, T) with Φ = φ + ξφ , T = t + ξt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='34) The infinitesimal transformations are given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The first expression is familiar and exponentiates via the Schwarzian derivative and the second expression is easily handled to yield ˆL(φ) = (∂φΦ)2 ˆL0 − 1 2{Φ(φ), φ} ˆK(φ) = ∂φΦ ˆK0 + ∂φT(Φ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='35) where the constant values ( ˆL0, ˆK0) serve as parameters labelling the orbit under considera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Translating to unhatted variables, as appear in the charge expression Q = � dφ(Lξφ + Kξt) we have L(φ) = (∂φΦ)2L0 − k 4π{Φ, φ} + ∂φΦ∂φTK0 + k 8π(∂φT)2 K(φ) = ∂φΦK0 + k 4π∂φT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='36) To extract the boundary symplectic potential Θ we use the momentum method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The 44 momentum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' generator of φ translations, is P = H[ξ = ∂φ] = � 2π 0 L(φ)dφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='37) To apply the momentum method we are instructed to write P in the form P = � (PΦΦ′ + PTT ′)dφ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='38) which is achieved by taking26 PΦ = L0Φ′ + k 8π � 1 Φ′ �′′ PT = K0Φ′ + k 8πT ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='39) The symplectic potential is then Θ = � dφ �� L0Φ′ + k 8π � 1 Φ′ �′′� δΦ + � K0Φ′ + k 8πT ′ � δT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='40) The Hamiltonian corresponding to time translations is H = Q[ξ = ∂t] = � Kdφ = 2πK0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='41) so the kinetic term in the action is obtained by making the replacements (δΦ, δT)→( ˙Φ, ˙T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The fact that this H is a constant on phase space reflects the fact the solutions in this theory are time independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' More interesting dynamics are obtained by choosing a Hamiltonian that generates translations along the vector field ∂t+Ω∂φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is appropriate for computing a partition function of the form Tr e−β(H+ΩP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Such partition functions are the appropriate ones to consider for two reasons [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' First, unitary representations of the warped current algebra have H unbounded from below while P is positive definite, so Ω > 0 is required for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Second, the black hole solutions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) (with constant ( ˆL, ˆK)) have horizon 26We should note that L(φ) has several terms which contain only first derivatives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' as discussed in section (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1), since they are purely quadratic their contribution to Ω is easily found by directly solving iV Ω = −δQV for such terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 45 Killing vector field (defined to be the vector field that vanishes at the bifurcation surface) ξH = ∂t + Ω∂φ = ∂t + 1 2 � ˆL + ˆK ∂φ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='42) Ω must be positive for a smooth horizon, as can be seen from the expression for the surface gravity κ = � ˆL � ˆL + ˆK/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='43) As in all of our examples, the boundary action captures all of the information regarding the warped spacetimes that is dictated by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For instance, it can be used to compute correlators of the currents, as well one loop corrections to the partition function coming from the boundary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 5 Emergent Boundary Modes As developed so far, boundary field theories arise due to the imposition of boundary con- ditions that are preserved by some infinite dimensional group of transformations that act nontrivially at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' However, situations can arise in which the boundary in question is not a sharp boundary in the sense of terminating the space on which the theory lives, but rather an interface that connects the original region to an “outer region”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The question is whether or not the boundary modes survive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' that is, are they effectively transported to the outer region?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This situation arises naturally in the AdS/CFT context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For example, one might have an AdS3 solution which appears as the near horizon limit of some non-asymptotically AdS3 solution M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Is the asymptotic symmetry group of the AdS3 region visible at the boundary of the larger spacetime M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' From the dual CFT point of view, the AdS3 region encodes physics in the deep IR, so one is asking about how to these IR degrees of freedom are realized in terms of the UV description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We will flesh out how this works in a particular example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Five-dimensional Einstein- Maxwell theory (with Λ < 0) admits an asymptotically AdS5 solution that has a near horizon AdS3×R2 factor which we compactify to T 2 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The T 2 is supported by a nonzero Maxwell field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We show how the boundary graviton phase space is visible at the AdS5 boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Including also a CS term A∧F ∧F, the AdS3 region supports boundary photons, 46 and we again show how these appear in the full description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' These results are consistent with the low energy correlators computed in [41], which are sensitive to the emergent boundary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1 Background solution We first briefly summarize the relevant features of the solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' more details may be found in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The action is S = − 1 16πG5 � d5x√g � R + F MNFMN − 12 � + k 12πG5 � A ∧ F ∧ F (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1) We consider a solution of the form ds2 = dr2 L(r)2 + 2L(r)dx+dx− + e2V (r)dxidxi , i = 1, 2 F = bdx1 ∧ dx2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) The function L can be found in terms of V , but numerics are required to find V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We will only need the asymptotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As r→∞ we have AdS5 asymptotics, L(r) = 2r + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' , e2V (r) = cV r + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) for some constant cV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As r→0 we have AdS3 × T 2 asymptotics, L(r) = 2br + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' , e2V (r) = 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) Fluctuations of the metric and gauge field with polarizations and spacetime dependence restricted to the AdS3 directions are governed by an effective 3d Einstein-CS theory, so if we place a boundary in this region we will find the usual 2d boundary photons and gravitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' On the other hand, the asymptotic symmetry group at the AdS5 boundary is finite dimensional, which leads to the aforementioned question of how the near horizon boundary modes are visible in terms of observables computed at the AdS5 boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 47 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2 Boundary photons 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1 Linearized solutions We considered a linearized gauge field perturbation of the form δA = a+(r, x+)dx+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) It obeys Le2V ∂a+ ∂r + −2kba+ = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6) The solution which is smooth at the origin (assuming k > 0) is a+(r, x+) = ϵ(x+)e −2kb � ∞ r dr′ L(r′)e2V (r′) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='7) This has asymptotics: r→0 : a+(r, x+) = ϵ(x+)rk + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' r→∞ : a+(r, x+) = ϵ(x+) � 1 − kb cV r + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8) The 1/r falloff term implies a nonzero boundary current J+ ∼ ϵ(x+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This solution is not quite what we want however, since normalizable solutions, corresponding to vanishing source in the dual CFT, should vanish in the large r limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We can remedy this by performing a gauge transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We first of all write ϵ(x+) = −∂+λ(x+) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='9) and then perform a gauge transformation with parameter Λ(r, x+) = f(r)λ(x+) where lim r→0 f(r) = 0 , lim r→∞ f(r) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='10) After the gauge transformation we have a+ + ∂+Λ = ∂+λ(x+) � f(r) − e −2kb � ∞ r dr′ L(r′)e2V (r′) � ar + ∂rΛ = λ(x+)∂rf(r) a− + ∂−Λ = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='11) 48 such that all components vanish at both small and large r, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It is apparent that these modes are not pure gauge, as they carry a nonzero field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2 Symplectic form We now wish to compute the symplectic form restricted to the space of these linearized solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In particular, we compute the full D = 4 + 1 symplectic form for these non-pure gauge solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We will see that the result agrees with what we would get by considering pure gauge modes living in the near horizon AdS3, with a boundary imposed in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since all polarizations and spacetime dependence of the fluctuations is confined to three dimensions it is convenient to dimensionally reduce the action (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Keeping the gauge field terms, after integrating over the T 2 the action can be written as S = bV2 4πG5 � M3 L (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='12) where V2 is the coordinate volume of T 2 and L = 1 2ΦF ∧ ⋆F + kA ∧ F (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='13) where Φ = b−1e2V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We will consider S = � L and tack on the bV2 4πG5 prefactor at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Proceeding as usual, we vary the action with respect to the gauge field and write the result as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='13), δL = EAδA + dθ yielding the field equation d[Φ ⋆ dA + 2kA] = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='14) and the symplectic form Ω = � Σ ω with ω = δθ = −ΦδA ∧ ⋆dδA − kδA ∧ δA (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15) The fluctuation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='11) takes the form A = a + dΛ where a obeys Φ ⋆ da + 2ka = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='16) Using this we have Ω = k � Σ � δa ∧ δa − dδΛ ∧ dδΛ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='17) In the case of interest a = a+dx+ and so the first term vanishes, while the second terms is 49 an exact form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Restoring the prefactor, we then obtain the symplectic form Ω = − kbV2 4πG5 � ∂Σ δΛ ∧ dδΛ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='18) This symplectic form lives on the AdS5 boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' However, and this is the main point, the form of the result is precisely the same as would have been obtained by just considering the pure CS action kb 4πG5 � A∧F defined on the near horizon AdS3 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It is furthermore easy to verify that the boundary currents take the same form as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This shows very explicitly how the near horizon boundary photon degrees of freedom are effectively transported to the AdS5 boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The mechanism for this is not entirely trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' in particular, note that the fluctuation modes we used in this analysis are not pure gauge in the near horizon region, yet the symplectic form defined on them agrees with that of pure gauge modes in AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 Boundary gravitons 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1 Linearized solutions We now consider a fluctuation around the metric in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) by considering ds2 = dr2 L2 + 2Ldx+dx− + M(dx+)2 + e2V dxidxi (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='19) where we will work to linear order in M = M(r, x+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For r ≪ 1 background metric contains the AdS3 factor ds2 3 = dr2 4b2r2 + 4brdx+dx−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In this near horizon region, a boundary graviton is obtained by applying a diff xµ→xµ + ξµ with ξ = ϵ(x+)∂+ − ∂+ϵ(x+)r∂r − 1 8b3r∂2 +ϵ(x+)∂− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='20) This yields a fluctuation of the form M = − 1 2b2∂3 +ϵ(x+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='21) The fluctuation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='21) is not a solution of the linearized Einstein equations outside the near horizon region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Our task is to extend (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='21) to a solution in the full spacetime that respects the AdS5 asymptotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As shown in [41] such a solution is obtained by multiplying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='21) by −2bLc(r), where Lc(r) = L(r) � r ∞ dr′ L(r′)2e2V (r′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='22) 50 The asymptotics of this function are r→0 Lc 0(r) ∼ − 1 2b r→∞ Lc 0(r) ∼ − 1 4cV r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='23) So the desired fluctuation mode is M = b−1Lc(r)∂3 +ϵ(x+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='24) This non-pure gauge solution carries a nonzero AdS5 boundary stress tensor (c = 3/2G3 is the near horizon Brown-Henneaux central charge), T++ = − cV c 96π∂3 +ϵ(x+) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) which is the same form as the contribution to the AdS3 boundary stress tensor coming from the pure gauge mode (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2 Symplectic form The fluctuation modes found above are not yet in a form suitable for computing the sym- plectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Indeed, contracting two such perturbations with the general gravitational symplectic form will give zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' this follows from symmetry considerations, as each perturba- tion carries two lower + indices and there are no available upper + indices to soak these up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The issue is that the perturbation is singular at r = 0 due to the breakdown of coordinates there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We can fix this by performing a compensating diff that zeroes out the perturbation as r→0 but acts trivially at large r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Let f(r) interpolate smoothly between −1 and 0 in going from small to large r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We then act with a diff of the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='20) except with ϵ(x+) replaced by f(r)ϵ(x+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We write the combined perturbation as φ + φf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='26) By construction, it vanishes as r→0 and is equivalent to our original perturbation at large r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The symplectic form contracted against two such perturbations is27 Ω(φ1 + φ1f, φ2 + φ2f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='27) 27Here we are using the notation Ω(φ1, φ2) = iV2iV1Ω, where Vi denotes the phase space vector field corresponding to the linearized perturbation φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 51 Now, Ω(φ1, φ2) = 0 as already noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Also Ω(φ1f, φ2 + φ2f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This follows since φ1f is a pure diff mode, and Ω contracted against a pure diff mode localizes to the boundary of Σ, but φ2 + φ2f vanishes as r→0 and φ1f vanishes as r→∞, so both boundary terms are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' All that survives is therefore Ω(φ1, φ2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This again localizes to the boundary, but now we get a nonzero boundary term at small r, Since φ1 = −φ1f at small r, this boundary term is precisely the same expression as appears in the symplectic form of pure diff modes in AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='28 So we once again find that the perturbations defined on the full spacetime carry the same symplectic form as the pure gauge near horizon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Acknowledgements We thank Ruben Monten for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' is supported in part by the National Science Foundation grant PHY-2209700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' A Forms Conventions For convenience we collect some conventions and useful expressions involving forms which are used elsewhere in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For a general p-form, ω = 1 p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='ωµ1···µpdxµ1 ∧ · · · ∧ dxµp, dω = 1 p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='∂νωµ1···µpdxν ∧ dxµ1 ∧ · · · ∧ dxµp = 1 p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='∇νωµ1···µpdxν ∧ dxµ1 ∧ · · · dxµp (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1) so long as the connection ∇ is torsionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This also implies 1 p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='∂[νωµ1···µp] = 1 (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (dω)νµ1···µp (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) where the antisymmetrization is defined to include a division by the order of the symmetric group29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The contraction is defined by iξω = 1 (p − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='ξνωνµ2···µpdxµ2 ∧ · · · ∧ dxµp, (iξω)µ2···µp = ξνωνµ2···µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) 28This relative minus sign is cancelled by the minus sign that occurs when we switch from taking the boundary to be inner versus outer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 29For example these conventions imply (dω)αβ = ∂αωβ − ∂βωα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 52 We often find it useful to define (dD−px)µ1···µp = 1 (D − p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='ϵµ1···µpν1···νD−pdxν1 ∧ · · · ∧ dxνD−p (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) and it is useful to note iξ(dD−px)µ1···µp = ξν(dD−(p+1)x)µ1···µpν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) With these notations the Hodge star is defined by ⋆ω = � |g| p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (D − p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='ωµ1···µpϵµ1···µpνp+1···νDdxνp+1 ∧ · · · ∧ dxνD = � |g| p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' ωµ1···µp(dD−px)µ1···µp (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6) where ϵµ1···µD is the totally antisymmetric numeric array with entries ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' With this definition ⋆2 = sgn(g)(−1)p(D−p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' One may also show, assuming torsionless connection, that for p + 1 ̸= D ∇νXµ1···µD−(p+1)ν(dp+1x)µ1···µD−(p+1) = d � 1 D − (p + 1)Xµ1···µD−(p+1)ν(dpx)µ1···µD−(p+1)ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='7) Note there is no factorial in the coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' A useful special case of this is p = D − 2 (so dω is a (D − 1)-form current) is ∇νXµν(dD−1x)ν = d �1 2Xµν(dD−2x)µν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8) For the special case p = D, if ω = ∇µjµ� |g|dDx, then ω = dJ with J = ⋆j = � |g|jµ(dD−1x)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Finally, suppose Σ is a codimension-1 (non-null) surface in M and let φ∗ be the pullback to Σ, as would appear in an integral over Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Then φ∗ �� |g|(dD−1x)µ � = σˆnµ Vol(Σ) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='9) where ˆnµ is the unit normal to Σ and σ = ˆnµˆnµ depends on whether Σ is timelike or lightlike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The orientation choice here is Vol(M) = ˆn ∧ Vol(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 53 B Identically Closed Forms The Poincar´e lemma ensures that if J is any closed form on spacetime then, at least locally, there exists a potential Q satisfying J = dQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Though such conserved currents are common in physics due to Noether’s theorem, the Poincar´e lemma is not typically useful because it carries no guarantee that Q can be constructed locally from the fields in our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Indeed, for a general current J, Q will indeed be non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The exception to this, of course, are the Noether currents associated to gauge symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' These are currents Jξ which are closed for every possible set of free functions ξ parametrizing the gauge transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This additional ingredient, closure despite depending on an arbitrary function, is what we need to ensure the potential Qξ can be constructed as a local functional of ξ and the other fields in the theory, assuming Jξ was local to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This result has appeared in the physics literature in [42] and [43,44], the latter referring to the older mathematical literature on the bivariational complex30 where this result is established by finding a homotopy operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' While [42] is clear on the recursive algorithm for constructing the potential Qξ, insights from the more mathematical literature on the utility of higher Euler operators allow us solve the recursion explicitly in a natural way31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' As most of this work only relies on the existence of an algorithm to compute a local Qξ and not on its details, we state the result as a theorem here32: Theorem 1 As elsewhere, suppose M is a D-dimensional spacetime, φ are some collections of fields over M, and ξ are some functions over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Let J be a p-form over M (p < D) and a local functional of φ and ξ, meaning at each point x ∈ M it depends on only φ and ξ, and finitely many of their derivatives, at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Suppose further that dJ = 0 for all free functions ξ and that J = J0 when ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Then there exists a local functional Q of φ and ξ such that J = J0 + dQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For the interested reader, we have included a short but pedagogical review of how the algorithm can be derived, by way of a simple example, in section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The statement of the algorithmic procedure can be found in section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2, along with a small notational dictionary to some other locations in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 30See [45,46] for textbook treatments on this perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 31This is essentially deriving the homotopy operator, though we will only show that it constructs local potentials and not that it obeys the stronger property of defining a homotopy operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 32The original statement in [42] is slightly more general as the ξ are allowed to be sections of an arbitrary bundle rather than functions, but this is all we will require and allows us to include the statement about ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 54 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1 Motivation The idea underlying this algorithm, in this presentation, relies on a simple observation which can be demonstrated by considering the special case where J is a 2-form which depends only linearly on the functions ξ up to the first derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It is also conceptually simpler to start with a potential for J, say ˜Q, rather than J itself33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Writing ˜Q as an expansion in the derivative order of ξ we have ˜Q = � ˜Qkαξk + ˜Qµ kα∂µξk� dxα = � ξkEk( ˜Qα) + ∂µ[ξkEµ k ( ˜Qα)] � dxα (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1) where in the second line we have integrated by parts until no derivatives act directly on the ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We will refer to the coefficients Eµ1···µr k ( ˜Qα) as the Euler coefficients34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Here we would have Ek( ˜Qα) = ˜Qkα − ∂µ ˜Qµ kα, Eµ k ( ˜Qα) = ˜Qµ kα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) Since the ξk are free functions, the coefficients in the derivative expansion are unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since there are only finitely many derivatives present we can always convert between the derivative and Euler expansions by integration by parts identities, so the Euler expansion must also be unique35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Of course, this is merely a rewriting of ˜Q, but it’s a rewriting with the advantage that when we take the differential of ˜Q we find d ˜Q = 1 1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' � 0 + ∂ρ[ξkEk( ˜Qα)] + ∂ρ∂µ[ξkEµ k ( ˜Qα)] � dxρ ∧ dxα = 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' � ξkEk(d ˜Qρα) + ∂µ[ξkEµ k (d ˜Qρα)] + ∂µ∂ν[ξkEµν k (d ˜Qρα)] � dxρ ∧ dxα (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) where in the second line we have written the generic Euler expansion for d ˜Q = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' But 33This ˜Q will be related to the potential Q we construct in the end, but in general will be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The actual algorithm makes no reference to this ˜Q, so it’s only purpose here is as a useful intermediary to motivate the key ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 34This language is inherited from the mathematical literature where the Eµ1···µr k would be called the higher Euler operators as they are generalizations of the Euler-Lagrange operator which produces the equations of motion from a Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 35This is the step which requires the “finite derivative order” part of locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In [42] it was that the recursion started with the highest derivative order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 55 because the Euler expansion is unique, the aligned terms must match so 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='Ek(Jρα) = 0, 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='Eµ k (Jρα) = δµ [ρEk( ˜Qα]), 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='Eµν k (Jρα) = δ(µ [ρ Eν) k ( ˜Qα]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) Throughout this section, antisymmetrization will not include the k index as it’s generically of a different type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' If we were able to invert these equations for the Euler coefficients of ˜Q, we would have determined ˜Q in terms of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Unfortunately, we cannot invert these equations, but we don’t need to because J doesn’t have a unique potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' So really since our goal is to construct a potential for J, we only need to reconstruct ˜Q up to a closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To do this we want to eliminate the Kronecker deltas, and we may do so by taking a trace between µ and ρ in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We find 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='Eµ k (Jµα) = 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' � DEk( ˜Qα) − Ek( ˜Qα) � , 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='Eµν k (Jµα) = 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' � DEν k( ˜Qα) − Eν k( ˜Qα) + Eν k( ˜Qα) − δν αEµ k ( ˜Qµ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) The overall factors come from the (anti-)symmetrization and we see that we can classify the types of terms which appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' If either µ or ρ (or both) appear on the delta, we obtain a term proportional to an Euler coefficient of ˜Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' If both indices do not appear on the delta, then we obtain a term which still contains a delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In the case both µ and ρ appear on the delta we find the coefficient to be the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' If only µ is on the delta we obtain a minus sign36, if only ρ is on the delta then we obtain a plus sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We may rewrite (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) in the form Ek( ˜Qα) = 1 D − 1Eµ k (Jµα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Eν k( ˜Qα) = 2 DEµν k (Jµα) + 1 Dδν αEµ k ( ˜Qµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6) The second equation does not specify the Euler coefficient37 of ˜Q in terms of only J, but observe that if we try to build ˜Q back from these expressions we find ˜Q = � 1 D − 1ξkEµ(Jµα) + ∂ν � ξk � 2 DEµν k (Jµα) + 2δν αEµ k ( ˜Qµ) ��� dxα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='7) 36That this observation generalizes to higher forms and more derivatives of ξ is not completely obvious, but is nonetheless true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The proof is essentially an exercise in combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 37In fact, the inability to invert these equations is related to the fact that we are really defining a homotopy operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' If the homotopy operator is denoted h, then ˜Q = hd ˜Q + dh ˜Q = hJ + dh ˜Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The “delta terms” in our organization of the contraction collects into the second term here as we argue shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [45] for a proof that these terms organize into the particular form dh ˜Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' These extra terms are not important to our particular application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 56 So the term preventing us from completely determining the Euler coefficients of ˜Q in terms of those of J has collected itself into an exact form!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Furthermore, we can see that this will be a generic conclusion because all of the upper spacetime indices (besides µ which is already contracted) have to contract on derivatives when we construct the form from the Euler coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In the “delta type” terms, at least one of these derivatives will therefore contract on the delta and hence on a form index, making the term a total differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This means we can always write ˜Q = Q+dα where α is constructed from all of the “delta type” terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since Q, which is thus constructed entirely from the Euler coefficients of J, is related to ˜Q by an exact form, both are equally good potentials for J and so there is no loss in taking Q over ˜Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The generalization to higher form degrees and arbitrary derivative orders in ξ is now mostly a matter of making sure we get the coefficients on the various terms in the contraction correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is not completely trivial, but a straightforward combinatorics problem, the main difficulty resting with the result mentioned in Footnote 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2 General algorithm Suppose that J is an identically closed p-form (p < D) depending on a set of free functions ξk and finitely many derivatives thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We give the algorithm in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' First we state it for the special case where J depends only linearly on ξ and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We then point out the simple generalization to other derivative operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Finally, we state how the algorithm for the linear case also gives the solution to the non-linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since J is linear in ξ and its derivatives, it can always be written in the form J = � Jkα1···αpξk + Jµ kα1···αp∂µξk + · · · � dxα1 ∧ · · · ∧ dxαp = � ξkEk(Jα1···αp) + ∂µ[ξkEµ k (Jα1···αp)] + · · · � dxα1 ∧ · · · ∧ dxαp (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8) where in the second line we have integrated by parts to remove all derivatives from ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The Euler coefficients, Eµ1···µr k (Jα1···µp), are all the data we need to construct a local potential for J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Note that the upper indices of the Euler coefficients are assumed to be symmetrized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' With this, the potential is given by Q = 1 (p − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' � r=0 r + 1 D − p + r + 1∂µ1 · · · ∂µr[ξkEνµ1···µr k (Jνα2···αp)]dxα2 ∧ · · · dxαp (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='9) where the upper bound on the sum over r depends on the derivative order J as it’s a sum over all of J’s Euler coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We also take the convention where r = 0 indicates no derivatives 57 should be taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The factor of (p−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' can be absorbed into a contraction against the vector ∂ν to write Q = � r=0 r + 1 D − p + r + 1∂µ1 · · · ∂µr[ξkEνµ1···µr k (i∂νJ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='10) This formula appears elsewhere in the literature using slightly different notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In [44,47] a multi-index notation is used (multi-index notation is also employed in [45,46], but the conventions are slightly different which makes coefficients look different) wherein (µ) is a tuple of indices, |µ| is its length, and ((µ)ν) is the concatenation of the tuple (µ) with the additional index ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' If we further denote Eµ1···µr k = δ δξk (µ) and i∂νJ = ∂J ∂dxν then (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='10) may be written compactly as Q = |µ| + 1 D − p + |µ| + 1∂(µ) � ξk δ δξk (ν(µ)) ∂J ∂dxν � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='11) where a summation over the length of the tuple (µ) is to be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For example, this is equation (A36) in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The generalization from partial derivatives to covariant derivatives of arbitrary type is described in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In the proof of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='9) the only property of the derivatives used is that they obey the product rule and that the upper indices of the Euler coefficients are completely symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Symmetry of the indices follows trivially for partial derivatives, but for covariant derivatives we can always work with the symmetrized derivatives at the cost of introducing field strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' So if we assume that all derivatives have first been symmetrized, we can write38 J = � ξkEk(Jα1···αp) + ∇µ[ξkEµ k (Jα1···αp)] + · · · � dxα1 ∧ · · · ∧ dxαp (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='12) with potential Q = 1 (p − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' � r=0 r + 1 D − p + r + 1∇µ1 · · · ∇µr[ξkE(νµ1···µr) k (Jνα2···αp)]dxα2 ∧ · · · ∧ dxαp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='13) For the generalization to non-linear dependence of J on ξ we consider an arbitrary 1- parameter path ξ(λ) through ξ-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Denote by Jλ the evaluation of J on ξ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This Jλ is 38Though we use the same symbol for the Euler coefficient, the Euler coefficients computed using symmetrized covariant derivatives and using partial derivatives will, of course, not generally be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 58 still identically closed and so it’s λ derivative is as well: d ˙Jλ = 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='14) where we have denoted the λ derivative by a dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Now ˙Jλ is an identically closed form which depends only linearly on ˙ξ, which is arbitrary and independent of ξ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Hence we may apply the (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='13) for the linear case to ˙J using ˙ξ as our free function to find Qλ = � r=0 r + 1 D − p + r + 1∇µ1 · · · ∇µr[ ˙ξkE(νµ1···µr) k (i∂νJ)] (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15) which satisfies ˙Jλ = dQλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The Euler coefficients now, of course, must be understood to have been computed from integrating by parts on ˙ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Taking the λ integral we find J1 = J0 + d � 1 0 Qλdλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='16) If we choose the flow ξ(λ) such that ξ(0) = 0 and ξ = ξ(1), then theorem 1 is an immediate consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' A useful choice of path is the contracting flow39 ξ(λ) = λξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In this case the potential for J is Q = � 1 0 dλ � r=0 r + 1 D − p + r + 1∇µ1 · · · ∇µr[ξkE(νµ1···µr) k (i∂νJ(λξ))] (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='17) where it should be understood that the Euler coefficients used here should be those of ˙Jλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Making the notational changes described above (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='11), we see that this expression is (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='9) in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 39Much like the Poincar´e contracting homotopy, this choice is not invariant under redefinitions of the ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Furthermore, there may be cases where this path is not suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We refer the reader to the discussion in [42] for some considerations in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The contracting flow is always possible when the ξ are functions on spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 59 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3 Examples As a simple first example we may derive the Komar term (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='41) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='42) we find Jξ = √−g 16πG (∇ν(∇µξν + ∇νξµ) − 2∇µ∇νξν) (ddx)µ − iξL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='18) The potential is simple to compute in this case without using the heavy machinery introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We need only write ∇µ∇νξν = −Rµ νξν + ∇ν∇µξν (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='19) so Jξ = �√−g 8πG Rµ νξν(ddx)µ − iξL � + √−g 16πG∇ν(∇νξµ − ∇µξν)(ddx)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='20) The first pair of terms, linear in ξ with no derivatives thereof, vanish on the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The latter pair of terms can be identified as yielding the Komar term via (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To demonstrate the heavy machinery, note first that we only need the linear case (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='10) and that only Euler derivatives with at least one upper index (meaning at least one derivative in the Euler expansion of Jξ) contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' So any terms linear in ξ with no derivatives, after putting Jξ in Euler form, will not contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Indeed, we can see in the example (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) that these terms must vanish when Jξ is closed40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In particular, this means the iξL term will not contribute to Qξ, which we also found in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since only the iVξθ terms in Jξ will contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In the remaining terms we need to symmetrize the covariant derivatives so the Euler coefficients will have symmetrized upper derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Here the derivatives are all 2nd order, so the anti-symmetrization will pro- duce Riemann curvatures with no additional derivatives on ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' These terms will again not contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The result is now Jξ = √−g 16πG∇(α∇β) �� gβµδα k + gαβδµ k − 2gαµδβ k � ξk� (ddx)µ + (· · · )kξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='21) The relevant Euler coefficients are then Eα k (Jξ) = 0, Eαβ k (Jξ) = 1 2 � 2gαβδµ k − gαµδβ k − gβµδα k � (ddx)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='22) 40This is actually a familiar statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Note that the zeroth Euler coefficient is precisely the Euler- Lagrange operator acting on Jξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is then the standard statement that the Euler-Lagrange equations annihilate total derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 60 The reconstruction (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='10) now produces Qξ = √−g 16πG � 0 + 1 + 1 D − (D − 1) + 1 + 1∇α � ξkEαβ k (i∂βJξ) �� = √−g 16πG 2 3 1 2∇α � ξk � gαβδµ k − gαµδβ k − gβµδα k � (dd−1x)µβ � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='23) which is again the Komar term, though this algorithm is obviously not unnecessary for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Another example, this time non-linear in ξ, would be the calculation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This example is again simple enough that we can simply obtain the result (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='63) without applying the general algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To do this we need only (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='14) to find 16πGδkξ = 4Λ D − 2 √−g [ξν ∧ ∇νξµ + ∇νξν ∧ ξµ] (ddx)µ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='24) which is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='63) upon collecting the total derivative and using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' To find this result using the general algorithm, we first take a derivative of δkξ to obtain 16πGδ ˙kξ = 4Λ D − 2 √−g∇ν � ˙ξν ∧ ξµ + ξν ∧ ˙ξµ� (ddx)µ = 4Λ D − 2 √−g∇ν � ˙ξk ∧ (δν kξµ − δµ kξν) � (ddx)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) From this we extract the Euler coefficient Eν k(δ ˙kξ) = 1 16πG 4Λ D − 2 √−g ∧ (δν kξµ − δµ kξν) (ddx)µ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='26) where we have included a dangling wedge to remind ourselves that when we perform the potential reconstruction the ˙ξ appears to the left of this coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since δkξ = 0 when ξ = 0, we have (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='16) with J0 = 0 if we use the contracting flow (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The reconstruction (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='17) then yields 16πG˜kξ = � 1 0 dλ 0 + 1 D − (D − 1) + 0 + 1 4Λ D − 2 √−gξk ∧ λ(δν kξµ − δµ kξν)(dd−1x)µν = − 2Λ D − 2 √−gξµ ∧ ξν(dd−1x)µν, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='27) as we have already shown should be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 61 C Diffeomorphism Charges for Generally non-Covariant Lagrangians In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2, we reviewed a general procedure for computing the Noether charge associated to an arbitrary gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In the special case where the gauge transformation is a diffeomorphism, additional simplifications are possible [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The transformation of the Lagrangian under a diffeomorphism ξ can generally be sepa- rated into covariant and non-covariant pieces as kξ = iξL + Yξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1) We may do the same for the action of the diffeomorphism on θ: LVξθ = Lξθ + ˜Πξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) With these separations together, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='24) becomes dΠξ = diξθ + (˜Πξ − δYξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) That is, the covariant part of the transformations collect automatically into a total derivative, so the task of computing Πξ is reduced to just finding a potential Σξ satisfying dΣξ = ˜Πξ − δYξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) The computation of Cξ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='26) is now δCξ = iξθ + Σξ − iVξδB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) These considerations only alter the computations we need to find Cξ and do not change the identification of the charge in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='27) as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' D Non-Abelian CS Here we consider a non-Abelian CS theory in D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is an important example and stress test of our techniques because of the CS/WZW correspondence which, at first glance, might seem to be in tension with our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In particular, recall the well known fact that one must supply an explicit parametrization in order to reduce a WZW model to an explicit 62 boundary theory, and in particular the theory cannot be reduced explicitly to the boundary in terms of the gauge element g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' On the other hand, the charges for CS theory can be explicitly constructed on the boundary from g and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The construction (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25) would then seem to suggest that Ω can also be constructed explicitly from g, and we will see that this is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In order for our construction (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='29) of the boundary action to be consistent with this known property of WZW models, it must be the case that while Ω can be explicitly constructed from g, the canonical 1-form Θ cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This is a non-trivial requirement and we will see that it is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' So then to define the theory we take the background manifold to be M = R × D where the spatial slices are disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The Lagrangian and some useful quantities are L = tr � A ∧ dA + 2 3A3 � , E = 2F, θ = − tr(A ∧ δA), kλ = tr(λdA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1) We choose the chiral boundary conditions At = Aφ so there is no need for a boundary contribution to the action, B = ℓ = 0, in light of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The Noether current associated to gauge transformations is Jλ = tr [d(λA) − 2λF] = d tr(λA) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) so Qλ = tr(λA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Thus one may check that the full Noether charges are H[λ] = 2 � 2π 0 dφ tr(λA), Ht = � 2π 0 dφ tr(A2 φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) Before attempting to apply our techniques to this theory, we should use the WZW correspondence to see that the boundary action produced by standard techniques should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since we will need an explicit parametrization, and hence gauge group, we wil choose for simplicity SU(2) with the parametrization g = eiα1σ1eiα2σ2eiα3σ3, z = 2 sin(2α2) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) where σk are the Pauli matrices and z turns out to be a convenient definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' By solving the Lagrange multiplier constraint, the action produced by the Lagrangian (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1) becomes the WZW model S = � M tr �� g−1dg �3� − � tr � g−1∂tgg−1∂φg � dt ∧ dφ + � ∂M tr � A2 φ � dt ∧ dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) 63 Using tr(g−1dg)3 = d(zdα1 ∧ dα3) the boundary action may be written S = −2 � ∂M dtdφ [(α′ 1 − zα′ 3)( ˙α1 − α′ 1) + α′ 2( ˙α2 − α′ 2) + α′ 3( ˙α3 − α′ 3)] (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6) where the minus sign comes from identifying the boundary orientation to be dφ ∧ dt, as commented on in footnote 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' With this result in mind, since we already have the full Noether charges in hand, the simplest approach is to compute the symplectic form via (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This produces Ω = � 2π 0 tr � δAφ ∧ g−1δg � dφ = � 2π 0 tr � δ(g−1∂φg) ∧ g−1δg � dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='7) As promised at the beginning of this section, due to the g−1 factor, this cannot be written as δ of something directly in terms of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Instead we must go to the explicit parametrization (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) in order to find the canonical 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' But this is a straightforward calculation to perform, the result being Ω = −2δ � 2π 0 [α′ 1δα1 + α′ 2δα2 + (α′ 3 − zα′ 1)δα3] dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8) Together with Ht = −2 � 2π 0 � α′2 1 + α′2 2 + (α′ 3 − zα′ 1)α′ 3 � dφ (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='9) we find the phase space action S = −2 � dt � ∂Σ dφ [α′ 1( ˙α1 − α′ 1) + α′ 2( ˙α2 − α′ 2) + (α′ 3 − zα′ 1)( ˙α3 − α′ 3)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='10) We could also find this result by the methods in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Since we already have Qλ from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2), we only need to compute the potential for δkw with w = g−1δg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In the case of diffeomorphisms it’s simple to use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='12) as our free function with respect to which the potential ˜kw can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Here, however, w = g−1δg is not a completely free function and we must use δα as our free function instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This unfortunately means that we cannot write down a potential ˜kw without specifying the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Using the parametrization (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) we find ˜kw = δz ∧ δα3dα1 + δα3 ∧ δα1dz + δα1 ∧ δzdα3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='11) 64 Furthermore we may compute δQw = δz ∧ (dα3δα1 + dα1δα3) + z(dδα3 ∧ δα1 + dδα1 ∧ δα3) + 2(δα1 ∧ dδα1 + δα2 ∧ dδα2 + δα3 ∧ dδα3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='12) Adding these forms we find δQw + ˜kw = −2δ [dα1δα1 + dα2δα2 + (dα3 − zdα2)δα2] + d(zδα2 ∧ δα1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='13) Integrating this over the φ circle the total derivative term does not contribute and we evidently reproduce (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8), from which the phase space action (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='10) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' E Relation to Schwazrzian Action for JT gravity From the perspective of the current paper, it is most illuminating to view JT gravity and its Schwarzian action description [48] as a special case of our approach to 3D gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' The 3D origin of the JT/Schwarzian is discussed in [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In 3D we have the Euclidean action S3 = − 1 16πG � d3x√g3(R3 + 2) − 1 8πG � ∂M d2x � h3(K3 − 1) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1) The 3D class of metrics we consider are ds2 3 = dz2 z2 + � 1 z2 + z2 4 LL � dwdw − 1 2Ldw2 − 1 2Ldw2 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) with w = φ + it, w = φ − it, (L, L) are functions of (φ, t) that take the form L = {F(φ, t), φ} + κ 2F ′2 L = {F(φ, t), φ} + κ 2F ′2 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3) where (κ, κ) are constants and ′ = ∂φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' At fixed t, the functions (F, F) are elements of diff(S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In general, the metrics (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) are off-shell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' to be on-shell the functions (F, F) must be, respectively, holomorphic and anti-holomorphic in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' It is rather tricky to obtain the off-shell action governing (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) by direct substitution into (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1), in part because the coordinates are not globally smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Instead, we apply phase space methods, viewing (F, F) at fixed time as points on phase space and building an action out of 65 the gravitational symplectic form and Hamiltonian on this phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' This procedure [14] leads to the Alekseev-Shatashvili action [30], SAS = − 1 16πG � d2x � κF ′∂ ¯wF − � 1 F ′ �′′ ∂ ¯wF + ¯κ ¯F ′∂w ¯F − � 1 ¯F ′ �′′ ∂w ¯F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) We now turn to the 2D story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' One approach is to KK reduce the 3D action by considering metrics ds2 3 = ds2 2 + Φ2dt2 , t ∼= t + 2π (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) taking the metric components to be t independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Using � M3 d3x√g3(R3 − Λ) = � M2 d2x√g2Φ(R2 − Λ) − 2 � ∂M2 dx √ hnµ∂µΦ (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='6) where the metric on the 2d boundary is h and nµ is the outward pointing unit normal to the boundary, along with √g3K3 = √ hΦK2 + √ hnν∂µΦ (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='7) we find that the 3D action (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2) reduces to the JT action S3 = − 1 16πG2 � d2x √ hΦ(R2 + 2) − 1 8πG2 � dφ √ hΦ(K2 − 1) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='8) with G2 = G/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Coming back to our 3D picture, to be in accordance with (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5) we should take F = F(φ) and F = F(φ), along with κ = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' We are thus considering 3D metrics (generically off-shell) of the form ds2 3 = dz2 z2 + 1 z2 � 1 − Lz2 2 �2 dφ2 + 1 z2 � 1 + Lz2 2 �2 dt2 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='9) with L(φ) = {F(φ), φ}+ κ 2F ′2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' For this restricted class of off-shell metrics it is straightforward to evaluate the 3D action S3 on the form (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' In particular, we have R2 = −2 so only the boundary term survives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Cutting off the space at z = zc and taking Φ|z=zc = z−1 c , we find the action as zc→0, S3 = − 1 8πG2 � L(φ)dφ + constant (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='10) 66 which is the Schwarzian action (supplemented with the κ term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='41 Alternatively, we can apply the reduction to the AS action (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' After integrating over t and doing some integration by parts, we find that (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='4) is equal to (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Witten, Quantum Field Theory and the Jones Polynomial, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 121 (1989) 351–399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Elitzur, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Moore, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Schwimmer, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' B 326 (1989) 108–134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Wald and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Zoupas, A General definition of ’conserved quantities’ in general relativity and other theories of gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' D 61 (2000) 084027, arXiv:gr-qc/9911095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [4] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Donnelly and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Wall, Entanglement entropy of electromagnetic edge modes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 114 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 11, (2015) 111603, arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1895 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Ghosh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Soni, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Trivedi, On The Entanglement Entropy For Gauge Theories, JHEP 09 (2015) 069, arXiv:1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='02593 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [6] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Donnelly and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102, arXiv:1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='04744 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Blommaert, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Mertens, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Verschelde, Edge dynamics from the path integral — Maxwell and Yang-Mills, JHEP 11 (2018) 080, arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='07585 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Crnkovic and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Witten, Covariant description of canonical formalism in geometrical theories, Three hundred years of gravitation (1986) 676–684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Lee and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Wald, Local symmetries and constraints, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 31 (1990) 725–743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Achucarro and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Townsend, A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' B 180 (1986) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' B 311 (1988) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 41The original approach to obtaining this action involved considering cutouts of AdS2 with wiggly boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Here the boundary is at fixed coordinate location z = zc, and wiggles are instead encoded in fluctuations of the metric via the function L(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 67 [12] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Coussaert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Henneaux, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' van Driel, The Asymptotic dynamics of three- dimensional Einstein gravity with a negative cosmological constant, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 12 (1995) 2961–2966, arXiv:gr-qc/9506019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Cotler and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Jensen, A theory of reparameterizations for AdS3 gravity, JHEP 02 (2019) 079, arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='03263 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [14] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Kraus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Monten, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Myers, 3D Gravity in a Box, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 11 (2021) 070, arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='13398 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Ebert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Hijano, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Kraus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Monten, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Myers, Field Theory of Interacting Boundary Gravitons, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 13 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 2, (2022) 038, arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='01780 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Anninos, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Padi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Song, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Strominger, Warped AdS(3) Black Holes, JHEP 03 (2009) 130, arXiv:0807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3040 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Deser, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Jackiw, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Templeton, Topologically Massive Gauge Theories, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 140 (1982) 372–411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [Erratum: Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 185, 406 (1988)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Kraus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Monten, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Roumpedakis, Refining the cutoff 3d gravity/TT correspondence, JHEP 10 (2022) 094, arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='00674 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Nutku, Exact solutions of topologically massive gravity with a cosmological constant, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 10 (1993) 2657–2661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' G¨urses, Perfect Fluid Sources in 2+1 Dimensions, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 11 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 10, (1994) 2585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Hofman and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Rollier, Warped Conformal Field Theory as Lower Spin Gravity, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' B 897 (2015) 1–38, arXiv:1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='0672 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Maloney and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029, arXiv:0712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='0155 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [23] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Comp`ere and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Fiorucci, Advanced Lectures on General Relativity, arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='07064 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Harlow and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Wu, Covariant phase space with boundaries, JHEP 10 (2020) 146, arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='08616 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Wald, Black hole entropy is the Noether charge, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' D 48 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 8, (1993) R3427–R3431, arXiv:gr-qc/9307038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 68 [26] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Tachikawa, Black hole entropy in the presence of Chern-Simons terms, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 24 (2007) 737–744, arXiv:hep-th/0611141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Coleman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Wess, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Zumino, Structure of phenomenological Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 177 (1969) 2239–2247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Callan, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Coleman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Wess, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Zumino, Structure of phenomenological Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 177 (1969) 2247–2250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [29] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Comp`ere, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Song, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Strominger, New Boundary Conditions for AdS3, JHEP 05 (2013) 152, arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='2662 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Alekseev and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Shatashvili, Path Integral Quantization of the Coadjoint Orbits of the Virasoro Group and 2D Gravity, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' B 323 (1989) 719–733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Ba˜nados, Three-dimensional quantum geometry and black holes, AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 484 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 1, (1999) 147–169, arXiv:hep-th/9901148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [32] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Iyer and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' D 50 (1994) 846–864, arXiv:gr-qc/9403028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [33] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Nazaroglu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Nutku, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Tekin, Covariant Symplectic Structure and Con- served Charges of Topologically Massive Gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' D 83 (2011) 124039, arXiv:1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3404 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Bouchareb and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Clement, Black hole mass and angular momentum in topologically massive gravity, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 24 (2007) 5581–5594, arXiv:0706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='0263 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [35] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Compere and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Detournay, Semi-classical central charge in topologically massive gravity, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 26 (2009) 012001, arXiv:0808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1911 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [Erratum: Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 26, 139801 (2009)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [36] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Compere and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Detournay, Boundary conditions for spacelike and timelike warped AdS3 spaces in topologically massive gravity, JHEP 08 (2009) 092, arXiv:0906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1243 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Blagojevic and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Cvetkovic, Asymptotic structure of topologically massive gravity in spacelike stretched AdS sector, JHEP 09 (2009) 006, arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='0950 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [38] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Azeyanagi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Detournay, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Riegler, Warped Black Holes in Lower-Spin Gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' D 99 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 2, (2019) 026013, arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='07263 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [39] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Detournay, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Hartman, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Hofman, Warped Conformal Field Theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' D 86 (2012) 124018, arXiv:1210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='0539 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 69 [40] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' D’Hoker and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Kraus, Magnetic Brane Solutions in AdS, JHEP 10 (2009) 088, arXiv:0908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3875 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [41] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' D’Hoker, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Kraus, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Shah, RG Flow of Magnetic Brane Correlators, JHEP 04 (2011) 039, arXiv:1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='5072 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [42] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Wald, On identically closed forms locally constructed from a field, Journal of Math- ematical Physics 31 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 10, (1990) 2378–2384, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='528839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='528839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [43] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Barnich, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Brandt, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Henneaux, Local BRST cohomology in gauge theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 338 (2000) 439–569, arXiv:hep-th/0002245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [44] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Barnich and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' B 633 (2002) 3–82, arXiv:hep-th/0111246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [45] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Olver, Applications of Lie Groups to Differential Equations, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Graduate Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Springer New York, NY, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [46] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Anderson, The Variational Bicomplex, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Utah State University, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [47] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Compere, Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' PhD thesis, Brussels U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=', 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' arXiv:0708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='3153 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [48] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Maldacena, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Stanford, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP 2016 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 12, (2016) 12C104, arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='01857 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' [49] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' Mertens, The Schwarzian theory — origins, JHEP 05 (2018) 036, arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content='09605 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} +page_content=' 70' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfKgMk/content/2301.02964v1.pdf'} diff --git a/b9AzT4oBgHgl3EQfLftx/content/tmp_files/load_file.txt b/b9AzT4oBgHgl3EQfLftx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e456b858cdcf82372a83d7d0599ac3b992ae0ad2 --- /dev/null +++ b/b9AzT4oBgHgl3EQfLftx/content/tmp_files/load_file.txt @@ -0,0 +1,1075 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf,len=1074 +page_content='Supercurrent in Bi4Te3 Topological Material-Based Three-Terminal Junctions Jonas K¨olzer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2 Abdur Rehman Jalil,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2 Daniel Rosenbach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2 Lisa Arndt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='3 Gregor Mussler,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2 Peter Sch¨uffelgen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2 Detlev Gr¨utzmacher,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2 Hans L¨uth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2 and Thomas Sch¨apers1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' ∗ 1Peter Gr¨unberg Institut (PGI-9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Forschungszentrum J¨ulich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Wilhelm-Johnen-Straße,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 52425 J¨ulich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Germany 2JARA-Fundamentals of Future Information Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' J¨ulich-Aachen Research Alliance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Forschungszentrum J¨ulich and RWTH Aachen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 52425 J¨ulich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Germany 3JARA Institute for Quantum Information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' RWTH Aachen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Germany (Dated: January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2023) In an in-situ prepared three-terminal Josephson junction based on the topological insulator Bi4Te3 and the superconductor Nb the transport properties are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The differential resistance maps as a function of two bias currents reveal extended areas of Josephson supercurrent including coupling effects between adjacent superconducting electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The observed dynamics for the coupling of the junctions is interpreted using a numerical simulation of a similar geometry based on a resistively and capacitively shunted Josephson junction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The temperature dependency indicates that the device behaves similar to prior experiments with single Josephson junctions comprising topological insulators weak links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Irradiating radio frequencies to the junction we find a spectrum of integer Shapiro steps and an additional fractional step, which is interpreted by a skewed current-phase relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In a perpendicular magnetic field we observe Fraunhofer-like interference patterns of the switching currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' INTRODUCTION Hybrid structures comprising three-dimensional topo- logical insulator nanoribbons combined with supercon- ductors are a very promising platform for realizing cir- cuits for fault-tolerant topological quantum computing [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' For its operation Majorana bound states are em- ployed, which are formed by aligning an external mag- netic field with a nanoribbon proximitized with an s-type superconductor [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' For the braiding of different pairs of Majorana states for qubit operation multi-terminal structures are required [2, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Braiding can be per- formed by adjusting the superconducting phase of the superconducting electrodes to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Multi-terminal Josephson junctions are the backbone of Majorana braiding mechanism in a topological qubit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' where a three-terminal Josephson junction acts as a basic building block [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Understanding the superconducting transport in such a device holds a key importance for the realization of a topological quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Generally, the use of hybrid devices with multiple connections leads to rich physics in terms of transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Indeed, theoretical studies have investigated singularities, such as Weyl nodes, in the Andreev spectra of multi-terminal Josephson junctions [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Moreover, multi-terminal Josephson junctions with topologically trivial supercon- ducting leads may lead to realizations where the junction itself can be regarded as an artificial topological material [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Furthermore, three-terminal junctions also allow transport via the quartet mechanism and non-local An- dreev processes [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' On the experimental side, multi-terminal Josephson junctions were fabricated with different materials for the ∗ th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='schaepers@fz-juelich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='de weak link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In three-terminal Josephson junctions with a Cu or InAs nanowire subgap states [18, 19] and half- integer Shapiro steps [20] were observed, indicating trans- port via quartets of entangled Cooper pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Supercur- rent flow affected by dissipative currents in an adjacent junction was studied on graphene-based junctions [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Moreover, the higher-dimensional phase space was found to lead to fractional Shapiro steps in this type of junctions due to the inverse AC Josephson effect [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' By combin- ing a multi-terminal junction with a top gate, the effect of gate voltage and magnetic field on the critical current contour has been studied [3, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Recently, flakes of the topological insulator Bi2Se3 were also used as a weak link in an interferometer structure, and evidence for a non-sinusoidal current-phase relationship was observed [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In flux-controlled three-terminal junctions based on Bi2Te3, the opening and closing of a minigap was studied using normal probes [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Here, we report on the transport properties of a three- terminal Josephson junction based on the Bi4Te3 mate- rial system as the weak link and Nb as the supercon- ductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' To fabricate the samples, we used selective-area growth for the Bi4Te3 layer in combination with an in- situ bridge technology to define the superconducting elec- trodes [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Bi4Te3 is a natural superlattice of alternating Bi2 bilayers and Bi2Te3 quintuple layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Initially, Bi4Te3 has been reported to be a semimetal with zero band gap and a Dirac cone at the Γ point [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' However, recent band structure calculations in conjunction with scanning tunneling spectroscopy and angular photoemission spec- troscopy measurements suggest that the material is a semimetal with topological surface states [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In par- ticular, advanced GW-band structure calculations have shown that a band gap of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='2 eV opens at the Γ point, which significantly reduces the density of the bulk state in this energy range [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Bi4Te3 is classified as a dual topological insulator, a strong topological insulator arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='01115v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='supr-con] 3 Jan 2023 2 with a non-zero mirror Chern number, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' a topological crystalline insulator phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Though Bi4Te3 does not ex- hibit the proposed Dirac semimetal phase, it is still a very interesting material as it resides in close proximity to the critical point of band crossing in the topological phase diagram of BixTey alloys [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Such a transition is pro- posed by Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [34] where a topological crystalline insulator (Bi2Te3) [35] can be topologically transformed into a topological Dirac semimetal through alloying it with other materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' On our multi-terminal junctions, we first investigated the DC properties and related the results to simulations based on the resistively and capac- itively shunted Josephson junction (RCSJ) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' We then measured the radio frequency (rf) response, finding evidence for coupling of adjacent junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Finally, the behavior of our three-terminal junctions when an out-of- plane magnetic field is applied is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' EXPERIMENTAL Using the previously introduced technologies of topo- logical insulator selective-area growth and in-situ bridge technology we fabricated three-terminal Josephson junc- tions, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 1(a) [2, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The geome- try of the nanoribbon T-shaped junction for selective- area growth is defined by trenches in a SiO2/Si3N4 (5 nm/15 nm) layer on a highly-resistive Si (111) sub- strate [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' First, the 600-nm-wide nanotrenches are etched into the top Si3N4 layer using a combination of electron beam lithography and reactive ion etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Sub- sequently, a second set of layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' a 100-nm-thick SiO2 layer and a 300-nm-thick Si3N4 layer, is deposited on top to define the stencil mask for the in-situ Nb deposition [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' After patterning the structures for the stencil mask into Si3N4, SiO2 is etched in hydrofluoric acid (HF) form- ing the free-hanging bridge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Simultaneously, the Si(111) surface in the selective-area growth trenches is released in the bottom SiO2 layer defined by the Si3N4 layer on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The Bi4Te3 layer is selectively grown within these trenches, while the Si3N4 bridge structures are em- ployed to define the geometry of the in situ deposited su- perconducting electrodes [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The Bi4Te3 layer is grown at a temperature of 310◦C using molecular beam epitaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Subsequently, the 50-nm-thick superconducting Nb elec- trodes are deposited by electron beam evaporation fol- lowed by covering the whole structure with a 5-nm-thick Al2O3 dielectric capping layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Our processing scheme ensured a high-quality crystalline topological insulator material with clean superconductor interfaces [2, 38], as reported in previous transmission electron microscopy studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' An electron microscopy image of the investi- gated device is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The measurements of the three-terminal Josephson junction were carried out in a dilution refrigerator with base temperature of T = 25 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' containing a 1 - 1 - 6 T vector magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' As indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 1(b), the left, right, and bottom junction electrodes are labeled as ”L”, ”R”, and ”B”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Two current sources supply cur- rents ILB and IRB from L and R to the bottom electrode, respectively, with the according voltages VLB and VRB measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The differential resistances are measured by adding an ac current of 10 nA to the DC current bias us- ing a lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The rf-irradiation for the Shapiro step measurements was provided by an antenna placed in close vicinity to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' RESULTS AND DISCUSSION DC characteristics Information about the basic junction characteristics is obtained by measuring the differential resistances RLB = ∆VLB/∆ILB and RRB = ∆VRB/∆IRB as a function of the bias currents ILB and IRB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Starting with the left junction we find that RLB shown in Figs 2(a) and (b) contains a superconducting region in the center when ILB and IRB are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The observed critical current contour is similar to what has been observed in induced supercon- ducting nano junctions made of high mobility materials such as InAs/Al [3, 24] or graphene [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The supercon- ducting region extends along an inclined line indicated by the dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The switching to the super- conducting state can be seen in the line cuts at fix values IRB = 0 and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='7µA provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The exten- sion of the superconducting state originates from a part of IRB which flows via R to L through the junction between L and B compensating the current ILR partly and by that reducing the total current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' For our three-terminal device no reduced differential resistance is observed along the line ILB = IRB, which would indicate the presence of a Josephson supercurrent between the junction formed be- tween electrodes L and R [3, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' We attribute this to the fact that the distance between these electrodes is slightly larger than for the other junctions so that no Josephson supercurrent is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' However, the junction between L and R acts as a shunt resistor taking care that the switching to the superconducting state is non-hysteretic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The differential resistance RRB measured between R and B electrodes, depicted in Figs 2(c) and (d), shows a sim- ilar behaviour as RLB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' featuring also an extended superconducting range due a compensation provided by part of ILR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The tilt of the superconducting range indi- cated by the dashed line in Figure 2(c) is lower compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2(a) since now ILR is the compensating current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Simulations The experimental results are modeled by assuming a network of two resistively and capacitively shunted Josephson (RCSJ) junctions coupled by a resistor RC, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Solving the related system of differential equations numerically, in analogy to what was presented in previous works [3, 4], we simulate the 3 400nm ILB IRB a b B L R VLB VRB FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rendering of a selective-area grown three-terminal Josephson junction and false color scanning electron micrograph with circuit: (a) The three-terminal junction is composed of the silicon substrate (gray bottom layer), the first hard mask composed out of a silicon oxide (white)/ silicon nitride (blue) layer (as indicated by the labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' On top of this another hard mask layer composed of silicon oxide (white) and silicon nitride (blue) is deposited and patterned as a shadow mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The topological insulator (red) is grown selectively into the first hard mask trench and the shadow mask is used for the definition of the junction in the metal deposition (silver) step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' (b) False-color scanning electron micrograph of the in-situ prepared three- terminal junction device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Niobium contacts (cyan) are deposited on top of the TI (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The measurement configuration is also indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Differential resistance maps: (a) shows RLB as a function of the bias currents ILB and IRB at 25 mK with cor- responding line cuts given in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In (c) the differential re- sistance map of RRB is depicted with a selection of line cuts given in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The dashed lines in a and (c) indicate the super- conducting regions of compensating bias currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The differ- ential resistances was measured by using lock-in technique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' RLB = ∆VLB/∆ILB and RRB = ∆VRB/∆IRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' behaviour of the experimental system (information about the procedure see Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The results of the simulations are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 3(b) to (e), where the differential resistance RLB is given as a function of the bias currents ILB and IRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The model describes the experiment well by reproduc- ing the Josephson supercurrent along the inclined lines originating from compensating currents from both elec- trodes with a superconducting region at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The inclination is determined by the coupling resistance RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 3(b) and (c), the coupling resistance was taken as RC = 4 · RLB, with RLB = 40 Ω which results in the same tilt as observed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Taking these val- ues into account the normal state resistance is given by RN = 6/5 · RLB = 48 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In our simulations for the crit- ical current and for the Steward-McCumber parameter we assumed Ic = 538 nA and βc = (2e/ℏ)IcR2 NC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='1, respectively, with c the junction capacitance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' We found that the superconducting state in the junction between R and B leads to some weak feature as a similar line in- clined towards horizontal orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Note, that for this line RLB is non-zero, as the supercurrent in the other junction only partly reduces the current in the junction between L and B and hence only partially reduces the voltage drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' A noticeable difference between experiment and simulation is that in the measurements the extension of the superconducting state observed along the inclined line (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2(a) is decreased compared to the simula- tion depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' As discussed by Draelos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [21], this effect can be explained by dissipation in the neighboring junction being in the normal state resulting in an effective heating, in particular for junctions with small dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In our simulation the direct coupling between the different junctions was neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' As shown by Arnault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [4], including coupling results in a more complex contour of the critical current area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' If the cou- pling resistance becomes very small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' RC → 0, the observed lines in the differential resistance shift towards the diagonal (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 3(d) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Thus, both junctions are maximally correlated to both current biases ILB and IRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Temperature dependence In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 4(a) to (f) the differential resistance maps are shown for RLB and RRB measured at temperatures of b a 150 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 (μA) (U) 60 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 RLB RLB B 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 20 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 ILB (μA) ILB (μA) d c 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 (U) (μA) 60 (U) 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 RRB RRB RB 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 20 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 ILB (μA) IRB (μA)200 nm* EHT = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='00 kV Signal A = InLens Date :1 Feb 2020 HNF Mag = 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='02 K X WD= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0mm Tilt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 ° LE01550VPSiN4 JLB JRB RN IRB VRB J C RN ILB VLB J C RC a b c d e FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Numerical simulation of different coupling scenar- ios: (a) The three-terminal circuit is modeled by two RCSJ shunted Josephson junctions JLB and JRB (green and blue dashed line boxes), which are each modeled by a resistor RN, a capacitor c, and an ideal Josephson junction J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Currents ILB and IRB are supplied via current sources while the volt- age drops VLB and VRB across the junctions are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Both junctions are coupled via a coupling resistance RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' (b) Differential resistance RLB as a function of current biases for a realistic scenario for RC close to the one extracted in the experiment: RN = 40 Ω, RC = 160 Ω, Ic = 538 nA, βc = (2e/ℏ)IcR2 NC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The zero resistance range is ob- served as a tilted line due to a compensation by a part of IRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Additionally, the influence of the second junction is ob- served as a similar line close to horizontal orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The corresponding line cuts indicated in (b) are presented in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The scenario for a very small coupling resistance (RC → 0) is shown as a color map of RLB and selected line cuts in (d) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 100 mK, 200 mK, and 800 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' One finds that with in- creasing temperature the area of the central supercon- ducting region shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' This is in accordance with the temperature dependence of the critical current of a sin- gle Nb/Bi4Te3/Nb reference junction, as shown in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' It is noteworthy that the super- conducting feature along the inclined lines basically does not change with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' This can be ex- plained by the fact, that the dissipation in the neighbor- ing junction already leads to an increased temperature being larger than the substrate temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 ILB ( A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 IRB ( A) a T=100 mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 ILB ( A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 IRB ( A) b T=100 mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 ILB ( A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 IRB ( A) c T=200 mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 ILB ( A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 IRB ( A) d T=200 mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 ILB ( A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 IRB ( A) e T=800 mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 ILB ( A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 IRB ( A) f T=800 mK 0 50 100 RLB ( ) 0 50 100 RRB ( ) 50 150 250 RLB ( ) 0 50 100 150 RRB ( ) 10 20 30 RLB ( ) 10 20 RRB ( ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Differential resistance maps at various temperatures: Left column (a), (c), (e) shows the differential resistance RLB, right column (b), (d), (f) RRB, accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The temper- atures displayed in the rows from up to down are 100 mK, 200 mK, and 800 mK, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' rf characteristics Next the radio frequency response of the system is in- vestigated in order to confirm that the experiment is de- scribed well by Josephson junction physics and to analyze the rf response of the Josephson current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' This is done by first choosing a frequency and an amplitude for the rf ir- radiation so that both junctions show a large rf response in the differential resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Subsequently the same DC bias sweeps are performed as in the prior experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Figures 5(a) and (b) show Shapiro step measurements of the differential resistances RLB and RRB, respectively, as a function of bias currents ILB and IRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The differential resistances are calculated by numerical differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Differential resistances obtained by lock-in amplifier mea- surements can be found in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The rf frequency frf and the according power was set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='8 GHz and 0 dBm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The differential resis- tances show clear intercrossing stripe-like patterns which can be attributed to the presence of Shapiro steps con- firming the presence of a Josephson supercurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The intercrossing parallel stripes indicating a coupling of both junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' By calculating the related voltage drop we find that for both junctions the Shapiro steps are located at integer multiples, n = 1, 2, 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' , of the characteristic 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 ILB ( A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 IRB ( A) a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 ILB ( A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 IRB ( A) b 20 40 60 RRB ( ) 0 20 40 60 RLB ( ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Shapiro step measurements at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='8 GHz: (a) Numer- ically determined differential resistance RLB as a function of ILB and IRB at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='8 GHz and rf power of 0 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' (b) Corre- sponding map of the differential resistance RRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' voltage V0 = hfrf/2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 6(a) and (b) the differential resistance maps of RLB and RLB, now taken at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 GHz at 0 dBm, are depicted, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Here, the color maps are plot- ted as a function of the normalized voltages VLB/V0 and VRB/V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' On first sight one finds that the Shapiro step pattern is more pronounced in RLB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' We attribute this to a stronger coupling of the rf signal compared to the neigh- bouring junction due to spatial variations of the rf field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' As for the measurements at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='8 GHz a coupling of both junctions, although weaker, is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Our experimen- tal results of Shapiro step measurements are supported by comparison to simulations based on the previously in- troduced RCSJ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In Supplementary Figures 4(a) and (b) maps of the simulated values of RLB and RLB as a function of the normalized bias voltages are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' There, one finds that the coupling by RC results in a weak cross coupling of the Shapiro signal resulting in in- tercrossing stripe-like patterns of different contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' A closer inspection of the resistance map presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 6(a) reveals that apart from the integer Shapiro steps also half-integer Shapiro steps, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' at n = 1/2, are ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The half integer steps are also clearly resolved in the averaged value of RLB along VLB/V0 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In single Josephson junctions such fractional steps are interpreted by assuming a skewed current-phase relationship [39–41] (a simulation for this case using our model is provided in the Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' More specifically for multi-terminal junctions the rf response of superconductivity induced into normal metal has been studied previously by Duvauchelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Here, half- integer steps have been found and interpreted as a feature due to the presence of coherent quartet states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' How- ever, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 2 we did not find indications of quartet states, which would be visible by a feature in the dif- ferential resistance at opposite voltage drops on the left and right terminal [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Other experimental observations of such fractional steps in multi-terminal junctions are in- terpreted on the basis of highly connected nonlinear net- works of Josephson junctions, where (due to the higher phase space) different transitions of the phase particle in the washboard potential are possible [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' However, since fractional Shapiro steps were observed in single junctions made with similar materials [42], we favor the explana- tion based on a skewed current-phase relationship, which can be attributed to contributions of quasi-ballistic trans- port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In our measurements under rf radiation we did not find indications of missing odd Shapiro steps, as pre- dicted when Majorana bound states are present in topo- logical junctions [2, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Probably, for our samples the narrow width of the Bi4Te3 ribbons prevents the forma- tion of these states, since due to the finite Berry phase a magnetic field along the junctions is required to gain a gap closure for the coherent surface states around the nanoribbon cross section [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Magnetic field response The junction characteristics were also analyzed in a perpendicularly oriented magnetic field B⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 7(a) the magnetic field dependence RLB is plotted as a func- tion of B⊥ and ILB, while IRB is kept at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' One clearly observes a Fraunhofer-like interference pattern of the switching current, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' the boundary between the red superconducting areas and the areas with finite resis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 7(a) indicates the according fitting based on the Fraunhofer interference relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The close resemblance of the experimental data to an ideal Fraunhofer pattern points towards a relatively homoge- neous distribution of the supercurrent density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' From the fit we extract a period of about ∆B =14 mT, which corre- sponds to a junction area of 152×103 nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Relating these values to the dimensions of the left junction JLB one finds that the period is about a factor of ten smaller than ex- pected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Based on the actual junction size of 200×72 nm2 a period of 144 mT is expected for a h/2e flux periodic- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' We attribute the discrepancy to the experimental period to a pronounced flux focusing effect, where the magnetic field is expelled from the edge regions of the superconducting electrodes and bundled in the junction area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' As a matter of fact, a comparably large flux fo- cusing effect was previously observed in similar planar Josephson junctions based on topological insulators and Nb superconducting electrodes [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 7(b) the magnetic field dependence RRB is shown as a function of B⊥ and IRB at ILB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Once again, a Fraunhofer-like interference is observed, al- though with a smaller period, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' an larger effective area where the magnetic flux is picked up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The reason for the difference compare to the measurements shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 7(a) might be some inhomogeneity in the super- current density in the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Finally, the RLB maps are scanned diagonally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' ILB = IRB, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 7(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Here, once again a regular Fraunhofer pat- tern is observed, which is almost identical to the pat- tern shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 7(d), indication, that the current IRB through the neighboring junction basically has not effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Shapiro step measurements at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 GHz: (a) Numerically determined differential resistance RLB as a function of the normalized voltage drops VLB/V0 and VRB/V0 at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 GHz and rf power of 0 dBm, with V0 = hfrf/2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The blue curves represent the averaged signal along VLB/V0 and VRB/V0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The dashed lines indicate the half-integer steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' (b) Corresponding map of the differential resistance RRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' a b c FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Differential resistances under perpendicular magnetic field sweep: (a) shows a map of RLB as a function of B⊥ and ILB for IRB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' (b) represents the corresponding map of RRB as a function of B⊥ and IRB for ILB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In (c) RLB is plotted with the sweep current chosen to be ILB = IRB, which corresponds a sweep along the diagonal in the current plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In all cases a standard Fraunhofer pattern is fitted indicated as blue lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' CONCLUSION We have succeeded in extending the previously devel- oped in situ fabrication technology for Josephson junc- tions to a working more complex design of a three- terminal junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Analysis of the transport experiments shows that our system indeed behaves like a coupled net- work of Josephson junctions in DC transport, rf response, as well as magnetic field response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' This is the first re- port on the topological multi-terminal devices where an interaction between the individual Josephson junctions is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Moreover, the observation of fractional steps in the rf response opens a window that provides a first insight into the novel physics of this type of device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' On a more technical level, our results demonstrate the re- alization of more complex devices required for network structures in topological quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Further investigations and detailed understanding of such a system are crucial for the realization of complex topological quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In future, similar exper- iments with more intricate circuit designs and super- conducting phase controlled measurements will be per- formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The complexities in the junction characteristics arose from the selected weak-link material Bi4Te3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In future experiments we plan to incorporate conventional three-dimensional topological insulators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Bi2Te3, Sb2Te3, Bi2Se3, and the topological Dirac semimetal ex- hibited by the correctly tuned BixTey stoichiometric al- loy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' ACKNOWLEDGEMENT We thank Herbert Kertz for technical assistance as well as Kristof Moors and Roman Riwar for fruitful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' This work was partly funded by the Deutsche Forschungsgemeinschaft (DFG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' German Research Foun- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='dation) under Germany’s Excellence Strategy—Cluster ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='of Excellence Matter and Light for Quantum Computing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='← 80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='fss ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='BA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='B7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='(ML4Q) EXC 2004/1—390534769,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' the German Federal Ministry of Education and Research (BMBF) via the Quantum Futur project “MajoranaChips” (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 13N15264) within the funding program Photonic Research Germany, as well as the Bavarian Ministry of Economic Affairs, Regional Development and Energy within Bavaria’s High-Tech Agenda Project “Bausteine f¨ur das Quantencomputing auf Basis topologischer Materialien mit experimentellen und theoretischen Ans¨atzen“ (grant allocation no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 07 02/686 58/1/21 1/22 2/23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Hyart, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' van Heck, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Fulga, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Burrello, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Akhmerov, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Beenakker, Flux-controlled quan- tum computation with Majorana fermions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 88, 035121 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Aasen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Hell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Mishmash, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Higginbotham, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Danon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Leijnse, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Jespersen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Folk, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Marcus, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Flensberg, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Alicea, Milestones toward Majorana-based quantum computing, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' X 6, 031016 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Manousakis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Altland, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Bagrets, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Egger, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Ando, Majorana qubits in a topological insulator nanoribbon architecture, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 95, 165424 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Cook and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Franz, Majorana fermions in a topological-insulator nanowire proximity-coupled to an s- wave superconductor, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 84, 201105 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Cook, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Vazifeh, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Franz, Stability of Ma- jorana fermions in proximity-coupled topological insulator nanowires, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 86, 155431 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Legg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Loss, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Klinovaja, Majorana bound states in topological insulators without a vortex, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 104, 165405 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Fu and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Kane, Superconducting proximity effect and Majorana fermions at the surface of a topological in- sulator, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 100, 096407 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Stenger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Hatridge, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Frolov, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Pekker, Braiding quantum circuit based on the 4π Josephson ef- fect, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 99, 035307 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Yokoyama and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Nazarov, Singularities in the Andreev spectrum of a multiterminal Josephson junction, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 92, 155437 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Eriksson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Riwar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Houzet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Meyer, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Nazarov, Topological transconductance quantization in a four-terminal Josephson junction, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 95, 075417 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Xie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Vavilov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Levchenko, Weyl nodes in Andreev spectra of multiterminal Josephson junctions: Chern numbers, conductances, and supercurrents, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 97, 035443 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Riwar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Houzet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Meyer, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Nazarov, Multi-terminal Josephson junctions as topological matter, Nature Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 7, 1 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Houzet and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Samuelsson, Multiple Andreev reflec- tions in hybrid multiterminal junctions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 82, 060517 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Freyn, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Dou¸cot, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Feinberg, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' M´elin, Produc- tion of nonlocal quartets and phase-sensitive entanglement in a superconducting beam splitter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 106, 257005 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Nowak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Wimmer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Akhmerov, Super- current carried by nonequilibrium quasiparticles in a mul- titerminal Josephson junction, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 99, 075416 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Melo, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Fatemi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Akhmerov, Multiplet su- percurrent in Josephson tunneling circuits, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 12, 17 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Pfeffer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Duvauchelle, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Courtois, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' M´elin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Feinberg, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Lefloch, Subgap structure in the con- ductance of a three-terminal Josephson junction, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 90, 075401 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Cohen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Ronen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Kang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Heiblum, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Fein- berg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' M´elin, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Shtrikman, Nonlocal supercurrent of quartets in a three-terminal Josephson junction, Pro- ceedings of the National Academy of Sciences 115, 6991 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Duvauchelle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Pfeffer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Courtois, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Lefloch, Quantum coherence of the quartet scheme ob- served by Shapiro resonance under radio-frequency irradi- ation in three terminal Josephson junctions, IEEE Trans- actions on Applied Superconductivity 26, 1 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Draelos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Wei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Seredinski, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Mehta, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Taniguchi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Borzenets, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Amet, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Finkelstein, Supercurrent flow in multi- terminal graphene Josephson junctions, Nano Letters 19, 1039–1043 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Arnault, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Larson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Seredinski, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Idris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' McConnell, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Taniguchi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Borzenets, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Amet, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Finkelstein, Multiterminal inverse ac Josephson effect, Nano Letters 21, 9668 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Graziano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Pendharkar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Palm- strøm, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Pribiag, Transport studies in a gate- tunable three-terminal Josephson junction, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 101, 054510 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [24] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Pankratova, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Lee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Kuzmin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Wickramasinghe, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Mayer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Yuan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Vavilov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Shabani, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Manucharyan, Multiterminal Josephson effect, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' X 10, 031051 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [25] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Graziano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Gupta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Pendharkar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Dong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Dempsey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Palmstrøm, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Pribiag, Selective control of conductance modes in multi-terminal Josephson junctions, arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='01373 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Kurter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Finck, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Hor, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Van Harlin- gen, Evidence for an anomalous current–phase relation in topological insulator Josephson junctions, Nature Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 6, 1 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [27] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Lyu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Ying, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Shen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Fan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Ji, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Jing, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Qu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Lu, Pro- tected gap closing in Josephson trijunctions constructed on Bi2Te3 , Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 100, 180501 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Sch¨uffelgen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rosenbach, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Schmitt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Schleenvoigt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Jalil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Schmitt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' K¨olzer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Bennemann, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Parlak, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Kibkalo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Trel- lenkamp, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Grap, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Meertens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Luysberg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Mus- sler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Berenschot, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Tas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Golubov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Brinkman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Sch¨apers, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Gr¨utzmacher, Selective area growth 8 and stencil lithography for in situ fabricated quantum de- vices, Nature Nanotechnology 14, 825 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [29] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Saito, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Fons, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Makino, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Mitrofanov, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Ue- sugi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Takeguchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Kolobov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Tominaga, Compositional tuning in sputter-grown highly-oriented Bi–Te films and their optical and electronic structures, Nanoscale 9, 15115 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [30] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Chagas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Ribeiro, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Gon¸calves, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Calil, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Silva, ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Malachias, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Mazzoni, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Ma- galh˜aes-Paniago, Bi2:Bi2Te3 stacking influence on the surface electronic response of the topological insulator Bi4Te3, Electronic Structure 2, 015002 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [31] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Chagas, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Ashour, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Ribeiro, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Silva, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Louie, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Magalh˜aes Paniago, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Petroff, Multiple strong topological gaps and hexagonal warping in Bi4Te3, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 105, L081409 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Nabok, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Tas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Kusaka, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Durgun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Friedrich, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Bihlmayer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Bl¨ugel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Hirahara, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Aguilera, Bulk and surface electronic structure of Bi4Te3 from GW calculations and photoemission experiments, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Materials 6, 034204 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Jalil, Engineering topological superlattices and their epitaxial integration in selectively grown hybrid nanostruc- tures via MBE, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' thesis, RWTH Aachen University (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [34] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Yang and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Nagaosa, Classification of stable three- dimensional dirac semimetals with nontrivial topology, Nature Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 5, 1 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [35] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rauch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Flieger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Henk, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Mertig, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Ernst, Dual topological character of chalcogenides: Theory for Bi2Te3, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 112, 016802 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [36] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rosenbach, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Schmitt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Sch¨uffelgen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Stehno, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Schleenvoigt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Jalil, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Mus- sler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Trellenkamp, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Golubov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Brinkman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Gr¨utzmacher, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Sch¨apers, Reap- pearance of first Shapiro step in narrow topological Josephson junctions, Science Advances 7, eabf1854 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [37] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Schmitt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Connolly, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Schleenvoigt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Liu, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Kennedy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Ch´avez-Garcia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Jalil, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Ben- nemann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Trellenkamp, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Lentz, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=', Integration of topological insulator Josephson junctions in superconduct- ing qubit circuits, Nano Letters 22, 2595 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' K¨olzer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Moors, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Jalil, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Zimmermann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rosenbach, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Kibkalo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Sch¨uffelgen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Mussler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Gr¨utzmacher, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Schmidt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' L¨uth, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Sch¨apers, In-plane magnetic field-driven symmetry breaking in topo- logical insulator-based three-terminal junctions, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Materials 2, 116 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [39] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Snyder, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Trimble, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Folkes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Taylor, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Williams, Weak-link Josephson junctions made from topological crystalline insulators, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 121, 097701 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [40] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Panghotra, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Raes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' de Souza Silva, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Cools, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Keijers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Scheerder, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Moshchalkov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Van de Vondel, Giant fractional Shapiro steps in anisotropic Josephson junction arrays, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Physics 3, 53 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [41] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Raes, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Tubsrinuan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Sreedhar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Guala, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Panghotra, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Dausy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' de Souza Silva, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Van de Vondel, Fractional Shapiro steps in resistively shunted Josephson junctions as a fingerprint of a skewed current-phase relationship, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 102, 054507 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [42] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rosenbach, Quantum transport and induced super- conductivity in selectively deposited topological insulator devices, Dissertation, RWTH Aachen University, Aachen (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [43] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Dom´ınguez, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Kashuba, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Bocquillon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Wieden- mann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Deacon, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Klapwijk, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Platero, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Molenkamp, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Trauzettel, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Hankiewicz, Joseph- son junction dynamics in the presence of 2π- and 4π- periodic supercurrents, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 95, 195430 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 1 Supplementary Material: Supercurrent in Bi4Te3 Topological Material-Based Three-Terminal Junctions SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' SINGLE JUNCTION MEASUREMENTS As a reference a single Nb/Bi4Te3/Nb junction was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The junction has a length of 140 nm and a width of 500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In Supplementary Figure S1(a) the current voltage characteristics is shown at temperatures in the range from 30 mK to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='77 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' At lowest temperature a critical current of 750 nA is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In contrast to the three terminal junction, here, a hysteretic behaviour is observed, which can be explained by the missing shunt for the single Josephson junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' We attribute the hysteresis to heating resulting in a lower return current Ir compared to Ic [S1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The critical current monotonously decreases with temperature with some kink around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The latter might be attributed to a switching from a more diffusive to a more ballistic transport in the weak link [S2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Supplementary Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Current-voltage characteristics of a single Nb/Bi4Te3/Nb junction: (a) Current-voltage charac- teristics at temperatures ranging from 30 mK to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='77 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' (b) Critical current Ic as well as return current Ir as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' SII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' RCSJ MODEL FOR A THREE-TERMINAL JUNCTION The characteristics of our three-terminal junctions is simulated by employing a two-dimensional resistively and capacitively shunted Josephson junction (RCSJ) Ansatz in analogy to what was presented in previous works [S3, S4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' 3(a) in the main text the corresponding network is depicted including two resistively and capacitively shunted Josephson junctions with the normal state resistance RN and the capacitance C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' We assume two identical junctions each having a critical current of Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The junctions are connected by a coupling resistor RC representing the non- superconducting junction between electrodes L and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Following the RCSJ Ansatz the characteristics of the three- terminal junction can be described by a set of coupled differential equations of the form: ILB Ic = sin(ϕLB) + dϕLB d�τ + βc d2ϕLB d�τ 2 + RN RC �dϕRB d�τ − dϕLB d�τ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' (S1) IRB Ic = sin(ϕRB) + dϕRB d�τ + βc d2ϕRB d�τ 2 − RN RC �dϕRB d�τ − dϕLB d�τ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' (S2) with ϕLB and ϕRB the phase differences between junctions JLB and JRB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' �τ = t/τJ the normalized time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' τJ = Φ0/(2πIcRN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' with Φ0 = h/2e the magnetic flux quantum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' and βc = (2e/ℏ)IcR2 NC the Stewart-McCumber parameter [S5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The equations are similar to the standard RCSJ model for a single junction, except of the last term, which introduces the current through the resistor, coupling the two junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' This current is a result of the voltage difference between the two junctions and the coupling resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' For RC → ∞ the coupling term goes to zero, leading to two individual junctions (decoupled system) and for RC → 0 the system is dominated by the coupling term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' a b 40 750 X X 500 Ic 20 X 250 (Λr) 0 0 20 250 X X X X X X 40 500 750 C 60 —500 250 0 250 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='8 1000 -750 T (K) I (nA)2 SIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' SHAPIRO STEPS IN THREE-TERMINAL JUNCTION EXPERIMENTS The differential resistances RLB and RRB exposed to an rf radiation with a frequency of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='8 GHz at 0 dBm recorded as a function of the applied DC currents are presented in Supplementary Figures S2(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In contrast to the corresponding figure, which was gained by numerical differentiation, here, the resistance is directly taken using a lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In Supplementary Figures S3 the corresponding measurements at a frequency of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='8 GHz at 0 dBm are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Supplementary Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Shapiro Step response at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='8 GHz: (a) shows the measured differential resistance across the first junction RLB as a function of the direct current ILB and IRB across the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' (b) shows RRB for the same current constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Supplementary Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Shapiro Step response at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 GHz: a shows the measured differential resistance across the first junction RLB as a function of the direct current ILB and IRB across the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' b shows RRB for the same current constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' SIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' SHAPIRO STEPS IN THREE-TERMINAL JUNCTION SIMULATION Using the model described in Supplementary Information SII the Shapiro response was simulated by adding an oscillation contribution ij,rf sin(2πfrft), j = LB, RB, to the dc bias currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The simulated differential resistances RLB and RRB as a function of the normalized voltage drops at a frequency of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 GHz are presented in Supplementary Figures S4(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' One finds that the Shapiro response is strong in the corresponding junctions, while the coupling from the neighboring junction is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In order to simulate the appearance of the fractional Shapiro steps a non-sinusoidal current-phase relationship was assumed for the Josephson junction by including a sin(2ϕ) contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' In Supplementary Figures S5(a) and (b) the b a 1000 1000 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 72 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='8 64 500 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='6 500 56 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='4 48 (nA) (nA) (nA) (nA) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='2 40 0 0 RB 9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 32 R R 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='8 24 500 500 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='6 16 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='4 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content="2 1000 1000- 0 1000 -500 0 500 1000 1000 -500 0 500 1000 'LB(nA) 'Lb(nA)b a 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 60 60 IRB(nA) (nA) C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 40 LB B R R 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 ILB (nA) ILB (nA)3 Supplementary Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Shapiro step simulations at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 GHz: (a) Numerically determined differential resistance RLB as a function of the normalized voltage drops VLB/V0 and VRB/V0 at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' The blue curves represent the averaged signal along VLB/V0 and VRB/V0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' (b) Corresponding map of the differential resistance RRB with the blue curves representing the averaged differential resistance along VLB/V0 and VRB/V0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' respective simulation outcomes RLB and RRB for junctions JLB and JRB are shown as a function of bias currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' One finds that by increasing the sin 2ϕ contribution fractional steps appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [S1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Courtois, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Meschke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Peltonen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Pekola, Origin of hysteresis in a proximity Josephson junction, Physical Review Letters 101, 067002 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [S2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Sch¨uffelgen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rosenbach, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Schmitt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Schleenvoigt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Jalil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Schmitt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' K¨olzer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Bennemann, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Parlak, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Kibkalo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Trellenkamp, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Grap, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Meertens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Luysberg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Mussler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Berenschot, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Tas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Golubov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Brinkman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Sch¨apers, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Gr¨utzmacher, Selective area growth and stencil lithography for in situ fabricated quantum devices, Nature Nanotechnology 14, 825 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [S3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Graziano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Pendharkar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Palmstrøm, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Pribiag, Transport studies in a gate-tunable three-terminal Josephson junction, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' B 101, 054510 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [S4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Arnault, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Larson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Seredinski, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Idris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' McConnell, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Taniguchi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Borzenets, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Amet, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Finkelstein, Multiterminal inverse ac Josephson effect, Nano Letters 21, 9668 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' [S5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Tinkham, Introduction to Superconductivity (Dover Publications, New York, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' e b 100 100 80 80 2 2 (5) 60 60 1 L RB B Lo R 0 0 m 40 B 40 V 1 1 20 20 2 2 0 0 2 0 2 2 0 2 VB / Vo VLB / Vo4 Supplementary Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' Simulation of Shapiro maps at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='5 GHz with 2φ term: (a) differential resistance of the first junction Shapiro steps as a function of ILB, IRB with an equal contribution of a sin 2φ-term in the system, (b) the same for the second junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' (c) and (d) show the same after doubling the sin 2φ contribution in the system and (e) and (f) show the same after doubling the contribution of (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=" a e 1000 1000 1000 (n' ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=' (nA) (nA) 0 0 0 9 R R 1000 1000 1000 1000 1000 1000 1000 1000 0 0 0 1000 ILB(nA) ILB(nA) I LB(nA) b d 1000 1000 1000 RRB(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} +page_content=') IRB(nA) (nA) IRB(nA) 0 0 0 R R 1000 1000 1000 1000 0 1000 1000 0 1000 1000 0 1000 ILB(nA) ILB(nA) ILB(nA)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfLftx/content/2301.01115v1.pdf'} diff --git a/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf b/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..75c6db3ada711b21f219b7fca743c7085fee4ed4 --- /dev/null +++ b/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c84bb0d326a1b0d5d0173204b10ad659dd7b8921743d3a00ee3d927030832a84 +size 618818 diff --git a/bNAyT4oBgHgl3EQfXPcY/vector_store/index.pkl b/bNAyT4oBgHgl3EQfXPcY/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..ce9a74c13dbeff07506b765ce1f81f6ed9fa2a0d --- /dev/null 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sha256:402ed49b16ec45dcc4d6ec0b2c30498a6d3acc84f7e9e081573d7628d62d8ba6 +size 402574 diff --git a/etE2T4oBgHgl3EQfxghO/vector_store/index.faiss b/etE2T4oBgHgl3EQfxghO/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e6651f7cd4f9e712ad0440e62fa978c0f9c19133 --- /dev/null +++ b/etE2T4oBgHgl3EQfxghO/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ac30cf1cc991a74d0a31473f4b06c5201bed4b961fac1d7608577d3768f889ff +size 4587565 diff --git a/fNE1T4oBgHgl3EQfegS8/content/tmp_files/2301.03208v1.pdf.txt b/fNE1T4oBgHgl3EQfegS8/content/tmp_files/2301.03208v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3aff7a655e754b3efc7b8174da8ea79c0fc6674 --- /dev/null +++ b/fNE1T4oBgHgl3EQfegS8/content/tmp_files/2301.03208v1.pdf.txt @@ -0,0 +1,1256 @@ +Symmetry classes of dissipative topological insulators with edge dark state +Fei Yang,1 Zheng Wei,1 Xianqi Tong,1 Kui Cao,1 and Su-Peng Kou1 +1Center for Advanced Quantum Studies, Department of Physics, +Beijing Normal University, Beijing 100875, China +We classify the dissipative topological insulators (TIs) with edge dark states (EDS) by using the +38-fold way of non-Hermitian systems in this paper. The dissipative dynamics of these quadratic +open fermionic systems is captured by a non-Hermitian single-particle matrix which contains both +the internal dynamics and the dissipation, refereed to as damping matrix X. And the dark states +in these systems are the eigenmodes of X which the eigenvalues’ imaginary part vanishes. However, +there is a constraint on X, namely that the modes in which the eigenvalues’ imaginary parts are +positive are forbidden. In other words, the imaginary line-gap of X is ill-defined, so the topological +band theory classifying the dark states can not be applied to X. To reveal the topological protection +of EDS, we propose the double damping matrix ˜ +X = diag (X, X∗), where the imaginary line-gap is +well defined. Thus, the 38-fold way can be applied to ˜ +X, and the topological protection of the EDS +is uncovered. Different from previous studies of EDS in purely dissipative dynamics, the EDS in +the dissipative TIs are robust against the inclusion of Hamiltonians. Furthermore, the topological +classification of ˜ +X not only reflects the topological protection of EDS in dissipative TIs but also +provides a paradigm to predict the appearance of EDS in other open free fermionic systems. +I. +INTRODUCTION +Symmetry and topology play the central role in mod- +ern physics, which results in many interesting phenom- +ena and future applications, such as robust edge mode +in topological insulators (TIs) and topological supercon- +ductors (TSs)[1–3]. +And the systematic classifications +of those phases are explained by the ten-fold way of free +fermions, or Altland-Zirnbauer (AZ) symmetry classes[4]. +The AZ class provides not only a framework to analyze +the topological behavior of system with different sym- +metries but also gives a paradigm to explore new topo- +logical phases. +However, those are for the close Her- +mitian systems, many physical systems in nature expe- +rience dissipation associated with gain and loss, such +as atomic, molecular, and optical physics[5–7], the dy- +namics of these systems are effectively described with a +non-hermitian (NH) Hamiltonian. Dissipation in these +NH systems would give rise to many interesting effect +that do not have Hermitian counterpart, such as NH +skin effects[8–11], PT -symmetric physics[12, 13], and +the breakdown of bulk-boundary correspondence [14–17]. +Moreover, the question that how the dissipation influ- +ences the topology of a system attracts much attention, +which has been explored in many papers[18–24]. +The +fundamental interests in studying the topological proper- +ties of NH systems are to expand the symmetry classes, +which had been settled by Bernard and LeClair based +on four fundamental symmetries, resulting in a total of +43 symmetry classes, that is known as Bernard-LeClair +class[25]. While Kawabata et al. discovered that only 38 +of 43 symmetry classes are topologically inequivalent[26], +which is known as 38-fold way. +The 38-fold way pro- +vides a paradigm to explore the topology of a NH system, +which has been applied in many NH systems, such as the +NH Sachdev-Ye-Kitaev Model[27], and symmetry classes +of the open quantum system[28, 29]. +As we have mentioned, the non-Hermiticity is ubiqui- +tous in nonequilibrium open systems, but most of them +are hard to solve especially if the interaction is pre- +sented. Fortunately, the open free fermionic systems are +exactly solvable, in which the dissipative dynamics are +completely captured by the so called damping matrix X, +which is a NH single-particle matrix that contains the in- +ternal dynamics as well as the dissipation. The topolog- +ical phenomena in the dissipative dynamics of quadratic +open fermionic systems can be understood with the topo- +logical classification of complex spectra of X (or Lindbla- +dian spectra) by using the 38-fold way, in which the sym- +metries of X play the central role. The typical future of +non-trivial bulk-topology is robust gapless edge-modes. +Notably, there are two types of topological edge modes +in the Lindbladian spectra (the eigenvalues of X, denote +as {λ}), the edge zero-frequency modes (Re (λedge) = 0) +and edge dark states (EDS, Im (λedge) = 0). The edge +zero-frequency modes are the dynamical signature of +topological order, which forces the damping behavior of +the bulk and the edge becomes different[30]. While the +EDS is usually related to the non-trivial steady-state of +system[31, 32]. And the relationship between the topol- +ogy of X and the edge zero-frequency modes has been +uncovered by Lieu et al. based on the 38-fold way [28], +however, the relationship between the topology of X and +the EDS is not revealed yet, which is the topic of this +paper. +Previous studies of EDS in open free fermionic sys- +tems are mainly focused on purely dissipative case, in +which the EDS is fragile once the Hamiltonian terms are +included[31, 32]. +In this paper, we study the EDS in +the case of full dynamics, of which both the Hamiltonian +and the dissipation are presented. We find that the EDS +are protected by the topology of double damping matrix +˜X = diag (X, X∗), and it is robust against the including +of Hamiltonian. In our scheme, the artificial degree of +freedom X∗ has no physical counterpart, it’s an auxiliary +system which is used to support the imaginary line-gap in +arXiv:2301.03208v1 [cond-mat.mes-hall] 9 Jan 2023 + +2 +˜X. In such way, we can classify ˜X topologically with the +38-fold way and the EDS become the in-gap zero modes +of ˜X. Additionally, the symmetry classes of two dissipa- +tive TIs of 1D and 2D cases are presented by using the +38-fold way, these examples revealed that the EDS which +are protected by the topology of ˜X is robust against the +inclusion of Hamiltonian. It is expected that the double +damping matrix that built from damping matrix of dis- +sipative dynamics in an open free fermionic system can +apply to the dissipative TSs as well, which is our future +direction. +This paper is organized in the following way. In Sec. +II, we introduce the damping dynamics of open free +fermionic system and the concepts of dark state in these +systems. In Sec. III, we briefly review the 38-fold way +which gives the symmetry classes of NHRM. In Sec. IV, +a double damping matrix which determines the symme- +try class of EDS is developed. In Sec. V, two examples +of dissipative TIs with EDS is studied, the dissipative +1D Su-Schrieffer-Heeger (SSH) model and dissipative 2D +Qi-Wu-Zhang (QWZ) model. The symmetry classes of +those two models are given, and the robustness of EDS +is checked. Those results revealed that EDS of dissipative +TIs is protected by the topology of the double damping +matrix. +We conclude our results and future potential +directions in Sec. VI. +II. +DAMPING DYNAMICS OF QUADRATIC +OPEN FERMIONIC SYSTEMS +A. +The damping dynamics +The Liouville dynamics of an open quantum system is +usually described by an Lindblad master equation[33–36] +d +dtρ = −i[H, ρ] + +� +µ +� +2L† +µρLµ − {L† +µLµ, ρ} +� +, +(1) +that the time evolution of density matrix ρ is governed +by two parts, the unitary dynamics and the non-unitary +dynamics. The Hamiltonian of system is responsible for +the unitary evolution, and the Lindblad operator Lµ that +describes the adding or removing of particles via a Marko- +vian bath is responsible for the non-unitary evolution. +In these open free fermionic system, we consider the +Lindblad operators as +Lg +µ = +� +s +Dµ,sc† +µ,s, +Ll +µ = +� +s +Dµ,scµ,s +(2) +where µ is the index of the lattice site and s denotes +the internal degree of freedom. And when the pairing +term in the Hamiltonian is absent, we can formulate the +density matrix with a Gaussian state in terms of one- +point correlation function for these quadratic systems[37] +ρ ∝ exp +� +i +2 +� +m,n +� +ln +C +I − C +� +mn +c† +mcn +� +, +(3) +where C is single-particle correlation function, Cmn = +Tr(c† +mcn ρ). Thus the time evolution of the density ma- +trix of a open free fermionic system is fully characterized +by it’s correlation function [11] +i d +dtC = [−hT , C] − {i +� +Mg + M T +l +� +, C} + 2iMg, += XC − CX† + 2iMg. +(4) +And X is known as the damping matrix, which is a NH +single-particle matrix that contains both the Hamiltonian +and the dissipation +X = −hT − i +� +Mg + M T +l +� +, +(5) +where H = � +m,n c† +ihijcj. And the bath matrix Mg and +Ml are caused by the dissipation, which are hermitian +matrix +(Mg)ij = +� +µ +Dg∗ +µiDg +µj, +(Ml)ij = +� +µ +Dl∗ +µiDl +µj. +(6) +Damping matrix X provides a complete description of +dissipative dynamics, it’s becomes more obvious when +we consider the speed that an initial state converging to +the steady state, i.e. +we focus on ˜C(t) = C(t) − Cs, +where Cs,ij = Tr(c† +icj ρs) is the steady state correlation +function. +Then, we find that ˜C(t) is governed by the +following equation +˜C(t) = e−iXt · ˜C0 · e−iX†t, += +� +m,n +ei(−λn+λ∗ +m)t|uR +n ⟩⟨uL +n| ˜C0|uL +m⟩⟨uR +m|. +(7) +The second step in Eq.(7) is obtained using eigen- +decomposition method, where {λn} are the eigenvalues +of X, that is Lindbladian spectra of the system, and |uR +n ⟩ +and |uL +n⟩ satisfy biorthogonal condition, ⟨uL +n|uR +n ⟩ = δmn, +X|uR +n ⟩ = λn|uR +n ⟩, X†|uL +n⟩ = λ∗ +n|uL +n⟩. It’s obvious that +Eq.(7) is coincide with the Schr¨odinger equation in quan- +tum mechanics, of which the dynamic generator is a NH +single-particle matrix, and ˜C(t) is analogous to the den- +sity matrix. +So the topological property of the dissi- +pative dynamics is captured by the NH damping ma- +trix X. The same dissipative dynamics described by the +damping matrix when the pairing-term is included in the +open free fermionic system by using the method of third +quantization[38, 39], while the quansi-particle is changed +into Marjorana fermion. +B. +Lindbladian spectra and the dark state +In the dissipative dynamics, the dynamic information +is hidden in the Lindbladian spectra (or rapidity spec- +tra), which is the eigenvalues’ spectra of damping matrix +X in the complex plane (denote as {λ}) in the quadratic +open free fermionic system, as seen in Fig.1. In which +the imaginary parts of λ specify the speed that the initial + +3 +state converges to the steady state, the smaller the Im (λ) +the quicker it converges to the steady state. Such that +the spectral gap is defined as ∆ = 2 max [Im (λ)][32, 38]. +In the Lindblidian spectra, the general modes are those +with negative imaginary parts (Im (λ) < 0), which de- +cays over time. And if there only have general modes +in the spectra, then there is a unique steady state of +the Liouville dynamics[38, 40]. The modes with positive +imaginary part are forbidden (Im (λ) > 0), which are un- +physical because they are amplified over time. The dark +state is the eigenmode which it’s eigenvalue’s imaginary +part vanishes (Im (λ) = 0), these modes are neither de- +cay nor amplify over time. +Because the dark state is +decoupled from the dissipative dynamics, so it is also a +steady-state of the system governed by Eq.(1), such that +the dark state implies the non-unique steady state of Li- +ouville dynamics[38]. +And the zero frequency mode is +the eigenmode which it’s eigenvalue’s real part vanishes +(Re (λ) = 0). +FIG. 1. The lindbladian spectra of open free fermionic system +in the complex plane. Where the physical modes are those +Im (λ) < 0 (green), the dark states are those Im (λ) = 0 (yel- +low), and the edge zero frequency modes are those Re (λ) = 0 +(light blue). However, The modes that Im (λ) > 0 (gray) are +the amplifying mode, which is unphysical. +Similar to the gapless edge mode which protected +by the topology of Hamiltonian, EDS and edge zero- +frequency mode in the dissipative dynamics are related +to the topology of damping matrix. From the point view +topological band theory, these edge modes are in-gap +states. +There are three kinds of bulk-gap in the com- +plex spectra, the point-gap, real line-gap, and imaginary +line-gap, which support different kinds of in-gap edge +modes. The closing of point gap are those λ = 0, and the +closing of real (imaginary) line-gap are those Re (λ) = 0 +(Im (λ) > 0). So the topological classification of dissipa- +tive dynamics is to study the complex spectra of damping +matrix. S. Lieu and et al. firstly apply such principle in +the open free fermionic system, and classify the dissi- +pative dynamics with edge zero-frequency mode by us- +ing the 38-fold way, which leads to the Ten-fold way for +quadratic lindbladians, denote as LMC class[28]. How- +ever, it’s not known whether the dissipative dynamics +with EDS can be classified with a similar scheme. +III. +TOPOLOGICAL CLASSIFICATION OF +NON-HERMITIAN RANDOM MATRIX +There are 43 non-equivalent symmetry classes of +NHRM, which is known as Bernard-LeClair class [25]. +The topological classification of those NHRM based on +the AZ scheme is proposed by K. Kawabata and et al., +that there only have 38 of topological inequivalent sym- +metry classes, which is the 38-fold way [26]. The main +principle of the classification is to flatten the spectra of +a NH matrix, it is accomplished by the unitary flatten +of point-gap, hermitian flatten of real line-gap and anti- +hermitian flatten of imaginary line-gap. +These flatten +procedures keep the symmetries and the bulk-gap of com- +plex spectra, such that the topological classification is +identical to it’s hermitian (or anti-hermitian) counter- +part, in which the principle in the AZ scheme is used. +There are three fundamental symmetries in the AZ class, +time-reversal symmetry (TRS), particle-hole symmetry +(PHS), chiral symmetry (CS), +TRS : +Ut · H∗ · U −1 +t += H, +(8) +PHS : +Uc · HT · U −1 +c += −H, +(9) +CS : +S · H† · S−1 = −H, +(10) +where Ut and Uc are unitary matrices, and square to +1 or to -1, i.e., we have Uc,tU ∗ +c,t = ±1. +And CS is a +combination of TRS and PHS, such that S = UtU ∗ +c . So +we have 3×3 = 9 kinds of symmetry classes, and another +symmetry class is that only the CS is satisfied, which +gives total of 10 symmetry classes. +In contrast to the +Hermitian case, there are a variant of TRS and PHS for +NH matrix, which is defined as +TRS† : +Ut · HT · U −1 +t += H, +(11) +PHS† : +Uc · H∗ · U −1 +c += −H. +(12) +These symmetry class is denoted as AZ† class. Compare +to AZ class, there only have 6 independent symmetry +classes. Furthermore, there is an additional symmetry +for NH matrix, the sub-lattice symmetry (SLS), +Us · H · U −1 +s += −H, +U 2 +s = 1, +(13) +which is equivalent to CS for Hermitian random ma- +trix, while it is an additional symmetry of NH matrix +since H† ̸= H. Such additional second-order symmetry +would alter the classification space[41], then symmetry +classes is enriched for NHRM, and gives another 22 sym- +metry classes that specified by the commutation/anti- +commutation relations of Us with TRS or PHS. And re- +sults, we have 10 + 6 + 22 = 38 symmetry classes, this is +the 38-fold way of NHRM. Another internal symmetry of +NHRM is the pseudo-hermitian [42, 43], ηH†η−1 = H, +which is a second-order symmetry that gives the same +symmetry classes as SLS. + +.. +4 +.. +Amplifying mode +0 +. +Dark state +zero +General mode +frequency +mode4 +IV. +THE DOUBLE DAMPING MATRIX AND +THE SYSTEM CLASS OF EDS +In the topological band theory, if there are gapless +modes that closing gap at the edge of system, then the +bulk-gap which has non-trivial topology is unavoidable. +Which means that if the EDS in the dissipative dynamics +are protected by the topology of system, then we must +have two bands in the system of which the imaginary +parts with opposite sign, one is negative, the other one +is positive. +FIG. 2. The double Lindbladian spectra of open free fermionic +system, that the modes which Im(λn) > 0 and Im(λn) < 0 are +allowed. However, the modes which Im(λn) > 0 are unphysi- +cal, the including of those modes is only for the convenience +of symmetry classification. +However, the eigenmodes that Im(λn) > 0 is forbidden +in the damping matrix, so we can’t define the imaginary +line-gap. Which means that the dark states of an open +quantum system might have no topological protection. +To reveal the topological protection of EDS, we com- +bine X with it’s complex conjugate X∗ to form a double +damping matrix +˜X = +� +X +0 +0 +X∗ +� +. +(14) +That the redundant freedom X∗ is only for the conve- +nience of topological classification, it is unphysical and +should be discarded in the dissipative dynamics. Both +the positive and the negative imaginary parts are present +in the spectra of ˜X, as seen in Fig.2, so the imaginary +line-gap is well defined in ˜X. Such that nontrivial topol- +ogy of ˜X would indicates the appearance of in-gap edge- +mode that closing the imaginary line-gap, which is the +EDS of dissipative dynamics. +Therefore, if Im (λ) ̸= 0, then the imaginary line-gap of +non-hermitian matrix ˜X can be defined as 2 max [Im (λ)], +which is the spectra gap of Liouville dynamics. And the +corresponding symmetry class is identified with the sym- +metries that ˜X obeys. We find that TRS is automati- +cally satisfied in ˜X, Ut ˜X∗U −1 +t += ˜X, where Ut = τx ⊗ I +or τy ⊗ I, and τx, τy is acting on the artificial degree of +freedom. However, both τx ⊗ I and τy ⊗ I are allowed for +˜X, which means that the symmetry classes of ˜X is non- +unique. Unfortunately, the ambiguity in the symmetry +classes can’t be removed, the interesting thing is that the +robustness of EDS against the perturbations is enhanced +if the symmetry classes which give non-trivial topology +is non-unique. The robustness of EDS can be checked by +considering the perturbation which corresponds to the +coupling between the degrees of freedom that belongs to +X∗ and X. If the coupling that satisfy the symmetry +constraint can gap out the EDS, then the corresponding +topology is trivial, otherwise, the topology is non-trivial. +To determine the robustness of EDS, we study the cou- +pled damping matrix +˜Xc = +�X +C +C† X∗ +� +. +(15) +The coupling C can breaks or preserves the symmetry +of ˜X, this is depends on the symmetry operation. For +example, if C = CT , the TRS is preserved if Ut = τx ⊗ I, +while the TRS is broken if Ut = τy ⊗ I. On the contrary, +C = −CT preserve TRS for Ut = τy ⊗ I and break TRS +for Ut = τx ⊗ I. Consequently, if the symmetry classes +of ˜X corresponds to the symmetry operation τx ⊗ I or +τy ⊗ I is topological non-trivial in a given spatial dimen- +sion, then one can check that whether the EDS are exist +or not in the spectra of ˜Xc to verify the robustness of +EDS. If the EDS still present in the spectra of ˜Xc, that +is to say the the EDS can’t be removed by the pertur- +bation which respect the symmetry constraint, therefore +topological protection of EDS is proved. Instead, if the +EDS is disappear in the spectra of ˜Xc, then the corre- +sponding topology is trivial, and the EDS is a trivial +mode of dissipative dynamics. In this paper, we consid- +ering the perturbation which preserve the symmetry as +C1 = c · τx ⊗ σx for Ut = τx ⊗ I and C2 = c · τx ⊗ σy for +Ut = τy ⊗ I. +In a word, the topological classification of double +damping matrix ˜X = diag (X, X∗) can reveal the topol- +ogy of the dissipative dynamics with EDS. If the topology +of ˜X is non-trivial in a given spatial dimension, then there +is EDS in the corresponding open quantum system which +described by X. And the ambiguities in the symmetry +classes of ˜X can enhance the robustness of EDS against +the perturbation. +V. +SOME EXAMPLES +A. +dissipative SSH model +The SSH model describes spinless fermions hopping on +a one-dimensional (1D) lattice with staggered hopping +amplitudes, and it’s Bloch Hamiltonian is written as[44, +45] +Hssh(k) = (v + w cos k) σx + w sin k σy. +(16) +Where σx,y,z represents the internal degree of freedom, +and denote as sub-lattice A and B. In order to have EDS, + +X*, +0 +Dark state +(an)5 +we consider the dissipators per lattice site as +Lg +µ = √γgc† +µ,A, +Ll +µ = √γlc† +µ,A, +(17) +where subscript µ represents the index of lattice site. +With simple derivation, we get damping matrix of the +dissipative SSH model as +Xssh(k) = − (v + w cos k) σx + w sin k σy − iγ σz − iγI, +(18) +where γ = γg+γl +2 +. The spectrum of Xssh(k) is +λssh(k) = −iγ ± +� +v2 + w2 + 2vw cos k − γ2. +(19) +It’s obvious that Im (λssh) ≤ 0, and ”=” is only for the +real gap closing point where v = w. +In other words, +the closing of imaginary line-gap is accompanied by the +closing of real gap for ˜Xssh(k), which means the edge +modes of ˜Xssh(k) are those that closing the point-gap. +Furthermore, if the winding number of Hssh is non-trivial +in the bulk, then the point-gap is closed at the edge of +system. So that the critical point of Xssh(k) to has EDS is +identical to the topological phase transition point of Hssh, +which means that the EDS are the dynamic signature of +the topological order of internal dynamics. +Next, we see the topological classification of dissipative +SSH model with EDS. The symmetries of Xssh(k) are as +follows +PHS† : +σzX∗ +ssh(k)σz = −Xssh(−k), +(20) +TRS† : +XT +ssh(k) = Xssh(−k), +(21) +then, Xssh(k) is belongs to the class BDI† with a Z +classification from LMC classes in the 1D case, which +means there are edge-modes that closing the real line- +gap (Re (λedge) = 0). However, one of such edge-mode is +the EDS (Im (λedge) = 0), that can’t be specified by the +class BDI†. To reveal the topological protection of EDS +in the dissipative SSH model, we considering the double +damping matrix, which has the following symmetry +SLS : +Us · ˜Xssh(k) · U −1 +s += − ˜Xssh(k), +(22) +PHS : +Uc · ˜XT +ssh(k) · U −1 +c += − ˜Xssh(−k), +(23) +TRS : +Ut · ˜X∗ +ssh(k) · U −1 +t += ˜Xssh(−k), +(24) +where Uc = Us = τx ⊗ σz or τy ⊗ σz, and Ut = τx ⊗ I or +τy ⊗ I. The ambiguities of symmetry operation leads to +four possible symmetry classes, which is obtained with +the permutation of Us and Ut, as seen in Table.I. +When the symmetry classes are identified, then the +corresponding classifying space with the point-gap and +it’s topological properties at a certain spatial dimension +is revealed, as seen in Table.II. In the 1D case, the model +falls into class S++BDI with Z classification or S++CII +with 2Z classification. The label 2Z indicates the topo- +logical number is an even integer, which is isomorphic +to the Z classification, in a word, the topological num- +ber of dissipative SSH model with EDS is characterize +by an integer Z. This can be understood with the sym- +metry analysis, the coulpling C1 respects TRS and PHS +Ut \ Us +τx ⊗ σz +τy ⊗ σz +τx ⊗ I +S++ BDI S−+ CI +τy ⊗ I +S−+ DIII S++ CII +TABLE I. The symmetry classes of ˜ +Xssh(k) with sub-lattice +symmetry (SLS), where the first and second subscripts of S±± +specifies the commutation (+) and anti-commutation (−) re- +lation to the time-reversal symmetry (TRS) and particle-hole +symmetry (PHS) correspondingly. +That UsUt = ±ϵtUtU ∗ +s , +UsUc = ±ϵcUcU ∗ +s , where UtU ∗ +t = ϵt and UcU ∗ +c = ϵs. +of class S++BDI, and the coulpling C2 respects TRS and +PHS of class S++CII, which means that the EDS in ˜Xssh +are robust both to the coupling C1 and C2, which can +be seen in Fig.3(c,d). However, the couplings C1 and C2 +break the symmetries of class S−+DIII and class S−+CI, +then gives trivial topological classification. +classifying space d = 0 d = 1 d = 2 d = 3 +S++ BDI +R1 +Z2 +Z +0 +0 +S−+ CI +C0 +Z +0 +Z +0 +S−+ DIII +C0 +Z +0 +Z +0 +S++ CII +R5 +0 +2Z +0 +Z2 +TABLE II. The classifying spaces and the topological numbers +of symmetry classes of ˜ +Xssh(k) with point-gap, where d is the +number of dimensions. +-3 +-2 +-1 +0 +1 +2 +3 +Re(λ) +-0.4 +-0.2 +0 +0.2 +0.4 +Im(λ) +-3 +-2 +-1 +0 +1 +2 +3 +Re(λ) +-0.4 +-0.2 +0 +0.2 +0.4 +Im(λ) +-3 +-2 +-1 +0 +1 +2 +3 +Re(λ) +-0.4 +-0.2 +0 +0.2 +0.4 +Im(λ) +-3 +-2 +-1 +0 +1 +2 +3 +Re(λ) +-0.4 +-0.2 +0 +0.2 +0.4 +Im(λ) +(c) +(a) +(d) +(b) +C1 +PBC +C2 +OBC +FIG. 3. The double Lindbladian spectra of dissipative SSH +model with edge dark state (EDS) in periodic boundary con- +dition (a, PBC) and open boundary condition (b, OBC) for +L = 100, the EDS is marked in green. Where v = 1, w = 1.3, +and γg = γl = 0.2. And the spectra of coupled damping ma- +trix ˜ +Xc,ssh for C1 (c) and C2 (d), where c = 0.2, it is obvious +that EDS is robust both to the C1 and C2. +The Lindbladian spectra of dissipative SSH mode in +the periodic boundary condition (PBC) and open bound- +ary condition (OBC) are presented in Fig.3, as well as + +6 +coupled damping matrix ˜Xc,ssh. The EDS is robust both +to the coupling C1 and C2, this is because that both class +S++CII and class S++BDI can characterize the topolog- +ical protection of EDS, as seen in Fig.3(c,d). +B. +dissipative QWZ model +The QWZ model is a two-dimensional (2D) Chern in- +sulator that describes the spinless fermions hopping in +2D lattice. The degree of freedoms (the orbital) for each +unit cell are 2, which we denote them as sub-lattice A and +B, then Bloch Hamiltonian of QWZ model is [44, 45] +Hqwz(k) = sin kx σx +sin ky σy +(u + cos kx + cos ky) σz, +(25) +Similarly, σx,y,z represents the internal degree of freedom. +We consider the following dissipators per lattice site +Lg +µ = √γg +� +c† +µ,A + ic† +µ,B +� +, +Ll +µ = √γl (cµ,A − icµ,B) , +(26) +where subscript µ respresents the index of lattice. Then +the damping matrix of dissipative QWZ model is +Xqwz(k) = sin kx σx + (sin ky − iγ) σy +− (u + cos kx + cos ky) σz − iγI, +(27) +and the spectrum of Xqwz(k) is obtained as +Eqwz(k) = −iγ +(28) +± +� +(u + cos kx + cos ky)2 + sin2 kx + (sin ky − iγ)2, +it’s obvious that Im (Eqwz) ≤ 0, and ”=” is only that +(u + cos kx + cos ky)2 +sin2 kx = 0, which is also the gap +closing points where u = −2, 0, 2. Amazingly, the closing +of imaginary line-gap can also be satisfied at the edge of +system when the Chern number of Hqwz is non-zero. So, +the EDS of dissipative QWZ model ware the dynamic +signature of the topological order of internal dynamics. +Ut \ Us τx ⊗ σx τy ⊗ σx +τx ⊗ I +S+ AI +S− AI +τy ⊗ I +S− AII S+ AII +TABLE III. The symmetry classes of ˜ +Xqwz(k) with sub-lattice +symmetry (SLS) S, where subscripts of S± specifies the com- +mutation (+) and anti-commutation (−) relation to the time- +reversal symmetry (TRS). That UsUt = ±ϵtUtU ∗ +s , where +UtU ∗ +t = ϵt. +Then, we study the topological protection of EDS. For +Xqwz, it’s symmetry is +PHS† : +σxX∗ +qwz(−k)σx = −Xqwz(k), +(29) +In the 2D case, it belongs to class D† with Z classifica- +tion in the LMC class, which means there are edge-modes +that closing the real line-gap. However, the EDS in the +dissipative QWZ model can’t be specified by class D†. +The symmetry protection of EDS can be revealed in the +double damping matrix, ˜Xqwz(k) has the following sym- +metry +SLS : +Us · ˜Xqwz(k) · U −1 +s += − ˜Xqwz(k), +(30) +TRS : +Ut · ˜X∗ +qwz(k) · U −1 +t += ˜Xqwz(−k), +(31) +where Us = τx ⊗ σx or τy ⊗ σx, and Ut = τx ⊗ I or τy ⊗ I. +Identical to the SSH model, the ambiguities of symmetry +operation leads to four possible symmetry classes, as seen +in Table.III +classifying space d = 0 d = 1 d = 2 d = 3 +S+ AI +R1 +Z2 +Z +0 +0 +S− AI +R3 +0 +Z2 +Z2 +Z +S− AII +R7 +0 +0 +0 +2Z +S+ AII +R5 +0 +2Z +0 +Z2 +TABLE IV. The classifying spaces and the topological num- +bers of symmetry classes of ˜ +Xqwz(k) with imaginary line-gap, +where d is the number of dimensions. +-4 +-2 +0 +2 +4 +Re(λ) +-0.4 +-0.2 +0 +0.2 +0.4 +Im(λ) +-4 +-2 +0 +2 +4 +Re(λ) +-0.4 +-0.2 +0 +0.2 +0.4 +Im(λ) +-4 +-2 +0 +2 +4 +Re(λ) +-0.4 +-0.2 +0 +0.2 +0.4 +Im(λ) +-4 +-2 +0 +2 +4 +Re(λ) +-0.4 +-0.2 +0 +0.2 +0.4 +Im(λ) +-4 0 +×10-5 +-2 +0 +2 +×10-4 +C2 +C1 +OBC +PBC +(a) +(b) +(d) +(c) +FIG. 4. The double Lindbladian spectra of dissipative QWZ +model with edge dark state (EDS) in periodic boundary con- +dition (a, PBC) and open boundary condition (b, OBC) for +Lx = 20 and Ly = 10, where OBC is that the boundary along +the x-direction is open, the EDS is marked in green. Where +u = −1, γg = γl = 0.2. And the spectra of coupled damping +matrix ˜ +Xc,qwz for C1 and C2 where c = 0.8. +One can see +that EDS is robust against the perturbation C1 (c), while it’s +gaped out by the perturbation C2(d). +From 38-fold way of NHRM, the topological classifica- +tion of ˜Xqwz(k) in different spatial dimensions is obtained +in Table.IV. In the 2D case, only the class S−AI with Z2 +classification is non-trivial, the other three classes are +trivial. The TRS in class S−AI is satisfied by the cou- +pling C1 while violated by C2, so it would expect that + +7 +the EDS might be gaped out when the coupling C2 is +introduced. +In Fig.4, the spectra of double damping matrix ˜Xqwz in +the PBC and OBC are presented. Which OBC is that the +boundary condition along the x-direction is open. This +is because that the imaginary momentum along the y- +direction will induce the NH skin effect, that forces edge- +modes of Xqwz becomes pure imaginary, such that the +EDS is absent for such boundary condition. Furthermore, +The EDS are gaped out by coupling C2 in Fig.4(d), while +they are robust to the coupling C1 in Fig.4(c), that is to +say the EDS are protected by the TRS Ut = τx ⊗ I. +VI. +CONCLUSION AND DISCUSSION +In this paper, a framework to understand the topolog- +ical protection of EDS in the presence of both dissipa- +tion and internal dynamics is provided. We make use of +the 38-fold way to classify the damping dynamics with +EDS in dissipative TIs, of which the dissipative dynamics +of these sysytems are completely captured by a single- +particle NH matrix X. It turns out that the symmetry +classification of X with EDS is ill defined, the right clas- +sification scheme is based on the double damping matrix +˜X = diag (X, X∗). In our scheme, the double damping +matrix ˜X is classified topologically by using the 38-fold +way, the edge modes that close the imaginary line-gap of +˜X are the EDS of X. Different from previous studies of +EDS in purely dissipative systems [31, 32], the EDS in +this work are robust against the including of the Hamilto- +nian terms. As the matter of fact, in the two explanation +examples of dissipative SSH model and dissipative QWZ +model, the appearance of EDS are associated with non- +trivial topology of internal dynamics, such that the EDS +are also a dynamic signature of topological order. +In 38-fold way, the imaginary line-gap can be defined +in the system of which TRS, PHS† or CS is satisfied, how- +ever these symmetries are equivalent when we flatten the +spectra[26]. Which means the double damping matrix ˜X +which TRS is automatically satisfied is universal for the +open free fermionic system. So it can be also applied to +the dissipative TSs, the only difference is that the quansi- +particle is changed into the Majorana fermions, and the +EDS become the Majorana zero-damping modes, that is +vital for the dissipative braiding[31]. We expect our pro- +posal can also be used in the dissipative TSs, which is +our future direction. +VII. +ACKNOWLEDGMENT +[1] Xiao-Liang Qi, and Shou-Cheng Zhang, Topological in- +sulators and superconductors, Rev. Mod. Phys. 83, 1057 +(2011).doi: 10.1103/RevModPhys.83.1057 +[2] Charles Day, Quantum spin Hall effect shows up in a +quantum well insulator, just as predicted, Phys. Today +61, 19 (2008).doi: 10.1063/1.2835139 +[3] B. Andrei Bernevig, and Shou-Cheng Zhang, Quantum +Spin Hall Effect, Phys. Rev. Lett. 96, 106802 (2006). doi: +10.1103/PhysRevLett.96.106802 +[4] Alexander Altland, and Martin R. Zirnbauer, Non- +standard +symmetry +classes +in +mesoscopic +normal- +superconducting hybrid structures, Phys. Rev. B 55, +1142 (1997). doi: 10.1103/PhysRevB.55.1142 +[5] S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, and +et al., Topologically protected bound states in pho- +tonic parity–time-symmetric crystals, Nat. Mater 16, 133 +(2017). doi: 10.1038/nmat4811 +[6] Miguel A. Bandres, Steffen Wittek, Gal Harari, Midya +Parto, Jinhan Ren, Mordechai Segev, Demetrios N. +Christodoulides, and Mercedeh Khajavikhan, Topolog- +ical insulator laser: Experiments, Science 359, eaar4005 +(2018). doi: 10.1126/science.aar4005 +[7] Julia M. Zeuner, Mikael C. Rechtsman, Yonatan Plotnik, +Yaakov Lumer, Stefan Nolte, Mark S. Rudner, Mordechai +Segev, and Alexander Szameit, Observation of a Topolog- +ical Transition in the Bulk of a Non-Hermitian System, +Phys. Rev. Lett. 115, 040402 (2015). doi: 10.1103/Phys- +RevLett.115.040402 +[8] Ching Hua Lee, and Ronny Thomale, +Anatomy of +skin modes and topology in non-Hermitian systems, +Phys. Rev. B 99, 201103(R) (2018).doi: 10.1103/Phys- +RevB.99.201103 +[9] Stefano Longhi, Probing non-Hermitian skin effect and +non-Bloch phase transitions, Phys. Rev. Research 1, +023013 (2019).doi: 10.1103/PhysRevResearch.1.023013 +[10] Hui Jiang, Li-Jun Lang, Chao Yang, Shi-Liang Zhu, +and Shu Chen, Interplay of non-Hermitian skin effects +and Anderson localization in nonreciprocal quasiperi- +odic lattices, Phys. Rev. B 100, 054301 (2019).doi: +10.1103/PhysRevB.100.054301 +[11] Fei Song, Shunyu Yao, and Zhong Wang, Non-Hermitian +Skin Effect and Chiral Damping in Open Quantum +Systems, Phys. Rev. Lett. 123, 170401 (2019). doi: +10.1103/PhysRevLett.123.170401 +[12] C. M. Bender, Making sense of non-Hermitian Hamiltoni- +ans, Rep. Prog. Phys. 70, 947 (2007). doi: 10.1088/0034- +4885/70/6/R03 +[13] M.-A. Miri, and A. Al`u, Exceptional points in optics +and photonics, Sci. 363, 6422 (2019). doi: 10.1126/sci- +ence.aar7709 +[14] Shun-Yu Yao, +and Zhong Wang, +Edge States and +Topological Invariants of Non-Hermitian Systems, Phys. +Rev. +Lett. +121, +086803 +(2018).doi: +10.1103/Phys- +RevLett.121.086803 +[15] Shun-Yu +Yao, +Fei +Song, +and +Zhong +Wang, +Non- +Hermitian Chern Bands, Phys. Rev. Lett. 121, 136802 +(2018).doi: 10.1103/PhysRevLett.121.136802 +[16] Lo¨Ic Herviou, Jens H. Bardarson, and Nicolas Reg- +nault, Defining a bulk-edge correspondence for non- +Hermitian Hamiltonians via singular-value decomposi- +tion, Phys. Rev. A. 99, 052118 (2019).doi: 10.1103/Phys- +RevA.99.052118 + +8 +[17] Xiao-Ran Wang, Cui-xian Guo, and Su-Peng Kou, Defec- +tive edge states and number-anomalous bulk-boundary +correspondence in non-Hermitian topological systems, +Phys. Rev. B. 101, 121116(R) (2020).doi: 10.1103/Phys- +RevB.101.121116 +[18] Alexander Altland, Michael Fleischhauer, and Sebas- +tian Diehl, Symmetry Classes of Open Fermionic Quan- +tum Matter, Phys. Rev. X 11, 021037 (2021). doi: +10.1103/PhysRevX.11.021037 +[19] A. Rivas, and O. Viyuela, Density-matrix Chern in- +sulators: Finite-temperature generalization of topolog- +ical insulators, Phys. Rev. B 88, 155141 (2013). doi: +10.1103/PhysRevB.88.155141 +[20] D. Linzner, L. Wawer, F. Grusdt, and M. Fleischhauer, +Reservoir-induced Thouless pumping and symmetry- +protected topological order in open quantum chains, +Phys. Rev. B 94, 201105(R) (2016). doi: 10.1103/Phys- +RevB.94.201105 +[21] F. Grusdt, Topological order of mixed states in correlated +quantum many-body systems, Phys. Rev. B 95, 075106 +(2017). doi: 10.1103/PhysRevB.95.075106 +[22] Felix Dangel, Marcel Wagner, Holger Cartarius, J¨org +Main, and G¨unter Wunner, Topological invariants in dis- +sipative extensions of the Su-Schrieffer-Heeger model, +Phys. Rev. A 98, 013628 (2018). doi: +10.1103/Phys- +RevA.98.013628 +[23] Clemens +Gneiting, +Akshay +Koottandavida, +A. +V. +Rozhkov, and Franco Nori, Unraveling the topology of +dissipative quantum systems, Phys. Rev. Research 4, +023036 (2022). doi: 10.1103/PhysRevResearch.4.023036 +[24] Jiewei Cheng, Xiujuan Zhang, Ming-Hui Lu, and Yan- +Feng Chen, Competition between band topology and +non-Hermiticity, Phys. Rev. B 105, 094103 (2022). doi: +10.1103/PhysRevB.105.094103 +[25] D. Bernard, and A. LeClair, A classification of non- +Hermitian random matrices, in Statistical Field Theo- +ries, edited by A. Cappelli and G. Mussardo (Springer +Netherlands, Dordrecht, 2002), pp. 207–214. +[26] Kohei Kawabata, Ken Shiozaki, Masahito Ueda, and +Masatoshi +Sato, +Symmetry +and +Topology +in +Non- +Hermitian Physics, Phys. Rev. X 9, 041015 (2019). doi: +10.1103/PhysRevX.9.041015 +[27] Antonio M. Garc´ıa-Garc´ıa, Lucas S´a, and Jacobus J. +M. Verbaarschot, Symmetry Classification and Univer- +sality in Non-Hermitian Many-Body Quantum Chaos by +the Sachdev-Ye-Kitaev Model, Phys. Rev. X 12, 021040 +(2022).doi: 10.1103/PhysRevX.12.021040 +[28] Simon Lieu, +Max McGinley, +and Nigel R. Cooper, +Tenfold +Way +for +Quadratic +Lindbladians, +Phys. +Rev. +Lett. +124, +040401 +(2020).doi: +10.1103/Phys- +RevLett.124.040401 +[29] Makio Kawasaki, Ken Mochizuki, and Hideaki Obuse, +Topological phases protected by shifted sublattice sym- +metry in dissipative quantum systems, Phys. Rev. B 106, +035408 (2022). doi: 10.1103/PhysRevB.106.035408 +[30] Moos T. van Caspel, Sergio Enrique Tapias Arze, and +Isaac P´erez Castillo, Dynamical signatures of topologi- +cal order in the driven-dissipative Kitaev chain, SciPost +Phys. 6, 026 (2019).doi: 10.21468/SciPostPhys.6.2.026 +[31] Sebastian Diehl, Enrique Rico, Mikhail A. Baranov, and +Peter Zoller, Topology by dissipation in atomic quantum +wires, Nat. Phys. 7, 971 (2011). doi: 10.1038/nphys2106 +[32] C-E Bardyn, M. A. Baranov, C. V. Kraus, E Rico, A. +˙Imamo˘glu, P. Zoller, and S. Diehl, Topology by dissipa- +tion, New. J. Phys. 15, 085001 (2013). doi: 10.1088/1367- +2630/15/8/085001 +[33] G. Lindblad, On the generators of quantum dynamical +semigroups, Commu. Math. Phys. 48, 119 (1976).doi: +10.1007/BF01608499 +[34] V. Gorini, A. Kossalowski, E. C. G. Sudarshan, E. Rico, +A. ˙Imamo˘glu, P. Zoller, and S. Diehl, Completely positive +dynamical semigroups of N-level systems, J. Math. Phys. +(N.Y.) 17, 821 (1976). doi: 10.1063/1.522979 +[35] M. Wolf, and J. I. Cirac, Dividing Quantum Chan- +nels, +Commu. +Math. +Phys. +279, +147 +(2008). +doi: +10.1007/s00220-008-0411-y +[36] H. -P. Breuer, and F. Petruccione, The theory of Open +Quantum Systems, Oxford University Press, Oxford, +2007. +[37] Siew-Ann Cheong and C. L. Henley, Many-body den- +sity matrices for free fermions, Phys. Rev. B 69, 075111 +(2004). doi: +10.1103/PhysRevB.69.075111 +[38] Tomaˇz Prosen, Third quantization: a general method to +solve master equations for quadratic open Fermi systems, +New. J. of Phys. 10, 043026 (2008). doi: 10.1088/1367- +2630/10/4/043026 +[39] Tomaˇz +Prosen, +Spectral +theorem +for +the +Lindblad +equation +for +quadratic +open +fermionic +systems, +J. +Stat. Mech. 07, +07020 (2010). doi: +10.1088/1742- +5468/2010/07/P07020 +[40] B. Kraus, H. P. B¨uchler, S. Diehl, A. Kantian, A. Micheli, +and P. Zoller, Preparation of entangled states by quan- +tum Markov processes, Phys. Rev. A 78, 042307 (2008). +doi: 10.1103/PhysRevA.78.042307 +[41] Ken Shiozaki and Masatoshi Sato, Topology of crystalline +insulators and superconductors, Phys. Rev. B 90, 165114 +(2014).doi: 10.1103/PhysRevB.90.165114 +[42] A. Mostafazadeh, Pseudo-Hermiticity versus PT symme- +try: The necessary condition for the reality of the spec- +trum of a non-Hermitian Hamiltonian, J. Math. Phys. +43, 205 (2002);doi: 10.1063/1.1418246 A. Mostafazadeh, +Pseudo-Hermiticity versus PT symmetry II: The nec- +essary condition for the reality of the spectrum of a +non-Hermitian Hamiltonian, J. Math. Phys. 43, 2814 +(2002);doi: 10.1063/1.1461427 A. Mostafazadeh, Pseudo- +Hermiticity versus PT symmetry III: The necessary con- +dition for the reality of the spectrum of a non-Hermitian +Hamiltonian, +J. +Math. +Phys. +43, +3944 +(2002);doi: +10.1063/1.14890727 +[43] A. Mostafazadeh, Exact PT-symmetry is equivalent to +Hermiticity, J. Phys. A: Math. Gen. 36, 7081 (2003).doi: +10.1088/0305-4470/36/25/312 +[44] J´anos K. Asb´oth, L´aszl´o Oroszl´any, and Andr´as P´alyi, A +Short Course on Topological Insulators, Springer Press, +Switzerland, 2016. +[45] B. Andrei Bernevig, Taylor L. Hughes, and Andr´as P´alyi, +Topological Insulators and topological superconductors, +Princeton University Press, Princeton and Oxford, 2013. + diff --git a/fNE1T4oBgHgl3EQfegS8/content/tmp_files/load_file.txt b/fNE1T4oBgHgl3EQfegS8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..19ae10da495eceae617f37b61a73699c04796d99 --- /dev/null +++ b/fNE1T4oBgHgl3EQfegS8/content/tmp_files/load_file.txt @@ -0,0 +1,604 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf,len=603 +page_content='Symmetry classes of dissipative topological insulators with edge dark state Fei Yang,1 Zheng Wei,1 Xianqi Tong,1 Kui Cao,1 and Su-Peng Kou1 1Center for Advanced Quantum Studies, Department of Physics, Beijing Normal University, Beijing 100875, China We classify the dissipative topological insulators (TIs) with edge dark states (EDS) by using the 38-fold way of non-Hermitian systems in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The dissipative dynamics of these quadratic open fermionic systems is captured by a non-Hermitian single-particle matrix which contains both the internal dynamics and the dissipation, refereed to as damping matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And the dark states in these systems are the eigenmodes of X which the eigenvalues’ imaginary part vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' However, there is a constraint on X, namely that the modes in which the eigenvalues’ imaginary parts are positive are forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In other words, the imaginary line-gap of X is ill-defined, so the topological band theory classifying the dark states can not be applied to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' To reveal the topological protection of EDS, we propose the double damping matrix ˜ X = diag (X, X∗), where the imaginary line-gap is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Thus, the 38-fold way can be applied to ˜ X, and the topological protection of the EDS is uncovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Different from previous studies of EDS in purely dissipative dynamics, the EDS in the dissipative TIs are robust against the inclusion of Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Furthermore, the topological classification of ˜ X not only reflects the topological protection of EDS in dissipative TIs but also provides a paradigm to predict the appearance of EDS in other open free fermionic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' INTRODUCTION Symmetry and topology play the central role in mod- ern physics, which results in many interesting phenom- ena and future applications, such as robust edge mode in topological insulators (TIs) and topological supercon- ductors (TSs)[1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And the systematic classifications of those phases are explained by the ten-fold way of free fermions, or Altland-Zirnbauer (AZ) symmetry classes[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The AZ class provides not only a framework to analyze the topological behavior of system with different sym- metries but also gives a paradigm to explore new topo- logical phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' However, those are for the close Her- mitian systems, many physical systems in nature expe- rience dissipation associated with gain and loss, such as atomic, molecular, and optical physics[5–7], the dy- namics of these systems are effectively described with a non-hermitian (NH) Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Dissipation in these NH systems would give rise to many interesting effect that do not have Hermitian counterpart, such as NH skin effects[8–11], PT -symmetric physics[12, 13], and the breakdown of bulk-boundary correspondence [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Moreover, the question that how the dissipation influ- ences the topology of a system attracts much attention, which has been explored in many papers[18–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The fundamental interests in studying the topological proper- ties of NH systems are to expand the symmetry classes, which had been settled by Bernard and LeClair based on four fundamental symmetries, resulting in a total of 43 symmetry classes, that is known as Bernard-LeClair class[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' While Kawabata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' discovered that only 38 of 43 symmetry classes are topologically inequivalent[26], which is known as 38-fold way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The 38-fold way pro- vides a paradigm to explore the topology of a NH system, which has been applied in many NH systems, such as the NH Sachdev-Ye-Kitaev Model[27], and symmetry classes of the open quantum system[28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' As we have mentioned, the non-Hermiticity is ubiqui- tous in nonequilibrium open systems, but most of them are hard to solve especially if the interaction is pre- sented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Fortunately, the open free fermionic systems are exactly solvable, in which the dissipative dynamics are completely captured by the so called damping matrix X, which is a NH single-particle matrix that contains the in- ternal dynamics as well as the dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The topolog- ical phenomena in the dissipative dynamics of quadratic open fermionic systems can be understood with the topo- logical classification of complex spectra of X (or Lindbla- dian spectra) by using the 38-fold way, in which the sym- metries of X play the central role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The typical future of non-trivial bulk-topology is robust gapless edge-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Notably, there are two types of topological edge modes in the Lindbladian spectra (the eigenvalues of X, denote as {λ}), the edge zero-frequency modes (Re (λedge) = 0) and edge dark states (EDS, Im (λedge) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The edge zero-frequency modes are the dynamical signature of topological order, which forces the damping behavior of the bulk and the edge becomes different[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' While the EDS is usually related to the non-trivial steady-state of system[31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And the relationship between the topol- ogy of X and the edge zero-frequency modes has been uncovered by Lieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' based on the 38-fold way [28], however, the relationship between the topology of X and the EDS is not revealed yet, which is the topic of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Previous studies of EDS in open free fermionic sys- tems are mainly focused on purely dissipative case, in which the EDS is fragile once the Hamiltonian terms are included[31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In this paper, we study the EDS in the case of full dynamics, of which both the Hamiltonian and the dissipation are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' We find that the EDS are protected by the topology of double damping matrix ˜X = diag (X, X∗), and it is robust against the including of Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In our scheme, the artificial degree of freedom X∗ has no physical counterpart, it’s an auxiliary system which is used to support the imaginary line-gap in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='03208v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='mes-hall] 9 Jan 2023 2 ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In such way, we can classify ˜X topologically with the 38-fold way and the EDS become the in-gap zero modes of ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Additionally, the symmetry classes of two dissipa- tive TIs of 1D and 2D cases are presented by using the 38-fold way, these examples revealed that the EDS which are protected by the topology of ˜X is robust against the inclusion of Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' It is expected that the double damping matrix that built from damping matrix of dis- sipative dynamics in an open free fermionic system can apply to the dissipative TSs as well, which is our future direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' This paper is organized in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' II, we introduce the damping dynamics of open free fermionic system and the concepts of dark state in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' III, we briefly review the 38-fold way which gives the symmetry classes of NHRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' IV, a double damping matrix which determines the symme- try class of EDS is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' V, two examples of dissipative TIs with EDS is studied, the dissipative 1D Su-Schrieffer-Heeger (SSH) model and dissipative 2D Qi-Wu-Zhang (QWZ) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The symmetry classes of those two models are given, and the robustness of EDS is checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Those results revealed that EDS of dissipative TIs is protected by the topology of the double damping matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' We conclude our results and future potential directions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' DAMPING DYNAMICS OF QUADRATIC OPEN FERMIONIC SYSTEMS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The damping dynamics The Liouville dynamics of an open quantum system is usually described by an Lindblad master equation[33–36] d dtρ = −i[H, ρ] + � µ � 2L† µρLµ − {L† µLµ, ρ} � , (1) that the time evolution of density matrix ρ is governed by two parts, the unitary dynamics and the non-unitary dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The Hamiltonian of system is responsible for the unitary evolution, and the Lindblad operator Lµ that describes the adding or removing of particles via a Marko- vian bath is responsible for the non-unitary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In these open free fermionic system, we consider the Lindblad operators as Lg µ = � s Dµ,sc† µ,s, Ll µ = � s Dµ,scµ,s (2) where µ is the index of the lattice site and s denotes the internal degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And when the pairing term in the Hamiltonian is absent, we can formulate the density matrix with a Gaussian state in terms of one- point correlation function for these quadratic systems[37] ρ ∝ exp � i 2 � m,n � ln C I − C � mn c† mcn � , (3) where C is single-particle correlation function, Cmn = Tr(c† mcn ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Thus the time evolution of the density ma- trix of a open free fermionic system is fully characterized by it’s correlation function [11] i d dtC = [−hT , C] − {i � Mg + M T l � , C} + 2iMg, = XC − CX† + 2iMg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' (4) And X is known as the damping matrix, which is a NH single-particle matrix that contains both the Hamiltonian and the dissipation X = −hT − i � Mg + M T l � , (5) where H = � m,n c† ihijcj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And the bath matrix Mg and Ml are caused by the dissipation, which are hermitian matrix (Mg)ij = � µ Dg∗ µiDg µj, (Ml)ij = � µ Dl∗ µiDl µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' (6) Damping matrix X provides a complete description of dissipative dynamics, it’s becomes more obvious when we consider the speed that an initial state converging to the steady state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' we focus on ˜C(t) = C(t) − Cs, where Cs,ij = Tr(c† icj ρs) is the steady state correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Then, we find that ˜C(t) is governed by the following equation ˜C(t) = e−iXt · ˜C0 · e−iX†t, = � m,n ei(−λn+λ∗ m)t|uR n ⟩⟨uL n| ˜C0|uL m⟩⟨uR m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' (7) The second step in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' (7) is obtained using eigen- decomposition method, where {λn} are the eigenvalues of X, that is Lindbladian spectra of the system, and |uR n ⟩ and |uL n⟩ satisfy biorthogonal condition, ⟨uL n|uR n ⟩ = δmn, X|uR n ⟩ = λn|uR n ⟩, X†|uL n⟩ = λ∗ n|uL n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' It’s obvious that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' (7) is coincide with the Schr¨odinger equation in quan- tum mechanics, of which the dynamic generator is a NH single-particle matrix, and ˜C(t) is analogous to the den- sity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' So the topological property of the dissi- pative dynamics is captured by the NH damping ma- trix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The same dissipative dynamics described by the damping matrix when the pairing-term is included in the open free fermionic system by using the method of third quantization[38, 39], while the quansi-particle is changed into Marjorana fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Lindbladian spectra and the dark state In the dissipative dynamics, the dynamic information is hidden in the Lindbladian spectra (or rapidity spec- tra), which is the eigenvalues’ spectra of damping matrix X in the complex plane (denote as {λ}) in the quadratic open free fermionic system, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In which the imaginary parts of λ specify the speed that the initial 3 state converges to the steady state, the smaller the Im (λ) the quicker it converges to the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Such that the spectral gap is defined as ∆ = 2 max [Im (λ)][32, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In the Lindblidian spectra, the general modes are those with negative imaginary parts (Im (λ) < 0), which de- cays over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And if there only have general modes in the spectra, then there is a unique steady state of the Liouville dynamics[38, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The modes with positive imaginary part are forbidden (Im (λ) > 0), which are un- physical because they are amplified over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The dark state is the eigenmode which it’s eigenvalue’s imaginary part vanishes (Im (λ) = 0), these modes are neither de- cay nor amplify over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Because the dark state is decoupled from the dissipative dynamics, so it is also a steady-state of the system governed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' (1), such that the dark state implies the non-unique steady state of Li- ouville dynamics[38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And the zero frequency mode is the eigenmode which it’s eigenvalue’s real part vanishes (Re (λ) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The lindbladian spectra of open free fermionic system in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Where the physical modes are those Im (λ) < 0 (green), the dark states are those Im (λ) = 0 (yel- low), and the edge zero frequency modes are those Re (λ) = 0 (light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' However, The modes that Im (λ) > 0 (gray) are the amplifying mode, which is unphysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Similar to the gapless edge mode which protected by the topology of Hamiltonian, EDS and edge zero- frequency mode in the dissipative dynamics are related to the topology of damping matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' From the point view topological band theory, these edge modes are in-gap states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' There are three kinds of bulk-gap in the com- plex spectra, the point-gap, real line-gap, and imaginary line-gap, which support different kinds of in-gap edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The closing of point gap are those λ = 0, and the closing of real (imaginary) line-gap are those Re (λ) = 0 (Im (λ) > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' So the topological classification of dissipa- tive dynamics is to study the complex spectra of damping matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Lieu and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' firstly apply such principle in the open free fermionic system, and classify the dissi- pative dynamics with edge zero-frequency mode by us- ing the 38-fold way, which leads to the Ten-fold way for quadratic lindbladians, denote as LMC class[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' How- ever, it’s not known whether the dissipative dynamics with EDS can be classified with a similar scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' TOPOLOGICAL CLASSIFICATION OF NON-HERMITIAN RANDOM MATRIX There are 43 non-equivalent symmetry classes of NHRM, which is known as Bernard-LeClair class [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The topological classification of those NHRM based on the AZ scheme is proposed by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Kawabata and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=', that there only have 38 of topological inequivalent sym- metry classes, which is the 38-fold way [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The main principle of the classification is to flatten the spectra of a NH matrix, it is accomplished by the unitary flatten of point-gap, hermitian flatten of real line-gap and anti- hermitian flatten of imaginary line-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' These flatten procedures keep the symmetries and the bulk-gap of com- plex spectra, such that the topological classification is identical to it’s hermitian (or anti-hermitian) counter- part, in which the principle in the AZ scheme is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' There are three fundamental symmetries in the AZ class, time-reversal symmetry (TRS), particle-hole symmetry (PHS), chiral symmetry (CS), TRS : Ut · H∗ · U −1 t = H, (8) PHS : Uc · HT · U −1 c = −H, (9) CS : S · H† · S−1 = −H, (10) where Ut and Uc are unitary matrices, and square to 1 or to -1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=', we have Uc,tU ∗ c,t = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And CS is a combination of TRS and PHS, such that S = UtU ∗ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' So we have 3×3 = 9 kinds of symmetry classes, and another symmetry class is that only the CS is satisfied, which gives total of 10 symmetry classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In contrast to the Hermitian case, there are a variant of TRS and PHS for NH matrix, which is defined as TRS† : Ut · HT · U −1 t = H, (11) PHS† : Uc · H∗ · U −1 c = −H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' (12) These symmetry class is denoted as AZ† class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Compare to AZ class, there only have 6 independent symmetry classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Furthermore, there is an additional symmetry for NH matrix, the sub-lattice symmetry (SLS), Us · H · U −1 s = −H, U 2 s = 1, (13) which is equivalent to CS for Hermitian random ma- trix, while it is an additional symmetry of NH matrix since H† ̸= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Such additional second-order symmetry would alter the classification space[41], then symmetry classes is enriched for NHRM, and gives another 22 sym- metry classes that specified by the commutation/anti- commutation relations of Us with TRS or PHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And re- sults, we have 10 + 6 + 22 = 38 symmetry classes, this is the 38-fold way of NHRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Another internal symmetry of NHRM is the pseudo-hermitian [42, 43], ηH†η−1 = H, which is a second-order symmetry that gives the same symmetry classes as SLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='. 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='. Amplifying mode 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Dark state zero General mode frequency mode4 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' THE DOUBLE DAMPING MATRIX AND THE SYSTEM CLASS OF EDS In the topological band theory, if there are gapless modes that closing gap at the edge of system, then the bulk-gap which has non-trivial topology is unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Which means that if the EDS in the dissipative dynamics are protected by the topology of system, then we must have two bands in the system of which the imaginary parts with opposite sign, one is negative, the other one is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The double Lindbladian spectra of open free fermionic system, that the modes which Im(λn) > 0 and Im(λn) < 0 are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' However, the modes which Im(λn) > 0 are unphysi- cal, the including of those modes is only for the convenience of symmetry classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' However, the eigenmodes that Im(λn) > 0 is forbidden in the damping matrix, so we can’t define the imaginary line-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Which means that the dark states of an open quantum system might have no topological protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' To reveal the topological protection of EDS, we com- bine X with it’s complex conjugate X∗ to form a double damping matrix ˜X = � X 0 0 X∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' (14) That the redundant freedom X∗ is only for the conve- nience of topological classification, it is unphysical and should be discarded in the dissipative dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Both the positive and the negative imaginary parts are present in the spectra of ˜X, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2, so the imaginary line-gap is well defined in ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Such that nontrivial topol- ogy of ˜X would indicates the appearance of in-gap edge- mode that closing the imaginary line-gap, which is the EDS of dissipative dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Therefore, if Im (λ) ̸= 0, then the imaginary line-gap of non-hermitian matrix ˜X can be defined as 2 max [Im (λ)], which is the spectra gap of Liouville dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And the corresponding symmetry class is identified with the sym- metries that ˜X obeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' We find that TRS is automati- cally satisfied in ˜X, Ut ˜X∗U −1 t = ˜X, where Ut = τx ⊗ I or τy ⊗ I, and τx, τy is acting on the artificial degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' However, both τx ⊗ I and τy ⊗ I are allowed for ˜X, which means that the symmetry classes of ˜X is non- unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Unfortunately, the ambiguity in the symmetry classes can’t be removed, the interesting thing is that the robustness of EDS against the perturbations is enhanced if the symmetry classes which give non-trivial topology is non-unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The robustness of EDS can be checked by considering the perturbation which corresponds to the coupling between the degrees of freedom that belongs to X∗ and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' If the coupling that satisfy the symmetry constraint can gap out the EDS, then the corresponding topology is trivial, otherwise, the topology is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' To determine the robustness of EDS, we study the cou- pled damping matrix ˜Xc = �X C C† X∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' (15) The coupling C can breaks or preserves the symmetry of ˜X, this is depends on the symmetry operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' For example, if C = CT , the TRS is preserved if Ut = τx ⊗ I, while the TRS is broken if Ut = τy ⊗ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' On the contrary, C = −CT preserve TRS for Ut = τy ⊗ I and break TRS for Ut = τx ⊗ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Consequently, if the symmetry classes of ˜X corresponds to the symmetry operation τx ⊗ I or τy ⊗ I is topological non-trivial in a given spatial dimen- sion, then one can check that whether the EDS are exist or not in the spectra of ˜Xc to verify the robustness of EDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' If the EDS still present in the spectra of ˜Xc, that is to say the the EDS can’t be removed by the pertur- bation which respect the symmetry constraint, therefore topological protection of EDS is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Instead, if the EDS is disappear in the spectra of ˜Xc, then the corre- sponding topology is trivial, and the EDS is a trivial mode of dissipative dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In this paper, we consid- ering the perturbation which preserve the symmetry as C1 = c · τx ⊗ σx for Ut = τx ⊗ I and C2 = c · τx ⊗ σy for Ut = τy ⊗ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In a word, the topological classification of double damping matrix ˜X = diag (X, X∗) can reveal the topol- ogy of the dissipative dynamics with EDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' If the topology of ˜X is non-trivial in a given spatial dimension, then there is EDS in the corresponding open quantum system which described by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And the ambiguities in the symmetry classes of ˜X can enhance the robustness of EDS against the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' SOME EXAMPLES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' dissipative SSH model The SSH model describes spinless fermions hopping on a one-dimensional (1D) lattice with staggered hopping amplitudes, and it’s Bloch Hamiltonian is written as[44, 45] Hssh(k) = (v + w cos k) σx + w sin k σy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' (16) Where σx,y,z represents the internal degree of freedom, and denote as sub-lattice A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In order to have EDS, X*, 0 Dark state (an)5 we consider the dissipators per lattice site as Lg µ = √γgc† µ,A, Ll µ = √γlc† µ,A, (17) where subscript µ represents the index of lattice site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' With simple derivation, we get damping matrix of the dissipative SSH model as Xssh(k) = − (v + w cos k) σx + w sin k σy − iγ σz − iγI, (18) where γ = γg+γl 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The spectrum of Xssh(k) is λssh(k) = −iγ ± � v2 + w2 + 2vw cos k − γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' (19) It’s obvious that Im (λssh) ≤ 0, and ”=” is only for the real gap closing point where v = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In other words, the closing of imaginary line-gap is accompanied by the closing of real gap for ˜Xssh(k), which means the edge modes of ˜Xssh(k) are those that closing the point-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Furthermore, if the winding number of Hssh is non-trivial in the bulk, then the point-gap is closed at the edge of system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' So that the critical point of Xssh(k) to has EDS is identical to the topological phase transition point of Hssh, which means that the EDS are the dynamic signature of the topological order of internal dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Next, we see the topological classification of dissipative SSH model with EDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The symmetries of Xssh(k) are as follows PHS† : σzX∗ ssh(k)σz = −Xssh(−k), (20) TRS† : XT ssh(k) = Xssh(−k), (21) then, Xssh(k) is belongs to the class BDI† with a Z classification from LMC classes in the 1D case, which means there are edge-modes that closing the real line- gap (Re (λedge) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' However, one of such edge-mode is the EDS (Im (λedge) = 0), that can’t be specified by the class BDI†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' To reveal the topological protection of EDS in the dissipative SSH model, we considering the double damping matrix, which has the following symmetry SLS : Us · ˜Xssh(k) · U −1 s = − ˜Xssh(k), (22) PHS : Uc · ˜XT ssh(k) · U −1 c = − ˜Xssh(−k), (23) TRS : Ut · ˜X∗ ssh(k) · U −1 t = ˜Xssh(−k), (24) where Uc = Us = τx ⊗ σz or τy ⊗ σz, and Ut = τx ⊗ I or τy ⊗ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The ambiguities of symmetry operation leads to four possible symmetry classes, which is obtained with the permutation of Us and Ut, as seen in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' When the symmetry classes are identified, then the corresponding classifying space with the point-gap and it’s topological properties at a certain spatial dimension is revealed, as seen in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In the 1D case, the model falls into class S++BDI with Z classification or S++CII with 2Z classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The label 2Z indicates the topo- logical number is an even integer, which is isomorphic to the Z classification, in a word, the topological num- ber of dissipative SSH model with EDS is characterize by an integer Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' This can be understood with the sym- metry analysis, the coulpling C1 respects TRS and PHS Ut \\ Us τx ⊗ σz τy ⊗ σz τx ⊗ I S++ BDI S−+ CI τy ⊗ I S−+ DIII S++ CII TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The symmetry classes of ˜ Xssh(k) with sub-lattice symmetry (SLS), where the first and second subscripts of S±± specifies the commutation (+) and anti-commutation (−) re- lation to the time-reversal symmetry (TRS) and particle-hole symmetry (PHS) correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' That UsUt = ±ϵtUtU ∗ s , UsUc = ±ϵcUcU ∗ s , where UtU ∗ t = ϵt and UcU ∗ c = ϵs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' of class S++BDI, and the coulpling C2 respects TRS and PHS of class S++CII, which means that the EDS in ˜Xssh are robust both to the coupling C1 and C2, which can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='3(c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' However, the couplings C1 and C2 break the symmetries of class S−+DIII and class S−+CI, then gives trivial topological classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' classifying space d = 0 d = 1 d = 2 d = 3 S++ BDI R1 Z2 Z 0 0 S−+ CI C0 Z 0 Z 0 S−+ DIII C0 Z 0 Z 0 S++ CII R5 0 2Z 0 Z2 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The classifying spaces and the topological numbers of symmetry classes of ˜ Xssh(k) with point-gap, where d is the number of dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 3 2 1 0 1 2 3 Re(λ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 Im(λ) 3 2 1 0 1 2 3 Re(λ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 Im(λ) 3 2 1 0 1 2 3 Re(λ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 Im(λ) 3 2 1 0 1 2 3 Re(λ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 Im(λ) (c) (a) (d) (b) C1 PBC C2 OBC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The double Lindbladian spectra of dissipative SSH model with edge dark state (EDS) in periodic boundary con- dition (a, PBC) and open boundary condition (b, OBC) for L = 100, the EDS is marked in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Where v = 1, w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='3, and γg = γl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And the spectra of coupled damping ma- trix ˜ Xc,ssh for C1 (c) and C2 (d), where c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2, it is obvious that EDS is robust both to the C1 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The Lindbladian spectra of dissipative SSH mode in the periodic boundary condition (PBC) and open bound- ary condition (OBC) are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='3, as well as 6 coupled damping matrix ˜Xc,ssh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The EDS is robust both to the coupling C1 and C2, this is because that both class S++CII and class S++BDI can characterize the topolog- ical protection of EDS, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='3(c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' dissipative QWZ model The QWZ model is a two-dimensional (2D) Chern in- sulator that describes the spinless fermions hopping in 2D lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The degree of freedoms (the orbital) for each unit cell are 2, which we denote them as sub-lattice A and B, then Bloch Hamiltonian of QWZ model is [44, 45] Hqwz(k) = sin kx σx +sin ky σy +(u + cos kx + cos ky) σz, (25) Similarly, σx,y,z represents the internal degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' We consider the following dissipators per lattice site Lg µ = √γg � c† µ,A + ic† µ,B � , Ll µ = √γl (cµ,A − icµ,B) , (26) where subscript µ respresents the index of lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Then the damping matrix of dissipative QWZ model is Xqwz(k) = sin kx σx + (sin ky − iγ) σy − (u + cos kx + cos ky) σz − iγI, (27) and the spectrum of Xqwz(k) is obtained as Eqwz(k) = −iγ (28) ± � (u + cos kx + cos ky)2 + sin2 kx + (sin ky − iγ)2, it’s obvious that Im (Eqwz) ≤ 0, and ”=” is only that (u + cos kx + cos ky)2 +sin2 kx = 0, which is also the gap closing points where u = −2, 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Amazingly, the closing of imaginary line-gap can also be satisfied at the edge of system when the Chern number of Hqwz is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' So, the EDS of dissipative QWZ model ware the dynamic signature of the topological order of internal dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Ut \\ Us τx ⊗ σx τy ⊗ σx τx ⊗ I S+ AI S− AI τy ⊗ I S− AII S+ AII TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The symmetry classes of ˜ Xqwz(k) with sub-lattice symmetry (SLS) S, where subscripts of S± specifies the com- mutation (+) and anti-commutation (−) relation to the time- reversal symmetry (TRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' That UsUt = ±ϵtUtU ∗ s , where UtU ∗ t = ϵt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Then, we study the topological protection of EDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' For Xqwz, it’s symmetry is PHS† : σxX∗ qwz(−k)σx = −Xqwz(k), (29) In the 2D case, it belongs to class D† with Z classifica- tion in the LMC class, which means there are edge-modes that closing the real line-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' However, the EDS in the dissipative QWZ model can’t be specified by class D†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The symmetry protection of EDS can be revealed in the double damping matrix, ˜Xqwz(k) has the following sym- metry SLS : Us · ˜Xqwz(k) · U −1 s = − ˜Xqwz(k), (30) TRS : Ut · ˜X∗ qwz(k) · U −1 t = ˜Xqwz(−k), (31) where Us = τx ⊗ σx or τy ⊗ σx, and Ut = τx ⊗ I or τy ⊗ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Identical to the SSH model, the ambiguities of symmetry operation leads to four possible symmetry classes, as seen in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='III classifying space d = 0 d = 1 d = 2 d = 3 S+ AI R1 Z2 Z 0 0 S− AI R3 0 Z2 Z2 Z S− AII R7 0 0 0 2Z S+ AII R5 0 2Z 0 Z2 TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The classifying spaces and the topological num- bers of symmetry classes of ˜ Xqwz(k) with imaginary line-gap, where d is the number of dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 4 2 0 2 4 Re(λ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 Im(λ) 4 2 0 2 4 Re(λ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 Im(λ) 4 2 0 2 4 Re(λ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 Im(λ) 4 2 0 2 4 Re(λ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4 Im(λ) 4 0 ×10-5 2 0 2 ×10-4 C2 C1 OBC PBC (a) (b) (d) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The double Lindbladian spectra of dissipative QWZ model with edge dark state (EDS) in periodic boundary con- dition (a, PBC) and open boundary condition (b, OBC) for Lx = 20 and Ly = 10, where OBC is that the boundary along the x-direction is open, the EDS is marked in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Where u = −1, γg = γl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' And the spectra of coupled damping matrix ˜ Xc,qwz for C1 and C2 where c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' One can see that EDS is robust against the perturbation C1 (c), while it’s gaped out by the perturbation C2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' From 38-fold way of NHRM, the topological classifica- tion of ˜Xqwz(k) in different spatial dimensions is obtained in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In the 2D case, only the class S−AI with Z2 classification is non-trivial, the other three classes are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' The TRS in class S−AI is satisfied by the cou- pling C1 while violated by C2, so it would expect that 7 the EDS might be gaped out when the coupling C2 is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4, the spectra of double damping matrix ˜Xqwz in the PBC and OBC are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Which OBC is that the boundary condition along the x-direction is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' This is because that the imaginary momentum along the y- direction will induce the NH skin effect, that forces edge- modes of Xqwz becomes pure imaginary, such that the EDS is absent for such boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Furthermore, The EDS are gaped out by coupling C2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4(d), while they are robust to the coupling C1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4(c), that is to say the EDS are protected by the TRS Ut = τx ⊗ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' CONCLUSION AND DISCUSSION In this paper, a framework to understand the topolog- ical protection of EDS in the presence of both dissipa- tion and internal dynamics is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' We make use of the 38-fold way to classify the damping dynamics with EDS in dissipative TIs, of which the dissipative dynamics of these sysytems are completely captured by a single- particle NH matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' It turns out that the symmetry classification of X with EDS is ill defined, the right clas- sification scheme is based on the double damping matrix ˜X = diag (X, X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In our scheme, the double damping matrix ˜X is classified topologically by using the 38-fold way, the edge modes that close the imaginary line-gap of ˜X are the EDS of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Different from previous studies of EDS in purely dissipative systems [31, 32], the EDS in this work are robust against the including of the Hamilto- nian terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' As the matter of fact, in the two explanation examples of dissipative SSH model and dissipative QWZ model, the appearance of EDS are associated with non- trivial topology of internal dynamics, such that the EDS are also a dynamic signature of topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' In 38-fold way, the imaginary line-gap can be defined in the system of which TRS, PHS† or CS is satisfied, how- ever these symmetries are equivalent when we flatten the spectra[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Which means the double damping matrix ˜X which TRS is automatically satisfied is universal for the open free fermionic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' So it can be also applied to the dissipative TSs, the only difference is that the quansi- particle is changed into the Majorana fermions, and the EDS become the Majorana zero-damping modes, that is vital for the dissipative braiding[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' We expect our pro- posal can also be used in the dissipative TSs, which is our future direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' ACKNOWLEDGMENT [1] Xiao-Liang Qi, and Shou-Cheng Zhang, Topological in- sulators and superconductors, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 83, 1057 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1057 [2] Charles Day, Quantum spin Hall effect shows up in a quantum well insulator, just as predicted, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Today 61, 19 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2835139 [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Andrei Bernevig, and Shou-Cheng Zhang, Quantum Spin Hall Effect, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 96, 106802 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='106802 [4] Alexander Altland, and Martin R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Zirnbauer, Non- standard symmetry classes in mesoscopic normal- superconducting hybrid structures, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B 55, 1142 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1142 [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Weimann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Kremer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Plotnik, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Lumer, and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=', Topologically protected bound states in pho- tonic parity–time-symmetric crystals, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Mater 16, 133 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1038/nmat4811 [6] Miguel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Bandres, Steffen Wittek, Gal Harari, Midya Parto, Jinhan Ren, Mordechai Segev, Demetrios N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Christodoulides, and Mercedeh Khajavikhan, Topolog- ical insulator laser: Experiments, Science 359, eaar4005 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='aar4005 [7] Julia M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Zeuner, Mikael C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rechtsman, Yonatan Plotnik, Yaakov Lumer, Stefan Nolte, Mark S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rudner, Mordechai Segev, and Alexander Szameit, Observation of a Topolog- ical Transition in the Bulk of a Non-Hermitian System, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 115, 040402 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/Phys- RevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='040402 [8] Ching Hua Lee, and Ronny Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B 99, 201103(R) (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/Phys- RevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='201103 [9] Stefano Longhi, Probing non-Hermitian skin effect and non-Bloch phase transitions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Research 1, 023013 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevResearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='023013 [10] Hui Jiang, Li-Jun Lang, Chao Yang, Shi-Liang Zhu, and Shu Chen, Interplay of non-Hermitian skin effects and Anderson localization in nonreciprocal quasiperi- odic lattices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B 100, 054301 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='054301 [11] Fei Song, Shunyu Yao, and Zhong Wang, Non-Hermitian Skin Effect and Chiral Damping in Open Quantum Systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 123, 170401 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='170401 [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Bender, Making sense of non-Hermitian Hamiltoni- ans, Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 70, 947 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1088/0034- 4885/70/6/R03 [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Miri, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Al`u, Exceptional points in optics and photonics, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 363, 6422 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1126/sci- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='aar7709 [14] Shun-Yu Yao, and Zhong Wang, Edge States and Topological Invariants of Non-Hermitian Systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 121, 086803 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/Phys- RevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='086803 [15] Shun-Yu Yao, Fei Song, and Zhong Wang, Non- Hermitian Chern Bands, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 121, 136802 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='136802 [16] Lo¨Ic Herviou, Jens H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Bardarson, and Nicolas Reg- nault, Defining a bulk-edge correspondence for non- Hermitian Hamiltonians via singular-value decomposi- tion, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 99, 052118 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/Phys- RevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='052118 8 [17] Xiao-Ran Wang, Cui-xian Guo, and Su-Peng Kou, Defec- tive edge states and number-anomalous bulk-boundary correspondence in non-Hermitian topological systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 101, 121116(R) (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/Phys- RevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='121116 [18] Alexander Altland, Michael Fleischhauer, and Sebas- tian Diehl, Symmetry Classes of Open Fermionic Quan- tum Matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' X 11, 021037 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='021037 [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rivas, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Viyuela, Density-matrix Chern in- sulators: Finite-temperature generalization of topolog- ical insulators, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B 88, 155141 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='155141 [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Linzner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Wawer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Grusdt, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Fleischhauer, Reservoir-induced Thouless pumping and symmetry- protected topological order in open quantum chains, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B 94, 201105(R) (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/Phys- RevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='201105 [21] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Grusdt, Topological order of mixed states in correlated quantum many-body systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B 95, 075106 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='075106 [22] Felix Dangel, Marcel Wagner, Holger Cartarius, J¨org Main, and G¨unter Wunner, Topological invariants in dis- sipative extensions of the Su-Schrieffer-Heeger model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' A 98, 013628 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/Phys- RevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='013628 [23] Clemens Gneiting, Akshay Koottandavida, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rozhkov, and Franco Nori, Unraveling the topology of dissipative quantum systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Research 4, 023036 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevResearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='023036 [24] Jiewei Cheng, Xiujuan Zhang, Ming-Hui Lu, and Yan- Feng Chen, Competition between band topology and non-Hermiticity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B 105, 094103 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='094103 [25] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Bernard, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' LeClair, A classification of non- Hermitian random matrices, in Statistical Field Theo- ries, edited by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Cappelli and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Mussardo (Springer Netherlands, Dordrecht, 2002), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 207–214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' [26] Kohei Kawabata, Ken Shiozaki, Masahito Ueda, and Masatoshi Sato, Symmetry and Topology in Non- Hermitian Physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' X 9, 041015 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='041015 [27] Antonio M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Garc´ıa-Garc´ıa, Lucas S´a, and Jacobus J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Verbaarschot, Symmetry Classification and Univer- sality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' X 12, 021040 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='021040 [28] Simon Lieu, Max McGinley, and Nigel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Cooper, Tenfold Way for Quadratic Lindbladians, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 124, 040401 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/Phys- RevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='040401 [29] Makio Kawasaki, Ken Mochizuki, and Hideaki Obuse, Topological phases protected by shifted sublattice sym- metry in dissipative quantum systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B 106, 035408 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='035408 [30] Moos T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' van Caspel, Sergio Enrique Tapias Arze, and Isaac P´erez Castillo, Dynamical signatures of topologi- cal order in the driven-dissipative Kitaev chain, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 6, 026 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='21468/SciPostPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='026 [31] Sebastian Diehl, Enrique Rico, Mikhail A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Baranov, and Peter Zoller, Topology by dissipation in atomic quantum wires, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 7, 971 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1038/nphys2106 [32] C-E Bardyn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Baranov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Kraus, E Rico, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' ˙Imamo˘glu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Zoller, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Diehl, Topology by dissipa- tion, New.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 15, 085001 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1088/1367- 2630/15/8/085001 [33] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Lindblad, On the generators of quantum dynamical semigroups, Commu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 48, 119 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1007/BF01608499 [34] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Gorini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Kossalowski, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Sudarshan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rico, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' ˙Imamo˘glu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Zoller, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Diehl, Completely positive dynamical semigroups of N-level systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=') 17, 821 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='522979 [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Wolf, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Cirac, Dividing Quantum Chan- nels, Commu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 279, 147 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1007/s00220-008-0411-y [36] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' -P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Breuer, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Petruccione, The theory of Open Quantum Systems, Oxford University Press, Oxford, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' [37] Siew-Ann Cheong and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Henley, Many-body den- sity matrices for free fermions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B 69, 075111 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='075111 [38] Tomaˇz Prosen, Third quantization: a general method to solve master equations for quadratic open Fermi systems, New.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' of Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 10, 043026 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1088/1367- 2630/10/4/043026 [39] Tomaˇz Prosen, Spectral theorem for the Lindblad equation for quadratic open fermionic systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 07, 07020 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1088/1742- 5468/2010/07/P07020 [40] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Kraus, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B¨uchler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Diehl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Kantian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Micheli, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Zoller, Preparation of entangled states by quan- tum Markov processes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' A 78, 042307 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='042307 [41] Ken Shiozaki and Masatoshi Sato, Topology of crystalline insulators and superconductors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' B 90, 165114 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='165114 [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Mostafazadeh, Pseudo-Hermiticity versus PT symme- try: The necessary condition for the reality of the spec- trum of a non-Hermitian Hamiltonian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 43, 205 (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1418246 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Mostafazadeh, Pseudo-Hermiticity versus PT symmetry II: The nec- essary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 43, 2814 (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1461427 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Mostafazadeh, Pseudo- Hermiticity versus PT symmetry III: The necessary con- dition for the reality of the spectrum of a non-Hermitian Hamiltonian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 43, 3944 (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='14890727 [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Mostafazadeh, Exact PT-symmetry is equivalent to Hermiticity, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' 36, 7081 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content='1088/0305-4470/36/25/312 [44] J´anos K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Asb´oth, L´aszl´o Oroszl´any, and Andr´as P´alyi, A Short Course on Topological Insulators, Springer Press, Switzerland, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' [45] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Andrei Bernevig, Taylor L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} +page_content=' Hughes, and Andr´as P´alyi, Topological Insulators and topological superconductors, Princeton University Press, Princeton and Oxford, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE1T4oBgHgl3EQfegS8/content/2301.03208v1.pdf'} diff --git a/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf b/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6271af9b48ac454dfca77ec8420994f3d371937b --- /dev/null +++ b/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf @@ -0,0 +1,3 @@ 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+1. School of Physics and Optoelectronic, +Yangtze University, Jingzhou 434023, China; +2. Institute for Frontiers in Astronomy and Astrophysics, +Beijing Normal University, Beijing 102206, China; +3. Department of Astronomy, Beijing Normal University, Beijing 100875, China; +4. Beijing Academy, Kangyuan Road, Dongba, Beijing 100018, China; +5. Department of Physics, and Collaborative Innovation +Center for Quantum Effects and Applications, +Hunan Normal University, Changsha, Hunan 410081, China. +Abstract +As an exact result required by the Etherington reciprocity theorem, the cosmic distance duality relation +(CDDR), η(z) = DL(z)(1 + z)−2/DA(z) = 1 plays an essential part in modern cosmology. In this paper, +we present a new method (η(zi)/η(zj)) to use the measurements of ultra-compact structure in radio quasars +(QSO) and the latest observations of type Ia supernova (SN Ia) to test CDDR. By taking the observations +directly from SN Ia and QSOs, one can completely eliminate the uncertainty caused by the calibration of the +absolute magnitudes of standard candles (MB) and the linear sizes of standard rulers (lm). Benefit from the +absence of nuisance parameters involved in other currently available methods, our analysis demonstrates no +evidence for the deviation and redshift evolution of CDDR up to z = 2.3. The combination of our method- +ology and the machine learning Artificial Neural Network (ANN) would produce 10−3 level constraints on +the violation parameter at high redshifts. Our results indicate perfect agreement between observations and +predictions, supporting the persisting claims that the Etherington reciprocity theorem could still be the best +description of our universe. +PACS numbers: 98.80.-k,98.54.-h,98.54.Aj,06.30.Bp +∗ Corresponding author: caoshuo@bnu.edu.cn +† Corresponding author: jcwang@hunnu.edu.cn +1 + +I. +INTRODUCTION +The cosmic distance duality relation (CDDR), as a fundamental relation in modern cos- +mology, correlates the luminosity distance DL(z) with the angular diameter distance DA(z). +More specifically, the CDDR indicates that DL(z) and DA(z) should satisfy the relation of +DL(z) = DA(z)(1 + z)2 at the same redshift [1, 2]. However, the validity of the CDDR depends +on three basic assumptions: i) the space-time is described by metric; ii) the light travels along the +null geodesics between the observer and the source; iii) the photon number is conserved, i.e., the +CDDR will be violated if the number of photons in the universe is not conserved. Therefore, the +validity test of CDDR is, to some extent, an indirect verification of the accelerated expansion of +the universe [3–6]. In addition, as a fundamental relationship in cosmology, the CDDR has been +widely used in various fields of astronomy, such as the observations of large-scale distribution of +galaxies and the near-uniformity of the cosmic microwave background radiation (CMBR) temper- +ature [7], the determination of gas mass density profile and temperature profile of galaxy clusters +[8, 9], as well as the measurements of cosmic curvature with strong gravitational lensing systems +[10–12]. +On the other hand, the accumulation of precise astrophysical observations allow us to test the +validity of CDDR at different redshifts. From the theoretical point of view, two types of cosmo- +logical distances are usually required in developing CDDR tests, i.e., angular diameter distance +DA and luminosity distance DL. For the observations of luminosity distances, many works turn to +luminous sources with known (or standardizable) intrinsic luminosity in the universe like type-Ia +supernova (SN Ia), while the angular diameter distances are inferred from baryon acoustic oscil- +lations (BAO), Sunyaev-Zeldovich (SZ) effect of galaxy clusters [13–15], gas mass fraction mea- +surements in galaxy clusters [16, 17], and strong gravitational lensing systems [18–21]. However, +it is worth noting that angular diameter distances derived from BAO is puzzled by the so-called +fitting problem, which is a major challenge confronted by the standard BAO peak location with a +fixed comoving ruler of about 105 h−1 Mpc [22]. Therefore, such distance estimation is model- +dependent to some extent, which inevitably brings systematic uncertainties and further affects the +validity of testing CDDR with BAO. Meanwhile, galaxy clusters alone are not able to provide +a competitive source of angular diameter distance at different redshifts, suffering from the large +observational uncertainties arising from radio observations of the SZ effect of galaxy clusters to- +gether with X-ray emission. In addition, based on the observations of SN Ia, it was argued that +the nuisance parameters characterizing SN Ia light-curves also introduce considerable uncertain- +ties to the final results [23, 24]. Therefore, in order to perform the validity of testing CDDR, +one needs to eliminate the effects and uncertanties caused by the nuisance parameters in both two +types of observational data sets (angular diameter distance DA and luminosity distance DL). In +2 + +this paper we further analyse the most updated QSO and SNe Ia data sets. Specially, we present a +new approach that harnesses the ratio η(zi)/η(zj) as cosmic observations, based on the luminosity +distance inferred from the latest observations of type Ia supernova and angular diameter distances +obtained from the measurements of ultra-compact structure in radio quasars. All of the quantities +used in the CDDR test come directly from observations, i.e., the absolute magnitudes of SN Ia and +the linear size of the compact structure in radio quasars need not to be calibrated. In this way, the +effects and uncertainties caused by the nuisance parameters are eliminated completely. More inter- +estingly, our methodology will also benefit from the consistent redshift coverage of both samples +that can reach a high redshift range of z ∼ 2.3. +With the increase both in the depth and quality of cosmological measurements, new techniques +and methods for CDDR tests have also been developed, focusing on different machine learning +algorithms [25–28]. In this paper, we will use the Artificial Neural Network (ANN) algorithm to +reconstruct the possible evolution of CDDR with redshifts. Note that such data-driven approach +has no assumptions about the observational data, suggesting its advantage of being completely +model-independent. Summarizing, we will propose an improved approach to test CDDR directly, +which not only effectively avoids the influence of nuisance parameter on CDDR, but also achieves +more stringent constraints on CDDR in the case of small data samples. The outline of this paper is +given as follow: in Section II we briefly introduce the observations of ultra-compact structure of +radio quasars acting standard rulers and SN Ia acting as standard candles. The improved method- +ology of testing CDDR and the corresponding results are presented in Section III. +II. +OBSERVATIONAL DATA +A. +Angular diameter distances from radio quasars +We consider extracting angular diameter distance from angular redshift relation of compact +structure of radio quasar. As the most distant and brightest objects in the Universe, quasars exhibit +great potential in studying early cosmology beyond the limitation of supernovae. Unfortunately +quasars exhibit large dispersion in luminosities at all wavelengths, which makes them unusable as +standard probes for measuring cosmological distances. In the past decades, great effort have been +made to make use of quasars as standard candles or standard rulers in modern cosmology, such as +the Baldwin effect [29], the Broad Line Region radius-luminosity relation [30], the properties of +highly accreting quasars [31], and the non-linear relation between the ultraviolet and X-ray fluxes +of the quasar to construct the Hubble diagram [10, 32–35]. According to the unified model of +active galactic nuclei and quasars, ultra-compact radio sources are identified as cases in which the +jets are moving relativistically and are close to the line of sight. At any given frequency, the core is +3 + +believed to be located in the region of the jet corresponding to unit optical depth with synchrotron +self-absorption being the dominating process. In the original work of [36], an interesting possi- +bility was discussed that compact radio sources (especially quasars) constitute another potential +class of standard rulers that could be observed by very long baseline interferometry (VLBI). The +VLBI with high precision can not only accurately locate the radio source, but also measure the ten- +sion angle of the compact radio source at the magnitude of mas. Based on the subsequent works +of [37, 38], the linear size of the compact structures in radio sources are related to the intrinsic +luminosity L and the redshift z of the background source +lm = lLβ(1 + z)n, +(1) +where l represents the linear size scaling factor which describes the apparent distribution of radio +brightness within the core, β and n denotes the possible dependence of the intrinsic size on the +luminosity and the redshift, respectively. However, the application of radio sources in cosmology +still suffered from the high dispersion in the observed relations or the limitation of a poor statistics. +With the gradually refined selection technique and observations, a key step forward was made in +the work of [37], which showed that the linear size dispersion in radio source with a flat spectral +index (−0.38 < α < 0.18) is greatly reduced. Based on a sample of 2.29 GHz VLBI survey +with 613 milliarcsecond compact radio sources, [39, 40] selected 120 intermediate-luminosity +(1027W/Hz< L < 1028W/Hz) quasars (ILQSOs) with reliable measurements on the angular size +of the compact structure. The final results demonstrated that ILQSOs are almost independent +from redshift and luminosity (|n| ≃ 10−3, β ≃ 10−4), which means they meet the requirements +expected from standard rulers. However, the crucial question is what is the intrinsic metric linear +size of the quasar source? The previous analysis roughly estimated that the lm parameter is robustly +of the scale of ∼ 11 pc [39]. For the sake of the following description, we take a prior value +lm = 11.03±0.25 pc determined by in a cosmological-model-independent method [40]. However, +in our work the value of lm does not affect the CDDR test, and we will later propose an improved +CDDR test to eliminate the bias and additional systematic errors associated with the lm value of +calibration. +The angular size in compact structure and cosmic distance relation for cosmological inference +was first proposed in [36] +DA(z) = lm +θ(z), +(2) +where DA is the angular diameter distance, lm is the intrinsic metric linear size of the source, and +θ(z) is the observed angular size, which is defined by the modulus of visibility Γ = Sc/St in the +literature [37]. The specific definition of angular size is θ(z) = 2 +√ +− ln Γ ln 2/πBθ, where Bθ is +interferometer baseline measured in wavelengths, Sc and St are correlated flux density and total +flux density, respectively [40]. The sample of raido quasars used in this work is the one described +4 + +FIG. 1: The scatter plot of the observed angular sizes of 120 radio quasars (left panel) and the apparent +magnitudes of 1048 Pantheon SN Ia (right panel). The red dotted line denotes the angular sizes calculated +from the fiducial ΛCDM model (H0 = 70.0 km/s/Mpc, Ωm = 0.30). +in [39] with the redshift range between z = 0.462 and z = 2.73. These compact radio sources +come from a well-known 2.29 GHz VLBI survey [41] (hereafter called P85) with 1398 detected +candidates and 917 selected sources. The P85 sample was updated with respect to redshift [42], +which includes 613 compact radio sources that cover the redshift range up to 0.0035 ≤ z ≤ 3.787 +(http://nrl.northumbria.ac.uk/13109/). These 120 radio quasars have been carefully selected for +cosmological studies and we refer to [39] for a detailed description of the selection procedure +used to turn them into standard rulers and for an explanation of the calibration method used to +include them in the extensive cosmological analysis [43–46]. The scatter diagram of the observed +angular sizes for 120 radio quasars is shown in Fig. 1. +B. +Luminosity distances from Type Ia Supernova +In order to carry out the test of CDDR, we need to find another cosmological probe that can +directly provide luminosity distances and satisfy the following criteria, i.e., the probe should be +able to cover roughly the redshift range of the compact radio quasars. In this work, we seek for +SN Ia as a reasonably empirically well-understood cosmological probe, the usefulness of which +to modern cosmology is well known in revealing the accelerated expansion of the Universe and +placing constraints on cosmological parameters to break parameter degeneracies. With the rapid +growth in the sample size of SN Ia distance measurements, the analysis and mitigation of system- +atic uncertainties of Type Ia Supernova has been considerably improved. However, the application +of SN Ia for cosmology involves so-called ”nuisance” parameters, which need to be optimized +5 + +0.0 +0'2 +T'O +T'2 +J4 +Je +18 +50 +SS +S4 +se5 +0.0 +0'2 +1'O +12 +5'2 +-0'2 +0.0 +0'2 +J'O +1'2 +(26m) +5'O +5'2 +0.E +3'2 +020 +VCDW +4'0FIG. 2: The CDDR parameter η(z) from the observations of radio quasars and SN Ia. +along with the unknown variables in cosmological models and could potentially affect reliable +constraints on cosmological model parameters. +Fortunately, the recent SN Ia sample called Pantheon has been released by the Pan-STARRS1 +(PS1) Medium Deep Survey, which contains 1048 SN Ia measurements spanning the redshift +range 0.01 < z < 2.3 [47]. Here, we only summarise the crucial points required by the present +work. Benefit from richness and depth of the sample,the Pantheon catalogue combines the subset +of 279 PS1 SN Ia [48, 49] and useful distance estimations of SN Ia from SDS, SNLS, various +low redshift and HST samples [47]. More importantly, compared with the previous SN Ia data +sets [50], the Pantheon sample applies a new approach called BEAMS with Bias Corrections +(BBC) [51], in which the apparent magnitude is replaced with the corrected apparent magnitude +mB,corr = mB + α⋆ · X1 − β · C for all the SN Ia [47]. Here, mB is the observed peak magnitude +in rest-frame B band, while X1 and C are the color and light-curve shape parameters. The two +nuisance parameters α⋆ and β should be fitted simultaneously with the cosmological parameters. +It should be noted that the stretch luminosity parameter α⋆ and the color-luminosity parameter +β are set to zero for the Pantheon sample. Therefore, the observed distance modulus of SN Ia +provides the luminosity distance as +DL,SN(z) = 10(mB,corr(z)−MB)/5−5(Mpc), +(3) +where MB is the absolute magnitude in B band. For the uncertainty of the luminosity distance +in Pantheon data set, the contribution from photometric error, distance bias correction, and the +peculiar velocity are included in this analysis [47]. The apparent B-band magnitude for 1048 +Pantheon SN Ia of is also illustrated in Fig. 1. +6 + +5 +a.0 +8.0 +J'O +I'S +1'4 +00.0 +0'52- +0'20- +0'12 +1'00 +J'52 +J'20- +J'J2 - +V=J +5'00III. +METHODOLOGY AND RESULTS +From the theoretical point of view, in order to directly test the DDR from observations, the +following parameterized form is commonly used +η(z) = +DL(z) +DA(z)(1 + z)2, +(4) +the likelihood of which is expected to peak at one in order to satisfy the CDDR. By combining +Eqs. (2) and (3) to Eq. (4), one can rewrite the above expression as +η(z) = θ(z)10(mB,corr(z)−MB)/5−5 +lm(1 + z)2 +. +(5) +The difficulty of testing CDDR lies in the fact that the angular diameter distance from an radio +quasar should be observed at the same redshift as SN Ia. In the previous work for example in +[15], it was pointed out that the CDDR test could be significantly affected by the particular choice +of the selection criteria for a given pair of data sets. Following the redshift selection criterion +widely used in the literature (within the redshift range of 0.01 < z < 2.3) [52–54], the redshifts of +SN Ia sample are carefully chosen to coincide with the associated quasar sample demanding that +the difference in redshift is smaller than 0.005. By performing such selection criterion that could +hopefully ease the systematic errors brought by redshift inconsistency, only 37 pairs of data sets +are obtained from the Pantheon and ILQSO sample. Combining these quasar data together with the +Pantheon SN Ia sample, we obtain the CDDR parameter η(z) shown in Fig. 2. More specifically, +the total uncertainties of η(z) are calculated from the standard uncertainty propagation formula, +based on the uncorrelated uncertainties of observables including the observed angular size errors +σθ, corrected apparent magnitude errors σmB,corr, as well as additional systematic errors introduced +from the calibrations of absolute magnitude (MB) of SN Ia and linear size (lm) of radio quasars. To +better illustrate the statistical significance of our results, we first use the weighted mean statistics +[55] to evaluate +η = Σi +� +ηi/σ2 +ηi +� +Σi +� +1/σ2ηi +� , +σ2 +η = +1 +Σi +� +1/σ2ηi +�, +(6) +where η stands for the weighted mean and ση is its corresponding uncertainty of CDDR parameter. +Such statistical method has been widely applied in meta-analysis to integrate the results of inde- +pendent measurements [56]. Our assessments for weighted mean and corresponding uncertainty +are Mean(η(z)) = 0.991(±0.147), which is in perfect agreement with the results of previous +works [23–27], indicates that there is no evidence for the CDDR violation. Given the possible +invalidity of Gaussian distribution of the errors, we also use a robust median statistics [57] to +evaluate the measurements of η(z). Moreover, if there are extreme values and outliers in the se- +quence, it is better to use the median as the representative value. When making a total number of +7 + +FIG. 3: The η(zi)/η(zj) two-point diagnostics calculated on the observations of radio quasars and SN Ia +(left panel). The reconstructed η(zi)/η(zj) two-point diagnostics with ANN machine learning algorithm +(right panel). +0.90 +0.93 +0.96 +0.99 +η +0 +−0.1 +0.0 +0.1 +0.2 +η +1 +−0.1 +0.0 +0.1 +0.2 +η +1 +FIG. 4: The scatter plot of the CDDR parameter η1/η0 (left panel) and constraints on the CDDR parameters +(η0, η1) (right panel), in the framework of η(zi)/η(zj) two-point diagnostics. +N measurements, one might naturally expect that there is a 50% chance that each measurement is +higher/lower than the true median. Therefore, the probability that n-th observation is higher than +the median follows the binomial distribution: P = 2−NN!/[n!(N−n)!] [58]. Similarly, we can de- +fine the 68.3% confidence interval with median statistics. In the framework of such non-parametric +approach, the resulting constraint on the CDDR parameter becomes Med(η(z)) = 1.117(±0.328) +with the median value and the absolute deviation. Therefore, the conclusion of CDDR validity +(η(z) = 1) seems robust within 1σ confidence interval. +Due to the ambiguous interpretation of the compact structure size in radio quasars and the +8 + +VS +00 +O1 +05 +03 +04 +0.2 +oe +03 +8.0 +-12 +1O: +-2 +UTI +-0 +2 +TO += +12SA +00 +OT +0'5 +0.3 +04 +02 +oe +03 +0:8 +-S +U(sU(s) +WWAVs +A.0 +oe +0:8 +1O +U(SU(s)absolute B-band magnitude of SN Ia whose value is determined by the host stellar mass, the +linear size parameter lm and the absolute magnitude MB are hard to determine precisely. In fact, +the uncertainty of CDDR measurements shown in Fig. 2 is dominated by the calibration of two +nuisance parameters. In order to eliminate the influence of these two nuisance parameters, we +propose an improved approach by introducing the ratio of CDDR parameter +η(zi)/η(zj) = θ(zi)(1 + zj)2 +θ(zj)(1 + zi)210∆mB,corr/5, +(7) +where ∆mB,corr = mB,corr(zi) − mB,corr(zj) is the difference of corrected apparent magnitude +between arbitrary two SN Ia data points. If one defines the ratio η(zi)/η(zj), where i, j denote +the order numbers of the radio quasars and SN Ia, then such quantity does not depend on the +nuisance parameters and it does not introduce any uncertainty to the results. Note that if we have +observational data at n different redshifts, then we can get n(n − 1)/2 data pairs. The uncertainty +of ηij = η(zi)/η(zj) is calculated using the standard error propagation formula, which is related +to the uncorrelated uncertainties of the observed angular size σθ and corrected apparent magnitude +σmB,corr. More importantly, our approach successfully eliminate the nuisance parameters MB and +lm, which brings benefits in alleviating the systematics caused by precise determination of these +parameters. These are the apparent merits of our methodology. Our approach was inspired by +the two-point diagnostic approach, which has been extensively applied to quantify the difference +between the cosmological constant (ΛCDM) and other dark energy models (including evolving +dark energy) [59–61]. +In order to gain insight concerning the two-point diagnostics calculated for every combination +of pairs taken from the full QSO+SN Ia data. We display these diagnostics together with their +uncertainties as a function of redshift difference ∆z = |zi − zj| in the left panel of Fig. 3. Benefit +from the improved methodology, the QSO/SN Ia pairs satisfying irrespective of the redshift selec- +tion criteria have a massive growth. One can see that there are some interesting features regarding +the uncertainties of the two-point diagnostics, i.e., they are apparently non-Gaussian. In order to +test further the validity and efficiency of our method, we use two approaches to produce a sum- +mary statistics of two-point diagnostics calculated on the data sets. The first is to use the weighted +mean statistical method. In order to ensure that each data point is uncorrelated, the weighted mean +formula for the η(zi)/η(zj) diagnostic should be rewritten as [60] +ηij = +Σn−1 +i=1 Σn +j=i+1 +� +ηij/σ2 +ηij +� +Σn−1 +i=1 Σn +j=i+1 +� +1/σ2 +ηij +� , +σ2 +ηij = +1 +Σn−1 +i=1 Σn +j=i+1 +� +1/σ2 +ηij +�. +(8) +The weighted mean value and corresponding uncertainty is Mean(η(zi)/η(zj)) = 0.968 ± 0.031, +which suggests that the weighted mean of this diagnostic is compatible with CDDR within the +observational uncertainty. Actually, benefit from the absence of nuisance parameters involved +9 + +in other currently available methods, our methodology produces more stringent constraints on +CDDR (with the precision of 10−2) at the current observational data level. The second approach is +the median statistics method, which is an appropriate measure in light of the non-Gaussian error +distribution. The validity of CDDR at z ∼ 2.3, with the 68% confidence intervals of the median +Med(η(zi)/η(zj)) = 0.998(±0.436), seems much more justified than the previous one drawn +from the weighted mean. Therefore, the results of η(zi)/η(zj) showed in this paper demonstrate +no evidence for the deviation from CDDR irrespective of the statistical method used. This is one of +the unambiguous conclusions in our work. However, one should also be aware of the disadvantage +of the above method, i.e., the ratio of CDDR parameter η(zi)/η(zj) should be constant and exactly +equal to one if the CDDR is the true one. However, the CDDR can be violated even if the ratio +is exactly equal to one. In order to fully explore the consequences of our proposed η(zi)/η(zj) +diagnostics, we adopt an explicit parameterization η(z) = η0 + η1z to better illustrate what our +results imply for the redshift-evolution of CDDR parameter. Thus, the ratio of CDDR parameter +can be rewritten as +η1 +η0 += ( +∆z +1 − θ(zi)(1+zj)2 +θ(zj)(1+zi)210∆mB,corr/5 − zj)−1, +(9) +which should be equal to zero if there is no redshift evolution of CDDR. The measurements of +these diagnostics as a function of redshift difference ∆z are shown in Fig. 4. Furthermore, we also +use a Python Markov Chain Monte Carlo (MCMC) module [62] to obtain fits on the two CDDR +parameters, by minimizing the χ2 objective function +χ2 = +2 +n(n − 1) +n−1 +� +i=1 +n +� +j=i+1 +(ηth +ij − ηobs +ij ) +σ2 +ηij +. +(10) +In Fig. 4 we also plot the one-dimensional marginalized distributions and two-dimensional con- +straint contours for the CDDR parameters, with the best-fit values of η0 = 0.952+0.019 +−0.019 and +η1 = 0.023+0.053 +−0.054, respectively. It is worth to comment that on the one hand, our methodology +produces a possible deviation from the expected value of CDDR parameter (η0 = 1) up to z ∼ 2.3. +However, our results are still marginally consistent with the CDDR validity within 2σ C.L., which +is in full agreement with other recent tests involving cosmological data. A summary of the cur- +rent constraints on the η0 from different cosmological observables can be found in Ref. [63]. On +the other hand, the CDDR remains redshift independent (η1 = 0) within 1σ C.L., supporting the +persisting claims that the Etherington reciprocity theorem could still be the best description of our +universe. +There are many ways the above findings could be improved. For instance, it is still interesting +to see whether those conclusions may be changed with machine learning algorithms, which have +shown their excellent potential in addressing cosmological issues and constraining cosmological +parameters [64–67]. More importantly, as a completely data driven approach, the Artificial Neural +10 + +FIG. 5: The CDDR parameters η(z) and η(zi)/η(zj) calculated from the two statistical methods as weighted +mean and the median statistics. Bands display the 68.3% confidence regions. +Network (ANN) method does not assume random variables that satisfy the Gaussian distribution. +The main purpose of an ANN (which consists of an input layer, one or more hidden layers and +an output layer) is to construct an approximate function fW,b(x) (in which W and b are linear +weights matrix and the offset vector) that correlates the input vector x with the output vector y +[68]. According to the difference between the predicted value fW,b(x) of the current network and +the target value y, the weight matrix of each layer needs to be constantly updated for minimize +the difference, which is defined by a loss function L [69]. An issue that needs clarification is the +achievable 1σ confidence region for the reconstructed function, which depends on both the actual +errors and the cost function. Following the detailed discussion in [70], a complete artificial neural +network has the following parts: firstly, the weight is randomly initialized in the neural network; +Secondly, the output value is compared with the expected output value, and the cost function is +used to calculate the error; Thirdly, the error is propagated back to the neural network and the +weight is set according to this information; Fourthly, repeat steps two to four for each input value +11 + +169M +n6ibgM +ae.0 +8e.0 +U(≤I)\U(s) +J'00 +J'OS +J'04169M +nsibgM +0'4 +a.0 +8.0 +U(sl)\u(sl) +T'O +I'S +J'4 +J'e169M +nsibgM +a.0 +8.0 +J'O +J'S +1'4- +I'ein the training set; Finally, when the entire training set is sent to the neural network, the entire +training is complete. The recent analysis has demonstrated the effectiveness of ANN acting as +“universal approximator” to produce representative uncertainties of the observations, especially +in high-precision test of CDDR in both electromagnetic and gravitational wave domain [54]. In +particular, Euclid collaboration improved the precision of CDDR test by approximately a factor of +six, based on machine learning reconstruction using genetic algorithms [71]. +Using the publicly released code called Reconstruct Functions with ANN [72], we perform the +reconstruction of the parameter η(zi)/η(zj) based on the current η(zi)/η(zj) two-point diagnos- +tics. The reconstructed functions with corresponding 1σ uncertainties, which can be considered +as the average level of observational error are given in right panel of Fig. 3. Working on the re- +constructed 1000 η(zi)/η(zj) points with ANN, we obtain Mean(η(zi)/η(zj)) = 0.998(±0.003) +and Med(η(zi)/η(zj)) = 0.998(±0.004) in the framework of weighted mean and median statis- +tics. Therefore, with ANN algorithm one could expect the parameter η(zi)/η(zj) to be estimated +at the precision of 10−3, which is more stringent than other results based on currently available +observational data. In order to facilitate comparison between the inferred values of CDDR pa- +rameters obtained from two statistical approaches, we display the results in Fig. 5. As a final +remark, possible violations of such fundamental relation (cosmic distance duality relation) might +have profound implications for the understanding of fundamental physics and natural laws. Based +on better uv-coverage in the future, we pin our hope on multi-frequency VLBI observations of +more compact radio quasars with higher angular resolution, smaller statistical and systematic un- +certainties. Meanwhile, considering the variety of different machine learning algorithms, we may +also be optimistic in detecting possible deviation from the CDDR with much higher precision. +IV. +ACKNOWLEDGMENTS +This work was supported by the National Natural Science Foundation of China under Grants +Nos. +12203009, 12122504, 12021003, 11875025, 11633001, 11920101003, and 62202469; +the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. +XDB23000000; Beijing Natural Science Foundation (Grant No. 4224091); the Interdiscipline +Research Funds of Beijing Normal University; and the China Manned Space Project (Nos. CMS- +CSST-2021-B01 and CMS-CSST-2021-A01). +[1] I. M. H. Etherington. On the Definition of Distance in General Relativity. Philosophical Magazine, +15(18):761, January 1933. +12 + +[2] I. M. H. Etherington. Republication of: LX. On the definition of distance in general relativity. General +Relativity and Gravitation, 39(7):1055–1067, July 2007. +[3] Adam G. Riess, Alexei V. Filippenko, Peter Challis, Alejandro Clocchiatti, Alan Diercks, Peter M. +Garnavich, Ron L. Gilliland, Craig J. Hogan, Saurabh Jha, Robert P. Kirshner, B. Leibundgut, M. M. +Phillips, David Reiss, Brian P. Schmidt, Robert A. Schommer, R. Chris Smith, J. Spyromilio, Christo- +pher Stubbs, Nicholas B. Suntzeff, and John Tonry. Observational Evidence from Supernovae for an +Accelerating Universe and a Cosmological Constant. Astron. J., 116(3):1009–1038, September 1998. +[4] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, +A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, +C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, +N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, +H. J. M. Newberg, W. J. Couch, and The Supernova Cosmology Project. Measurements of Ω and Λ +from 42 High-Redshift Supernovae. Astrophys. J. , 517(2):565–586, June 1999. +[5] Adam G. Riess, Louis-Gregory Strolger, John Tonry, Stefano Casertano, Henry C. Ferguson, Bahram +Mobasher, Peter Challis, Alexei V. Filippenko, Saurabh Jha, Weidong Li, Ryan Chornock, Robert P. +Kirshner, Bruno Leibundgut, Mark Dickinson, Mario Livio, Mauro Giavalisco, Charles C. Steidel, +Txitxo Ben´ıtez, and Zlatan Tsvetanov. Type Ia Supernova Discoveries at z ¿ 1 from the Hubble Space +Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution. Astrophys. J. , +607(2):665–687, June 2004. +[6] R. A. Knop, G. Aldering, R. Amanullah, P. Astier, G. Blanc, M. S. Burns, A. Conley, S. E. Deustua, +M. Doi, R. Ellis, S. Fabbro, G. Folatelli, A. S. Fruchter, G. Garavini, S. Garmond, K. Garton, R. Gib- +bons, G. Goldhaber, A. Goobar, D. E. Groom, D. Hardin, I. Hook, D. A. Howell, A. G. Kim, B. C. Lee, +C. Lidman, J. Mendez, S. Nobili, P. E. Nugent, R. Pain, N. Panagia, C. R. Pennypacker, S. Perlmut- +ter, R. Quimby, J. Raux, N. Regnault, P. Ruiz-Lapuente, G. Sainton, B. Schaefer, K. Schahmaneche, +E. Smith, A. L. Spadafora, V. Stanishev, M. Sullivan, N. A. Walton, L. Wang, W. M. Wood-Vasey, +and N. Yasuda. New Constraints on ΩM, ΩΛ, and w from an Independent Set of 11 High-Redshift +Supernovae Observed with the Hubble Space Telescope. Astrophys. J. , 598(1):102–137, November +2003. +[7] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Bal- +lardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, R. Battye, K. Benabed, J. P. Bernard, +M. Bersanelli, P. Bielewicz, J. J. Bock, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, M. Bucher, +13 + +C. Burigana, R. C. Butler, E. Calabrese, J. F. Cardoso, J. Carron, A. Challinor, H. C. Chiang, J. Chluba, +L. P. L. Colombo, C. Combet, D. Contreras, B. P. Crill, F. Cuttaia, P. de Bernardis, G. de Zotti, +J. Delabrouille, J. M. Delouis, E. Di Valentino, J. M. Diego, O. Dor´e, M. Douspis, A. Ducout, +X. Dupac, S. Dusini, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, Y. Fantaye, M. Farhang, +J. Fergusson, R. Fernandez-Cobos, F. Finelli, F. Forastieri, M. Frailis, A. A. Fraisse, E. Franceschi, +A. Frolov, S. Galeotta, S. Galli, K. Ganga, R. T. G´enova-Santos, M. Gerbino, T. Ghosh, J. Gonz´alez- +Nuevo, K. M. G´orski, S. Gratton, A. Gruppuso, J. E. Gudmundsson, J. Hamann, W. Handley, F. K. +Hansen, D. Herranz, S. R. Hildebrandt, E. Hivon, Z. Huang, A. H. Jaffe, W. C. Jones, A. Karakci, +E. Keih¨anen, R. Keskitalo, K. Kiiveri, J. Kim, T. S. Kisner, L. Knox, N. Krachmalnicoff, M. Kunz, +H. Kurki-Suonio, G. Lagache, J. M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, M. Le Je- +une, P. Lemos, J. Lesgourgues, F. Levrier, A. Lewis, M. Liguori, P. B. Lilje, M. Lilley, V. Lindholm, +M. L´opez-Caniego, P. M. Lubin, Y. Z. Ma, J. F. Mac´ıas-P´erez, G. Maggio, D. Maino, N. Mandolesi, +A. Mangilli, A. Marcos-Caballero, M. Maris, P. G. Martin, M. Martinelli, E. Mart´ınez-Gonz´alez, +S. Matarrese, N. Mauri, J. D. McEwen, P. R. Meinhold, A. Melchiorri, A. Mennella, M. Migliac- +cio, M. Millea, S. Mitra, M. A. Miville-Deschˆenes, D. Molinari, L. Montier, G. Morgante, A. Moss, +P. Natoli, H. U. Nørgaard-Nielsen, L. Pagano, D. Paoletti, B. Partridge, G. Patanchon, H. V. Peiris, +F. Perrotta, V. Pettorino, F. Piacentini, L. Polastri, G. Polenta, J. L. Puget, J. P. Rachen, M. Reinecke, +M. Remazeilles, A. Renzi, G. Rocha, C. Rosset, G. Roudier, J. A. Rubi˜no-Mart´ın, B. Ruiz-Granados, +L. Salvati, M. Sandri, M. Savelainen, D. Scott, E. P. S. Shellard, C. Sirignano, G. Sirri, L. D. Spencer, +R. Sunyaev, A. S. Suur-Uski, J. A. Tauber, D. Tavagnacco, M. Tenti, L. Toffolatti, M. Tomasi, T. Trom- +betti, L. Valenziano, J. Valiviita, B. Van Tent, L. Vibert, P. Vielva, F. Villa, N. Vittorio, B. D. Wandelt, +I. K. Wehus, M. White, S. D. M. White, A. Zacchei, and A. Zonca. Planck 2018 results. VI. Cosmo- +logical parameters. Astron. Astrophys., 641:A6, September 2020. +[8] Shuo Cao and Zonghong Zhu. The distance duality relation and the temperature profile of galaxy +clusters. Science China Physics, Mechanics, and Astronomy, 54(12):2260–2264, December 2011. +[9] Shuo Cao, Marek Biesiada, Xiaogang Zheng, and Zong-Hong Zhu. Testing the gas mass density +profile of galaxy clusters with distance duality relation. Mon. Not. Roy. Astron. Soc., 457(1):281–287, +March 2016. +[10] Tonghua Liu, Shuo Cao, Jia Zhang, Marek Biesiada, Yuting Liu, and Yujie Lian. Testing the cosmic +curvature at high redshifts: the combination of LSST strong lensing systems and quasars as new +standard candles. Mon. Not. Roy. Astron. Soc., 496(1):708–717, July 2020. +14 + +[11] Jun-Qing Xia, Hai Yu, Guo-Jian Wang, Shu-Xun Tian, Zheng-Xiang Li, Shuo Cao, and Zong-Hong +Zhu. Revisiting Studies of the Statistical Property of a Strong Gravitational Lens System and Model- +Independent Constraint on the Curvature of the Universe. Astrophys. J. , 834(1):75, January 2017. +[12] Jing-Zhao Qi, Shuo Cao, Sixuan Zhang, Marek Biesiada, Yan Wu, and Zong-Hong Zhu. The distance +sum rule from strong lensing systems and quasars - test of cosmic curvature and beyond. Mon. Not. +Roy. Astron. Soc., 483(1):1104–1113, February 2019. +[13] Puxun Wu, Zhengxiang Li, Xiaoliang Liu, and Hongwei Yu. Cosmic distance-duality relation test +using type Ia supernovae and the baryon acoustic oscillation. Phys. Rev. D , 92(2):023520, July 2015. +[14] R. F. L. Holanda, J. A. S. Lima, and M. B. Ribeiro. Testing the Distance-Duality Relation with Galaxy +Clusters and Type Ia Supernovae. Astrophys. J. Lett., 722(2):L233–L237, October 2010. +[15] Shuo Cao and Nan Liang. Testing the distance-duality relation with a combination of cosmological +distance observations. Research in Astronomy and Astrophysics, 11(10):1199–1208, October 2011. +[16] Kamal Bora and Shantanu Desai. A test of cosmic distance duality relation using SPT-SZ galaxy +clusters, Type Ia supernovae, and cosmic chronometers. JCAP, 2021(6):052, June 2021. +[17] Xiao-Lei Meng, Tong-Jie Zhang, Hu Zhan, and Xin Wang. Morphology of Galaxy Clusters: A Cosmo- +logical Model-independent Test of the Cosmic Distance-Duality Relation. Astrophys. J. , 745(1):98, +January 2012. +[18] Cheng-Zong Ruan, Fulvio Melia, and Tong-Jie Zhang. Model-independent Test of the Cosmic Dis- +tance Duality Relation. Astrophys. J. , 866(1):31, October 2018. +[19] Kai Liao. The Cosmic Distance Duality Relation with Strong Lensing and Gravitational Waves: An +Opacity-free Test. Astrophys. J. , 885(1):70, November 2019. +[20] Yu-Bo Ma, Shuo Cao, Jia Zhang, Jingzhao Qi, Tonghua Liu, Yuting Liu, and Shuaibo Geng. Testing +Cosmic Opacity with the Combination of Strongly Lensed and Unlensed Supernova Ia. Astrophys. J. +, 887(2):163, December 2019. +[21] Kai Liao, Marek Biesiada, and Zong-Hong Zhu. Strongly Lensed Transient Sources: A Review. +Chinese Physics Letters, 39(11):119801, November 2022. +[22] G. F. R. Ellis and W. Stoeger. The ’fitting problem’ in cosmology. Classical and Quantum Gravity, +4(6):1697–1729, November 1987. +[23] Zhengxiang Li, Guo-Jian Wang, Kai Liao, and Zong-Hong Zhu. Model-independent Estimations for +the Curvature from Standard Candles and Clocks. Astrophys. J. , 833(2):240, December 2016. +[24] Jun-Jie Wei and Xue-Feng Wu. An Improved Method to Measure the Cosmic Curvature. Astrophys. +15 + +J. , 838(2):160, April 2017. +[25] Rub´en Arjona, Hai-Nan Lin, Savvas Nesseris, and Li Tang. +Machine learning forecasts of the +cosmic distance duality relation with strongly lensed gravitational wave events. +Phys. Rev. D , +103(10):103513, May 2021. +[26] N. B. Hogg, M. Martinelli, and S. Nesseris. Constraints on the distance duality relation with standard +sirens. JCAP, 2020(12):019, December 2020. +[27] Purba Mukherjee and Ankan Mukherjee. Assessment of the cosmic distance duality relation using +Gaussian process. Mon. Not. Roy. Astron. Soc., 504(3):3938–3946, June 2021. +[28] Ariadna Montiel, Ruth Lazkoz, Irene Sendra, Celia Escamilla-Rivera, and Vincenzo Salzano. Non- +parametric reconstruction of the cosmic expansion with local regression smoothing and simulation +extrapolation. Phys. Rev. D , 89(4):043007, February 2014. +[29] Jack A. Baldwin. +Luminosity Indicators in the Spectra of Quasi-Stellar Objects. +Astrophys. J. , +214:679–684, June 1977. +[30] D. Watson, K. D. Denney, M. Vestergaard, and T. M. Davis. A New Cosmological Distance Measure +Using Active Galactic Nuclei. Astrophys. J. Lett., 740(2):L49, October 2011. +[31] Jian-Min Wang, Pu Du, David Valls-Gabaud, Chen Hu, and Hagai Netzer. Super-Eddington Accret- +ing Massive Black Holes as Long-Lived Cosmological Standards. Phys. Rev. Lett. , 110(8):081301, +February 2013. +[32] G. Risaliti and E. Lusso. Cosmological Constraints from the Hubble Diagram of Quasars at High +Redshifts. Nature Astronomy, 3:272–277, January 2019. +[33] G. Risaliti and E. Lusso. A Hubble Diagram for Quasars. Astrophys. J. , 815(1):33, December 2015. +[34] G. Risaliti and E. Lusso. Cosmology with AGN: can we use quasars as standard candles? Astronomis- +che Nachrichten, 338(329):329–333, March 2017. +[35] Tonghua Liu, Shuo Cao, Marek Biesiada, Yuting Liu, Shuaibo Geng, and Yujie Lian. Testing the +Cosmic Opacity at Higher Redshifts: Implication from Quasars with Available UV and X-Ray Obser- +vations. Astrophys. J. , 899(1):71, August 2020. +[36] K. I. Kellermann. The cosmological deceleration parameter estimated from the angular-size/redshift +relation for compact radio sources. Nature (London) , 361(6408):134–136, January 1993. +[37] L. I. Gurvits. +Apparent Milliarcsecond Sizes of Active Galactic Nuclei and the Geometry of the +Universe. Astrophys. J. , 425:442, April 1994. +[38] R. G. Vishwakarma. Consequences on variable Λ-models from distant type Ia supernovae and compact +16 + +radio sources. Classical and Quantum Gravity, 18(7):1159–1172, April 2001. +[39] Shuo Cao, Marek Biesiada, John Jackson, Xiaogang Zheng, Yuhang Zhao, and Zong-Hong Zhu. +Measuring the speed of light with ultra-compact radio quasars. JCAP, 2017(2):012, February 2017. +[40] Shuo Cao, Xiaogang Zheng, Marek Biesiada, Jingzhao Qi, Yun Chen, and Zong-Hong Zhu. Ultra- +compact structure in intermediate-luminosity radio quasars: building a sample of standard cosmologi- +cal rulers and improving the dark energy constraints up to z 3. Astron. Astrophys., 606:A15, September +2017. +[41] R. A. Preston, D. D. Morabito, J. G. Williams, J. Faulkner, D. L. Jauncey, and G. Nicolson. A VLBI +survey at 2.29 GHz. Astron. J., 90:1599–1603, September 1985. +[42] J. C. Jackson. Tight cosmological constraints from the angular-size/redshift relation for ultra-compact +radio sources. JCAP, 2004(11):007, November 2004. +[43] Xiaolei Li, Shuo Cao, Xiaogang Zheng, Jingzhao Qi, Marek Biesiada, and Zong-Hong Zhu. Testing +and selecting cosmological models with ultra-compact radio quasars. European Physical Journal C, +77(8):677, August 2017. +[44] Yubo Ma, Jia Zhang, Shuo Cao, Xiaogang Zheng, Tengpeng Xu, and Jingzhao Qi. The general- +ized cosmic equation of state: a revised study with cosmological standard rulers. European Physical +Journal C, 77(12):891, December 2017. +[45] Jing-Zhao Qi, Shuo Cao, Marek Biesiada, Xiaogang Zheng, and Zong-Hong Zhu. New observational +constraints on f( T) cosmology from radio quasars. European Physical Journal C, 77(8):502, August +2017. +[46] Xiaogang Zheng, Kai Liao, Marek Biesiada, Shuo Cao, Tong-Hua Liu, and Zong-Hong Zhu. Mul- +tiple Measurements of Quasars Acting as Standard Probes: Exploring the Cosmic Distance Duality +Relation at Higher Redshift. Astrophys. J. , 892(2):103, April 2020. +[47] D. M. Scolnic, D. O. Jones, A. Rest, Y. C. Pan, R. Chornock, R. J. Foley, M. E. Huber, R. Kessler, +G. Narayan, A. G. Riess, S. Rodney, E. Berger, D. J. Brout, P. J. Challis, M. Drout, D. Finkbeiner, +R. Lunnan, R. P. Kirshner, N. E. Sanders, E. Schlafly, S. Smartt, C. W. Stubbs, J. Tonry, W. M. Wood- +Vasey, M. Foley, J. Hand, E. Johnson, W. S. Burgett, K. C. Chambers, P. W. Draper, K. W. Hodapp, +N. Kaiser, R. P. Kudritzki, E. A. Magnier, N. Metcalfe, F. Bresolin, E. Gall, R. Kotak, M. McCrum, +and K. W. Smith. The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from +Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample. Astrophys. J. , +859(2):101, June 2018. +17 + +[48] A. Rest, D. Scolnic, R. J. Foley, M. E. Huber, R. Chornock, G. Narayan, J. L. Tonry, E. Berger, +A. M. Soderberg, C. W. Stubbs, A. Riess, R. P. Kirshner, S. J. Smartt, E. Schlafly, S. Rodney, M. T. +Botticella, D. Brout, P. Challis, I. Czekala, M. Drout, M. J. Hudson, R. Kotak, C. Leibler, R. Lunnan, +G. H. Marion, M. McCrum, D. Milisavljevic, A. Pastorello, N. E. Sanders, K. Smith, E. Stafford, +D. Thilker, S. Valenti, W. M. Wood-Vasey, Z. Zheng, W. S. Burgett, K. C. Chambers, L. Denneau, +P. W. Draper, H. Flewelling, K. W. Hodapp, N. Kaiser, R. P. Kudritzki, E. A. Magnier, N. Metcalfe, +P. A. Price, W. Sweeney, R. Wainscoat, and C. Waters. Cosmological Constraints from Measurements +of Type Ia Supernovae Discovered during the First 1.5 yr of the Pan-STARRS1 Survey. Astrophys. J. +, 795(1):44, November 2014. +[49] D. Scolnic, A. Rest, A. Riess, M. E. Huber, R. J. Foley, D. Brout, R. Chornock, G. Narayan, J. L. +Tonry, E. Berger, A. M. Soderberg, C. W. Stubbs, R. P. Kirshner, S. Rodney, S. J. Smartt, E. Schlafly, +M. T. Botticella, P. Challis, I. Czekala, M. Drout, M. J. Hudson, R. Kotak, C. Leibler, R. Lunnan, +G. H. Marion, M. McCrum, D. Milisavljevic, A. Pastorello, N. E. Sanders, K. Smith, E. Stafford, +D. Thilker, S. Valenti, W. M. Wood-Vasey, Z. Zheng, W. S. Burgett, K. C. Chambers, L. Denneau, +P. W. Draper, H. Flewelling, K. W. Hodapp, N. Kaiser, R. P. Kudritzki, E. A. Magnier, N. Metcalfe, +P. A. Price, W. Sweeney, R. Wainscoat, and C. Waters. Systematic Uncertainties Associated with +the Cosmological Analysis of the First Pan-STARRS1 Type Ia Supernova Sample. Astrophys. J. , +795(1):45, November 2014. +[50] M. Betoule, R. Kessler, J. Guy, J. Mosher, D. Hardin, R. Biswas, P. Astier, P. El-Hage, M. Konig, +S. Kuhlmann, J. Marriner, R. Pain, N. Regnault, C. Balland, B. A. Bassett, P. J. Brown, H. Campbell, +R. G. Carlberg, F. Cellier-Holzem, D. Cinabro, A. Conley, C. B. D’Andrea, D. L. DePoy, M. Doi, +R. S. Ellis, S. Fabbro, A. V. Filippenko, R. J. Foley, J. A. Frieman, D. Fouchez, L. Galbany, A. Goo- +bar, R. R. Gupta, G. J. Hill, R. Hlozek, C. J. Hogan, I. M. Hook, D. A. Howell, S. W. Jha, L. Le +Guillou, G. Leloudas, C. Lidman, J. L. Marshall, A. M¨oller, A. M. Mour˜ao, J. Neveu, R. Nichol, +M. D. Olmstead, N. Palanque-Delabrouille, S. Perlmutter, J. L. Prieto, C. J. Pritchet, M. Richmond, +A. G. Riess, V. Ruhlmann-Kleider, M. Sako, K. Schahmaneche, D. P. Schneider, M. Smith, J. Soller- +man, M. Sullivan, N. A. Walton, and C. J. Wheeler. Improved cosmological constraints from a joint +analysis of the SDSS-II and SNLS supernova samples. Astron. Astrophys., 568:A22, August 2014. +[51] R. Kessler and D. Scolnic. Correcting Type Ia Supernova Distances for Selection Biases and Contam- +ination in Photometrically Identified Samples. Astrophys. J. , 836(1):56, February 2017. +[52] Zhengxiang Li, Puxun Wu, and Hongwei Yu. +Cosmological-model-independent Tests for the +18 + +Distance-Duality Relation from Galaxy Clusters and Type Ia Supernova. +Astrophys. J. Lett., +729(1):L14, March 2011. +[53] Kai Liao, Zhengxiang Li, Shuo Cao, Marek Biesiada, Xiaogang Zheng, and Zong-Hong Zhu. The +Distance Duality Relation from Strong Gravitational Lensing. Astrophys. J. , 822(2):74, May 2016. +[54] Tonghua Liu, Shuo Cao, Sixuan Zhang, Xiaolong Gong, Wuzheng Guo, and Chenfa Zheng. Revisiting +the cosmic distance duality relation with machine learning reconstruction methods: the combination +of HII galaxies and ultra-compact radio quasars. European Physical Journal C, 81(10):903, October +2021. +[55] Philip R. Bevington, D. Keith Robinson, J. Morris Blair, A. John Mallinckrodt, and Susan McKay. +Data Reduction and Error Analysis for the Physical Sciences. Computers in Physics, 7(4):415, January +1993. +[56] Shuo Cao, Jingzhao Qi, Zhoujian Cao, Marek Biesiada, Jin Li, Yu Pan, and Zong-Hong Zhu. Direct +test of the FLRW metric from strongly lensed gravitational wave observations. Scientific Reports, +9:11608, August 2019. +[57] Eric D. Feigelson and G. Jogesh Babu. Modern Statistical Methods for Astronomy. 2012. +[58] III Gott, J. Richard, Michael S. Vogeley, Silviu Podariu, and Bharat Ratra. Median Statistics, H0, and +the Accelerating Universe. Astrophys. J. , 549(1):1–17, March 2001. +[59] Arman Shafieloo, Varun Sahni, and Alexei A. Starobinsky. New null diagnostic customized for re- +constructing the properties of dark energy from baryon acoustic oscillations data. Phys. Rev. D , +86(10):103527, November 2012. +[60] Xiaogang Zheng, Xuheng Ding, Marek Biesiada, Shuo Cao, and Zong-Hong Zhu. What Are the Omhˆ2 +(z1, z2) and Om (z1, z2) Diagnostics Telling us in Light of H(z) Data? Astrophys. J. , 825(1):17, July +2016. +[61] Xiaogang Zheng, Marek Biesiada, Xuheng Ding, Shuo Cao, Sixuan Zhang, and Zong-Hong Zhu. +Statistical analysis with cosmic-expansion-rate measurements and two-point diagnostics. European +Physical Journal C, 78(10):274, July 2018. +[62] Daniel Foreman-Mackey, David W. Hogg, Dustin Lang, and Jonathan Goodman. emcee: The MCMC +Hammer. Publications of the Astronomical Society of the Pacific, 125(925):306, March 2013. +[63] R. F. L. Holanda, F. S. Lima, Akshay Rana, and Deepak Jain. Strong lensing systems and galaxy +cluster observations as probe to the cosmic distance duality relation. European Physical Journal C, +82(2):115, February 2022. +19 + +[64] Janis Fluri, Tomasz Kacprzak, Alexandre Refregier, Adam Amara, Aurelien Lucchi, and Thomas +Hofmann. Cosmological constraints from noisy convergence maps through deep learning. Phys. Rev. +D , 98(12):123518, December 2018. +[65] Janis Fluri, Tomasz Kacprzak, Aurelien Lucchi, Alexandre Refregier, Adam Amara, Thomas Hof- +mann, and Aurel Schneider. Cosmological constraints with deep learning from KiDS-450 weak lens- +ing maps. Phys. Rev. D , 100(6):063514, September 2019. +[66] Dezs˝o Ribli, B´alint ´Armin Pataki, Jos´e Manuel Zorrilla Matilla, Daniel Hsu, Zolt´an Haiman, and +Istv´an Csabai. Weak lensing cosmology with convolutional neural networks on noisy data. Mon. Not. +Roy. Astron. Soc., 490(2):1843–1860, December 2019. +[67] Guo-Jian Wang, Si-Yao Li, and Jun-Qing Xia. ECoPANN: A Framework for Estimating Cosmological +Parameters Using Artificial Neural Networks. Astrophys. J., Suppl. Ser., 249(2):25, August 2020. +[68] Djork-Arn´e Clevert, Thomas Unterthiner, and Sepp Hochreiter. Fast and Accurate Deep Network +Learning by Exponential Linear Units (ELUs). arXiv e-prints, page arXiv:1511.07289, November +2015. +[69] Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization. arXiv e-prints, +page arXiv:1412.6980, December 2014. +[70] Guo-Jian Wang, Xiao-Jiao Ma, and Jun-Qing Xia. Machine learning the cosmic curvature in a model- +independent way. Mon. Not. Roy. Astron. Soc., 501(4):5714–5722, March 2021. +[71] M. Martinelli, C. J. A. P. Martins, S. Nesseris, D. Sapone, I. Tutusaus, A. Avgoustidis, S. Camera, +C. Carbone, S. Casas, S. Ili´c, Z. Sakr, V. Yankelevich, N. Auricchio, A. Balestra, C. Bodendorf, +D. Bonino, E. Branchini, M. Brescia, J. Brinchmann, V. Capobianco, J. Carretero, M. Castellano, +S. Cavuoti, R. Cledassou, G. Congedo, L. Conversi, L. Corcione, F. Dubath, A. Ealet, M. Frailis, +E. Franceschi, M. Fumana, B. Garilli, B. Gillis, C. Giocoli, F. Grupp, S. V. H. Haugan, W. Holmes, +F. Hormuth, K. Jahnke, S. Kermiche, M. Kilbinger, T. D. Kitching, B. Kubik, M. Kunz, H. Kurki- +Suonio, S. Ligori, P. B. Lilje, I. Lloro, O. Marggraf, K. Markovic, R. Massey, S. Mei, M. Meneghetti, +G. Meylan, L. Moscardini, S. Niemi, C. Padilla, S. Paltani, F. Pasian, V. Pettorino, S. Pires, G. Po- +lenta, M. Poncet, L. Popa, L. Pozzetti, F. Raison, J. Rhodes, M. Roncarelli, R. Saglia, P. Schneider, +A. Secroun, S. Serrano, C. Sirignano, G. Sirri, F. Sureau, A. N. Taylor, I. Tereno, R. Toledo-Moreo, +L. Valenziano, T. Vassallo, Y. Wang, N. Welikala, J. Weller, and A. Zacchei. Euclid: Forecast con- +straints on the cosmic distance duality relation with complementary external probes. Astron. Astro- +phys., 644:A80, December 2020. +20 + +[72] ReFANN; https://github.com/Guo-Jian-Wang/refann. +21 + diff --git a/i9E1T4oBgHgl3EQfNAPK/content/tmp_files/load_file.txt b/i9E1T4oBgHgl3EQfNAPK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..52a412871f6c7b0b902490a7dba2e1d583438005 --- /dev/null +++ b/i9E1T4oBgHgl3EQfNAPK/content/tmp_files/load_file.txt @@ -0,0 +1,1443 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf,len=1442 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='02997v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='CO] 8 Jan 2023 What are recent observations telling us in light of improved tests of distance duality relation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Tonghua Liu1,2, Shuo Cao2,3 ∗, Shuai Ma4, Yuting Liu2,3, Chenfa Zheng2,3, Jieci Wang5 † 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' School of Physics and Optoelectronic, Yangtze University, Jingzhou 434023, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Department of Astronomy, Beijing Normal University, Beijing 100875, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Beijing Academy, Kangyuan Road, Dongba, Beijing 100018, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Department of Physics, and Collaborative Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan 410081, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Abstract As an exact result required by the Etherington reciprocity theorem, the cosmic distance duality relation (CDDR), η(z) = DL(z)(1 + z)−2/DA(z) = 1 plays an essential part in modern cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In this paper, we present a new method (η(zi)/η(zj)) to use the measurements of ultra-compact structure in radio quasars (QSO) and the latest observations of type Ia supernova (SN Ia) to test CDDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' By taking the observations directly from SN Ia and QSOs, one can completely eliminate the uncertainty caused by the calibration of the absolute magnitudes of standard candles (MB) and the linear sizes of standard rulers (lm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Benefit from the absence of nuisance parameters involved in other currently available methods, our analysis demonstrates no evidence for the deviation and redshift evolution of CDDR up to z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The combination of our method- ology and the machine learning Artificial Neural Network (ANN) would produce 10−3 level constraints on the violation parameter at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Our results indicate perfect agreement between observations and predictions, supporting the persisting claims that the Etherington reciprocity theorem could still be the best description of our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' PACS numbers: 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='-k,98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='-h,98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='Aj,06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='Bp ∗ Corresponding author: caoshuo@bnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='cn † Corresponding author: jcwang@hunnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='cn 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' INTRODUCTION The cosmic distance duality relation (CDDR), as a fundamental relation in modern cos- mology, correlates the luminosity distance DL(z) with the angular diameter distance DA(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' More specifically, the CDDR indicates that DL(z) and DA(z) should satisfy the relation of DL(z) = DA(z)(1 + z)2 at the same redshift [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' However, the validity of the CDDR depends on three basic assumptions: i) the space-time is described by metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' ii) the light travels along the null geodesics between the observer and the source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' iii) the photon number is conserved, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', the CDDR will be violated if the number of photons in the universe is not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Therefore, the validity test of CDDR is, to some extent, an indirect verification of the accelerated expansion of the universe [3–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In addition, as a fundamental relationship in cosmology, the CDDR has been widely used in various fields of astronomy, such as the observations of large-scale distribution of galaxies and the near-uniformity of the cosmic microwave background radiation (CMBR) temper- ature [7], the determination of gas mass density profile and temperature profile of galaxy clusters [8, 9], as well as the measurements of cosmic curvature with strong gravitational lensing systems [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' On the other hand, the accumulation of precise astrophysical observations allow us to test the validity of CDDR at different redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' From the theoretical point of view, two types of cosmo- logical distances are usually required in developing CDDR tests, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', angular diameter distance DA and luminosity distance DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' For the observations of luminosity distances, many works turn to luminous sources with known (or standardizable) intrinsic luminosity in the universe like type-Ia supernova (SN Ia), while the angular diameter distances are inferred from baryon acoustic oscil- lations (BAO), Sunyaev-Zeldovich (SZ) effect of galaxy clusters [13–15], gas mass fraction mea- surements in galaxy clusters [16, 17], and strong gravitational lensing systems [18–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' However, it is worth noting that angular diameter distances derived from BAO is puzzled by the so-called fitting problem, which is a major challenge confronted by the standard BAO peak location with a fixed comoving ruler of about 105 h−1 Mpc [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Therefore, such distance estimation is model- dependent to some extent, which inevitably brings systematic uncertainties and further affects the validity of testing CDDR with BAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Meanwhile, galaxy clusters alone are not able to provide a competitive source of angular diameter distance at different redshifts, suffering from the large observational uncertainties arising from radio observations of the SZ effect of galaxy clusters to- gether with X-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In addition, based on the observations of SN Ia, it was argued that the nuisance parameters characterizing SN Ia light-curves also introduce considerable uncertain- ties to the final results [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Therefore, in order to perform the validity of testing CDDR, one needs to eliminate the effects and uncertanties caused by the nuisance parameters in both two types of observational data sets (angular diameter distance DA and luminosity distance DL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In 2 this paper we further analyse the most updated QSO and SNe Ia data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Specially, we present a new approach that harnesses the ratio η(zi)/η(zj) as cosmic observations, based on the luminosity distance inferred from the latest observations of type Ia supernova and angular diameter distances obtained from the measurements of ultra-compact structure in radio quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' All of the quantities used in the CDDR test come directly from observations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', the absolute magnitudes of SN Ia and the linear size of the compact structure in radio quasars need not to be calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In this way, the effects and uncertainties caused by the nuisance parameters are eliminated completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' More inter- estingly, our methodology will also benefit from the consistent redshift coverage of both samples that can reach a high redshift range of z ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' With the increase both in the depth and quality of cosmological measurements, new techniques and methods for CDDR tests have also been developed, focusing on different machine learning algorithms [25–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In this paper, we will use the Artificial Neural Network (ANN) algorithm to reconstruct the possible evolution of CDDR with redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Note that such data-driven approach has no assumptions about the observational data, suggesting its advantage of being completely model-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Summarizing, we will propose an improved approach to test CDDR directly, which not only effectively avoids the influence of nuisance parameter on CDDR, but also achieves more stringent constraints on CDDR in the case of small data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The outline of this paper is given as follow: in Section II we briefly introduce the observations of ultra-compact structure of radio quasars acting standard rulers and SN Ia acting as standard candles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The improved method- ology of testing CDDR and the corresponding results are presented in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' OBSERVATIONAL DATA A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Angular diameter distances from radio quasars We consider extracting angular diameter distance from angular redshift relation of compact structure of radio quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' As the most distant and brightest objects in the Universe, quasars exhibit great potential in studying early cosmology beyond the limitation of supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Unfortunately quasars exhibit large dispersion in luminosities at all wavelengths, which makes them unusable as standard probes for measuring cosmological distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In the past decades, great effort have been made to make use of quasars as standard candles or standard rulers in modern cosmology, such as the Baldwin effect [29], the Broad Line Region radius-luminosity relation [30], the properties of highly accreting quasars [31], and the non-linear relation between the ultraviolet and X-ray fluxes of the quasar to construct the Hubble diagram [10, 32–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' According to the unified model of active galactic nuclei and quasars, ultra-compact radio sources are identified as cases in which the jets are moving relativistically and are close to the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' At any given frequency, the core is 3 believed to be located in the region of the jet corresponding to unit optical depth with synchrotron self-absorption being the dominating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In the original work of [36], an interesting possi- bility was discussed that compact radio sources (especially quasars) constitute another potential class of standard rulers that could be observed by very long baseline interferometry (VLBI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The VLBI with high precision can not only accurately locate the radio source, but also measure the ten- sion angle of the compact radio source at the magnitude of mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Based on the subsequent works of [37, 38], the linear size of the compact structures in radio sources are related to the intrinsic luminosity L and the redshift z of the background source lm = lLβ(1 + z)n, (1) where l represents the linear size scaling factor which describes the apparent distribution of radio brightness within the core, β and n denotes the possible dependence of the intrinsic size on the luminosity and the redshift, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' However, the application of radio sources in cosmology still suffered from the high dispersion in the observed relations or the limitation of a poor statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' With the gradually refined selection technique and observations, a key step forward was made in the work of [37], which showed that the linear size dispersion in radio source with a flat spectral index (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='38 < α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='18) is greatly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Based on a sample of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='29 GHz VLBI survey with 613 milliarcsecond compact radio sources, [39, 40] selected 120 intermediate-luminosity (1027W/Hz< L < 1028W/Hz) quasars (ILQSOs) with reliable measurements on the angular size of the compact structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The final results demonstrated that ILQSOs are almost independent from redshift and luminosity (|n| ≃ 10−3, β ≃ 10−4), which means they meet the requirements expected from standard rulers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' However, the crucial question is what is the intrinsic metric linear size of the quasar source?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The previous analysis roughly estimated that the lm parameter is robustly of the scale of ∼ 11 pc [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' For the sake of the following description, we take a prior value lm = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='25 pc determined by in a cosmological-model-independent method [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' However, in our work the value of lm does not affect the CDDR test, and we will later propose an improved CDDR test to eliminate the bias and additional systematic errors associated with the lm value of calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The angular size in compact structure and cosmic distance relation for cosmological inference was first proposed in [36] DA(z) = lm θ(z), (2) where DA is the angular diameter distance, lm is the intrinsic metric linear size of the source, and θ(z) is the observed angular size, which is defined by the modulus of visibility Γ = Sc/St in the literature [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The specific definition of angular size is θ(z) = 2 √ − ln Γ ln 2/πBθ, where Bθ is interferometer baseline measured in wavelengths, Sc and St are correlated flux density and total flux density, respectively [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The sample of raido quasars used in this work is the one described 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 1: The scatter plot of the observed angular sizes of 120 radio quasars (left panel) and the apparent magnitudes of 1048 Pantheon SN Ia (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The red dotted line denotes the angular sizes calculated from the fiducial ΛCDM model (H0 = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='0 km/s/Mpc, Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' in [39] with the redshift range between z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='462 and z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' These compact radio sources come from a well-known 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='29 GHz VLBI survey [41] (hereafter called P85) with 1398 detected candidates and 917 selected sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The P85 sample was updated with respect to redshift [42], which includes 613 compact radio sources that cover the redshift range up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='0035 ≤ z ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='787 (http://nrl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='northumbria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='uk/13109/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' These 120 radio quasars have been carefully selected for cosmological studies and we refer to [39] for a detailed description of the selection procedure used to turn them into standard rulers and for an explanation of the calibration method used to include them in the extensive cosmological analysis [43–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The scatter diagram of the observed angular sizes for 120 radio quasars is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Luminosity distances from Type Ia Supernova In order to carry out the test of CDDR, we need to find another cosmological probe that can directly provide luminosity distances and satisfy the following criteria, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', the probe should be able to cover roughly the redshift range of the compact radio quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In this work, we seek for SN Ia as a reasonably empirically well-understood cosmological probe, the usefulness of which to modern cosmology is well known in revealing the accelerated expansion of the Universe and placing constraints on cosmological parameters to break parameter degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' With the rapid growth in the sample size of SN Ia distance measurements, the analysis and mitigation of system- atic uncertainties of Type Ia Supernova has been considerably improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' However, the application of SN Ia for cosmology involves so-called ”nuisance” parameters, which need to be optimized 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content="0 0'2 T'O T'2 J4 Je 18 50 SS S4 se5 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content="0 0'2 1'O 12 5'2 0'2 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content="0 0'2 J'O 1'2 (26m) 5'O 5'2 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content="E 3'2 020 VCDW 4'0FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 2: The CDDR parameter η(z) from the observations of radio quasars and SN Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' along with the unknown variables in cosmological models and could potentially affect reliable constraints on cosmological model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Fortunately, the recent SN Ia sample called Pantheon has been released by the Pan-STARRS1 (PS1) Medium Deep Survey, which contains 1048 SN Ia measurements spanning the redshift range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='01 < z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='3 [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Here, we only summarise the crucial points required by the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Benefit from richness and depth of the sample,the Pantheon catalogue combines the subset of 279 PS1 SN Ia [48, 49] and useful distance estimations of SN Ia from SDS, SNLS, various low redshift and HST samples [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' More importantly, compared with the previous SN Ia data sets [50], the Pantheon sample applies a new approach called BEAMS with Bias Corrections (BBC) [51], in which the apparent magnitude is replaced with the corrected apparent magnitude mB,corr = mB + α⋆ · X1 − β · C for all the SN Ia [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Here, mB is the observed peak magnitude in rest-frame B band, while X1 and C are the color and light-curve shape parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The two nuisance parameters α⋆ and β should be fitted simultaneously with the cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' It should be noted that the stretch luminosity parameter α⋆ and the color-luminosity parameter β are set to zero for the Pantheon sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Therefore, the observed distance modulus of SN Ia provides the luminosity distance as DL,SN(z) = 10(mB,corr(z)−MB)/5−5(Mpc), (3) where MB is the absolute magnitude in B band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' For the uncertainty of the luminosity distance in Pantheon data set, the contribution from photometric error, distance bias correction, and the peculiar velocity are included in this analysis [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The apparent B-band magnitude for 1048 Pantheon SN Ia of is also illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 6 5 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content="0 J'O I'S 1'4 00." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content="0 0'52- 0'20- 0'12 1'00 J'52 J'20- J'J2 - V=J 5'00III." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' METHODOLOGY AND RESULTS From the theoretical point of view, in order to directly test the DDR from observations, the following parameterized form is commonly used η(z) = DL(z) DA(z)(1 + z)2, (4) the likelihood of which is expected to peak at one in order to satisfy the CDDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' By combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' (2) and (3) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' (4), one can rewrite the above expression as η(z) = θ(z)10(mB,corr(z)−MB)/5−5 lm(1 + z)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' (5) The difficulty of testing CDDR lies in the fact that the angular diameter distance from an radio quasar should be observed at the same redshift as SN Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In the previous work for example in [15], it was pointed out that the CDDR test could be significantly affected by the particular choice of the selection criteria for a given pair of data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Following the redshift selection criterion widely used in the literature (within the redshift range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='01 < z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='3) [52–54], the redshifts of SN Ia sample are carefully chosen to coincide with the associated quasar sample demanding that the difference in redshift is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' By performing such selection criterion that could hopefully ease the systematic errors brought by redshift inconsistency, only 37 pairs of data sets are obtained from the Pantheon and ILQSO sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Combining these quasar data together with the Pantheon SN Ia sample, we obtain the CDDR parameter η(z) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' More specifically, the total uncertainties of η(z) are calculated from the standard uncertainty propagation formula, based on the uncorrelated uncertainties of observables including the observed angular size errors σθ, corrected apparent magnitude errors σmB,corr, as well as additional systematic errors introduced from the calibrations of absolute magnitude (MB) of SN Ia and linear size (lm) of radio quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' To better illustrate the statistical significance of our results, we first use the weighted mean statistics [55] to evaluate η = Σi � ηi/σ2 ηi � Σi � 1/σ2ηi � , σ2 η = 1 Σi � 1/σ2ηi �, (6) where η stands for the weighted mean and ση is its corresponding uncertainty of CDDR parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Such statistical method has been widely applied in meta-analysis to integrate the results of inde- pendent measurements [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Our assessments for weighted mean and corresponding uncertainty are Mean(η(z)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='991(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='147), which is in perfect agreement with the results of previous works [23–27], indicates that there is no evidence for the CDDR violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Given the possible invalidity of Gaussian distribution of the errors, we also use a robust median statistics [57] to evaluate the measurements of η(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Moreover, if there are extreme values and outliers in the se- quence, it is better to use the median as the representative value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' When making a total number of 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 3: The η(zi)/η(zj) two-point diagnostics calculated on the observations of radio quasars and SN Ia (left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The reconstructed η(zi)/η(zj) two-point diagnostics with ANN machine learning algorithm (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='99 η 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='2 η 1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='2 η 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 4: The scatter plot of the CDDR parameter η1/η0 (left panel) and constraints on the CDDR parameters (η0, η1) (right panel), in the framework of η(zi)/η(zj) two-point diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' N measurements, one might naturally expect that there is a 50% chance that each measurement is higher/lower than the true median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Therefore, the probability that n-th observation is higher than the median follows the binomial distribution: P = 2−NN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='/[n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='(N−n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='] [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Similarly, we can de- fine the 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='3% confidence interval with median statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In the framework of such non-parametric approach, the resulting constraint on the CDDR parameter becomes Med(η(z)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='117(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='328) with the median value and the absolute deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Therefore, the conclusion of CDDR validity (η(z) = 1) seems robust within 1σ confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Due to the ambiguous interpretation of the compact structure size in radio quasars and the 8 VS 00 O1 05 03 04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='2 oe 03 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content="0 12 1O: 2 UTI 0 2 TO = 12SA 00 OT 0'5 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='3 04 02 oe 03 0:8 S U(sU(s) WWAVs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='0 oe 0:8 1O U(SU(s)absolute B-band magnitude of SN Ia whose value is determined by the host stellar mass, the linear size parameter lm and the absolute magnitude MB are hard to determine precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In fact, the uncertainty of CDDR measurements shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 2 is dominated by the calibration of two nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In order to eliminate the influence of these two nuisance parameters, we propose an improved approach by introducing the ratio of CDDR parameter η(zi)/η(zj) = θ(zi)(1 + zj)2 θ(zj)(1 + zi)210∆mB,corr/5, (7) where ∆mB,corr = mB,corr(zi) − mB,corr(zj) is the difference of corrected apparent magnitude between arbitrary two SN Ia data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' If one defines the ratio η(zi)/η(zj), where i, j denote the order numbers of the radio quasars and SN Ia, then such quantity does not depend on the nuisance parameters and it does not introduce any uncertainty to the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Note that if we have observational data at n different redshifts, then we can get n(n − 1)/2 data pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The uncertainty of ηij = η(zi)/η(zj) is calculated using the standard error propagation formula, which is related to the uncorrelated uncertainties of the observed angular size σθ and corrected apparent magnitude σmB,corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' More importantly, our approach successfully eliminate the nuisance parameters MB and lm, which brings benefits in alleviating the systematics caused by precise determination of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' These are the apparent merits of our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Our approach was inspired by the two-point diagnostic approach, which has been extensively applied to quantify the difference between the cosmological constant (ΛCDM) and other dark energy models (including evolving dark energy) [59–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In order to gain insight concerning the two-point diagnostics calculated for every combination of pairs taken from the full QSO+SN Ia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' We display these diagnostics together with their uncertainties as a function of redshift difference ∆z = |zi − zj| in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Benefit from the improved methodology, the QSO/SN Ia pairs satisfying irrespective of the redshift selec- tion criteria have a massive growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' One can see that there are some interesting features regarding the uncertainties of the two-point diagnostics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', they are apparently non-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In order to test further the validity and efficiency of our method, we use two approaches to produce a sum- mary statistics of two-point diagnostics calculated on the data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The first is to use the weighted mean statistical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In order to ensure that each data point is uncorrelated, the weighted mean formula for the η(zi)/η(zj) diagnostic should be rewritten as [60] ηij = Σn−1 i=1 Σn j=i+1 � ηij/σ2 ηij � Σn−1 i=1 Σn j=i+1 � 1/σ2 ηij � , σ2 ηij = 1 Σn−1 i=1 Σn j=i+1 � 1/σ2 ηij �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' (8) The weighted mean value and corresponding uncertainty is Mean(η(zi)/η(zj)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='968 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='031, which suggests that the weighted mean of this diagnostic is compatible with CDDR within the observational uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Actually, benefit from the absence of nuisance parameters involved 9 in other currently available methods, our methodology produces more stringent constraints on CDDR (with the precision of 10−2) at the current observational data level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The second approach is the median statistics method, which is an appropriate measure in light of the non-Gaussian error distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The validity of CDDR at z ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='3, with the 68% confidence intervals of the median Med(η(zi)/η(zj)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='998(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='436), seems much more justified than the previous one drawn from the weighted mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Therefore, the results of η(zi)/η(zj) showed in this paper demonstrate no evidence for the deviation from CDDR irrespective of the statistical method used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' This is one of the unambiguous conclusions in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' However, one should also be aware of the disadvantage of the above method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', the ratio of CDDR parameter η(zi)/η(zj) should be constant and exactly equal to one if the CDDR is the true one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' However, the CDDR can be violated even if the ratio is exactly equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In order to fully explore the consequences of our proposed η(zi)/η(zj) diagnostics, we adopt an explicit parameterization η(z) = η0 + η1z to better illustrate what our results imply for the redshift-evolution of CDDR parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Thus, the ratio of CDDR parameter can be rewritten as η1 η0 = ( ∆z 1 − θ(zi)(1+zj)2 θ(zj)(1+zi)210∆mB,corr/5 − zj)−1, (9) which should be equal to zero if there is no redshift evolution of CDDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The measurements of these diagnostics as a function of redshift difference ∆z are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Furthermore, we also use a Python Markov Chain Monte Carlo (MCMC) module [62] to obtain fits on the two CDDR parameters, by minimizing the χ2 objective function χ2 = 2 n(n − 1) n−1 � i=1 n � j=i+1 (ηth ij − ηobs ij ) σ2 ηij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' (10) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 4 we also plot the one-dimensional marginalized distributions and two-dimensional con- straint contours for the CDDR parameters, with the best-fit values of η0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='952+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='019 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='019 and η1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='023+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='053 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='054, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' It is worth to comment that on the one hand, our methodology produces a possible deviation from the expected value of CDDR parameter (η0 = 1) up to z ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' However, our results are still marginally consistent with the CDDR validity within 2σ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', which is in full agreement with other recent tests involving cosmological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A summary of the cur- rent constraints on the η0 from different cosmological observables can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' On the other hand, the CDDR remains redshift independent (η1 = 0) within 1σ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', supporting the persisting claims that the Etherington reciprocity theorem could still be the best description of our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' There are many ways the above findings could be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' For instance, it is still interesting to see whether those conclusions may be changed with machine learning algorithms, which have shown their excellent potential in addressing cosmological issues and constraining cosmological parameters [64–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' More importantly, as a completely data driven approach, the Artificial Neural 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 5: The CDDR parameters η(z) and η(zi)/η(zj) calculated from the two statistical methods as weighted mean and the median statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Bands display the 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='3% confidence regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Network (ANN) method does not assume random variables that satisfy the Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The main purpose of an ANN (which consists of an input layer, one or more hidden layers and an output layer) is to construct an approximate function fW,b(x) (in which W and b are linear weights matrix and the offset vector) that correlates the input vector x with the output vector y [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' According to the difference between the predicted value fW,b(x) of the current network and the target value y, the weight matrix of each layer needs to be constantly updated for minimize the difference, which is defined by a loss function L [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' An issue that needs clarification is the achievable 1σ confidence region for the reconstructed function, which depends on both the actual errors and the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Following the detailed discussion in [70], a complete artificial neural network has the following parts: firstly, the weight is randomly initialized in the neural network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Secondly, the output value is compared with the expected output value, and the cost function is used to calculate the error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Thirdly, the error is propagated back to the neural network and the weight is set according to this information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Fourthly, repeat steps two to four for each input value 11 169M n6ibgM ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='0 8e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content="0 U(≤I)\\U(s) J'00 J'OS J'04169M nsibgM 0'4 a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content="0 U(sl)\\u(sl) T'O I'S J'4 J'e169M nsibgM a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content="0 J'O J'S 1'4- I'ein the training set;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Finally, when the entire training set is sent to the neural network, the entire training is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The recent analysis has demonstrated the effectiveness of ANN acting as “universal approximator” to produce representative uncertainties of the observations, especially in high-precision test of CDDR in both electromagnetic and gravitational wave domain [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In particular, Euclid collaboration improved the precision of CDDR test by approximately a factor of six, based on machine learning reconstruction using genetic algorithms [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Using the publicly released code called Reconstruct Functions with ANN [72], we perform the reconstruction of the parameter η(zi)/η(zj) based on the current η(zi)/η(zj) two-point diagnos- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The reconstructed functions with corresponding 1σ uncertainties, which can be considered as the average level of observational error are given in right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Working on the re- constructed 1000 η(zi)/η(zj) points with ANN, we obtain Mean(η(zi)/η(zj)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='998(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='003) and Med(η(zi)/η(zj)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='998(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='004) in the framework of weighted mean and median statis- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Therefore, with ANN algorithm one could expect the parameter η(zi)/η(zj) to be estimated at the precision of 10−3, which is more stringent than other results based on currently available observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' In order to facilitate comparison between the inferred values of CDDR pa- rameters obtained from two statistical approaches, we display the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' As a final remark, possible violations of such fundamental relation (cosmic distance duality relation) might have profound implications for the understanding of fundamental physics and natural laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Based on better uv-coverage in the future, we pin our hope on multi-frequency VLBI observations of more compact radio quasars with higher angular resolution, smaller statistical and systematic un- certainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Meanwhile, considering the variety of different machine learning algorithms, we may also be optimistic in detecting possible deviation from the CDDR with much higher precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China under Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 12203009, 12122504, 12021003, 11875025, 11633001, 11920101003, and 62202469;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' XDB23000000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Beijing Natural Science Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 4224091);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' the Interdiscipline Research Funds of Beijing Normal University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' and the China Manned Space Project (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' CMS- CSST-2021-B01 and CMS-CSST-2021-A01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Etherington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' On the Definition of Distance in General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Philosophical Magazine, 15(18):761, January 1933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 12 [2] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Etherington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Republication of: LX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' On the definition of distance in general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' General Relativity and Gravitation, 39(7):1055–1067, July 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [3] Adam G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Riess, Alexei V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Filippenko, Peter Challis, Alejandro Clocchiatti, Alan Diercks, Peter M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Garnavich, Ron L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Gilliland, Craig J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Hogan, Saurabh Jha, Robert P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kirshner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Leibundgut, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Phillips, David Reiss, Brian P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Schmidt, Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Schommer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Chris Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Spyromilio, Christo- pher Stubbs, Nicholas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Suntzeff, and John Tonry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 116(3):1009–1038, September 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Perlmutter, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Aldering, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Goldhaber, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Knop, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Nugent, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Castro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Deustua, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Fabbro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Goobar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Groom, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Hook, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lee, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Nunes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Pain, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Pennypacker, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Quimby, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lidman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Ellis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Irwin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' McMahon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Ruiz-Lapuente, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Walton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Schaefer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Boyle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Filippenko, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Matheson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Fruchter, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Panagia, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Newberg, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Couch, and The Supernova Cosmology Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Measurements of Ω and Λ from 42 High-Redshift Supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 517(2):565–586, June 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [5] Adam G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Riess, Louis-Gregory Strolger, John Tonry, Stefano Casertano, Henry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Ferguson, Bahram Mobasher, Peter Challis, Alexei V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Filippenko, Saurabh Jha, Weidong Li, Ryan Chornock, Robert P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kirshner, Bruno Leibundgut, Mark Dickinson, Mario Livio, Mauro Giavalisco, Charles C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Steidel, Txitxo Ben´ıtez, and Zlatan Tsvetanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Type Ia Supernova Discoveries at z ¿ 1 from the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 607(2):665–687, June 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Knop, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Aldering, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Amanullah, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Blanc, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Burns, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Conley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Deustua, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Doi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Ellis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Fabbro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Folatelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Fruchter, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Garavini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Garmond, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Garton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Gib- bons, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Goldhaber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Goobar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Groom, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Hardin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Hook, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Howell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kim, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lidman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Mendez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Nobili, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' E.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 598(1):102–137, November 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [7] Planck Collaboration, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Aghanim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Akrami, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Ashdown, J.' metadata={'source': 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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Shellard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Sirignano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Sirri, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Spencer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Sunyaev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Suur-Uski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Tauber, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Tavagnacco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Tenti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Toffolatti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Tomasi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Trom- betti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Valenziano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Valiviita, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Van Tent, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Vibert, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Vielva, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Villa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Vittorio, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Wandelt, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Wehus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' White, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' White, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Zacchei, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Zonca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Cosmo- logical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 641:A6, September 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [8] Shuo Cao and Zonghong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The distance duality relation and the temperature profile of galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Science China Physics, Mechanics, and Astronomy, 54(12):2260–2264, December 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [9] Shuo Cao, Marek Biesiada, Xiaogang Zheng, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Testing the gas mass density profile of galaxy clusters with distance duality relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 457(1):281–287, March 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [10] Tonghua Liu, Shuo Cao, Jia Zhang, Marek Biesiada, Yuting Liu, and Yujie Lian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Testing the cosmic curvature at high redshifts: the combination of LSST strong lensing systems and quasars as new standard candles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 496(1):708–717, July 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 14 [11] Jun-Qing Xia, Hai Yu, Guo-Jian Wang, Shu-Xun Tian, Zheng-Xiang Li, Shuo Cao, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Revisiting Studies of the Statistical Property of a Strong Gravitational Lens System and Model- Independent Constraint on the Curvature of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 834(1):75, January 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [12] Jing-Zhao Qi, Shuo Cao, Sixuan Zhang, Marek Biesiada, Yan Wu, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The distance sum rule from strong lensing systems and quasars - test of cosmic curvature and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 483(1):1104–1113, February 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [13] Puxun Wu, Zhengxiang Li, Xiaoliang Liu, and Hongwei Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Cosmic distance-duality relation test using type Ia supernovae and the baryon acoustic oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' D , 92(2):023520, July 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Holanda, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lima, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Ribeiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Testing the Distance-Duality Relation with Galaxy Clusters and Type Ia Supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 722(2):L233–L237, October 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [15] Shuo Cao and Nan Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Testing the distance-duality relation with a combination of cosmological distance observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Research in Astronomy and Astrophysics, 11(10):1199–1208, October 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [16] Kamal Bora and Shantanu Desai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A test of cosmic distance duality relation using SPT-SZ galaxy clusters, Type Ia supernovae, and cosmic chronometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' JCAP, 2021(6):052, June 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [17] Xiao-Lei Meng, Tong-Jie Zhang, Hu Zhan, and Xin Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Morphology of Galaxy Clusters: A Cosmo- logical Model-independent Test of the Cosmic Distance-Duality Relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 745(1):98, January 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [18] Cheng-Zong Ruan, Fulvio Melia, and Tong-Jie Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Model-independent Test of the Cosmic Dis- tance Duality Relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 866(1):31, October 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [19] Kai Liao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The Cosmic Distance Duality Relation with Strong Lensing and Gravitational Waves: An Opacity-free Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 885(1):70, November 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [20] Yu-Bo Ma, Shuo Cao, Jia Zhang, Jingzhao Qi, Tonghua Liu, Yuting Liu, and Shuaibo Geng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Testing Cosmic Opacity with the Combination of Strongly Lensed and Unlensed Supernova Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 887(2):163, December 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [21] Kai Liao, Marek Biesiada, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Strongly Lensed Transient Sources: A Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Chinese Physics Letters, 39(11):119801, November 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [22] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Ellis and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Stoeger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The ’fitting problem’ in cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Classical and Quantum Gravity, 4(6):1697–1729, November 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [23] Zhengxiang Li, Guo-Jian Wang, Kai Liao, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Model-independent Estimations for the Curvature from Standard Candles and Clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 833(2):240, December 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [24] Jun-Jie Wei and Xue-Feng Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' An Improved Method to Measure the Cosmic Curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 15 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 838(2):160, April 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [25] Rub´en Arjona, Hai-Nan Lin, Savvas Nesseris, and Li Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Machine learning forecasts of the cosmic distance duality relation with strongly lensed gravitational wave events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' D , 103(10):103513, May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [26] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Hogg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Martinelli, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Nesseris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Constraints on the distance duality relation with standard sirens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' JCAP, 2020(12):019, December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [27] Purba Mukherjee and Ankan Mukherjee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Assessment of the cosmic distance duality relation using Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 504(3):3938–3946, June 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [28] Ariadna Montiel, Ruth Lazkoz, Irene Sendra, Celia Escamilla-Rivera, and Vincenzo Salzano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Non- parametric reconstruction of the cosmic expansion with local regression smoothing and simulation extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' D , 89(4):043007, February 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [29] Jack A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Baldwin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Luminosity Indicators in the Spectra of Quasi-Stellar Objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 214:679–684, June 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [30] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Watson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Denney, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Vestergaard, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Davis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A New Cosmological Distance Measure Using Active Galactic Nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 740(2):L49, October 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [31] Jian-Min Wang, Pu Du, David Valls-Gabaud, Chen Hu, and Hagai Netzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Super-Eddington Accret- ing Massive Black Holes as Long-Lived Cosmological Standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 110(8):081301, February 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [32] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Risaliti and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lusso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Cosmological Constraints from the Hubble Diagram of Quasars at High Redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Nature Astronomy, 3:272–277, January 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [33] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Risaliti and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lusso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A Hubble Diagram for Quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 815(1):33, December 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [34] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Risaliti and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lusso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Cosmology with AGN: can we use quasars as standard candles?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astronomis- che Nachrichten, 338(329):329–333, March 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [35] Tonghua Liu, Shuo Cao, Marek Biesiada, Yuting Liu, Shuaibo Geng, and Yujie Lian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Testing the Cosmic Opacity at Higher Redshifts: Implication from Quasars with Available UV and X-Ray Obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 899(1):71, August 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [36] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kellermann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The cosmological deceleration parameter estimated from the angular-size/redshift relation for compact radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Nature (London) , 361(6408):134–136, January 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [37] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Gurvits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Apparent Milliarcsecond Sizes of Active Galactic Nuclei and the Geometry of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 425:442, April 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [38] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Vishwakarma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Consequences on variable Λ-models from distant type Ia supernovae and compact 16 radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Classical and Quantum Gravity, 18(7):1159–1172, April 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [39] Shuo Cao, Marek Biesiada, John Jackson, Xiaogang Zheng, Yuhang Zhao, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Measuring the speed of light with ultra-compact radio quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' JCAP, 2017(2):012, February 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [40] Shuo Cao, Xiaogang Zheng, Marek Biesiada, Jingzhao Qi, Yun Chen, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Ultra- compact structure in intermediate-luminosity radio quasars: building a sample of standard cosmologi- cal rulers and improving the dark energy constraints up to z 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 606:A15, September 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [41] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Preston, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Morabito, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Faulkner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Jauncey, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Nicolson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A VLBI survey at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='29 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 90:1599–1603, September 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Jackson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Tight cosmological constraints from the angular-size/redshift relation for ultra-compact radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' JCAP, 2004(11):007, November 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [43] Xiaolei Li, Shuo Cao, Xiaogang Zheng, Jingzhao Qi, Marek Biesiada, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Testing and selecting cosmological models with ultra-compact radio quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' European Physical Journal C, 77(8):677, August 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [44] Yubo Ma, Jia Zhang, Shuo Cao, Xiaogang Zheng, Tengpeng Xu, and Jingzhao Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The general- ized cosmic equation of state: a revised study with cosmological standard rulers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' European Physical Journal C, 77(12):891, December 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [45] Jing-Zhao Qi, Shuo Cao, Marek Biesiada, Xiaogang Zheng, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' New observational constraints on f( T) cosmology from radio quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' European Physical Journal C, 77(8):502, August 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [46] Xiaogang Zheng, Kai Liao, Marek Biesiada, Shuo Cao, Tong-Hua Liu, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Mul- tiple Measurements of Quasars Acting as Standard Probes: Exploring the Cosmic Distance Duality Relation at Higher Redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 892(2):103, April 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [47] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Scolnic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Jones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Rest, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Pan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Chornock, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Foley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Huber, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kessler, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Narayan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Riess, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Rodney, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Berger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Brout, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Challis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Drout, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Finkbeiner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lunnan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kirshner, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Sanders, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Schlafly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Smartt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Stubbs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Tonry, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Wood- Vasey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Foley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Hand, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Johnson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Burgett, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Chambers, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Draper, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Hodapp, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kaiser, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kudritzki, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Magnier, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Metcalfe, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Bresolin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Gall, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kotak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' McCrum, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Wood-Vasey, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Zheng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Burgett, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Chambers, L.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kudritzki, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Magnier, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Metcalfe, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Price, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Sweeney, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Wainscoat, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Waters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Cosmological Constraints from Measurements of Type Ia Supernovae Discovered during the First 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='5 yr of the Pan-STARRS1 Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 795(1):44, November 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [49] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Scolnic, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Rest, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Riess, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Huber, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Foley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Brout, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Chornock, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Narayan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' L.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Perlmutter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Prieto, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Pritchet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Richmond, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Riess, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Ruhlmann-Kleider, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Sako, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Schahmaneche, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Schneider, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Soller- man, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Sullivan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Walton, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Wheeler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 568:A22, August 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [51] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kessler and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Scolnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Correcting Type Ia Supernova Distances for Selection Biases and Contam- ination in Photometrically Identified Samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 836(1):56, February 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [52] Zhengxiang Li, Puxun Wu, and Hongwei Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Cosmological-model-independent Tests for the 18 Distance-Duality Relation from Galaxy Clusters and Type Ia Supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 729(1):L14, March 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [53] Kai Liao, Zhengxiang Li, Shuo Cao, Marek Biesiada, Xiaogang Zheng, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' The Distance Duality Relation from Strong Gravitational Lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 822(2):74, May 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [54] Tonghua Liu, Shuo Cao, Sixuan Zhang, Xiaolong Gong, Wuzheng Guo, and Chenfa Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Revisiting the cosmic distance duality relation with machine learning reconstruction methods: the combination of HII galaxies and ultra-compact radio quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' European Physical Journal C, 81(10):903, October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [55] Philip R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Bevington, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Keith Robinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Morris Blair, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' John Mallinckrodt, and Susan McKay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Data Reduction and Error Analysis for the Physical Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Computers in Physics, 7(4):415, January 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [56] Shuo Cao, Jingzhao Qi, Zhoujian Cao, Marek Biesiada, Jin Li, Yu Pan, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Direct test of the FLRW metric from strongly lensed gravitational wave observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Scientific Reports, 9:11608, August 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [57] Eric D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Feigelson and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Jogesh Babu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Modern Statistical Methods for Astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [58] III Gott, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Richard, Michael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Vogeley, Silviu Podariu, and Bharat Ratra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Median Statistics, H0, and the Accelerating Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 549(1):1–17, March 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [59] Arman Shafieloo, Varun Sahni, and Alexei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Starobinsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' New null diagnostic customized for re- constructing the properties of dark energy from baryon acoustic oscillations data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' D , 86(10):103527, November 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [60] Xiaogang Zheng, Xuheng Ding, Marek Biesiada, Shuo Cao, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' What Are the Omhˆ2 (z1, z2) and Om (z1, z2) Diagnostics Telling us in Light of H(z) Data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' , 825(1):17, July 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [61] Xiaogang Zheng, Marek Biesiada, Xuheng Ding, Shuo Cao, Sixuan Zhang, and Zong-Hong Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Statistical analysis with cosmic-expansion-rate measurements and two-point diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' European Physical Journal C, 78(10):274, July 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [62] Daniel Foreman-Mackey, David W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Hogg, Dustin Lang, and Jonathan Goodman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' emcee: The MCMC Hammer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Publications of the Astronomical Society of the Pacific, 125(925):306, March 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [63] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Holanda, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Lima, Akshay Rana, and Deepak Jain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Strong lensing systems and galaxy cluster observations as probe to the cosmic distance duality relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' European Physical Journal C, 82(2):115, February 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 19 [64] Janis Fluri, Tomasz Kacprzak, Alexandre Refregier, Adam Amara, Aurelien Lucchi, and Thomas Hofmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Cosmological constraints from noisy convergence maps through deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' D , 98(12):123518, December 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [65] Janis Fluri, Tomasz Kacprzak, Aurelien Lucchi, Alexandre Refregier, Adam Amara, Thomas Hof- mann, and Aurel Schneider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Cosmological constraints with deep learning from KiDS-450 weak lens- ing maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' D , 100(6):063514, September 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [66] Dezs˝o Ribli, B´alint ´Armin Pataki, Jos´e Manuel Zorrilla Matilla, Daniel Hsu, Zolt´an Haiman, and Istv´an Csabai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Weak lensing cosmology with convolutional neural networks on noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 490(2):1843–1860, December 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [67] Guo-Jian Wang, Si-Yao Li, and Jun-Qing Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' ECoPANN: A Framework for Estimating Cosmological Parameters Using Artificial Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 249(2):25, August 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [68] Djork-Arn´e Clevert, Thomas Unterthiner, and Sepp Hochreiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' arXiv e-prints, page arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='07289, November 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [69] Diederik P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Kingma and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Adam: A Method for Stochastic Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' arXiv e-prints, page arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='6980, December 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' [70] Guo-Jian Wang, Xiao-Jiao Ma, and Jun-Qing Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Machine learning the cosmic curvature in a model- independent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Soc.' metadata={'source': 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Weller, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Zacchei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Euclid: Forecast con- straints on the cosmic distance duality relation with complementary external probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' Astro- phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=', 644:A80, December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 20 [72] ReFANN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content='com/Guo-Jian-Wang/refann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfNAPK/content/2301.02997v1.pdf'} diff --git a/itFMT4oBgHgl3EQf4zH8/content/tmp_files/2301.12454v1.pdf.txt b/itFMT4oBgHgl3EQf4zH8/content/tmp_files/2301.12454v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0a2d9e45de0d6569a0d6fb5fb8cf1e73046a7c4a --- /dev/null +++ b/itFMT4oBgHgl3EQf4zH8/content/tmp_files/2301.12454v1.pdf.txt @@ -0,0 +1,760 @@ +LARGE-SCALE DATA MODELLING IN HIVE AND +DISTRIBUTED QUERY PROCESSING USING +MAPREDUCE AND TEZ +Abzetdin Adamov +Center for Data Analytics Research (CeDAR) +ADA University, Baku, Azerbaijan +aadamov@ada.edu.az + +Abstract +Huge amounts of data being generated continuously by digitally interconnected systems of +humans, organizations and machines. Data comes in variety of formats including structured, +unstructured and semi-structured, what makes it impossible to apply the same standard +approaches, techniques and algorithms to manage and process this data. Fortunately, the +enterprise level distributed platform named Hadoop Ecosystem exists. +This paper explores Apache Hive component that provides full stack data managements +functionality in terms of Data Definition, Data Manipulation and Data Processing. Hive is a data +warehouse system, which works with structured data stored in tables. Since, Hive works on top +the Hadoop HDSFS, it benefits from extraordinary feature of HDFS including Fault Tolerance, +Reliability, High Availability, Scalability, etc. In addition, Hive can take advantage of distributed +computing power of the cluster through assigning jobs to MapReduce, Tez and Spark engines to +run complex queries. The paper is focused on studying of Hive Data Model and analysis of +processing performance done by MapReduce and Tez. +Keywords +Data Model, Apache Hive, HDFS, MapReduce and Tez Distributed Computing. +INTRODUCTION +Hadoop is open source common platform that combines two main tasks of any operating +system: storing and processing data. Unlike to traditional systems, Hadoop accomplishes +that tasks towards Big Data. The popularity of Hadoop increases day by day, because of +simplicity, scalability and affordability that it enables thanks to its distributed +architecture. Although, the Hadoop Core consists of two main components (HDFS and +MapReduce) and has limited functionality, but thanks to many other components +available in the Hadoop Ecosystem under Apache Licence, this platform can cover any +requirements to manage and process data regardless to its size and format. +Data Analytics in scale that is enabled by Hadoop Ecosystem opens new horizons in +turning operational data of businesses into actionable knowledge and consequently into +value. In most cases, operational data is generated in structured format and stored in +RDBMS. These databases and warehouses are still important, but now in era of Big Data +businesses deal with amounts of data that can't feet into traditional RDBMS. Another + +challenge here is that in order to keep competitive advantage, businesses want to use for +analytics all available data including website clickstream data, text from call centers, +emails, instant messaging repositories, open data initiatives from public and private +entities. Its clear that this goal can't be achieved based on traditional RDBMS systems. In +contrast, Hadoop is a platform which consists of many components designed to +accomplish specific tasks using particular data format. Hadoop Ecosystem components +are classified into several categories to make it easier for user to choose appropriate +components in accordance to the functions they designed for. There are following +categories of the Hadoop components: +• Core Hadoop +• Governance, Integration + +• Data Access and Storage + +• Operations, Monitoring, Orchestration + +• Security + +• Data Intelligence +This paper explores Apache Hive component that provides full stack data +managements functionality in terms of Data Definition, Data Manipulation and Data +Processing. Hive is a data warehouse system, which works with structured data stored in +tables. Since, Hive works on top the Hadoop HDSFS, it benefits from extraordinary +feature of HDFS including Fault Tolerance, Reliability, High Availability, Scalability, +etc. (Zhou, W., Feng, D., Tan, Z., & Zheng, Y., 2017). In addition, Hive can take +advantage of distributed computing power of the cluster through assigning jobs to +MapReduce, Tez and Spark engines to run complex queries. The study is focussed on +studying of Hive Data Model and analysis of processing performance done by +MapReduce and Tez (Hortonworks Inc., 2017). +Hive is open-source software that is components of Hadoop Ecosystem. It designed to +query and analysis of huge amounts of data stored in HDFS using SQL-like language +HiveQL (Hive Query Language). Hive also can be considered as ETL and Data +warehousing tool for the large-scale data. +Hive can be also considered as alternative of the MapReduce with higher level of +abstraction. Since MapReduce applications are developed in Java or Python, its more +flexible, efficient and faster. It is designed to process structured data, so it suppose +creation of table with certain structure before loading data. To work with Hive there are +at least two options: Web GUI or more popular command line interface (CLI) using HQL +(for DDL, less for DML). Hive supports four file formats those are TEXTFILE, +SEQUENCEFILE, ORC and RCFILE (Record Columnar File). +Significant difference between DBMS and Hive is that: DBMS generally works on +"Schema on READ and Schema on Write", but Hive on "Schema on READ only" (latest +version on "WRITE Once READ Many Times"). + +LITERATURE REVIEW + +As the most popular platform for large-scale data management and analytics, Hadoop +ecosystem has attracted substantial interest from researchers. Hadoop and its multiple +components build unique system that allow to hide most of complexities staying focussed + +on real data analysis. This advantages inspires many researchers and us to to understand +key components of the ecosystem in depth. The following studies from authors are +devoted to different use-cases of the Hadoop Ecosystem. +Lee, Shao and Kang solved the problem of handling big graph that doesn't fit into memory +of traditional system. Authors offered to use distributed platform Hadoop HDFS to store +data physically and HBase for low latency data access (Lee H., Shao B., & Kang U., +2015). HBase is considered as an open source implementation of Google's Bigtable +technology. +Authors of following paper explain relation between cloud computing platforms widely +used by enterprises and Hadoop. Paper comprises the ways how enterprises can benefit +from Hadoop platform along with existing cloud computing systems. Author observe and +discuss primary sub-components of core Hadoop and how they related to each other +enabling execution and monitoring of jobs that process data stored on top of HDFS (Ghazi +M. R., & Gangodkar D., 2015). +Gadiraju, Verma, Davis and Talaga performed benchmark research comparing +performance of Apache Hive with traditional database management system MySQL. +Hive is a data warehouse platform that is member of Hadoop ecosystem and works on top +of distributed file system HDFS. Hive queries written in HiveQL (SQL-like language) +are executed as a MapReduse jobs using cumulative power of distributed Hadoop cluster +(Laboshin, L. U., Lukashin, A. A., & Zaborovsky, V. S., 2017). The authors also provide +evidence based on experiments that Hive loads the large datasets much faster than +MySQL, while it loses its advantage over MySQL when loading the smaller datasets +(Gadiraju K.K., Verma M., Davis K.C., & Talaga P.G., 2016). Same is true for query +execution as well: Hive is much faster when it comes to processing large amounts of data. +Using similar arguments, our paper paper states that the architecture of Hadoop and most +of its components is tuned to manage huge amounts of data, but not for random low- +latency data access to small chunks of data. + +HIVE AS A DISTRIBUTED ETL PLATFORM +Hive Arcitectire +Even Hive provides same DDL and DML services and acts like DBMS, but in reality it +is not a DBMS. Traditional DBMS is a software that encapsulates two main +subcomponents implemented internally: storage and query engine. Unlike DBMS, Hive +is just SQL Query Engine. Hive doesn't care about data storage, instead it relies on the +scalable and redundant HDFS. Furthermore, Hive doesn't process computation-intensive +queries itself, instead it consumes the MapReduce (or Tez and Spark) framework to use +the distributed power of the Hadoop cluster. General architecture and interaction between +it’s sub-components is demonstrated on Figure 1. + + + +Figure 1: General Architecture of Hive. + +Hive Data Model – Schema on Read +Unlike Database Systems, Hive enforces the Read schema rather than the Write schema. +Any DBMS is strictly checking the model of any data that pretended to be inserted into +database whether it follows to the predefined structure, and declines insertion if does not. +Opposite to this Hive does not check the model of new data, but instead just copies it into +HDFS without any control in order to improve writing speed. Hive checks the relevance +of the data and the structure just on read. +Look at the following example that demonstrate what can happen if the data uploaded +into table does not follow to the structure defined during the table creation (HiveQL script +on Figure 2.). + +Figure 2: Source code of HiveQL to create a table. + +Thrift +Application +Hive Thrift +Client +JDBC +Application +Hive JDBC +Driver +ODBC +Application +Hive ODBC +Driver +CLI +Metastore +Hive Server +FileSystem +Hive Web +Interface +Execution +Engine +Driver +Hadoop +Cluster +MapReduce +Tez +Spark +HDFS +Hive Clients +Hive Services +Hive Storage +and Computing +Metastore +Database +create table salaries (id INT, +rank STRING, +discipline STRING, +yrsphd INT, +yrsservice INT, +sex STRING, +salary DOUBLE) row format delimited fields terminated by ',' stored as textfile +tblproperties("skip.header.line.count"="1"); + + + +Figure 3: Source code of HiveQL to create a table. +As it has been clearly seen in Figure 3. the column “id” was declared as an integer. In this +particular example, even data associated with mentioned column was loaded into Hive, +but still the values are missing in output of the “select” query. The problem becomes clear +after screening the content of data source file. The values of column associated with +attribute “id” is surrounded by quotes, what means the type is character. The data type +conflict has not been revealed during data load, but comes clear on read. +Hive Metadata +The first implementation of Metadata in Hadoop Ecosystem has started with Hive that +used Metastore to store description of data model of Hive tables. Later it became clear +that other components of Hadoop need this technique as well. As a result, new +components, particularly HCatalog and WebHCat (REST API) appeared those enable +Hive metastore to other components of the Hadoop Ecosystem. +Hive storage consists of two categories: metadata and real data. Metadata of tables is +stored in "Meta storage database" (generally MySQL), while real data stored in HDFS. + +Data Model Components + +The Hive data models contain the following components: +• Databases +hive> describe salaries; +OK +id +int + +rank +string + +discipline +string + +yrsphd +int + +yrsservice +int + +sex +string + +salary +double + +Time taken: 0.814 seconds, Fetched: 7 row(s) + +hive> select * from salaries limit 4; +OK +NULL +"Prof" "B" +19 +18 +"Male" 139750.0 +NULL +"Prof" "B" +20 +16 +"Male" 173200.0 +NULL +"Prof" "B" +30 +23 +"Male" 175000.0 +NULL +"Prof" "B" +18 +18 +"Female" +129000.0 +Time taken: 1.043 seconds, Fetched: 4 row(s) +[hadoop@namenode ~]$ hdfs dfs -cat +/user/hive/warehouse/mydb.db/salaries/Salaries.csv | head -n 5 +"","rank","discipline","yrs.since.phd","yrs.service","sex","salary" +"1","Prof","B",19,18,"Male",139750 +"2","Prof","B",20,16,"Male",173200 +"3","Prof","B",30,23,"Male",175000 +"4","Prof","B",18,18,"Female",129000 + + +• Tables +• Partitions +• Denormalizing +• Buckets or clusters +Data partitioning is about splitting datasets into smaller pieces in order to avoid reading +huge volumes of data at once. The reason of doing this is to reduce the speed of read and +manipulation of any particular data. Unlike traditional DBMS systems, HDFS does not +support low latency transactions, instead it was designed to support the ingestion of huge +amounts of data at high speed. In terms of Hive, Data Partitioning enables breaking the +data into smaller subsets that allow to retrieve particular data enclosed into subset instead +of retrieving all data from the table. (Gwen Shapira, Jonathan Seidman, Ted Malaska, +Mark Grover, 2015). +To create a partitioned table in Hive, the certain instruction of HiveQL “PARTITIONED +BY” should be used while creating the table (as shown in Figure 4.). + +Figure 4: HiveQL code to Create and Manage Partitioned Table. + +After loading the data into partitioned table, the content of folder on HDFS associated +with the table may look similar to the structure demonstrated on Figure 5. + +Figure 5: The Structure of Partitioned Table’s Folder. + +HiveQL provides special command (SHOW PARTITIONS table;) to list all partitions +generated for the table. +Another technique that can increase data access and processing speed on expense of +storage usage effectiveness is Data Denormalization. Usually, tables in relational +databases follow to the requirements of the 3rd Normal Form (3NF), as well as 1st and +2nd. The 3NF states that if two tables are related between each other based on Primary +Key (example on Figure 6.), not any attribute of the master table can be included into the +second table except Primary Key itself. This approach helps to keep records smaller +saving a memory at the same time providing high consistency of data. +CREATE TABLE customer (id INT, name STRING, surname STRING) PARTITIONED BY +(city STRING); +LOAD DATA LOCAL INPATH '/data/customer.txt' INTO TABLE customer PARTITION +(city STRING); +/apps/hive/warehouse/customer +|---Baku/ +| |---file1 +| |---file2 +| |---file3 +| +|---Sheki/ +| |---file1 +| |---file2 +| + + + +Figure 6: Two Related Table those follow to 3rd Normal Form. + +When it comes to Hadoop, in reality each join-included query is accomplished by the +MapReduce operation that takes too much resources of cluster, especially if the query is +called frequently. The idea is to change the structure of table in advance that will eliminate +the need to join tables on run as it is shown in Figure 7. + +Figure 7: Same two table are Denormalized by joining in advance. + +DISTRIBUTED QUERY PROCESSING + +HiveServer2 (HS2) is essential component of Hive 2.x that in contrast with HiveServer1, +supports multiclient concurrency and authentication. Another important advantage of the +HiveServer2 is the fact that it provides JDBC and ODBC interface to interact with Hive. +Queries submitted to Hive are processed in the following way (look at Figure 8.): +1. Client send query to one of HiveServer2 instances connecting over JDBC/ODBC +interface; +2. Query is compiled and divided into sub-tasks by HS2; +3. Compiled query is submitted to Tez or MapReduce (depending on which +execution engine was set); +4. Coordinator (Tez/ApplicationMaster) asks YARN for allocation of the computing +resources (containers) across the cluster; +5. Tez/ApplicationMaster transfers tasks into containers; +6. Data that resides within HDFS in variety formats (text, ORC, AVRO, Parquet) is +read using HDFS interface; +7. Data is processed and result are returned over JDBC/ODBC interface. +FID | Name | Surname | City | Product | Qty | Date | +-----|-------|-----------|-------|----------|-----|------------| +1001 | Ali | Kerimov | Baku | HDD | 2 | 10.12.2017 | +1001 | Ali | Kerimov | Baku | Keyboard | 1 | 10.12.2017 | +1002 | Samir | Alasgarov | Sheki | CPU | 1 | 10.12.2017 | +1003 | Jemil | Mamadov | Baku | Printer | 1 | 10.12.2017 | + +ID | Name | Surname | City | +-----|-------|-----------|-------| +1001 | Ali | Kerimov | Baku | +1002 | Samir | Alasgarov | Sheki | +1003 | Jemil | Mamadov | Baku | + +FID | Product | Qty | Date | +-----|----------|-----|------------| +1001 | HDD | 2 | 10.12.2017 | +1001 | Keyboard | 1 | 10.12.2017 | +1002 | CPU | 1 | 10.12.2017 | +1003 | Printer | 1 | 10.12.2017 | + + +Tez is new high-performance batch processing framework for execution of complex Hive +queries that significantly outperforms traditional MapReduce framework (which is used +by Hive as a default execution engine). + +Figure 8: Query Execution Architecture of Hive. + +Hive queries are submitted to HiveServer2 server that generates Tez graph that in its turn +is transfered to YARN for processing. Each Hive query is monitored by their individual +Tez ApplicatiuonMaster. Number of simultaneous queries are limited with number of +allowed ApplicationMasters. +MapReduce and Tez have significant differences in computation models that effects their +performance. Looking to the architecture of computation model of MapReduce shown in +Figure 10., (while executing the query shown on Figure 9.) it obvious that following +elements increase cost and time of the MapReduce execution: +• To execute this query using MapReduce execution engine, Hive should launch 4 +MR jobs +• Generally, each MR job has its own start-up time and after processing writes result +to HDFS providing data to subsequent job for read. + +Figure 9: HiveQL code to retrieve data from three tables. + +SELECT a.occ_code, c.occ_name, COUNT(*) AS cnt, AVG(b.value) AS avg +FROM occup a +JOIN occupdata b ON (a.sid = b.sid) +JOIN jobs c ON (a.occ_code = c.occ_code) +GROUP BY a.occ_code, c.occ_name +ORDER BY avg DESC + +Hive Queries +HiveQL +HIVE +ETL Application +SPARK +LLAP +TEZ +MapReduce +Execution Engine +YARN +Resource Manager +Distributed File System +HDFS +Figure 10: Computation Model of MapReduce. + +At the same the following features of Tez make it’s computation model (look at Figure +11.) more efficient and consequently fast: +• In contrary to MapReduce, Tez performs complex query as a single execution +graph +• Tez doesn’t implement wasting intermediate IO operations with HDFS +• Vertexes in graph are processing jobs and edges are data streams +• Tez supports “hot containers” to start jobs immediately without wasting time for +start-up + + +Figure 11: Computation Model of Tez. + + +M +M +M +SELECT d.occname +SELECT a.occ code +M +M +R +R +R +R +JOIN (a, b) +JOIN(a,c) +GROUP BY a.occ code +COUNT（*) +AvG(b.value) +R +ORDER BYaVgM +M +M +SELEcT d.occ name +SELECT a.occ code +M +M +R +R +R +HDFS +M +M +HDFS +JOIN (a,b) +R +M +HDFS +JOIN (a,c) +M +GROUP BY a.occ code +COUNT（*） +AvG(b.value) +R +ORDERBYaVgLLAP is a new computation paradigm recently implemented in Hive. It consists of the set +of persistent daemons that execute fragments of Hive queries. This persistency allows to +start jobs much faster, since containers do not need warm-up. Query execution on LLAP +is very similar to Hive without LLAP, except that jobs run inside LLAP daemons, and +not within YARN containers. Both the Hive on Tez engine for batch queries and the +enhanced Hive on Tez LLAP-enhanced engine run on YARN nodes. The Hive LLAP +layer over Tez execution engine requires particular Hadoop YARN settings to consume +full potential of this new advancement in Hive (Hortonworks Community Connection, +2017). + +Computing Resources for Experiments + +In the framework of the Center for Data Analytics Research (CeDAR) the mid power +computing cluster has been launched (Figure 12.). + +Computing Cluster Hardware – the primary component of the CeDAR. This is +powerful, scalable and fault-tolerant computing cluster based on distributed architecture, +which operates totally on open-source software. Each computing node is equipped with +Intel Xeon E-5 processor, 96 GB memory, 8 LFF Hard Drives of 2 TB storage each and +1Gb Ethernet support. + +Characteristics of the cluster: +• Processing Cores: 102 +• RAM: 1,568 TB +• Storage: 136 TB + +Specifications of the cluster's components: + +• NameNode (1 unit): HP DL360 Gen9 4LFF CTO Server: 2 x Intel® Xeon® +E5-2603v4 (1.7GHz/6- core/15MB/85W), 128GB RAM, 4 x HP 2TB 12G SAS +7.2K rpm LFF HDD, HP 1U LFF Gen9 Mod Easy Install Rail Kit; +• DataNode (15 units): HP DL380 Gen9 12LFF CTO Server: Intel® Xeon® E5- +2603v4 (1.7GHz/6- core/15MB/85W), 96GB RAM, 8 x HP 2TB 12G SAS 7.2K +rpm LFF HDD, HP 2U LFF Easy Install Rail Kit; + + + +Figure 12: Distributed cluster at the CeDAR research center. + +Computing Cluster Software – the cluster is running on Apache Hadoop ecosystem. It +is deployed using Hortonworks HDP 2.6 distribution, which is 100% open source. + +EXPERIMENTAL RESULTS + +Many experiments have been accomplished on the CeDAR cluster to identify most +important parameters and criteria those have highest impact on the performance of the +query processing on Hive. +The following datasets listed in Table 1. publicly available at the Kaggle Datasets +repository (Kaggle Inc., 2018) were used to implement experiments. The table +“houses_part” with relatively large number of records was used to reveal the effect of the +file format on the performance. Other three tables “occup”, “occupdata” and “jobs” were +used in the complex HQL queries where 2-4 tables are merged using multiple join +instructions. +Table 1: Parameters of tables used in experiments. + +Name +Num. of Records +Num. of Fields +File size +1. +houses_part +22.489.349 +10 +2.4 GB +2. +occup +6.462.646 +15 +1.4 GB +3. +occupdata +6.462.646 +5 +417 MB +4. +jobs +1.090 +5 +0.05 MB + +Query Performance depending on Execution Engine +As it was stated above Tez outperforms MapReduce as a execution engine while +processing HiveQL queries. Computation architectures of both engines depicted on +Figures 10. and 11. displays key differences those affect the performance. As the Table +2. demonstrates, based on experimental query execution initiated with three different +queries it is clearly seen that Tez is faster. + +Table 2: Query performance dependence on Execution Engine. + +Query +Tez +(sec.) +MapReduce +(sec.) +1. select propertytype, count(*) from houses group by propertytype; +25 +34* +2. select PropertyType, sum(price) as sumprice, count(*) from houses +group by PropertyType; +28 +33* +3. set hive.input.format=org.apache.hadoop.hive.ql.io.HiveInputFormat; +set hive.merge.mapfiles=false; +select PropertyType, sum(price) as sumprice, count(*) from houses +group by PropertyType; +32 +154** + +* - with following settings +set hive.input.format=org.apache.hadoop.hive.ql.io.CombineHiveInputFormat; +set hive.merge.mapfiles=true; + +** - if execution engine is tuned using following settings, the execution time increases to +about 5 times (154 sec): +set hive.input.format=org.apache.hadoop.hive.ql.io.HiveInputFormat; +set hive.merge.mapfiles=false; + +Partitioning Effect on Query Performance + +As it was indicated above, Hive supports the partitioning of the data file by value of +specific column or several columns. This technique can significantly effect the query +performance. This impact can be explained by the fact that in the partitioned table the +query executor does not forced to read whole file from the HDFS, instead it reads just +particular partition(s) where the data of interest is located. +To create the partitioned table, the instruction PARTITIONED BY “columnName TYPE” +should be included into the table creation script (look at Figure 13.). To ingest the data +into the partitioned table, we need to load data firstly into non-partitioned intermediate +table, and after that insert into the partitioned table. + + +Figure 13: Storage Efficiency by Hive File Formats. + +CREATE TABLE houses_part ( +id STRING, price STRING, dateoftransfer STRING, oldNew STRING, duration +STRING, city STRING, district STRING, county STRING, ppd STRING, status +STRING) +PARTITIONED BY (propertytype STRING) +ROW FORMAT DELIMITED FIELDS TERMINATED BY ',' STORED AS TEXTFILE +TBLPROPERTIES ("skip.header.line.count"="1"); + +INSERT INTO TABLE houses_part +PARTITION (propertytype) +SELECT id, price, dateoftransfer, oldNew, duration, city, district, county, +ppd, status, propertytype +FROM houses; + +After ingesting the data into table, the exact number of data-files equal to the number of +unique values of column. These files (partitions) will appear in the folder associated with +partitioned table. In presented example (Figure 13.) there are five distinct values (“D”, +”F”, “O”, “S”, “T”) of the column and accordingly there are five files (the size of each +partition is on the left) in the directory. + +Figure 13: Computation Model of Tez. +The experimental results obtained after execution the same queries on the data that this +time stored in partitioned table reveals significant performance improvement, as shown +on Table 3., in comparison with results of unpartitioned table shown in Table 2. + +Table 3: Folder associated with partitioned table. + +Query +Tez +(sec.) +1. select PropertyType, count(*) as count from houses_part group by PropertyType; 5.5 +2. select PropertyType, sum(price) as sumprice, count(*) from houses_part group +by PropertyType; +13.2 + +File Format Effect on Query Performance +Hive as a many other components of the Hadoop Ecosystem is designed following to the +write-once concept, but not for low latency data access. By default, Hive that is +functioning on top of HDFS supports neither ACID nor OLTP transactions. Even so, low +latency data access can be enabled using particular file format to store the data and LLAP +(Low Latency Analytical Processing) daemons based on persistent query executors. +Hive supports four file formats those are TEXTFILE, SEQUENCEFILE, ORC and +RCFILE (Record Columnar File). Optimized Row Columnar (ORC) is a file format that +specially designed for storing the Hive data. ORC outperforms all other file formats +supported by Hive. The following chart in Figure 14. demonstrates the advantage of the +ORC file format in terms of storage efficiency. + +[root@nnode ~]# hdfs dfs -du /apps/hive/warehouse/bigdata.db/houses_part +543805800 /apps/hive/warehouse/bigdata.db/houses_part/propertytype=D +427515342 /apps/hive/warehouse/bigdata.db/houses_part/propertytype=F +10602972 /apps/hive/warehouse/bigdata.db/houses_part/propertytype=O +653235054 /apps/hive/warehouse/bigdata.db/houses_part/propertytype=S +725547877 /apps/hive/warehouse/bigdata.db/houses_part/propertytype=T + + +Figure 14: Storage Efficiency by Hive File Formats. + +Besides, ORC file format supports internal indexing that enables skipping large intervals +of rows those are out of interest. Default size of the ORC blocks is 256 MB what makes +sequential read highly effective and decrease the load on the NameNode (Blog by +Christian Prokopp, 2014). +In order to create the table that stores data in ORC format, it is enough to replace the +instruction “STORED AS TEXTFILE” to “STORED AS ORC” in HiveQL code shown +in Figure 13. +After loading the data into ORC-table, the following files will be created in the folder +associated with partitioned table. Simple comparison of total size of the folders associated +with two tables stored as TextFile and ORC, demonstrates that ORC file format is about +4 times more space-effective even without compression (look at Figure 15.). + +Figure 15: Folder associated with partitioned table stored as ORC file. +By being space-effective ORC file format has quite significant impact on overall +performance of the HiveQL execution. If same data occupies less space on the file system, +it means less time will be spent for HDFS read operations speeding up split and map jobs. +The Table 4. shows the time spent for query execution applied to the table stored as ORC +file. It is important to notice that the execution performance will increase in parallel with +growth of data size. +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +Text +RCF +ORC +RCF+Zlib +ORC+Zlib +Textual Data +Numerical Data +[root@nnode ~]# hdfs dfs -du /apps/hive/warehouse/bigdata.db/houses_part_orc +138576786 /apps/hive/warehouse/bigdata.db/houses_part_orc/propertytype=D +106081995 /apps/hive/warehouse/bigdata.db/houses_part_orc/propertytype=F +1016862 /apps/hive/warehouse/bigdata.db/houses_part_orc/propertytype=O +166501388 /apps/hive/warehouse/bigdata.db/houses_part_orc/propertytype=S +183961469 /apps/hive/warehouse/bigdata.db/houses_part_orc/propertytype=T + +[root@nnode ~]# hdfs dfs -du /apps/hive/warehouse/bigdata.db/ +2405685902 /apps/hive/warehouse/bigdata.db/houses +2360707045 /apps/hive/warehouse/bigdata.db/houses_part +596138500 /apps/hive/warehouse/bigdata.db/houses_part_orc + +Table 4: Query Performance on the ORC file format. + +Query +Tez +(sec.) +1. + +select PropertyType, count(*) as count from houses_part_orc group by +PropertyType; +4.8 +2. select PropertyType, sum(price) as sumprice, count(*) from houses_part_orc +group by PropertyType; +11.6 +CONCLUSION AND FUTURE WORK +The Hadoop Ecosystem becomes the de facto standard platform of choice for enterprises +that provides critical features like scalability, fault-tolerance, low TCO and high ROI +those are hardly available in traditional IT platforms. One of the most important +components of Hadoop Ecosystem Apaches Hive has been thoroughly observed, key +features those have highest impact on performance revealed and extensive experiments +conducted to demonstrate the truth of findings. +While observing the results it is important to keep in mind that Hive, as a many other +components of Hadoop Ecosystem running on top of HDFS, is not designed for low +latency random access to data. Real power of Hive can be seen while processing huge +chunks of structured data stored on HDFS. +Further research is needed to investigate the computation model of LLAP and its +advantage in terms of the query processing performance as compared to Tez without +LLAP. Even both Tez and Tez with LLAP are working on top on the YARN nodes, there +are some peculiarities implemented in the architecture of LLAP engine those make the +query processing about 25 times faster than performance offered by Hive without LLAP +(Nita Dembla, 2016). +ACKNOWLEDGEMENT +This research was supported by a grant from “Strengthening Teaching and Research +Capacity at ADA University” project funded by the European Union. For the research +experiments the distributed computing cluster of the Center for Data Analytics Research +(CeDAR) has been used. +REFERENCES +Lee, H., Shao, B., & Kang, U. (2015). Fast graph mining with HBase. Information Sciences, +315, 56–66. https://doi.org/10.1016/j.ins.2015.04.016 +Ghazi, M. R., & Gangodkar, D. (2015). Hadoop, MapReduce and HDFS: A Developers +Perspective. Procedia Computer Science, 48, 45–50. +https://doi.org/10.1016/j.procs.2015.04.108 +Gadiraju, K. K., Verma, M., Davis, K. C., & Talaga, P. G. (2016). Benchmarking performance +for migrating a relational application to a parallel implementation. Future Generation +Computer Systems, 63, 148–156. https://doi.org/10.1016/j.future.2015.12.015 + +Gwen Shapira, Jonathan Seidman, Ted Malaska, Mark Grover, 2015, Hadoop Application +Architectures, O'Reilly Media, Inc. +Blog by Christian Prokopp, 2014, ORC: An Intelligent Big Data file format for Hadoop and +Hive. [online] Available at [Accessed January 2018]. +Hortonworks Community Connection, 2017, Introduction: how does LLAP fit into Hive, +[online] Available at [Accessed Fevruary 2018]. +Hortonworks Inc., 2017. Apache Tez. [online] Available at + [Accessed January-March 2018]. +Laboshin, L. U., Lukashin, A. A., & Zaborovsky, V. S. (2017). The Big Data Approach to +Collecting and Analyzing Traffic Data in Large Scale Networks. Procedia Computer +Science, 103, 536–542. https://doi.org/10.1016/j.procs.2017.01.048 +Zhou, W., Feng, D., Tan, Z., & Zheng, Y. (2017). Improving big data storage performance in +hybrid environment. Journal of Computational Science. +https://doi.org/10.1016/j.jocs.2017.01.003 +Kaggle Inc., 2017, Publicly open datasets, [online] Available at + [Accessed December 2017 - March 2018]. +Nita Dembla, 2016, Announcing Apache Hive 2.1: 25X Faster Queries and Much More, [online] +Available at [Accessed March 2018] + diff --git a/itFMT4oBgHgl3EQf4zH8/content/tmp_files/load_file.txt b/itFMT4oBgHgl3EQf4zH8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..765836b1ac1cedf5d8fa066544bc246cb9accbf3 --- /dev/null +++ b/itFMT4oBgHgl3EQf4zH8/content/tmp_files/load_file.txt @@ -0,0 +1,485 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf,len=484 +page_content='LARGE-SCALE DATA MODELLING IN HIVE AND DISTRIBUTED QUERY PROCESSING USING MAPREDUCE AND TEZ Abzetdin Adamov Center for Data Analytics Research (CeDAR) ADA University, Baku, Azerbaijan aadamov@ada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='az Abstract Huge amounts of data being generated continuously by digitally interconnected systems of humans, organizations and machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Data comes in variety of formats including structured, unstructured and semi-structured, what makes it impossible to apply the same standard approaches, techniques and algorithms to manage and process this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Fortunately, the enterprise level distributed platform named Hadoop Ecosystem exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' This paper explores Apache Hive component that provides full stack data managements functionality in terms of Data Definition, Data Manipulation and Data Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive is a data warehouse system, which works with structured data stored in tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Since, Hive works on top the Hadoop HDSFS, it benefits from extraordinary feature of HDFS including Fault Tolerance, Reliability, High Availability, Scalability, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' In addition, Hive can take advantage of distributed computing power of the cluster through assigning jobs to MapReduce, Tez and Spark engines to run complex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The paper is focused on studying of Hive Data Model and analysis of processing performance done by MapReduce and Tez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Keywords Data Model, Apache Hive, HDFS, MapReduce and Tez Distributed Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' INTRODUCTION Hadoop is open source common platform that combines two main tasks of any operating system: storing and processing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Unlike to traditional systems, Hadoop accomplishes that tasks towards Big Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The popularity of Hadoop increases day by day, because of simplicity, scalability and affordability that it enables thanks to its distributed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Although, the Hadoop Core consists of two main components (HDFS and MapReduce) and has limited functionality, but thanks to many other components available in the Hadoop Ecosystem under Apache Licence, this platform can cover any requirements to manage and process data regardless to its size and format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Data Analytics in scale that is enabled by Hadoop Ecosystem opens new horizons in turning operational data of businesses into actionable knowledge and consequently into value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' In most cases, operational data is generated in structured format and stored in RDBMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=" These databases and warehouses are still important, but now in era of Big Data businesses deal with amounts of data that can't feet into traditional RDBMS." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Another challenge here is that in order to keep competitive advantage, businesses want to use for analytics all available data including website clickstream data, text from call centers, emails, instant messaging repositories, open data initiatives from public and private entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=" Its clear that this goal can't be achieved based on traditional RDBMS systems." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' In contrast, Hadoop is a platform which consists of many components designed to accomplish specific tasks using particular data format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hadoop Ecosystem components are classified into several categories to make it easier for user to choose appropriate components in accordance to the functions they designed for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' There are following categories of the Hadoop components: • Core Hadoop • Governance, Integration Data Access and Storage Operations, Monitoring, Orchestration Security Data Intelligence This paper explores Apache Hive component that provides full stack data managements functionality in terms of Data Definition, Data Manipulation and Data Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive is a data warehouse system, which works with structured data stored in tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Since, Hive works on top the Hadoop HDSFS, it benefits from extraordinary feature of HDFS including Fault Tolerance, Reliability, High Availability, Scalability, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' (Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', Feng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', Tan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', & Zheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' In addition, Hive can take advantage of distributed computing power of the cluster through assigning jobs to MapReduce, Tez and Spark engines to run complex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The study is focussed on studying of Hive Data Model and analysis of processing performance done by MapReduce and Tez (Hortonworks Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive is open source software that is components of Hadoop Ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' It designed to query and analysis of huge amounts of data stored in HDFS using SQL like language HiveQL (Hive Query Language).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive also can be considered as ETL and Data warehousing tool for the large scale data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive can be also considered as alternative of the MapReduce with higher level of abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Since MapReduce applications are developed in Java or Python, its more flexible, efficient and faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' It is designed to process structured data, so it suppose creation of table with certain structure before loading data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' To work with Hive there are at least two options: Web GUI or more popular command line interface (CLI) using HQL (for DDL, less for DML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive supports four file formats those are TEXTFILE, SEQUENCEFILE, ORC and RCFILE (Record Columnar File).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Significant difference between DBMS and Hive is that: DBMS generally works on "Schema on READ and Schema on Write", but Hive on "Schema on READ only" (latest version on "WRITE Once READ Many Times").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' LITERATURE REVIEW As the most popular platform for large-scale data management and analytics, Hadoop ecosystem has attracted substantial interest from researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hadoop and its multiple components build unique system that allow to hide most of complexities staying focussed on real data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' This advantages inspires many researchers and us to to understand key components of the ecosystem in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The following studies from authors are devoted to different use-cases of the Hadoop Ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=" Lee, Shao and Kang solved the problem of handling big graph that doesn't fit into memory of traditional system." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Authors offered to use distributed platform Hadoop HDFS to store data physically and HBase for low latency data access (Lee H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', Shao B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', & Kang U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=" HBase is considered as an open source implementation of Google's Bigtable technology." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Authors of following paper explain relation between cloud computing platforms widely used by enterprises and Hadoop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Paper comprises the ways how enterprises can benefit from Hadoop platform along with existing cloud computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Author observe and discuss primary sub-components of core Hadoop and how they related to each other enabling execution and monitoring of jobs that process data stored on top of HDFS (Ghazi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', & Gangodkar D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Gadiraju, Verma, Davis and Talaga performed benchmark research comparing performance of Apache Hive with traditional database management system MySQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive is a data warehouse platform that is member of Hadoop ecosystem and works on top of distributed file system HDFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive queries written in HiveQL (SQL-like language) are executed as a MapReduse jobs using cumulative power of distributed Hadoop cluster (Laboshin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', Lukashin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', & Zaborovsky, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The authors also provide evidence based on experiments that Hive loads the large datasets much faster than MySQL, while it loses its advantage over MySQL when loading the smaller datasets (Gadiraju K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', Verma M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', Davis K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', & Talaga P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Same is true for query execution as well: Hive is much faster when it comes to processing large amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Using similar arguments, our paper paper states that the architecture of Hadoop and most of its components is tuned to manage huge amounts of data, but not for random low- latency data access to small chunks of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' HIVE AS A DISTRIBUTED ETL PLATFORM Hive Arcitectire Even Hive provides same DDL and DML services and acts like DBMS, but in reality it is not a DBMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Traditional DBMS is a software that encapsulates two main subcomponents implemented internally: storage and query engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Unlike DBMS, Hive is just SQL Query Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=" Hive doesn't care about data storage, instead it relies on the scalable and redundant HDFS." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=" Furthermore, Hive doesn't process computation-intensive queries itself, instead it consumes the MapReduce (or Tez and Spark) framework to use the distributed power of the Hadoop cluster." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' General architecture and interaction between it’s sub-components is demonstrated on Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Figure 1: General Architecture of Hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive Data Model – Schema on Read Unlike Database Systems, Hive enforces the Read schema rather than the Write schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Any DBMS is strictly checking the model of any data that pretended to be inserted into database whether it follows to the predefined structure, and declines insertion if does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Opposite to this Hive does not check the model of new data, but instead just copies it into HDFS without any control in order to improve writing speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive checks the relevance of the data and the structure just on read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Look at the following example that demonstrate what can happen if the data uploaded into table does not follow to the structure defined during the table creation (HiveQL script on Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Figure 2: Source code of HiveQL to create a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Thrift Application Hive Thrift Client JDBC Application Hive JDBC Driver ODBC Application Hive ODBC Driver CLI Metastore Hive Server FileSystem Hive Web Interface Execution Engine Driver Hadoop Cluster MapReduce Tez Spark HDFS Hive Clients Hive Services Hive Storage and Computing Metastore Database create table salaries (id INT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' rank STRING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' discipline STRING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' yrsphd INT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' yrsservice INT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' sex STRING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=" salary DOUBLE) row format delimited fields terminated by '," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='\' stored as textfile tblproperties("skip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='header.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='count"="1");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Figure 3: Source code of HiveQL to create a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' As it has been clearly seen in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' the column “id” was declared as an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' In this particular example, even data associated with mentioned column was loaded into Hive, but still the values are missing in output of the “select” query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The problem becomes clear after screening the content of data source file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The values of column associated with attribute “id” is surrounded by quotes, what means the type is character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The data type conflict has not been revealed during data load, but comes clear on read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive Metadata The first implementation of Metadata in Hadoop Ecosystem has started with Hive that used Metastore to store description of data model of Hive tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Later it became clear that other components of Hadoop need this technique as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' As a result, new components, particularly HCatalog and WebHCat (REST API) appeared those enable Hive metastore to other components of the Hadoop Ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive storage consists of two categories: metadata and real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Metadata of tables is stored in "Meta storage database" (generally MySQL), while real data stored in HDFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Data Model Components The Hive data models contain the following components: • Databases hive> describe salaries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' OK id int rank string discipline string yrsphd int yrsservice int sex string salary double Time taken: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='814 seconds, Fetched: 7 row(s) hive> select * from salaries limit 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' OK NULL "Prof" "B" 19 18 "Male" 139750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='0 NULL "Prof" "B" 20 16 "Male" 173200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='0 NULL "Prof" "B" 30 23 "Male" 175000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='0 NULL "Prof" "B" 18 18 "Female" 129000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='0 Time taken: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='043 seconds, Fetched: 4 row(s) [hadoop@namenode ~]$ hdfs dfs -cat /user/hive/warehouse/mydb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/salaries/Salaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='csv | head -n 5 "","rank","discipline","yrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='since.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='phd","yrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='service","sex","salary" "1","Prof","B",19,18,"Male",139750 "2","Prof","B",20,16,"Male",173200 "3","Prof","B",30,23,"Male",175000 "4","Prof","B",18,18,"Female",129000 Tables Partitions Denormalizing Buckets or clusters Data partitioning is about splitting datasets into smaller pieces in order to avoid reading huge volumes of data at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The reason of doing this is to reduce the speed of read and manipulation of any particular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Unlike traditional DBMS systems, HDFS does not support low latency transactions, instead it was designed to support the ingestion of huge amounts of data at high speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' In terms of Hive, Data Partitioning enables breaking the data into smaller subsets that allow to retrieve particular data enclosed into subset instead of retrieving all data from the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' (Gwen Shapira, Jonathan Seidman, Ted Malaska, Mark Grover, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' To create a partitioned table in Hive, the certain instruction of HiveQL “PARTITIONED BY” should be used while creating the table (as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Figure 4: HiveQL code to Create and Manage Partitioned Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' After loading the data into partitioned table, the content of folder on HDFS associated with the table may look similar to the structure demonstrated on Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Figure 5: The Structure of Partitioned Table’s Folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' HiveQL provides special command (SHOW PARTITIONS table;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=') to list all partitions generated for the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Another technique that can increase data access and processing speed on expense of storage usage effectiveness is Data Denormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Usually, tables in relational databases follow to the requirements of the 3rd Normal Form (3NF), as well as 1st and 2nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The 3NF states that if two tables are related between each other based on Primary Key (example on Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' ), not any attribute of the master table can be included into the second table except Primary Key itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' This approach helps to keep records smaller saving a memory at the same time providing high consistency of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' CREATE TABLE customer (id INT, name STRING, surname STRING) PARTITIONED BY (city STRING);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=" LOAD DATA LOCAL INPATH '/data/customer." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content="txt' INTO TABLE customer PARTITION (city STRING);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' /apps/hive/warehouse/customer |---Baku/ | |---file1 | |---file2 | |---file3 | |---Sheki/ | |---file1 | |---file2 | Figure 6: Two Related Table those follow to 3rd Normal Form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' When it comes to Hadoop, in reality each join-included query is accomplished by the MapReduce operation that takes too much resources of cluster, especially if the query is called frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The idea is to change the structure of table in advance that will eliminate the need to join tables on run as it is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Figure 7: Same two table are Denormalized by joining in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' DISTRIBUTED QUERY PROCESSING HiveServer2 (HS2) is essential component of Hive 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='x that in contrast with HiveServer1, supports multiclient concurrency and authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Another important advantage of the HiveServer2 is the fact that it provides JDBC and ODBC interface to interact with Hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Queries submitted to Hive are processed in the following way (look at Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' ): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Client send query to one of HiveServer2 instances connecting over JDBC/ODBC interface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Query is compiled and divided into sub-tasks by HS2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Compiled query is submitted to Tez or MapReduce (depending on which execution engine was set);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Coordinator (Tez/ApplicationMaster) asks YARN for allocation of the computing resources (containers) across the cluster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Tez/ApplicationMaster transfers tasks into containers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Data that resides within HDFS in variety formats (text, ORC, AVRO, Parquet) is read using HDFS interface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Data is processed and result are returned over JDBC/ODBC interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' FID | Name | Surname | City | Product | Qty | Date | -----|-------|-----------|-------|----------|-----|------------| 1001 | Ali | Kerimov | Baku | HDD | 2 | 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2017 | 1001 | Ali | Kerimov | Baku | Keyboard | 1 | 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2017 | 1002 | Samir | Alasgarov | Sheki | CPU | 1 | 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2017 | 1003 | Jemil | Mamadov | Baku | Printer | 1 | 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2017 | ID | Name | Surname | City | -----|-------|-----------|-------| 1001 | Ali | Kerimov | Baku | 1002 | Samir | Alasgarov | Sheki | 1003 | Jemil | Mamadov | Baku | FID | Product | Qty | Date | -----|----------|-----|------------| 1001 | HDD | 2 | 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2017 | 1001 | Keyboard | 1 | 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2017 | 1002 | CPU | 1 | 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2017 | 1003 | Printer | 1 | 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2017 | Tez is new high-performance batch processing framework for execution of complex Hive queries that significantly outperforms traditional MapReduce framework (which is used by Hive as a default execution engine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Figure 8: Query Execution Architecture of Hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive queries are submitted to HiveServer2 server that generates Tez graph that in its turn is transfered to YARN for processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Each Hive query is monitored by their individual Tez ApplicatiuonMaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Number of simultaneous queries are limited with number of allowed ApplicationMasters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' MapReduce and Tez have significant differences in computation models that effects their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Looking to the architecture of computation model of MapReduce shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', (while executing the query shown on Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=') it obvious that following elements increase cost and time of the MapReduce execution: • To execute this query using MapReduce execution engine, Hive should launch 4 MR jobs • Generally, each MR job has its own start-up time and after processing writes result to HDFS providing data to subsequent job for read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Figure 9: HiveQL code to retrieve data from three tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' SELECT a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='occ_code, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='occ_name, COUNT(*) AS cnt, AVG(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='value) AS avg FROM occup a JOIN occupdata b ON (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='sid = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='sid) JOIN jobs c ON (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='occ_code = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='occ_code) GROUP BY a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='occ_code, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='occ_name ORDER BY avg DESC Hive Queries HiveQL HIVE ETL Application SPARK LLAP TEZ MapReduce Execution Engine YARN Resource Manager Distributed File System HDFS Figure 10: Computation Model of MapReduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' At the same the following features of Tez make it’s computation model (look at Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=') more efficient and consequently fast: • In contrary to MapReduce, Tez performs complex query as a single execution graph • Tez doesn’t implement wasting intermediate IO operations with HDFS • Vertexes in graph are processing jobs and edges are data streams • Tez supports “hot containers” to start jobs immediately without wasting time for start-up Figure 11: Computation Model of Tez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' M M M SELECT d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='occname SELECT a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='occ code M M R R R R JOIN (a, b) JOIN(a,c) GROUP BY a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='occ code COUNT（*) AvG(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='value) R ORDER BYaVgM M M SELEcT d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='occ name SELECT a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='occ code M M R R R HDFS M M HDFS JOIN (a,b) R M HDFS JOIN (a,c) M GROUP BY a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='occ code COUNT（*） AvG(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='value) R ORDERBYaVgLLAP is a new computation paradigm recently implemented in Hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' It consists of the set of persistent daemons that execute fragments of Hive queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' This persistency allows to start jobs much faster, since containers do not need warm-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Query execution on LLAP is very similar to Hive without LLAP, except that jobs run inside LLAP daemons, and not within YARN containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Both the Hive on Tez engine for batch queries and the enhanced Hive on Tez LLAP-enhanced engine run on YARN nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The Hive LLAP layer over Tez execution engine requires particular Hadoop YARN settings to consume full potential of this new advancement in Hive (Hortonworks Community Connection, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Computing Resources for Experiments In the framework of the Center for Data Analytics Research (CeDAR) the mid power computing cluster has been launched (Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Computing Cluster Hardware – the primary component of the CeDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' This is powerful, scalable and fault-tolerant computing cluster based on distributed architecture, which operates totally on open-source software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Each computing node is equipped with Intel Xeon E-5 processor, 96 GB memory, 8 LFF Hard Drives of 2 TB storage each and 1Gb Ethernet support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=" Characteristics of the cluster: Processing Cores: 102 RAM: 1,568 TB Storage: 136 TB Specifications of the cluster's components: NameNode (1 unit): HP DL360 Gen9 4LFF CTO Server: 2 x Intel® Xeon® E5 2603v4 (1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='7GHz/6 core/15MB/85W), 128GB RAM, 4 x HP 2TB 12G SAS 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2K rpm LFF HDD, HP 1U LFF Gen9 Mod Easy Install Rail Kit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' DataNode (15 units): HP DL380 Gen9 12LFF CTO Server: Intel® Xeon® E5 2603v4 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='7GHz/6 core/15MB/85W), 96GB RAM, 8 x HP 2TB 12G SAS 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2K rpm LFF HDD, HP 2U LFF Easy Install Rail Kit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Figure 12: Distributed cluster at the CeDAR research center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Computing Cluster Software – the cluster is running on Apache Hadoop ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' It is deployed using Hortonworks HDP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='6 distribution, which is 100% open source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' EXPERIMENTAL RESULTS Many experiments have been accomplished on the CeDAR cluster to identify most important parameters and criteria those have highest impact on the performance of the query processing on Hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The following datasets listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' publicly available at the Kaggle Datasets repository (Kaggle Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', 2018) were used to implement experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The table “houses_part” with relatively large number of records was used to reveal the effect of the file format on the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Other three tables “occup”, “occupdata” and “jobs” were used in the complex HQL queries where 2-4 tables are merged using multiple join instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Table 1: Parameters of tables used in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Name Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' of Records Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' of Fields File size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' houses_part 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='349 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='4 GB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' occup 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='646 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='4 GB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' occupdata 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='646 5 417 MB 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' jobs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='090 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='05 MB Query Performance depending on Execution Engine As it was stated above Tez outperforms MapReduce as a execution engine while processing HiveQL queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Computation architectures of both engines depicted on Figures 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' displays key differences those affect the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' As the Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' demonstrates, based on experimental query execution initiated with three different queries it is clearly seen that Tez is faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Table 2: Query performance dependence on Execution Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Query Tez (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=') MapReduce (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' select propertytype, count(*) from houses group by propertytype;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 25 34* 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' select PropertyType, sum(price) as sumprice, count(*) from houses group by PropertyType;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 28 33* 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' set hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='format=org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='apache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='hadoop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='ql.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='HiveInputFormat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' set hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='mapfiles=false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' select PropertyType, sum(price) as sumprice, count(*) from houses group by PropertyType;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 32 154** with following settings set hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='format=org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='apache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='hadoop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='ql.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='CombineHiveInputFormat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' set hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='mapfiles=true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' ** - if execution engine is tuned using following settings, the execution time increases to about 5 times (154 sec): set hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='format=org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='apache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='hadoop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='ql.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='HiveInputFormat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' set hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='mapfiles=false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Partitioning Effect on Query Performance As it was indicated above, Hive supports the partitioning of the data file by value of specific column or several columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' This technique can significantly effect the query performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' This impact can be explained by the fact that in the partitioned table the query executor does not forced to read whole file from the HDFS, instead it reads just particular partition(s) where the data of interest is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' To create the partitioned table, the instruction PARTITIONED BY “columnName TYPE” should be included into the table creation script (look at Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' To ingest the data into the partitioned table, we need to load data firstly into non-partitioned intermediate table, and after that insert into the partitioned table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Figure 13: Storage Efficiency by Hive File Formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' CREATE TABLE houses_part ( id STRING, price STRING, dateoftransfer STRING, oldNew STRING, duration STRING, city STRING, district STRING, county STRING, ppd STRING, status STRING) PARTITIONED BY (propertytype STRING) ROW FORMAT DELIMITED FIELDS TERMINATED BY \',\' STORED AS TEXTFILE TBLPROPERTIES ("skip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='header.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='count"="1");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' INSERT INTO TABLE houses_part PARTITION (propertytype) SELECT id, price, dateoftransfer, oldNew, duration, city, district, county, ppd, status, propertytype FROM houses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' After ingesting the data into table, the exact number of data-files equal to the number of unique values of column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' These files (partitions) will appear in the folder associated with partitioned table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' In presented example (Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=') there are five distinct values (“D”, ”F”, “O”, “S”, “T”) of the column and accordingly there are five files (the size of each partition is on the left) in the directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Figure 13: Computation Model of Tez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The experimental results obtained after execution the same queries on the data that this time stored in partitioned table reveals significant performance improvement, as shown on Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', in comparison with results of unpartitioned table shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Table 3: Folder associated with partitioned table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Query Tez (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' select PropertyType, count(*) as count from houses_part group by PropertyType;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' select PropertyType, sum(price) as sumprice, count(*) from houses_part group by PropertyType;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2 File Format Effect on Query Performance Hive as a many other components of the Hadoop Ecosystem is designed following to the write-once concept, but not for low latency data access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' By default, Hive that is functioning on top of HDFS supports neither ACID nor OLTP transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Even so, low latency data access can be enabled using particular file format to store the data and LLAP (Low Latency Analytical Processing) daemons based on persistent query executors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hive supports four file formats those are TEXTFILE, SEQUENCEFILE, ORC and RCFILE (Record Columnar File).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Optimized Row Columnar (ORC) is a file format that specially designed for storing the Hive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' ORC outperforms all other file formats supported by Hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The following chart in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' demonstrates the advantage of the ORC file format in terms of storage efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' [root@nnode ~]# hdfs dfs -du /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part 543805800 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part/propertytype=D 427515342 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part/propertytype=F 10602972 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part/propertytype=O 653235054 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part/propertytype=S 725547877 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part/propertytype=T Figure 14: Storage Efficiency by Hive File Formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Besides, ORC file format supports internal indexing that enables skipping large intervals of rows those are out of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Default size of the ORC blocks is 256 MB what makes sequential read highly effective and decrease the load on the NameNode (Blog by Christian Prokopp, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' In order to create the table that stores data in ORC format, it is enough to replace the instruction “STORED AS TEXTFILE” to “STORED AS ORC” in HiveQL code shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' After loading the data into ORC-table, the following files will be created in the folder associated with partitioned table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Simple comparison of total size of the folders associated with two tables stored as TextFile and ORC, demonstrates that ORC file format is about 4 times more space-effective even without compression (look at Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Figure 15: Folder associated with partitioned table stored as ORC file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' By being space-effective ORC file format has quite significant impact on overall performance of the HiveQL execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' If same data occupies less space on the file system, it means less time will be spent for HDFS read operations speeding up split and map jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' The Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' shows the time spent for query execution applied to the table stored as ORC file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' It is important to notice that the execution performance will increase in parallel with growth of data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2 Text RCF ORC RCF+Zlib ORC+Zlib Textual Data Numerical Data [root@nnode ~]# hdfs dfs -du /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part_orc 138576786 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part_orc/propertytype=D 106081995 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part_orc/propertytype=F 1016862 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part_orc/propertytype=O 166501388 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part_orc/propertytype=S 183961469 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part_orc/propertytype=T [root@nnode ~]# hdfs dfs du /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/ 2405685902 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses 2360707045 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part 596138500 /apps/hive/warehouse/bigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='db/houses_part_orc Table 4: Query Performance on the ORC file format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Query Tez (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' select PropertyType, count(*) as count from houses_part_orc group by PropertyType;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' select PropertyType, sum(price) as sumprice, count(*) from houses_part_orc group by PropertyType;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='6 CONCLUSION AND FUTURE WORK The Hadoop Ecosystem becomes the de facto standard platform of choice for enterprises that provides critical features like scalability, fault-tolerance, low TCO and high ROI those are hardly available in traditional IT platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' One of the most important components of Hadoop Ecosystem Apaches Hive has been thoroughly observed, key features those have highest impact on performance revealed and extensive experiments conducted to demonstrate the truth of findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' While observing the results it is important to keep in mind that Hive, as a many other components of Hadoop Ecosystem running on top of HDFS, is not designed for low latency random access to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Real power of Hive can be seen while processing huge chunks of structured data stored on HDFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Further research is needed to investigate the computation model of LLAP and its advantage in terms of the query processing performance as compared to Tez without LLAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Even both Tez and Tez with LLAP are working on top on the YARN nodes, there are some peculiarities implemented in the architecture of LLAP engine those make the query processing about 25 times faster than performance offered by Hive without LLAP (Nita Dembla, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' ACKNOWLEDGEMENT This research was supported by a grant from “Strengthening Teaching and Research Capacity at ADA University” project funded by the European Union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' For the research experiments the distributed computing cluster of the Center for Data Analytics Research (CeDAR) has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' REFERENCES Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', Shao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', & Kang, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Fast graph mining with HBase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Information Sciences, 315, 56–66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='016 Ghazi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', & Gangodkar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Hadoop, MapReduce and HDFS: A Developers Perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Procedia Computer Science, 48, 45–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='procs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='108 Gadiraju, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', Verma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', Davis, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=', & Talaga, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Benchmarking performance for migrating a relational application to a parallel implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Future Generation Computer Systems, 63, 148–156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content="015 Gwen Shapira, Jonathan Seidman, Ted Malaska, Mark Grover, 2015, Hadoop Application Architectures, O'Reilly Media, Inc." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' Blog by Christian Prokopp, 2014, ORC: An Intelligent Big Data file format for Hadoop and Hive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFMT4oBgHgl3EQf4zH8/content/2301.12454v1.pdf'} +page_content=' [online] Available at 0 is +known. In practice, however, σ would often need to be estimated from the data, and this +2 + +poses a particular challenge in the setting of this clustering problem. Without knowing the +true cluster structure of the data (since of course, this is exactly the question we are aiming +to test), it is difficult to obtain a reliable estimate of σ–indeed, we will see shortly that many +natural options lead to either substantial power loss or substantial loss of the Type I error +control. This motivates the need for the more general model that avoids the need to estimate +the true variance. In this work, we propose a method that avoids this obstacle, by allowing +for an unknown variance σ2 (or more generally, an unknown structured covariance matrix), +while guaranteeing Type I error control and maintaining high power. +The remainder of this paper is organized as follows. In Section 2, we review the selective +inference framework developed by Gao et al. [2022] for the setting where σ is known and +discuss motivations for allowing σ to be unknown. In Section 3, we present our new method +for performing inference on clustering in the setting of an unknown σ (with proofs deferred to +the Appendix). Empirical results are presented in Section 4 to demonstrate the performance +of the new method and compare against the existing framework. Finally, we conclude with +a discussion and some open questions in Section 5. +2 +Background: the known variance case +In this section, we will first give a brief overview of the selective inference method developed +in Gao et al. [2022]’s work, and discuss the challenges posed by unknown variance σ2. +2.1 +Gao et al. [2022]’s Method +Consider clusters Ck, Ck′, which are two disjoint subsets of [n]. If these clusters were chosen +ahead of time—that is, independently of the data—then it would be simple to test the null +hypothesis H0(Ck, Ck′) defined in (1)—specifically, we would naturally use the test statistic +1 +|Ck| +� +i∈Ck +Xi − +1 +|Ck′| +� +i∈Ck′ +Xi = X⊤v where v := 1Ck +|Ck| − 1Ck′ +|Ck′|. +Here X ∈ Rn×q is the matrix of observed data with i-th row Xi ∈ Rq, and where, for a subset +C ⊆ [n], 1C ∈ Rn represents the vector with ith entry equal to 1 for each i ∈ C and 0 for +i ̸∈ C. This test statistic follows a mean-zero normal distribution under the null hypothesis +H0(Ck, Ck′), and so its norm follows a rescaled χ distributed under the null, +∥X⊤v∥2 +H0(Ck,Ck′) +∼ +σ +� 1 +|Ck| + +1 +|Ck′| +�1/2 +· χq. +However, since the clusters were chosen in a data-dependent way, this distribution is not +the correct null distribution for ∥X⊤v∥2. To address this, we can rewrite X as +X = PvX + P⊥ +v X = ∥X⊤v∥2 +∥v∥2 +· +vv⊤X +∥vv⊤X∥F ++ P⊥ +v X, +which decomposes X into components lying in the span of v and its orthogonal complement, +with Pv = vv⊤ +∥v∥2 +2 denoting the projection matrix that projects to the span of v, and P⊥ +v = In − +3 + +vv⊤ +∥v∥2 +2 projecting to its orthogonal complement, and where ∥·∥2 denotes the Euclidean norm and +∥·∥F the Frobenius norm. Gao et al. [2022]’s insight into handling the data-dependent cluster +selection is to condition on the normalized matrix +vv⊤X +∥vv⊤X∥F and the orthogonal projection +P⊥ +v X, so that only the test statistic ∥X⊤v∥2 remains unknown, and moreover to condition +on the range of values of ∥X⊤v∥2 that agree with the clustering selection. +Specifically, +defining +x(φ) = +φ +∥v∥2 +· +vv⊤X +∥vv⊤X∥F ++ P⊥ +v X, +(3) +let +S = {φ > 0 : Cluster(X) = Cluster(x(φ))} , +where Cluster(·) refers to the outcome of the clustering procedure. In other words, S contains +all values of φ for which the same clustering outcome would have been obtained, if we plug +in φ in place of the observed test statistic value ∥X⊤v∥2. +Their main result establishes +that, even given the data-dependent clustering procedure, the re-scaled χ distribution is the +correct null distribution once truncated to this set S. +Theorem 1 (Gao et al. [2022, Theorem 1]). Let Xi +⊥⊥∼ N(µi, σ2Iq) where σ is known, and let +v be defined as above. Then, conditional on Cluster(X), +vv⊤X +∥vv⊤X∥F , and P⊥ +v X, under the null +hypothesis H0(Ck, Ck′) the test statistic ∥X⊤v∥2 follows a truncated rescaled χ distribution, +σ +� +1 +|Ck| + +1 +|Ck′| +�1/2 +· χq truncated to S. In particular, the p-value +P = 1 − Fχq +� +∥X⊤v∥2; σ +� 1 +|Ck| + +1 +|Ck′| +�1/2 +, S +� +is uniformly distributed under H0(Ck, Ck′), where Fχq(·; c, S) is the CDF of a c · χq random +variable truncated to the set S. +Gao et al. [2022] provide an algorithm for exactly computing the set S for the hierarchical +clustering algorithm with linkages for which the exact computation of this set is tractable, +along with an implementation of the importance sampling algorithm for clustering algorithms +where this set cannot be efficiently computed. +2.2 +Challenges in estimating σ +We next discuss motivations for allowing σ to be unknown. Continuing the discussion earlier +on the difficulty of estimating σ from the data, we consider a simple scenario where we are +aiming to determine whether data points X1, . . . , Xn arise from a single cluster or from two +clusters. To test this, we would choose a data-dependent clustering [n] = C1 ∪C2, and would +now need to estimate σ in order to run Gao et al. [2022]’s test. +• Suppose we estimate the variance by using the within-cluster means, for instance, +ˆσ2 +clustered = +� +i∈C1 ∥Xi − ¯XC1∥2 +2 + � +i∈C2 ∥Xi − ¯XC2∥2 +2 +(n − 2)q +, +4 + +Figure 1: The top row shows results under the null, and the bottom row shows results under the +alternative. In each row, the left plot shows one draw of the data, along with the estimated values +ˆσall and ˆσclustered, while the middle and right plots show results for Gao et al. [2022]’s method +applied with ˆσclustered or with ˆσall, respectively. (See Section 2.2 for discussion.) +where ¯XCk is the sample mean in cluster Ck. With this choice, we might substantially +underestimate the variance if the true data distribution only has a single cluster. The +middle column of Figure 1 demonstrates this problem in practice—we can see that, +when the null H0(C1, C2) is true, the variance may sometimes be vastly underestimated +and, as a result, the empirical distribution of the p-value is far from uniform, which +would lead to false positives. +• Alternatively, we might take a more conservative estimate of variance by treating the +data as a single cluster, e.g., +ˆσ2 +all = +� +i∈[n] ∥Xi − ¯X[n]∥2 +2 +(n − 1)q +. +Indeed, this is the estimator proposed in Gao et al. [2022, Section S3], and they prove +theoretically that, as this is asymptotically an over-estimate of σ2, Type I error control +is guaranteed. However, this choice can lead to a substantially over-conservative test, +as demonstrated in the right column of Figure 1—if the true data distribution arises +from two clusters, this estimate can massively over-estimate σ2 leading to a large loss +of power. +(See Section 4 for details on these simulations.) +Thus, in Figure 1, we clearly see a tradeoff between Type I error control and power. When +using the cluster-wise estimate ˆσclustered, we see that power is high under the alternative, with +the empirical power being as high as the case where the true σ is used, but Type I error +control is lost under the null. On the other hand, when using the estimate ˆσall that treats +5 + +1.00 +1.00 +Quantiles +Quantil +0.75 +0.75 +■ +0.50 +0.50 +0 +irical +irical +0 +0.25 +00.25 +Emr +Emp +-1 +0.00 +0.00 +0.75 +-2 +-1 +0 +0.00 +0.25 +0.50 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +1 +Theoretical Quantiles +Theoretical Quantiles +Empirical Power with Oall +1.00 +1.00 +Alternative +0.75 +Pow +0.75 +0 +. +0.00 +0.00 +. +3 +6 +2 +0 +0 +6 +0 +2 +4 +6 +4 +- Oracle (Gao et al.'s method with true o) +- Oracle (Gao et al's method with true o)the entire dataset as a single cluster, we see that it controls Type I error under the null but +incurs a loss in power under the alternative. +To avoid this tradeoff, in this work we propose a selective inference procedure for the +clustering problem that can handle an unknown variance σ2 > 0. This more general model +resolves the issue—the p-value distribution is uniform when the data is generated from a +single cluster, without sacrificing too much power in the scenario where the data is instead +generated from distinct clusters. +3 +Proposed method: the unknown variance case +We now introduce our proposed method for the setting where the variance is unknown. In +this new setting, we assume that the data is distributed as Xi +⊥⊥∼ N(µi, σ2Iq), where the +means µi ∈ Rq, as well as the (shared) variance σ2 > 0, are unknown. +Recall the null hypothesis +H0(Ck, Ck′) : +1 +|Ck| +� +i∈Ck +µi = +1 +|Ck′| +� +i∈Ck′ +µi +in Gao et al. [2022] and the corresponding test statistic ∥X⊤v∥2. +Unfortunately, in our +new setting where σ is unknown, the distribution of this test statistic cannot be computed. +However, we will see that we can overcome this obstacle if we restrict to a stronger null +hypothesis, +H′ +0(Ck, Ck′) : µi = µi′ ∀ i, i′ ∈ Ck ∪ Ck′. +(4) +In other words, H′ +0 assumes that each data point in clusters Ck and Ck′ has the same mean, +while H0 makes the weaker assumption that the sample mean of data points in cluster Ck +and in cluster Ck′ have the same mean. We can equivalently rewrite H′ +0(Ck, Ck′) as +H′ +0(Ck, Ck′) : +� +�In − ww⊤ +∥w∥2 − +� +i∈[n]\(Ck∪Ck′) +eie⊤ +i +� +� µ = 0 where w := +1Ck∪Ck′ +|Ck ∪ Ck′|. +(5) +3.1 +Decomposition of X +To define our test statistic, we begin by taking a decomposition of the observed data X. This +decomposition plays an analogous role to the decomposition (3) used by Gao et al. [2022], +but is more complex to allow us to handle unknown variance. We begin by writing +X = P0X + P1X + P2X, +where P0 = +vv⊤ +∥v∥2 +2 is the rank-one projection matrix that captures the difference in cluster +means for Ck and Ck′, while +P1 = +� +ICk − 1Ck1⊤ +Ck +|Ck| +� ++ +� +ICk′ − +1Ck′1⊤ +Ck′ +|Ck′| +� +, +6 + +where, for any subset C ⊆ [n], IC represents the diagonal matrix with entry (i, i) set to 1 if +i ∈ C and 0 if i ̸∈ C. Finally, +P2 = In − P0 − P1 +is the projection matrix to the orthogonal complement of P0 and P1. We can see that P0, +P1, and P2 project to subspaces of dimension 1, m − 2, and n − m + 1, respectively, where +m = |Ck| + |Ck′| is the number of data points in the two clusters. Intuitively, we can think +of this decomposition of the data as follows: +• P0X captures the difference in means between clusters Ck and Ck′; +• P1X captures differences among points within Ck, and among points within Ck′; +• P2X captures all other aspects of the data (i.e., the overall mean of the combined +clusters Ck ∪ Ck′, as well as information about data points not lying in Ck ∪ Ck′). +Figure 2 illustrates the roles of these three terms in the decomposition of the data X. +Figure 2: Left: visualization of a dataset with colors indicating the clusters formed by the clustering +algorithm with Ck represented in blue and Ck′ in red. The blue triangle represents the combined +mean of Ck ∪Ck′, and the black dots represent the cluster means. Middle: the original dataset with +∥P0X∥F scaled by a factor of 2, which pushes apart the clusters Ck and Ck′. Right: the original +dataset with ∥P1X∥F scaled by a factor of 2, which spreads points in Ck apart from each other +while preserving the cluster mean, and same for points in Ck′. +3.2 +The test statistic +In Gao et al. [2022]’s work, the test statistic they use is equivalent to ∥P0X∥F, which under +the null hypothesis H0, follows a χq distribution (rescaled by σ), truncated to a region S +that controls for the selection event. In our work, since σ is unknown, we will use a Beta +distribution in place of the χ. The test statistic we propose is given by the ratio +R = +∥P0X∥2 +F +∥P0X∥2 +F + ∥P1X∥2 +F +. +where the numerator is the same as the statistic used by Gao et al. [2022] (up to a transfor- +mation), while the denominator acts by rescaling with respect to an estimate of the noise. +7 + +2PoX+ P,X+ P2X +PoX+ PX+ P2X +PoX+2PX+ P2X +15 +15 +15 +10 +10 +10 +5 +5 +5 +0 +0 +0 +0 +10 +15 +-5 +0 +5 +10 +15 +-5 +5 +0 +5 +10 +15 +-5We next need to define the truncation set. First, we rewrite our decomposition as +X = ∥P0X∥F · +P0X +∥P0X∥F ++ ∥P1X∥F · +P1X +∥P1X∥F ++ P2X. +(6) +Our test will condition on: +• The total squared norm ∥P0X∥2 +F + ∥P1X∥2 +F for the first and second terms in the +decomposition; +• The normalized terms +P0X +∥P0X∥F and +P1X +∥P1X∥F for the first and second terms in the decom- +position; +• The third term P2X in the decomposition. +With these terms treated as known, the data X can then be fully determined by revealing +the value R = +∥P0X∥2 +F +∥P0X∥2 +F +∥P1X∥2 +F of the test statistic. For any r ∈ (0, 1), define +x′(r) = +�√r · +P0X +∥P0X∥F ++ +√ +1 − r · +P1X +∥P1X∥F +� +· +� +∥P0X∥2 +F + ∥P1X∥2 +F + P2X. +We can verify from the definition of R that X = x′(R) holds by definition. Finally, define +S′ = {r ∈ (0, 1) : Cluster(X) = Cluster(x′(r))} ⊆ (0, 1). +3.3 +Main result +Our main result, presented next, establishes that we can compute the exact post-selection +distribution of R, which thus allows us to perform valid selective inference in the unknown- +variance setting. +Theorem 2. Let Xi +⊥⊥∼ N(µi, σ2Iq) where σ is unknown, and let P0, P1, and P2 be defined +as above. Then, conditional on Cluster(X), ∥P0X∥2 +F + ∥P1X∥2 +F, +P0X +∥P0X∥F , +P1X +∥P1X∥F , and P2X, +under the null hypothesis H′ +0 (Ck, Ck′), the random variable R follows the Beta(q/2, (m − +2)q/2) distribution truncated to the set S′. In particular, the p-value +P ′ = 1 − FBeta(q/2,(m−2)q/2) (R; S′) +is uniformly distributed under H′ +0 (Ck, Ck′), where FBeta(q/2,(m−2)q/2)(·; S′) is the CDF of a +Beta(q/2, (m − 2)q/2) random variable truncated to the set S′. +The intuition is that, if P0 and P1 were fixed rather than data-dependent (i.e., if the clus- +ters Ck and Ck′ were chosen before viewing the data), then we would have ∥P0X∥2 +F ∼ χ2 +q and, +independently, ∥P1X∥2 +F ∼ χ2 +(m−2)q; thus R = +∥P0X∥2 +F +∥P0X∥2 +F +∥P1X∥2 +F would follow a Beta(q/2, (m − +2)q/2) distribution. After accounting for the selection event, the null distribution is instead +given by a truncated Beta distribution. +To implement the results of Theorem 2 in practice, we need to be able to compute this +p-value. In other words, we need to either explicitly characterize the set S′ that is consistent +with the selection event, or develop an empirical sampling strategy to estimate the p-value. +We next consider this computational question. +8 + +3.4 +Computing the p-value +To characterize the truncation set S′, we will split into two cases. In the general setting, +when the data is separated into an arbitrary number K ≥ 2 of clusters, we will handle the +truncation event via numerical approximation. For the special case K = 2, however, we will +show that S′ can potentially be computed explicitly, by relating the problem back to the +work of Gao et al. [2022] for the known-variance case. +3.4.1 +Special case: K = 2 +Rewriting Gao et al. [2022]’s procedure in our notation, the modified data is defined as +x(φ) = +φ +∥v∥2 +· +P0X +∥P0X∥F ++ P1X + P2X, +(7) +where v = +1Ck +|Ck| − +1Ck′ +|Ck′| so that P0 = vv⊤ +∥v∥2 +2 is projection onto the span of v, and their selection +set is given by S = {φ > 0 : Cluster(X) = Cluster(x(φ))}. In contrast, our method defines +x′(r) = +�√r · +P0X +∥P0X∥F ++ +√ +1 − r · +P1X +∥P1X∥F +� +· +� +∥P0X∥2 +F + ∥P1X∥2 +F + P2X +and S′ = {r ∈ (0, 1) : Cluster(X) = Cluster(x′(r))}. +The next result shows that, for the case K = 2, these two definitions can be related in a +simple way. +Proposition 1. Suppose that K = 2 (so that we have k = 1, k′ = 2). Assume that the +clustering procedure is location- and scale-invariant—that is, for any x ∈ Rn×q, +Cluster(x) = Cluster +� +a · x + 1n · b⊤� +, +for any a ∈ R and b ∈ Rq. Then, under the notation and definitions above, +S′ = +� +r ∈ (0, 1) : +� +r +1 − r · ∥P1X∥F · +� +1 +|C1| + +1 +|C2| ∈ S +� +. +The work of Gao et al. [2022] gives an explicit characterization of the set S for a family +of agglomerative hierarchical clustering algorithms, while Chen and Witten [2022] does the +same for k-means clustering. Moreover, both of these algorithms are location- and scale- +invariant. Consequently, for the case K = 2, we can explicitly characterize the truncation +set S′, and thus can compute the p-value P ′ constructed in Theorem 2 by leveraging Gao +et al. [2022]’s construction of the set S, for these two popular algorithms. +Of course, a major limitation of this result is that we can only handle K = 2. However, +in iterative procedures (e.g., hierarchical clustering), the test can be applied at the first step +(i.e., the first split, from a single cluster to K = 2 clusters). This hypothesis test can then +essentially be interpreted as testing the global null, i.e., whether the data should be split +into clusters at all or simply treated as a single cluster. +9 + +3.4.2 +General case: K ≥ 2 +Next we consider the general case. If K > 2, then the result of Proposition 1 no longer +applies—as we will see in the proof of this proposition, the case K = 2 leads to the result +specifically because P2 = 1n1⊤ +n +n +in that case, but this no longer holds for K > 2. Moreover, +even if K = 2, it might be the case that we are using a clustering algorithm for which S +does not have an explicit characterization and/or which is not location- and scale-invariant. +In any of these settings, we do not have an explicit characterization of S′, and thus the +truncated CDF needed in Theorem 2 cannot be computed exactly. +Nonetheless, the inference procedure can still be run in the general case, by computing +P ′ approximately through importance sampling. Specifically, in order to (approximately) +compute the p-value P ′, we need to be able to estimate +1 − FBeta(q/2,(m−2)q/2) (r; S′) = PR′∼Beta(q/2,(m−2)q/2) {R′ > r | R′ ∈ S′} += PR′∼Beta(q/2,(m−2)q/2) {R′ > r, R′ ∈ S′} +PR′∼Beta(q/2,(m−2)q/2) {R′ ∈ S′} +, +and then evaluate this probability at r = R. Since r = R may be extremely large, and/or +the selection set S′ may be very small, the events R′ > r and/or R′ ∈ S′ may have extremely +low probability under the distribution R′ ∼ Beta(q/2, (m − 2)q/2). We approximate this +probability with importance sampling, using a truncated normal distribution as the proposal +distribution. Further details are given in Appendix A.3.2. +4 +Empirical results +We now provide empirical results for our proposed method. We present simulation results +for Type I error control and empirical power for our proposed method, as well as results +on a small real dataset (the penguin dataset provided by [Horst et al., 2020], which was +also analyzed in Gao et al. [2022]). For all experiments, we use the hierarchical clustering +algorithm with average linkage for clustering.1 +4.1 +Type I error control +Theorem 2 states that P ′ follows the uniform distribution under the null, so it controls the +Type 1 error rate of the hypothesis test. We illustrate this empirically by plotting empirical +quantiles of a sample of P ′ against the theoretical quantiles of the uniform distribution. +We compare to the p-value P computed by Gao et al. [2022]’s method, with either oracle +knowledge of the true σ, or with plug-in estimates ˆσall or ˆσclustered. (The results for the latter +two variants of Gao et al. [2022]’s method were also shown in the top row of Figure 1 in +Section 2.2.) +We perform 2000 independent trials, with datasets generated as +Xi +⊥⊥∼ N(µi, σ2Iq) +1Code for reproducing all experiments is available online at https://github.com/yjyun97/cluster_inf_ +unknown_var. +10 + +for i ∈ [n], with µi = 0q for all i ∈ [n] so that the null hypothesis is true. We set σ = 1, +n = 30, and q = 2. Figure 3 presents results for two different settings, K = 2 and K = 3. We +use the exact computation method presented in Section 3.4.1 for K = 2, and the importance +sampling method discussed in 3.4.2 for K = 3. +For the K = 2 case, the p-values are +generated for the comparison between clusters k = 1 and k′ = 2. For the K = 3 case, in each +trial, the p-values are generated for the comparison between two randomly chosen clusters +k ̸= k′ ∈ {1, 2, 3}. +Figure 3 illustrates that the proposed method, along with Gao et al. [2022]’s method +with either the conservative estimate ˆσall or with oracle knowledge of the true σ, all result +in uniformly distributed p-values (and thus, control the Type I error rate) for both settings. +On the other hand, Gao et al. [2022]’s method applied with ˆσclustered fails to do so, with +a substantially anticonservative distribution of the p-values. +As expected, since ˆσclustered +is a more extreme underestimate for larger K, the nonuniformity of the p-values is more +substantial when K = 3 than when K = 2. +Figure 3: Simulation under the null (see Section 4.1 for details). The left plots show one draw of +the data, along with the result of hierarchical clustering for K = 2 (top) K = 3 (bottom). The +right plots show QQ plots comparing the p-values obtained via the four different methods. +4.2 +Empirical power +It is inevitable that not knowing σ will cause some loss in power, as we are forced to condition +on more components of X than we would otherwise do when computing the p-value. On +the other hand, using ˆσall in place of σ in P also causes a loss in power, due to ˆσall being an +overestimate of σ in settings where the alternative hypothesis is true. Here, we compute the +empirical power of the same four methods as above. +We perform 500 independent trials, with datasets generated as +Xi +⊥⊥∼ N(µi, σ2Iq) +11 + +Visualization for K = 2 +QQ Plot for K = 2 +1.00 +2 +0. +C + 0.75 +1 +uar +0.50 +0 +irical +00.25 +-1 +0.00 +-21 +0.00 +0.25 +0.50 +0.75 +-2 +-1 +0 +2 +1 +Theoretical Quantiles +Visualization for K - 3 +QQ Plot for K = 3 +2 +. Oracle (Gao et al.'s method with true o) +1.00 +les +1 +0 +0.50 +irical +00.25 +-1 +0.00 +-2 +2 +-2 +0 +0.00 +0.25 +0.50 +0.75 +1.00 +.1 +1 +Theoretical QuantilesFigure 4: Simulation under the alternative, for Setting 1 (top row), Setting 2 (middle row), and +Setting 3 (bottom row). (See Section 4.2 for details.) The left and middle column of plots show one +draw of the data for two different signal strengths δ, along with the result of hierarchical clustering. +The right column of plots shows power as a function of signal strength δ for the four different +methods. +for i ∈ [n] with σ = 1, n = 30, and q = 2. We test three different settings for generating the +means µi: +• Setting 1: K = 2 true clusters, with µi = (0, 0)⊤ for n/2 = 15 data points, and +µi = (δ, 0)⊤ for the remaining n/2 = 15 data points. (The results for the three variants +of Gao et al. [2022]’s method in Setting 1 were also shown in the bottom row of Figure 1 +in Section 2.2.) +• Setting 2: K = 3 true clusters, with µi = (0, 0)⊤ for n/3 = 10 data points, µi = (δ, 0)⊤ +for n/3 = 10 data points, and µi = (δ/2, δ +√ +3/2)⊤ for the remaining n/3 = 10 data +points. +• Setting 3: K = 3 true clusters, with µi = (0, 0)⊤ for n/3 = 10 data points, µi = (δ, 0)⊤ +for n/3 = 10 data points, and µi = (2δ, 0)⊤ for the remaining n/3 = 10 data points. +12 + +Visualization for = 5 +Visualization for S = 3 +Empirical Power +2 +2 +1.00 +1 +1 +0.75 +■ +■ +■ +06 +0 +P +. +tting +■ +0.50 +■ +-1 +-11 +e +S += 0.25 +-2 +-2 +■ +-3 +-3 +0.00 +2.5 +0 +0.0 +2.5 +5.0 +0.0 +5.0 +2 +4 +6Visualization for = 5 +Visualization for S = 3 +Empirical Power +61 +6 +1.00 +0.75 +4 +2 +etting +2 +2 +S += 0.25 +0 +0 +E +: +-2 +-21 +. +0.00 +0.0 +2.5 +5.0 +2.5 +0 +0.0 +2 +5.0 +4 +6Visualization for = 3 +Visualization for = 5 +Empirical Power +2 +1.00 +2/ +1 +1 +3 +6ume +一 +P +0 +0 +■ +S +！ +兰0.25 +-1 +-11 +. +-2 +0.00 +. +-2 +0 +4 +8 +12 +0 +4 +8 +12 +0 +2 +4 +6 +- Proposed method with unknown o +- Gao et al.'s method with Oall +- Oracle (Gao et al.'s method with true gFigure 5: Visualization of the penguin dataset. The output of the clustering algorithm corresponds +to that run on the centered and standardized dataset. +We then run hierarchical clustering with the true number of clusters K. In each setting, the +parameter δ ∈ {0, 1, . . . , 7} controls the signal strength. Note that δ = 0 corresponds to the +null(s) being true, as all data points have mean µi = 0. Setting 1 with δ = 0 is identical to +the first simulation under the null in Section 4.1 (where hierarchical clustering is run with +K = 2), while Settings 2 and 3 with δ = 0 are identical to the second simulation under +the null in Section 4.1 (where hierarchical clustering is run with K = 3). For the K = 2 +case (Setting 1), the p-values are generated for the comparison between clusters k = 1 and +k′ = 2. For the K = 3 case (Settings 2 and 3), in each trial, the p-values are generated for +the comparison between two randomly chosen clusters k ̸= k′ ∈ {1, 2, 3}. In all settings, the +threshold for rejecting a p-value is α = 0.05. +Figure 4 illustrates the power, as a function of δ, for the four methods in each of the +three settings. We see that the highest power is achieved by Gao et al. [2022]’s method +applied with the anticonservative estimate ˆσclustered, but this power comes at the cost of loss +of Type I error control (as we can see due to the high rejection rate at δ = 0, where the +null is true). Among the remaining methods, we see that the proposed method always has +power at least as high as Gao et al. [2022]’s method applied with the conservative estimate +ˆσclustered—sometimes approximately the same, and sometimes substantially higher, in the +various settings. +4.3 +Real dataset +We next compare the methods on a real dataset—the penguin dataset [Horst et al., 2020], +which contains information about the bill length (mm) and flipper length (mm) of three +different species of penguins, Adelie, Chinstrap, and Gentoo. (This dataset was also studied +by Gao et al. [2022, Section 6] to test their method for inference after clustering.) The data +is shown in Figure 5. +Table 1 shows the p-values computed by each of the three methods that does not require +knowledge of σ—our proposed method, along with Gao et al. [2022]’s with ˆσall or with +ˆσclustered. We test three different pairwise cluster comparisons (i.e., comparing Ck and Ck′, +13 + +Visualization of Data +Clusters Formed +1 +２３4５６ +True Clusters +40 +Adelie +Gentoo +Chinstrap +170 +180 +190 +200 +210 +220 +Flipper length (mm)(k, k′) +(1, 2) +(1, 5) +(4, 5) +Proposed method +0.5 +0.0045 +1.5e-08 +Gao et al. [2022]’s method with ˆσall +0.85 +0.13 +0.0014 +Gao et al. [2022]’s method with ˆσclustered +0.31 +1.6e-07 +4.2e-22 +Table 1: Results on the centered and standardized penguin dataset. +for three different choices of (k, k′)). +Since the true clustering (i.e., the penguin species) is known, we can see that (k, k′) = +(1, 2) corresponds to two estimated clusters that are not very well separated in terms of the +true species labeling, while (k, k′) = (1, 5) and (k, k′) = (4, 5) clearly correspond to a true +difference in species. In Table 1, overall we see that the proposed method produces p-values +that are substantially lower than those computed by Gao et al. [2022]’s method with the +conservative estimate ˆσall, corresponding to a gain in power. Gao et al. [2022]’s method +applied with ˆσclustered produces p-values even lower than our proposed method, but may not +be reliable in terms of Type I error control as we have seen in our simulations. +5 +Discussion +In this work, we have proposed an extension of Gao et al. [2022]’s framework for selective +inference for clustering to the case where the isotropic covariance matrix is unknown. Since +the method does not rely on plug-in empirical estimates of σ, we can avoid loss of power +and/or loss of Type I error control. For location- and scale-invariant clustering algorithms, we +have shown that the resulting p-value can be computed exactly in the case of K = 2 clusters +by leveraging connections to the findings of Gao et al. [2022]; more generally, standard +sampling strategies allow for accurate estimation of the p-value. +These results suggest a number of possible extensions and open questions. +First, in +our work we allow for unknown variance but assume an isotropic covariance structure, i.e., +σ2Iq, for the q-dimensional data points. It would be interesting to extend these techniques +to the setting of diagonal covariance, or even an arbitrary covariance matrix, to allow for +nonconstant variance along the q coordinates and/or nonzero correlation. Another important +question is whether these tools can be extended to the non-Gaussian setting, which would +offer further flexibility and robustness for real-data applications. +Acknowledgements +R.F.B. was supported by the National Science Foundation via grants DMS-1654076 and +DMS-2023109, and by the Office of Naval Research via grant N00014-20-1-2337. +References +Yoav Benjamini. Selective inference: The silent killer of replicability. Harvard Data Science +Review, 2(4):12, 2020. +14 + +Yiqun T Chen and Daniela M Witten. Selective inference for k-means clustering. arXiv +preprint arXiv:2203.15267, 2022. +William Fithian, Dennis Sun, and Jonathan Taylor. Optimal inference after model selection. +arXiv preprint arXiv:1410.2597, 2014. +Lucy L. Gao, Jacob Bien, and Daniela Witten. Selective inference for hierarchical clustering. +Journal of the American Statistical Association, 2022. +Benjamin Hivert, Denis Agniel, Rodolphe Thi´ebaut, and Boris P Hejblum. +Post- +clustering difference testing: valid inference and practical considerations. arXiv preprint +arXiv:2210.13172, 2022. +Allison Marie Horst, Alison Presmanes Hill, and Kristen B Gorman. palmerpenguins: Palmer +archipelago (antarctica) penguin data. R package version 0.1. 0, 2020. +Baibing Li and Elaine B Martin. An approximation to the f distribution using the chi-square +distribution. Computational statistics & data analysis, 40(1):21–26, 2002. +Bing Liu, Yiyuan Xia, and Philip S Yu. Clustering through decision tree construction. In +Proceedings of the ninth international conference on Information and knowledge manage- +ment, pages 20–29, 2000. +Joshua R Loftus. Selective inference after cross-validation. arXiv preprint arXiv:1511.08866, +2015. +Fionn Murtagh and Pedro Contreras. Algorithms for hierarchical clustering: an overview. +Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 2(1):86–97, 2012. +Douglas Steinley. K-means clustering: a half-century synthesis. British Journal of Mathe- +matical and Statistical Psychology, 59(1):1–34, 2006. +Jonathan Taylor and Robert J Tibshirani. Statistical learning and selective inference. Pro- +ceedings of the National Academy of Sciences, 112(25):7629–7634, 2015. +Ulrike Von Luxburg. A tutorial on spectral clustering. Statistics and computing, 17(4): +395–416, 2007. +Chihiro Watanabe and Taiji Suzuki. Selective inference for latent block models. Electronic +Journal of Statistics, 15(1):3137–3183, 2021. +Jesse M Zhang, Govinda M Kamath, and N Tse David. Valid post-clustering differential +analysis for single-cell rna-seq. Cell systems, 9(4):383–392, 2019. +15 + +A +Additional proofs and details +A.1 +Proof of Theorem 2 +First, suppose that clusters Ck and Ck′ were fixed, i.e., not data-dependent. We will show +that, in that case, we have +R | Z ∼ Beta(q/2, (m − 2)q/2) +under H′ +0 (Ck, Ck′), where +Z = +� +∥P0X∥2 +F + ∥P1X∥2 +F, +P0X +∥P0X∥F +, +P1X +∥P1X∥F +, P2X +� +. +Since X has isotropic covariance while P0, P1, P2 are orthogonal, we see that P0X, P1X, P2X +are mutually independent, and so we now only need to show +R | +� +∥P0X∥2 +F + ∥P1X∥2 +F, +P0X +∥P0X∥F +, +P1X +∥P1X∥F +� +∼ Beta(q/2, (m − 2)q/2). +Moreover, under H′ +0 (Ck, Ck′), we see that E [P0X] = 0 and E [P1X] = 0; therefore, by +properties of the Gaussian distribution (with mean zero and isotropic covariance), we have +(∥P0X∥F, ∥P1X∥F) independent from +� +P0X +∥P0X∥F , +P1X +∥P1X∥F +� +, and so now it suffices to show that +R | ∥P0X∥2 +F + ∥P1X∥2 +F ∼ Beta(q/2, (m − 2)q/2). +Finally, ∥P0X∥F and ∥P1X∥F are independent, with σ−2∥P0X∥2 +F ∼ χ2 +q and σ−2∥P1X∥2 +F ∼ +χ2 +(m−2)q (because P0 is a rank-1 projection matrix while P1 is a rank-(m − 2) projection +matrix). The desired statement follows since, for independent random variables A ∼ χ2 +a and +B ∼ χ2 +b, it follows from properties of the χ2 and Beta distributions that +A +A+B is independent +from A + B, and follows a Beta(a/2, b/2) distribution. +Next, we will account for the fact that clusters Ck and Ck′ are data-dependent, by con- +ditioning on Cluster(X). +Let f(R,Z)(r, z) denote the joint density of (R, Z) with respect +to the appropriate base measure, when we treat Cluster(X) (and thus Ck, Ck′) as fixed. +By the work above we can write f(R,Z)(r, z) = fR(r)fZ(z) where fR is the density of the +Beta(q/2, (m − 2)q/2) distribution. +We then need to calculate the conditional distribu- +tion of R, given the event (R, Z) ∈ E, where E is the subset of all values (r, z) such that +Cluster(x′(r)) = Cluster∗, for some particular clustering Cluster∗ (and then we will apply the +calculation with Cluster∗ = Cluster(X)). Let E∗ be the set of all such pairs (r, z) (and note +that, by the construction of clustering procedures, (R, Z) ∈ E∗ has positive probability). +The density of (R, Z) | (R, Z) ∈ E∗ is then given by +f(R,Z)|(R,Z)∈E∗(r, z) ∝ f(R,Z)(r, z) · 1(r,z)∈E∗ = fR(r)fZ(z) · 1(r,z)∈E∗, +and therefore the conditional density of R is given by +fR|Z;(R,Z)∈E∗(r | z) ∝ fR(r) · 1(r,Z)∈E∗. +16 + +Returning to our definitions, we see that, letting Cluster∗ = Cluster(X), +(r, Z) ∈ E∗ ⇔ r ∈ S′, +and therefore, we have calculated that the distribution of R conditional on Z and on +Cluster(X) is +fR|Z;(R,Z)∈E∗(r | z) ∝ fR(r) · 1r∈S′. +This proves the desired result about the distribution of R. The validity of the p-value P ′ +follows as an immediate consequence. +A.2 +Proof of Proposition 1 +Let v = +1C1 +|C1| − +1C2 +|C2|, specializing the construction from before to the case K = 2 (i.e., we are +comparing clusters k = 1 and k′ = 2). Define +φ = +� +r +1 − r · ∥v∥2 · ∥P1X∥F. +Then we can solve for r, +r = +φ2 +φ2 + ∥v∥2 +2 · ∥P1X∥2 +F +. +We compute +x′(r) = +�√r · +P0X +∥P0X∥F ++ +√ +1 − r · +P1X +∥P1X∥F +� +· +� +∥P0X∥2 +F + ∥P1X∥2 +F + P2X += +� +φ · +P0X +∥P0X∥F ++ ∥v∥2 · ∥P1X∥F · +P1X +∥P1X∥F +� +· +� +∥P0X∥2 +F + ∥P1X∥2 +F +φ2 + ∥v∥2 +2 · ∥P1X∥2 +F ++ P2X += x(φ) · ∥v∥2 · +� +∥P0X∥2 +F + ∥P1X∥2 +F +φ2 + ∥v∥2 +2 · ∥P1X∥2 +F +− P2X · +� +∥v∥2 · +� +∥P0X∥2 +F + ∥P1X∥2 +F +φ2 + ∥v∥2 +2 · ∥P1X∥2 +F +− 1 +� +, +where for the last step we apply the calculation (7). Furthermore, in the case K = 2, we +can verify that P2 = 1n1⊤ +n +n , the projection to the span of 1n. Therefore, we can write +x′(r) = a · x(φ) + 1n · b⊤, +where +a = ∥v∥2 · +� +∥P0X∥2 +F + ∥P1X∥2 +F +φ2 + ∥v∥2 +2 · ∥P1X∥2 +F +and +b = − +� +∥v∥2 · +� +∥P0X∥2 +F + ∥P1X∥2 +F +φ2 + ∥v∥2 +2 · ∥P1X∥2 +F +− 1 +� +· X⊤1n +n . +By our assumption on the clustering procedure, we therefore have +Cluster(x′(r)) = Cluster(x(φ)), +and so r ∈ S′ if and only if φ ∈ S, as desired. +17 + +A.3 +Implementation details +In our experiments, the three variants of Gao et al. [2022]’s method (i.e., using the true σ, +or the estimates ˆσall or ˆσclustered) are run using the code provided by Gao et al. [2022] in the +R package clusterpval. We now give details for implementation of our proposed method. +A.3.1 +Notes on computation in the K = 2 case +Computing the p-value +P ′ = 1 − FBeta(q/2,(m−2)q/2)(R; S′) +for our method often requires extremely precise calculations of the CDF of the Beta distri- +bution due to the truncation. +In order to approximate this p-value, we first convert the Beta distribution to an F dis- +tribution, then use Li and Martin [2002]’s Shrinkage Factor Approximation method, which +offers a chi-square approximation for the F distribution. Specifically, we first use the equiv- +alence +FBeta(k/2,ℓ/2)(t) = FFk,ℓ +� +t/k +(1 − t)/ℓ +� +, +where Fk,ℓ is the F distribution with k, ℓ degrees of freedom. Next, to estimate the CDF in +the tail of the F distribution, we use Li and Martin [2002]’s approximation, +FFk,ℓ(t) ≈ Fχ2 +k +� +2ℓ + kt +3 + k − 2 +2ℓ + 4kt +3 +· kt +� +. +Note that Li and Martin [2002]’s method is accurate in the regime where k is finite while +ℓ → ∞, which is appropriate to our setting as we apply the approximation with k = q and +ℓ = (m − 2)q. Combining all these calculations means that we can approximate P ′ via the +CDF of a truncated χ2 distribution, +P ′ ≈ 1 − Fχ2q +� +2(m − 2)q + (m−2)qR +3(1−R) + q − 2 +2(m − 2)q + 4(m−2)qR +3(1−R) +· (m − 2)qR +1 − R +; ˜S′ +� +where +˜S′ = +� +2(m − 2)q + (m−2)qr +3(1−r) + q − 2 +2(m − 2)q + 4(m−2)qr +3(1−r) +· (m − 2)qr +1 − r +: r ∈ S′ +� +. +Finally, we use the TChisqRatioApprox function from Gao et al. [2022]’s R package clusterpval +for this remaining calculation with the truncated χ2 distribution. +A.3.2 +Importance sampling algorithm for the K ≥ 2 case +For the setting where K > 2, or where K = 2 but we do not have an exact characterization +of the truncation set S′, we use importance sampling to approximate the p-value P ′. +18 + +In order to calculate this p-value, we need to be able to perform a calculation of the form +1 − FBeta(q/2,(m−2)q/2) (r; S′) = PR′∼Beta(q/2,(m−2)q/2) {R′ > r | R′ ∈ S′} += PR′∼Beta(q/2,(m−2)q/2) {R′ > r, R′ ∈ S′} +PR′∼Beta(q/2,(m−2)q/2) {R′ ∈ S′} +, +and then apply this calculation at the value r = R. We estimate the numerator and denom- +inator simultaneously, using importance sampling with the proposal distribution +TN(R, α2; 0, 1), +which is the truncated normal distribution—i.e., the N(R, α2) distribution truncated to the +interval [0, 1]. +Our procedure is: +• Draw R(1), . . . , R(N) iid∼ TN(R, α2; 0, 1), for N = 8000 draws. +• Compute importance weights +w(R(i)) = fBeta(q/2,(m−2)q/2)(R(i)) +fTN(R,α2;0,1)(R(i)) +for i ∈ [N], +where fBeta(q/2,(m−2)q/2) and fTN(R,α2;0,1) denote the densities of the respective distribu- +tions. +• Estimate +P ′ ≈ +�N +i=1 w(R(i)) · 1 +� +R(i) > R, R(i) ∈ S′� +�N +i=1 w(R(i)) · 1 {R(i) ∈ S′} +. +The tuning parameter α is adjusted based on empirical performance—specifically, we choose +α so that 1 +N +�N +i=1 1{R(i) ∈ S′} ≈ 0.5, to ensure that our proposal distribution TN(R, α2; 0, 1) +is neither too concentrated nor too dispersed to accurately approximate the truncated target +distribution. +19 + diff --git a/jNFPT4oBgHgl3EQfFTQk/content/tmp_files/load_file.txt b/jNFPT4oBgHgl3EQfFTQk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e79b5968bf31ecefbdb3cad69a8c61cc701aea52 --- /dev/null +++ b/jNFPT4oBgHgl3EQfFTQk/content/tmp_files/load_file.txt @@ -0,0 +1,526 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf,len=525 +page_content='Selective inference for clustering with unknown variance Young-Joo Yun∗ Rina Foygel Barber† January 31, 2023 Abstract In many modern statistical problems, the limited available data must be used both to develop the hypotheses to test, and to test these hypotheses—that is, both for exploratory and confirmatory data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Reusing the same dataset for both explo- ration and testing can lead to massive selection bias, leading to many false discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Selective inference is a framework that allows for performing valid inference even when the same data is reused for exploration and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In this work, we are interested in the problem of selective inference for data clustering, where a clustering procedure is used to hypothesize a separation of the data points into a collection of subgroups, and we then wish to test whether these data-dependent clusters in fact represent meaningful differences within the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Recent work by Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022] provides a framework for doing selective inference for this setting, where the hierarchical clustering algorithm is used for producing the cluster assignments, which was then extended to k-means clustering by Chen and Witten [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Both these works rely on assuming a known covariance structure for the data, but in practice, the noise level needs to be estimated— and this is particularly challenging when the true cluster structure is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In our work, we extend to the setting of noise with unknown variance, and provide a selective inference method for this more general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Empirical results show that our new method is better able to maintain high power while controlling Type I error when the true noise level is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 1 Introduction Data clustering is a popular method for summarizing trends in unlabeled data, and is a pow- erful tool for gaining understanding and interpretability in large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' However, it is well known that clustering can easily lead to false discoveries, in the sense that data from a single source (one true cluster) can be falsely separated into multiple clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' A core challenge in addressing this issue is that of selective inference—the problem of performing inference on a hypothesis that was developed using the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In-depth motivations for selective ∗Department of Statistics, University of Wisconsin-Madison †Department of Statistics, University of Chicago 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='12999v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='ME] 30 Jan 2023 inference can be found in Taylor and Tibshirani [2015] and Benjamini [2020], where the latter illustrates its importance from the perspective of replicability of experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' There has been extensive work on selective inference in supervised settings, such as the work by Fithian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2014] and Loftus [2015] that provide selective inference frameworks for doing valid inference after model selection, but less is known about the unsupervised setting, which is the context for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Some popular clustering methods include hierarchical clustering (Murtagh and Contreras [2012]), k-means clustering (Steinley [2006]), decision tree clustering (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2000]), and spectral clustering (Von Luxburg [2007]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Concretely, for the clustering problem, if the n data points X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' , Xn ∈ Rq are parti- tioned into clusters C1 ∪ · · · ∪ CK = [n] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' , n}, how can we determine whether the “discovered” clusters Ck and Ck′ are genuinely different using the observed data (the Xi’s for i ∈ Ck, and for i ∈ Ck′), when these same data values were used to choose the clusters themselves?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' To address this problem, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2019] propose a method based on data splitting, where the hyperplane separating two clusters is estimated using a portion of the data, and the fitted hyperplane, instead of the clustering algorithm, is used on the rest of the data for generating cluster assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' They then condition on the selection event—the event where the hyperplane is chosen—to account for the data dependency in the hypothesis test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022] provide an alternative solution to this problem, where they account for the clustering event by directly conditioning on it, specifically for the hierarchical clustering algorithm, and a recent work of Chen and Witten [2022] extends their work to the k-means clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Relatedly, Hivert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022] propose a set of valid inference procedures for three different null hypotheses testing whether two clusters estimated by a clustering algorithm are truly different, where one of the proposed procedures is an extension of the work of Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022] to testing whether a single feature plays a role in distinguishing the two clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' There have also been relevant work on data with structural assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' For example, the work of Watanabe and Suzuki [2021] provides a method for doing inference on a data matrix represented by a latent block model after choosing the cluster membership of each entry of the data matrix using the same data matrix, and conditions on this selection event to do a valid inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The aforementioned work of Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022] offers an elegant solution to this problem, providing a framework for performing selective inference to test the null hypothesis that states H0(Ck, Ck′) : 1 |Ck| � i∈Ck µi = 1 |Ck′| � i∈Ck′ µi (1) for each pair k ̸= k′, where µi = E [Xi] is the true mean of the i-th data point—in other words, is the mean of cluster Ck equal to the mean of cluster Ck′?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Note that this hypothesis is indeed data-dependent, since the clusters Ck and Ck′ are chosen based on the observed data, and therefore testing this null must account for this data dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s work, it is assumed that the data is distributed as Xi ⊥⊥∼ N(µi, σ2Iq) (2) for i ∈ [n], where the means µi ∈ Rq are unknown while the (shared) variance σ2 > 0 is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In practice, however, σ would often need to be estimated from the data, and this 2 poses a particular challenge in the setting of this clustering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Without knowing the true cluster structure of the data (since of course, this is exactly the question we are aiming to test), it is difficult to obtain a reliable estimate of σ–indeed, we will see shortly that many natural options lead to either substantial power loss or substantial loss of the Type I error control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' This motivates the need for the more general model that avoids the need to estimate the true variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In this work, we propose a method that avoids this obstacle, by allowing for an unknown variance σ2 (or more generally, an unknown structured covariance matrix), while guaranteeing Type I error control and maintaining high power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In Section 2, we review the selective inference framework developed by Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022] for the setting where σ is known and discuss motivations for allowing σ to be unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In Section 3, we present our new method for performing inference on clustering in the setting of an unknown σ (with proofs deferred to the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Empirical results are presented in Section 4 to demonstrate the performance of the new method and compare against the existing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Finally, we conclude with a discussion and some open questions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 2 Background: the known variance case In this section, we will first give a brief overview of the selective inference method developed in Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s work, and discuss the challenges posed by unknown variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1 Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s Method Consider clusters Ck, Ck′, which are two disjoint subsets of [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' If these clusters were chosen ahead of time—that is, independently of the data—then it would be simple to test the null hypothesis H0(Ck, Ck′) defined in (1)—specifically, we would naturally use the test statistic 1 |Ck| � i∈Ck Xi − 1 |Ck′| � i∈Ck′ Xi = X⊤v where v := 1Ck |Ck| − 1Ck′ |Ck′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Here X ∈ Rn×q is the matrix of observed data with i-th row Xi ∈ Rq, and where, for a subset C ⊆ [n], 1C ∈ Rn represents the vector with ith entry equal to 1 for each i ∈ C and 0 for i ̸∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' This test statistic follows a mean-zero normal distribution under the null hypothesis H0(Ck, Ck′), and so its norm follows a rescaled χ distributed under the null, ∥X⊤v∥2 H0(Ck,Ck′) ∼ σ � 1 |Ck| + 1 |Ck′| �1/2 χq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' However, since the clusters were chosen in a data-dependent way, this distribution is not the correct null distribution for ∥X⊤v∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' To address this, we can rewrite X as X = PvX + P⊥ v X = ∥X⊤v∥2 ∥v∥2 vv⊤X ∥vv⊤X∥F + P⊥ v X, which decomposes X into components lying in the span of v and its orthogonal complement, with Pv = vv⊤ ∥v∥2 2 denoting the projection matrix that projects to the span of v, and P⊥ v = In − 3 vv⊤ ∥v∥2 2 projecting to its orthogonal complement, and where ∥·∥2 denotes the Euclidean norm and ∥·∥F the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s insight into handling the data-dependent cluster selection is to condition on the normalized matrix vv⊤X ∥vv⊤X∥F and the orthogonal projection P⊥ v X, so that only the test statistic ∥X⊤v∥2 remains unknown, and moreover to condition on the range of values of ∥X⊤v∥2 that agree with the clustering selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Specifically, defining x(φ) = φ ∥v∥2 vv⊤X ∥vv⊤X∥F + P⊥ v X, (3) let S = {φ > 0 : Cluster(X) = Cluster(x(φ))} , where Cluster(·) refers to the outcome of the clustering procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In other words, S contains all values of φ for which the same clustering outcome would have been obtained, if we plug in φ in place of the observed test statistic value ∥X⊤v∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Their main result establishes that, even given the data-dependent clustering procedure, the re-scaled χ distribution is the correct null distribution once truncated to this set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Theorem 1 (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022, Theorem 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Let Xi ⊥⊥∼ N(µi, σ2Iq) where σ is known, and let v be defined as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Then, conditional on Cluster(X), vv⊤X ∥vv⊤X∥F , and P⊥ v X, under the null hypothesis H0(Ck, Ck′) the test statistic ∥X⊤v∥2 follows a truncated rescaled χ distribution, σ � 1 |Ck| + 1 |Ck′| �1/2 χq truncated to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In particular, the p-value P = 1 − Fχq � ∥X⊤v∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' σ � 1 |Ck| + 1 |Ck′| �1/2 , S � is uniformly distributed under H0(Ck, Ck′), where Fχq(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' c, S) is the CDF of a c · χq random variable truncated to the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022] provide an algorithm for exactly computing the set S for the hierarchical clustering algorithm with linkages for which the exact computation of this set is tractable, along with an implementation of the importance sampling algorithm for clustering algorithms where this set cannot be efficiently computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2 Challenges in estimating σ We next discuss motivations for allowing σ to be unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Continuing the discussion earlier on the difficulty of estimating σ from the data, we consider a simple scenario where we are aiming to determine whether data points X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' , Xn arise from a single cluster or from two clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' To test this, we would choose a data-dependent clustering [n] = C1 ∪C2, and would now need to estimate σ in order to run Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Suppose we estimate the variance by using the within-cluster means, for instance, ˆσ2 clustered = � i∈C1 ∥Xi − ¯XC1∥2 2 + � i∈C2 ∥Xi − ¯XC2∥2 2 (n − 2)q , 4 Figure 1: The top row shows results under the null, and the bottom row shows results under the alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In each row, the left plot shows one draw of the data, along with the estimated values ˆσall and ˆσclustered, while the middle and right plots show results for Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method applied with ˆσclustered or with ˆσall, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' (See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2 for discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=') where ¯XCk is the sample mean in cluster Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' With this choice, we might substantially underestimate the variance if the true data distribution only has a single cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The middle column of Figure 1 demonstrates this problem in practice—we can see that, when the null H0(C1, C2) is true, the variance may sometimes be vastly underestimated and, as a result, the empirical distribution of the p-value is far from uniform, which would lead to false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Alternatively, we might take a more conservative estimate of variance by treating the data as a single cluster, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', ˆσ2 all = � i∈[n] ∥Xi − ¯X[n]∥2 2 (n − 1)q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Indeed, this is the estimator proposed in Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022, Section S3], and they prove theoretically that, as this is asymptotically an over-estimate of σ2, Type I error control is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' However, this choice can lead to a substantially over-conservative test, as demonstrated in the right column of Figure 1—if the true data distribution arises from two clusters, this estimate can massively over-estimate σ2 leading to a large loss of power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' (See Section 4 for details on these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=') Thus, in Figure 1, we clearly see a tradeoff between Type I error control and power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' When using the cluster-wise estimate ˆσclustered, we see that power is high under the alternative, with the empirical power being as high as the case where the true σ is used, but Type I error control is lost under the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' On the other hand, when using the estimate ˆσall that treats 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 Quantiles Quantil 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='75 ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='50 0 irical irical 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='25 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='25 Emr Emp 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='75 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 1 Theoretical Quantiles Theoretical Quantiles Empirical Power with Oall 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 Alternative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='75 Pow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='75 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 3 6 2 0 0 6 0 2 4 6 4 Oracle (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=" 's method with true o) Oracle (Gao et al's method with true o)the entire dataset as a single cluster, we see that it controls Type I error under the null but incurs a loss in power under the alternative." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' To avoid this tradeoff, in this work we propose a selective inference procedure for the clustering problem that can handle an unknown variance σ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' This more general model resolves the issue—the p-value distribution is uniform when the data is generated from a single cluster, without sacrificing too much power in the scenario where the data is instead generated from distinct clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 3 Proposed method: the unknown variance case We now introduce our proposed method for the setting where the variance is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In this new setting, we assume that the data is distributed as Xi ⊥⊥∼ N(µi, σ2Iq), where the means µi ∈ Rq, as well as the (shared) variance σ2 > 0, are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Recall the null hypothesis H0(Ck, Ck′) : 1 |Ck| � i∈Ck µi = 1 |Ck′| � i∈Ck′ µi in Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022] and the corresponding test statistic ∥X⊤v∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Unfortunately, in our new setting where σ is unknown, the distribution of this test statistic cannot be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' However, we will see that we can overcome this obstacle if we restrict to a stronger null hypothesis, H′ 0(Ck, Ck′) : µi = µi′ ∀ i, i′ ∈ Ck ∪ Ck′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' (4) In other words, H′ 0 assumes that each data point in clusters Ck and Ck′ has the same mean, while H0 makes the weaker assumption that the sample mean of data points in cluster Ck and in cluster Ck′ have the same mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We can equivalently rewrite H′ 0(Ck, Ck′) as H′ 0(Ck, Ck′) : � �In − ww⊤ ∥w∥2 − � i∈[n]\\(Ck∪Ck′) eie⊤ i � � µ = 0 where w := 1Ck∪Ck′ |Ck ∪ Ck′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' (5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1 Decomposition of X To define our test statistic, we begin by taking a decomposition of the observed data X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' This decomposition plays an analogous role to the decomposition (3) used by Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022], but is more complex to allow us to handle unknown variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We begin by writing X = P0X + P1X + P2X, where P0 = vv⊤ ∥v∥2 2 is the rank-one projection matrix that captures the difference in cluster means for Ck and Ck′, while P1 = � ICk − 1Ck1⊤ Ck |Ck| � + � ICk′ − 1Ck′1⊤ Ck′ |Ck′| � , 6 where, for any subset C ⊆ [n], IC represents the diagonal matrix with entry (i, i) set to 1 if i ∈ C and 0 if i ̸∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Finally, P2 = In − P0 − P1 is the projection matrix to the orthogonal complement of P0 and P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We can see that P0, P1, and P2 project to subspaces of dimension 1, m − 2, and n − m + 1, respectively, where m = |Ck| + |Ck′| is the number of data points in the two clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Intuitively, we can think of this decomposition of the data as follows: P0X captures the difference in means between clusters Ck and Ck′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' P1X captures differences among points within Ck, and among points within Ck′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' P2X captures all other aspects of the data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', the overall mean of the combined clusters Ck ∪ Ck′, as well as information about data points not lying in Ck ∪ Ck′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Figure 2 illustrates the roles of these three terms in the decomposition of the data X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Figure 2: Left: visualization of a dataset with colors indicating the clusters formed by the clustering algorithm with Ck represented in blue and Ck′ in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The blue triangle represents the combined mean of Ck ∪Ck′, and the black dots represent the cluster means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Middle: the original dataset with ∥P0X∥F scaled by a factor of 2, which pushes apart the clusters Ck and Ck′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Right: the original dataset with ∥P1X∥F scaled by a factor of 2, which spreads points in Ck apart from each other while preserving the cluster mean, and same for points in Ck′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2 The test statistic In Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s work, the test statistic they use is equivalent to ∥P0X∥F, which under the null hypothesis H0, follows a χq distribution (rescaled by σ), truncated to a region S that controls for the selection event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In our work, since σ is unknown, we will use a Beta distribution in place of the χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The test statistic we propose is given by the ratio R = ∥P0X∥2 F ∥P0X∥2 F + ∥P1X∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' where the numerator is the same as the statistic used by Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022] (up to a transfor- mation), while the denominator acts by rescaling with respect to an estimate of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 7 2PoX+ P,X+ P2X PoX+ PX+ P2X PoX+2PX+ P2X 15 15 15 10 10 10 5 5 5 0 0 0 0 10 15 5 0 5 10 15 5 5 0 5 10 15 5We next need to define the truncation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' First, we rewrite our decomposition as X = ∥P0X∥F · P0X ∥P0X∥F + ∥P1X∥F · P1X ∥P1X∥F + P2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' (6) Our test will condition on: The total squared norm ∥P0X∥2 F + ∥P1X∥2 F for the first and second terms in the decomposition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The normalized terms P0X ∥P0X∥F and P1X ∥P1X∥F for the first and second terms in the decom- position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The third term P2X in the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' With these terms treated as known, the data X can then be fully determined by revealing the value R = ∥P0X∥2 F ∥P0X∥2 F +∥P1X∥2 F of the test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' For any r ∈ (0, 1), define x′(r) = �√r · P0X ∥P0X∥F + √ 1 − r · P1X ∥P1X∥F � � ∥P0X∥2 F + ∥P1X∥2 F + P2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We can verify from the definition of R that X = x′(R) holds by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Finally, define S′ = {r ∈ (0, 1) : Cluster(X) = Cluster(x′(r))} ⊆ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='3 Main result Our main result, presented next, establishes that we can compute the exact post-selection distribution of R, which thus allows us to perform valid selective inference in the unknown- variance setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Let Xi ⊥⊥∼ N(µi, σ2Iq) where σ is unknown, and let P0, P1, and P2 be defined as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Then, conditional on Cluster(X), ∥P0X∥2 F + ∥P1X∥2 F, P0X ∥P0X∥F , P1X ∥P1X∥F , and P2X, under the null hypothesis H′ 0 (Ck, Ck′), the random variable R follows the Beta(q/2, (m − 2)q/2) distribution truncated to the set S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In particular, the p-value P ′ = 1 − FBeta(q/2,(m−2)q/2) (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' S′) is uniformly distributed under H′ 0 (Ck, Ck′), where FBeta(q/2,(m−2)q/2)(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' S′) is the CDF of a Beta(q/2, (m − 2)q/2) random variable truncated to the set S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The intuition is that, if P0 and P1 were fixed rather than data-dependent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', if the clus- ters Ck and Ck′ were chosen before viewing the data), then we would have ∥P0X∥2 F ∼ χ2 q and, independently, ∥P1X∥2 F ∼ χ2 (m−2)q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' thus R = ∥P0X∥2 F ∥P0X∥2 F +∥P1X∥2 F would follow a Beta(q/2, (m − 2)q/2) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' After accounting for the selection event, the null distribution is instead given by a truncated Beta distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' To implement the results of Theorem 2 in practice, we need to be able to compute this p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In other words, we need to either explicitly characterize the set S′ that is consistent with the selection event, or develop an empirical sampling strategy to estimate the p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We next consider this computational question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='4 Computing the p-value To characterize the truncation set S′, we will split into two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In the general setting, when the data is separated into an arbitrary number K ≥ 2 of clusters, we will handle the truncation event via numerical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' For the special case K = 2, however, we will show that S′ can potentially be computed explicitly, by relating the problem back to the work of Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022] for the known-variance case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1 Special case: K = 2 Rewriting Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s procedure in our notation, the modified data is defined as x(φ) = φ ∥v∥2 P0X ∥P0X∥F + P1X + P2X, (7) where v = 1Ck |Ck| − 1Ck′ |Ck′| so that P0 = vv⊤ ∥v∥2 2 is projection onto the span of v, and their selection set is given by S = {φ > 0 : Cluster(X) = Cluster(x(φ))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In contrast, our method defines x′(r) = �√r · P0X ∥P0X∥F + √ 1 − r · P1X ∥P1X∥F � � ∥P0X∥2 F + ∥P1X∥2 F + P2X and S′ = {r ∈ (0, 1) : Cluster(X) = Cluster(x′(r))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The next result shows that, for the case K = 2, these two definitions can be related in a simple way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Suppose that K = 2 (so that we have k = 1, k′ = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Assume that the clustering procedure is location- and scale-invariant—that is, for any x ∈ Rn×q, Cluster(x) = Cluster � a · x + 1n · b⊤� , for any a ∈ R and b ∈ Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Then, under the notation and definitions above, S′ = � r ∈ (0, 1) : � r 1 − r · ∥P1X∥F · � 1 |C1| + 1 |C2| ∈ S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The work of Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022] gives an explicit characterization of the set S for a family of agglomerative hierarchical clustering algorithms, while Chen and Witten [2022] does the same for k-means clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Moreover, both of these algorithms are location- and scale- invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Consequently, for the case K = 2, we can explicitly characterize the truncation set S′, and thus can compute the p-value P ′ constructed in Theorem 2 by leveraging Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s construction of the set S, for these two popular algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Of course, a major limitation of this result is that we can only handle K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' However, in iterative procedures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', hierarchical clustering), the test can be applied at the first step (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', the first split, from a single cluster to K = 2 clusters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' This hypothesis test can then essentially be interpreted as testing the global null, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', whether the data should be split into clusters at all or simply treated as a single cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2 General case: K ≥ 2 Next we consider the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' If K > 2, then the result of Proposition 1 no longer applies—as we will see in the proof of this proposition, the case K = 2 leads to the result specifically because P2 = 1n1⊤ n n in that case, but this no longer holds for K > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Moreover, even if K = 2, it might be the case that we are using a clustering algorithm for which S does not have an explicit characterization and/or which is not location- and scale-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In any of these settings, we do not have an explicit characterization of S′, and thus the truncated CDF needed in Theorem 2 cannot be computed exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Nonetheless, the inference procedure can still be run in the general case, by computing P ′ approximately through importance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Specifically, in order to (approximately) compute the p-value P ′, we need to be able to estimate 1 − FBeta(q/2,(m−2)q/2) (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' S′) = PR′∼Beta(q/2,(m−2)q/2) {R′ > r | R′ ∈ S′} = PR′∼Beta(q/2,(m−2)q/2) {R′ > r, R′ ∈ S′} PR′∼Beta(q/2,(m−2)q/2) {R′ ∈ S′} , and then evaluate this probability at r = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Since r = R may be extremely large, and/or the selection set S′ may be very small, the events R′ > r and/or R′ ∈ S′ may have extremely low probability under the distribution R′ ∼ Beta(q/2, (m − 2)q/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We approximate this probability with importance sampling, using a truncated normal distribution as the proposal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Further details are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 4 Empirical results We now provide empirical results for our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We present simulation results for Type I error control and empirical power for our proposed method, as well as results on a small real dataset (the penguin dataset provided by [Horst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', 2020], which was also analyzed in Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' For all experiments, we use the hierarchical clustering algorithm with average linkage for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1 Type I error control Theorem 2 states that P ′ follows the uniform distribution under the null, so it controls the Type 1 error rate of the hypothesis test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We illustrate this empirically by plotting empirical quantiles of a sample of P ′ against the theoretical quantiles of the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We compare to the p-value P computed by Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method, with either oracle knowledge of the true σ, or with plug-in estimates ˆσall or ˆσclustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' (The results for the latter two variants of Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method were also shown in the top row of Figure 1 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=') We perform 2000 independent trials, with datasets generated as Xi ⊥⊥∼ N(µi, σ2Iq) 1Code for reproducing all experiments is available online at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='com/yjyun97/cluster_inf_ unknown_var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 10 for i ∈ [n], with µi = 0q for all i ∈ [n] so that the null hypothesis is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We set σ = 1, n = 30, and q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Figure 3 presents results for two different settings, K = 2 and K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We use the exact computation method presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1 for K = 2, and the importance sampling method discussed in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2 for K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' For the K = 2 case, the p-values are generated for the comparison between clusters k = 1 and k′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' For the K = 3 case, in each trial, the p-values are generated for the comparison between two randomly chosen clusters k ̸= k′ ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Figure 3 illustrates that the proposed method, along with Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method with either the conservative estimate ˆσall or with oracle knowledge of the true σ, all result in uniformly distributed p-values (and thus, control the Type I error rate) for both settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' On the other hand, Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method applied with ˆσclustered fails to do so, with a substantially anticonservative distribution of the p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' As expected, since ˆσclustered is a more extreme underestimate for larger K, the nonuniformity of the p-values is more substantial when K = 3 than when K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Figure 3: Simulation under the null (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The left plots show one draw of the data, along with the result of hierarchical clustering for K = 2 (top) K = 3 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The right plots show QQ plots comparing the p-values obtained via the four different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2 Empirical power It is inevitable that not knowing σ will cause some loss in power, as we are forced to condition on more components of X than we would otherwise do when computing the p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' On the other hand, using ˆσall in place of σ in P also causes a loss in power, due to ˆσall being an overestimate of σ in settings where the alternative hypothesis is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Here, we compute the empirical power of the same four methods as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We perform 500 independent trials, with datasets generated as Xi ⊥⊥∼ N(µi, σ2Iq) 11 Visualization for K = 2 QQ Plot for K = 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='75 1 uar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='50 0 irical 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='25 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='75 2 1 0 2 1 Theoretical Quantiles Visualization for K - 3 QQ Plot for K = 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Oracle (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=" 's method with true o) 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 les 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='50 irical 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='25 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 2 2 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1 1 Theoretical QuantilesFigure 4: Simulation under the alternative, for Setting 1 (top row), Setting 2 (middle row), and Setting 3 (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' (See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=') The left and middle column of plots show one draw of the data for two different signal strengths δ, along with the result of hierarchical clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The right column of plots shows power as a function of signal strength δ for the four different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' for i ∈ [n] with σ = 1, n = 30, and q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We test three different settings for generating the means µi: Setting 1: K = 2 true clusters, with µi = (0, 0)⊤ for n/2 = 15 data points, and µi = (δ, 0)⊤ for the remaining n/2 = 15 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' (The results for the three variants of Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method in Setting 1 were also shown in the bottom row of Figure 1 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=') Setting 2: K = 3 true clusters, with µi = (0, 0)⊤ for n/3 = 10 data points, µi = (δ, 0)⊤ for n/3 = 10 data points, and µi = (δ/2, δ √ 3/2)⊤ for the remaining n/3 = 10 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Setting 3: K = 3 true clusters, with µi = (0, 0)⊤ for n/3 = 10 data points, µi = (δ, 0)⊤ for n/3 = 10 data points, and µi = (2δ, 0)⊤ for the remaining n/3 = 10 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 12 Visualization for = 5 Visualization for S = 3 Empirical Power 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='75 ■ ■ ■ 06 0 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' tting ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='50 ■ 1 11 e S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='25 2 2 ■ 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='0 2 4 6Visualization for = 5 Visualization for S = 3 Empirical Power 61 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='75 4 2 etting 2 2 S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='25 0 0 E : 2 21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='0 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='0 4 6Visualization for = 3 Visualization for = 5 Empirical Power 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 2/ 1 1 3 6ume 一 P 0 0 ■ S ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 兰0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='25 1 11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 2 0 4 8 12 0 4 8 12 0 2 4 6 Proposed method with unknown o Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=" 's method with Oall Oracle (Gao et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=" 's method with true gFigure 5: Visualization of the penguin dataset." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The output of the clustering algorithm corresponds to that run on the centered and standardized dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We then run hierarchical clustering with the true number of clusters K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In each setting, the parameter δ ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' , 7} controls the signal strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Note that δ = 0 corresponds to the null(s) being true, as all data points have mean µi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Setting 1 with δ = 0 is identical to the first simulation under the null in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1 (where hierarchical clustering is run with K = 2), while Settings 2 and 3 with δ = 0 are identical to the second simulation under the null in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1 (where hierarchical clustering is run with K = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' For the K = 2 case (Setting 1), the p-values are generated for the comparison between clusters k = 1 and k′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' For the K = 3 case (Settings 2 and 3), in each trial, the p-values are generated for the comparison between two randomly chosen clusters k ̸= k′ ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In all settings, the threshold for rejecting a p-value is α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Figure 4 illustrates the power, as a function of δ, for the four methods in each of the three settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We see that the highest power is achieved by Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method applied with the anticonservative estimate ˆσclustered, but this power comes at the cost of loss of Type I error control (as we can see due to the high rejection rate at δ = 0, where the null is true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Among the remaining methods, we see that the proposed method always has power at least as high as Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method applied with the conservative estimate ˆσclustered—sometimes approximately the same, and sometimes substantially higher, in the various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='3 Real dataset We next compare the methods on a real dataset—the penguin dataset [Horst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', 2020], which contains information about the bill length (mm) and flipper length (mm) of three different species of penguins, Adelie, Chinstrap, and Gentoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' (This dataset was also studied by Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022, Section 6] to test their method for inference after clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=') The data is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Table 1 shows the p-values computed by each of the three methods that does not require knowledge of σ—our proposed method, along with Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s with ˆσall or with ˆσclustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We test three different pairwise cluster comparisons (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', comparing Ck and Ck′, 13 Visualization of Data Clusters Formed 1 ２３4５６ True Clusters 40 Adelie Gentoo Chinstrap 170 180 190 200 210 220 Flipper length (mm)(k, k′) (1, 2) (1, 5) (4, 5) Proposed method 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='0045 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='5e-08 Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method with ˆσall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='0014 Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method with ˆσclustered 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='6e-07 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2e-22 Table 1: Results on the centered and standardized penguin dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' for three different choices of (k, k′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Since the true clustering (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', the penguin species) is known, we can see that (k, k′) = (1, 2) corresponds to two estimated clusters that are not very well separated in terms of the true species labeling, while (k, k′) = (1, 5) and (k, k′) = (4, 5) clearly correspond to a true difference in species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In Table 1, overall we see that the proposed method produces p-values that are substantially lower than those computed by Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method with the conservative estimate ˆσall, corresponding to a gain in power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method applied with ˆσclustered produces p-values even lower than our proposed method, but may not be reliable in terms of Type I error control as we have seen in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 5 Discussion In this work, we have proposed an extension of Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s framework for selective inference for clustering to the case where the isotropic covariance matrix is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Since the method does not rely on plug-in empirical estimates of σ, we can avoid loss of power and/or loss of Type I error control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' For location- and scale-invariant clustering algorithms, we have shown that the resulting p-value can be computed exactly in the case of K = 2 clusters by leveraging connections to the findings of Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' more generally, standard sampling strategies allow for accurate estimation of the p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' These results suggest a number of possible extensions and open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' First, in our work we allow for unknown variance but assume an isotropic covariance structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', σ2Iq, for the q-dimensional data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' It would be interesting to extend these techniques to the setting of diagonal covariance, or even an arbitrary covariance matrix, to allow for nonconstant variance along the q coordinates and/or nonzero correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Another important question is whether these tools can be extended to the non-Gaussian setting, which would offer further flexibility and robustness for real-data applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Acknowledgements R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' was supported by the National Science Foundation via grants DMS-1654076 and DMS-2023109, and by the Office of Naval Research via grant N00014-20-1-2337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' References Yoav Benjamini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Selective inference: The silent killer of replicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Harvard Data Science Review, 2(4):12, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 14 Yiqun T Chen and Daniela M Witten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Selective inference for k-means clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='15267, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' William Fithian, Dennis Sun, and Jonathan Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Optimal inference after model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' arXiv preprint arXiv:1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2597, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Lucy L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Gao, Jacob Bien, and Daniela Witten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Selective inference for hierarchical clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Journal of the American Statistical Association, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Benjamin Hivert, Denis Agniel, Rodolphe Thi´ebaut, and Boris P Hejblum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Post- clustering difference testing: valid inference and practical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='13172, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Allison Marie Horst, Alison Presmanes Hill, and Kristen B Gorman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' palmerpenguins: Palmer archipelago (antarctica) penguin data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' R package version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 0, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Baibing Li and Elaine B Martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' An approximation to the f distribution using the chi-square distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Computational statistics & data analysis, 40(1):21–26, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Bing Liu, Yiyuan Xia, and Philip S Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Clustering through decision tree construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In Proceedings of the ninth international conference on Information and knowledge manage- ment, pages 20–29, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Joshua R Loftus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Selective inference after cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' arXiv preprint arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='08866, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Fionn Murtagh and Pedro Contreras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Algorithms for hierarchical clustering: an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 2(1):86–97, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Douglas Steinley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' K-means clustering: a half-century synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' British Journal of Mathe- matical and Statistical Psychology, 59(1):1–34, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Jonathan Taylor and Robert J Tibshirani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Statistical learning and selective inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Pro- ceedings of the National Academy of Sciences, 112(25):7629–7634, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Ulrike Von Luxburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' A tutorial on spectral clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Statistics and computing, 17(4): 395–416, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Chihiro Watanabe and Taiji Suzuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Selective inference for latent block models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Electronic Journal of Statistics, 15(1):3137–3183, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Jesse M Zhang, Govinda M Kamath, and N Tse David.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Valid post-clustering differential analysis for single-cell rna-seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Cell systems, 9(4):383–392, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 15 A Additional proofs and details A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1 Proof of Theorem 2 First, suppose that clusters Ck and Ck′ were fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', not data-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We will show that, in that case, we have R | Z ∼ Beta(q/2, (m − 2)q/2) under H′ 0 (Ck, Ck′), where Z = � ∥P0X∥2 F + ∥P1X∥2 F, P0X ∥P0X∥F , P1X ∥P1X∥F , P2X � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Since X has isotropic covariance while P0, P1, P2 are orthogonal, we see that P0X, P1X, P2X are mutually independent, and so we now only need to show R | � ∥P0X∥2 F + ∥P1X∥2 F, P0X ∥P0X∥F , P1X ∥P1X∥F � ∼ Beta(q/2, (m − 2)q/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Moreover, under H′ 0 (Ck, Ck′), we see that E [P0X] = 0 and E [P1X] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' therefore, by properties of the Gaussian distribution (with mean zero and isotropic covariance), we have (∥P0X∥F, ∥P1X∥F) independent from � P0X ∥P0X∥F , P1X ∥P1X∥F � , and so now it suffices to show that R | ∥P0X∥2 F + ∥P1X∥2 F ∼ Beta(q/2, (m − 2)q/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Finally, ∥P0X∥F and ∥P1X∥F are independent, with σ−2∥P0X∥2 F ∼ χ2 q and σ−2∥P1X∥2 F ∼ χ2 (m−2)q (because P0 is a rank-1 projection matrix while P1 is a rank-(m − 2) projection matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The desired statement follows since, for independent random variables A ∼ χ2 a and B ∼ χ2 b, it follows from properties of the χ2 and Beta distributions that A A+B is independent from A + B, and follows a Beta(a/2, b/2) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Next, we will account for the fact that clusters Ck and Ck′ are data-dependent, by con- ditioning on Cluster(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Let f(R,Z)(r, z) denote the joint density of (R, Z) with respect to the appropriate base measure, when we treat Cluster(X) (and thus Ck, Ck′) as fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' By the work above we can write f(R,Z)(r, z) = fR(r)fZ(z) where fR is the density of the Beta(q/2, (m − 2)q/2) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We then need to calculate the conditional distribu- tion of R, given the event (R, Z) ∈ E, where E is the subset of all values (r, z) such that Cluster(x′(r)) = Cluster∗, for some particular clustering Cluster∗ (and then we will apply the calculation with Cluster∗ = Cluster(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Let E∗ be the set of all such pairs (r, z) (and note that, by the construction of clustering procedures, (R, Z) ∈ E∗ has positive probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The density of (R, Z) | (R, Z) ∈ E∗ is then given by f(R,Z)|(R,Z)∈E∗(r, z) ∝ f(R,Z)(r, z) · 1(r,z)∈E∗ = fR(r)fZ(z) · 1(r,z)∈E∗, and therefore the conditional density of R is given by fR|Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='(R,Z)∈E∗(r | z) ∝ fR(r) · 1(r,Z)∈E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 16 Returning to our definitions, we see that, letting Cluster∗ = Cluster(X), (r, Z) ∈ E∗ ⇔ r ∈ S′, and therefore, we have calculated that the distribution of R conditional on Z and on Cluster(X) is fR|Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='(R,Z)∈E∗(r | z) ∝ fR(r) · 1r∈S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' This proves the desired result about the distribution of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The validity of the p-value P ′ follows as an immediate consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2 Proof of Proposition 1 Let v = 1C1 |C1| − 1C2 |C2|, specializing the construction from before to the case K = 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', we are comparing clusters k = 1 and k′ = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Define φ = � r 1 − r · ∥v∥2 · ∥P1X∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Then we can solve for r, r = φ2 φ2 + ∥v∥2 2 · ∥P1X∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We compute x′(r) = �√r · P0X ∥P0X∥F + √ 1 − r · P1X ∥P1X∥F � � ∥P0X∥2 F + ∥P1X∥2 F + P2X = � φ · P0X ∥P0X∥F + ∥v∥2 · ∥P1X∥F · P1X ∥P1X∥F � � ∥P0X∥2 F + ∥P1X∥2 F φ2 + ∥v∥2 2 · ∥P1X∥2 F + P2X = x(φ) · ∥v∥2 · � ∥P0X∥2 F + ∥P1X∥2 F φ2 + ∥v∥2 2 · ∥P1X∥2 F − P2X · � ∥v∥2 · � ∥P0X∥2 F + ∥P1X∥2 F φ2 + ∥v∥2 2 · ∥P1X∥2 F − 1 � , where for the last step we apply the calculation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Furthermore, in the case K = 2, we can verify that P2 = 1n1⊤ n n , the projection to the span of 1n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Therefore, we can write x′(r) = a · x(φ) + 1n · b⊤, where a = ∥v∥2 · � ∥P0X∥2 F + ∥P1X∥2 F φ2 + ∥v∥2 2 · ∥P1X∥2 F and b = − � ∥v∥2 · � ∥P0X∥2 F + ∥P1X∥2 F φ2 + ∥v∥2 2 · ∥P1X∥2 F − 1 � X⊤1n n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' By our assumption on the clustering procedure, we therefore have Cluster(x′(r)) = Cluster(x(φ)), and so r ∈ S′ if and only if φ ∈ S, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 17 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='3 Implementation details In our experiments, the three variants of Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', using the true σ, or the estimates ˆσall or ˆσclustered) are run using the code provided by Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022] in the R package clusterpval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We now give details for implementation of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='1 Notes on computation in the K = 2 case Computing the p-value P ′ = 1 − FBeta(q/2,(m−2)q/2)(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' S′) for our method often requires extremely precise calculations of the CDF of the Beta distri- bution due to the truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' In order to approximate this p-value, we first convert the Beta distribution to an F dis- tribution, then use Li and Martin [2002]’s Shrinkage Factor Approximation method, which offers a chi-square approximation for the F distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Specifically, we first use the equiv- alence FBeta(k/2,ℓ/2)(t) = FFk,ℓ � t/k (1 − t)/ℓ � , where Fk,ℓ is the F distribution with k, ℓ degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Next, to estimate the CDF in the tail of the F distribution, we use Li and Martin [2002]’s approximation, FFk,ℓ(t) ≈ Fχ2 k � 2ℓ + kt 3 + k − 2 2ℓ + 4kt 3 kt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Note that Li and Martin [2002]’s method is accurate in the regime where k is finite while ℓ → ∞, which is appropriate to our setting as we apply the approximation with k = q and ℓ = (m − 2)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Combining all these calculations means that we can approximate P ′ via the CDF of a truncated χ2 distribution, P ′ ≈ 1 − Fχ2q � 2(m − 2)q + (m−2)qR 3(1−R) + q − 2 2(m − 2)q + 4(m−2)qR 3(1−R) (m − 2)qR 1 − R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' ˜S′ � where ˜S′ = � 2(m − 2)q + (m−2)qr 3(1−r) + q − 2 2(m − 2)q + 4(m−2)qr 3(1−r) (m − 2)qr 1 − r : r ∈ S′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Finally, we use the TChisqRatioApprox function from Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' [2022]’s R package clusterpval for this remaining calculation with the truncated χ2 distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='2 Importance sampling algorithm for the K ≥ 2 case For the setting where K > 2, or where K = 2 but we do not have an exact characterization of the truncation set S′, we use importance sampling to approximate the p-value P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 18 In order to calculate this p-value, we need to be able to perform a calculation of the form 1 − FBeta(q/2,(m−2)q/2) (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' S′) = PR′∼Beta(q/2,(m−2)q/2) {R′ > r | R′ ∈ S′} = PR′∼Beta(q/2,(m−2)q/2) {R′ > r, R′ ∈ S′} PR′∼Beta(q/2,(m−2)q/2) {R′ ∈ S′} , and then apply this calculation at the value r = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' We estimate the numerator and denom- inator simultaneously, using importance sampling with the proposal distribution TN(R, α2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 0, 1), which is the truncated normal distribution—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=', the N(R, α2) distribution truncated to the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Our procedure is: Draw R(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' , R(N) iid∼ TN(R, α2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 0, 1), for N = 8000 draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Compute importance weights w(R(i)) = fBeta(q/2,(m−2)q/2)(R(i)) fTN(R,α2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='0,1)(R(i)) for i ∈ [N], where fBeta(q/2,(m−2)q/2) and fTN(R,α2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='0,1) denote the densities of the respective distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' Estimate P ′ ≈ �N i=1 w(R(i)) · 1 � R(i) > R, R(i) ∈ S′� �N i=1 w(R(i)) · 1 {R(i) ∈ S′} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' The tuning parameter α is adjusted based on empirical performance—specifically, we choose α so that 1 N �N i=1 1{R(i) ∈ S′} ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content='5, to ensure that our proposal distribution TN(R, α2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 0, 1) is neither too concentrated nor too dispersed to accurately approximate the truncated target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQfFTQk/content/2301.12999v1.pdf'} diff --git a/l9FJT4oBgHgl3EQfZixR/content/tmp_files/2301.11532v1.pdf.txt b/l9FJT4oBgHgl3EQfZixR/content/tmp_files/2301.11532v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..927b5ff1ba29628eab1a3f8215b3fd5e3b15d72f --- /dev/null +++ b/l9FJT4oBgHgl3EQfZixR/content/tmp_files/2301.11532v1.pdf.txt @@ -0,0 +1,2393 @@ +arXiv:2301.11532v1 [quant-ph] 27 Jan 2023 +On classical simulation algorithms for noisy Boson Sampling +Changhun Oh ∗1, Liang Jiang †1, and Bill Fefferman ‡2 +1Pritzker School of Molecular Engineering, University of Chicago, Chicago +2Department of Computer Science, University of Chicago, Chicago +January 30, 2023 +Abstract +We present a classical algorithm that approximately samples from the output distribution +of certain noisy Boson Sampling experiments. This algorithm is inspired by a recent result of +Aharonov, Gao, Landau, Liu and Vazirani and makes use of an observation originally due to +Kalai and Kindler that the output probability of Boson Sampling experiments with a Gaussian +noise model can be approximated by sparse low-degree polynomials. This observation alone does +not suffice for classical sampling, because its marginal probabilities might not be approximated +by sparse low-degree polynomials, and furthermore, the approximated probabilities might be +negative. We solve this problem by employing the first quantization representation to give an +algorithm for computing the marginal probabilities of these experiments. +We prove that when the overall noise rate is constant, the algorithm runs in time quasi- +polynomial in the number of input photons N and accuracy. When the overall noise rate scales +as 1 − xγ +1 for constant x1 and γ = Ω(log N), the running time becomes polynomial. +Furthermore, we study noisy Boson Sampling with practically relevant noise models such as +partial distinguishability and photon loss. We show that the same technique does not immediately +apply in these settings, leaving open the possibility of a scalable demonstration of noisy quantum +advantage for these noise models in certain parameter regimes. +1 +Introduction +We have recently seen the first claims of experimental quantum advantage using both the random +circuit sampling proposal implemented with superconducting qubits [AAB+19, WBC+21] as well +as the Gaussian Boson Sampling proposal implemented in a linear optical architecture [ZWD+20, +ZDQ+21, MLA+22]. Such quantum advantage is a necessary step on the path toward building scal- +able, fault-tolerant quantum computers. In addition quantum advantage is a fundamental milestone +in its own right, where it can be interpreted as providing an experimental violation to the Extended +Church-Turing thesis (see e.g., [BV93, AA11]). +With such an important milestone it is critical to analyze our evidence for believing that such +experiments are classically intractable. Here much is still unknown, and to improve this situation +we must both bolster the classical hardness arguments as well as develop new classical simulation +algorithms to challenge our assumptions. In this work, inspired by a recent algorithm for simulating +logarithmic depth noisy random quantum circuits due to Aharonov, Gao, Landau, Liu and Vazirani +∗changhun@uchicago.edu +†liangjiang@uchicago.edu +‡wjf@uchicago.edu + +[AGL+22] and earlier work due to Gao and Duan [GD18], we develop a classical algorithm to approx- +imately sample from the output distribution of certain noisy Boson Sampling experiments. Much +like the Aharonov et al. result, we do not expect that this algorithm is practical in its present form. +That is, it most likely will not “spoof” present day Boson Sampling experiments in a reasonable +amount of classical running time, due mainly to the inefficient scaling of the algorithm’s running +time with the noise rate. Nonetheless we are able to prove that our algorithm works for a Gaussian +noise model proposed in past work by Kalai and Kindler [KK14], which like depolarizing noise in the +Aharonov et al. algorithm has the property that the noisy output distribution eventually converges +to the uniform distribution. We then discuss the prospects for extending this algorithm to other +noise models, including photon loss and partial distinguishability. +1.1 +Putting recent simulation results in context +After more than a decade of research in this area, there now is a body of work to support the +classical intractability of these quantum advantage experiments. +This evidence comes primarily +from complexity theoretic arguments proving that no efficient classical algorithm can simulate these +experiments in the asymptotic regime as the system size increases under reasonable complexity +theoretical assumptions (see e.g., [TD04, BJS10, AA11, BFNV19, AC17, AG19, BFLL21, KMM21, +DMV+22, HE22]). +One potential challenge to these hardness arguments comes from uncorrected noise, which is +perhaps the defining characteristic of near-term quantum computational experiments. This noise +degrades the quantum signal as the system size increases. Consequently it is reasonable to expect that +classical algorithms could potentially take advantage of this weakness to simulate noisy experiments +at a sufficiently large system size. While this has been an active subject of research with many +results [KK14, BMS17, GD18, RSGP18, Shc19, GPRS19, NJF20, TTT21, QBQGP20, ONFJ21, +VNL+21], we have arguably not yet seen a classical algorithm that simulates state-of-the-art quantum +advantage experiments using a comparable amount of computational resources (see e.g., [Aar22] for +more discussion on this point). +Indeed, at the moment there is hope that near-term quantum +advantage experiments operate in a “Goldilocks” regime in which the system size is large enough +to be classically intractable to simulate, but not so large that uncorrected noise overwhelms the +quantum signal1. +Short of spoofing fixed size near-term experiments, one can ask if classical algorithms can effi- +ciently simulate noisy quantum advantage experiments in the asymptotic limit as the system size +scales. Such a classical algorithm would rule out a fully scalable demonstration of quantum advantage +with uncorrected noise. Indeed, such a scalable demonstration would be of great interest, but until +very recently was thought to be infeasible. This pessimism was mainly due to two reasons. The first +major reason came from a foundational result due to Aharonov et al. from the late 1990’s [ABOIN96] +showing that the total variation distance between the output distribution of a noisy quantum circuit +with circuit depth d and the uniform distribution is upper bounded by 2−O(d) 2. This early result +already rules out scalable quantum advantage for any depth that is super-logarithmic in the system +size. To make things worse, there is numerical evidence that the output distribution of most noisy +random quantum circuits converges to the uniform distribution at the even faster rate of 2−O(n·d) +(see [BSN17] and the corresponding discussion in [BFLL21]). This rapid convergence would rule out +scalable, noisy quantum advantage at any depth. +1This “Goldilocks” regime is also important to enable classical verification techniques such as the cross-entropy +benchmark, which currently requires exponential time on a classical computer. +2Strictly speaking this upper bound applies to any quantum circuit that is subject to depolarizing noise with constant +noise rate, although more recent results have clarified that it is widely applicable to a variety of reasonable noise models +(see e.g., [GD18, DNS+22]). +2 + +The second major reason for pessimism came from a statistical property of the output distribution +of random quantum experiments known as “anticoncentration”, which is useful in the theoretical +hardness analysis of these systems (see e.g., [AA11] for more discussion). Anticoncentration is known +to be a property of any ensemble of random quantum circuits that forms an approximate unitary +two-design (see e.g., [BHH16, BVHS+18, HBVSE18]). For D-dimensional local random quantum +circuits with Haar random gates this property first arises at depth n1/D and this is believed to +be optimal [BHH16, HM18]. Consequently, if the spatial locality is constant, then combining this +result together with the upper bound of Aharonov et al. [ABOIN96] we find that the noisy output +distribution of such circuits is inverse superpolynomially close to the uniform distribution, which +again rules out noisy, scalable quantum advantage in this regime. +However, in the last two years new results were proven which offered some brief hope that random +quantum circuits might be able to achieve such a scalable noisy advantage at precisely logarithmic +depth. First the results of Dalzell et al. and Barak et al. proved that random quantum circuits with +Haar random two-qubit gates anticoncentrate at logarithmic depth3 [DHJB22, BCG21]. Crucially, +these papers directly analyze the anticoncentration property of the ensemble of circuits without +relying on the approximate two-design property. Moreover, these results are optimal, in the sense +that sublogarithmic depth random quantum circuits with two-qubit Haar random gates are known +not to anticoncentrate [DHJB22, DNS+22]. +In addition, a result of Deshpande et al. proved that the total variation distance between the +output distribution of most random quantum circuits and the uniform distribution is lower bounded +by a quantity that scales as 2−O(d), matching the Aharonov et al. upper bound of 2−O(d) [DNS+22]. +Putting these two results together gave rise to the (as it turns out, fleeting) hope that logarithmic +depth random quantum circuits with Haar random gates could offer a “sweet-spot” regime in which +the depth was both sufficient to have anticoncentration yet shallow enough so that uncorrected noise +does not overwhelm. +1.2 +The Aharonov et al. random circuit simulation algorithm +This hope was very recently ruled out by a result of Aharonov et al. [AGL+22] which presents an +efficient algorithm for approximately sampling from the output distribution of noisy random circuit +ensembles that anticoncentrate, modulo the “gate-set orthogonality” constraint which is satisfied +e.g., by two qubit Haar random gates. This algorithm follows up on earlier work of Gao and Duan, +which achieved the same accuracy in quasi-polynomial time [GD18]. +Owing to the requirement of anticoncentration, these algorithms are useful for simulating random +quantum circuits with depth that scales at least logarithmically in the system size 4. In particular +at logarithmic depth the earlier Aharonov et al. result implies that sampling from the uniform dis- +tribution achieves total variation distance 1/2O(d) = 1/poly(n) [ABOIN96]. However, approximating +the noisy output distribution by the uniform sampler cannot reduce the total variation distance by +increasing the running time because the approximate sampler is fixed. By contrast this new result +is stronger and gives a classical algorithm that can achieve any total variation distance parameter ǫ +with a running time that scales as poly(1/ǫ). +The key observation behind this algorithm is that the output (or marginal) probabilities of +3Strictly speaking this is proven for 1D and all-to-all connectivities, but is believed to hold for intermediate regimes +such as a 2D grid. +4It still remains possible to prove hardness of sampling results for random quantum circuits with Haar random +gates at sublogarithmic depths without needing anti-concentration, although it is likely that new ideas will be required. +Additionally there exist ensembles of random circuits that anticoncentrate at constant depths [HHB+20] by using a +distribution over gates that is very different from Haar random. It remains unclear if the Aharonov et al. algorithm +can be adapted to simulate such ensembles in the presence of noise. +3 + +noisy random circuits with a constant rate of depolarizing noise per gate can be expressed as the +sum of polynomially many dominant Fourier coefficients with exponentially many other Fourier +coefficients that are highly suppressed due to the noise. In other words, the output probability of noisy +random circuits can be approximately represented by sparse Fourier coefficients with a small error +occurring by discarding other Fourier coefficients. Using sparsity of the Fourier coefficients involved +in the output (or marginal) probabilities, one can efficiently approximate the output (marginal) +probabilities, which enables us to sample from the distribution. We emphasize that it is crucial that +any output probability of a given circuit has to be described by the same polynomially many Fourier +coefficients to guarantee that all the marginals can also be efficiently computed. The latter is not +obvious because the marginal probabilities can be the sum of exponentially many probabilities, which +may eventually require an exponential number of Fourier coefficients even though each probability +has a sparse Fourier description. In addition, since the approximated distribution can be a quasi- +probability distribution, i.e., it can be negative, it was crucial to exploit a technique proposed in +[BMS17], which enables us to approximately sample from a proper probability distribution when the +quasi-probability distribution is sufficiently close to the noisy probability distribution. +1.3 +Noisy Boson Sampling +Let us turn our attention to Boson Sampling [AA11], which is our main focus in the present work. +The main question of the present work is whether the same type of Aharonov et al. classical algorithm +[AGL+22] works to simulate noisy Boson Sampling. Interestingly, even before studies on the sparsity +of Fourier coefficients in noisy random circuit sampling [GD18, AGL+22], Kalai and Kindler already +pointed out that low-degree polynomials can approximate the output probability of noisy Boson +Sampling with a particular choice of noise type, which transforms a given linear-optical circuit +U → √xU + √1 − xY , where Y is a random Gaussian matrix and 1 − x is the noise rate. +To +avoid confusion we emphasize that 1 − x is the noise rate not x, which is the case in [AGL+22]. +After Kalai and Kindler’s analysis on a mathematically appealing noise model, several subsequent +works studied more physical noise types such as partial distinguishability using similar techniques +[RMC+18, RSGP18, Shc19, MGPRT19]. However, the previous works did not provide a classical +sampler to exploit the low-degree polynomial approximation (See Sec. 1.4 for more details). +In this work, we present a classical algorithm that approximately simulates noisy Boson Sampling +with noise studied in [KK14] using sparsity of low-degree polynomials and the method in [BMS17]. In +particular, assuming Haar-random linear-optical circuits (instead of anticoncentration), the classical +algorithm’s running time is given by quasi-polynomial in the system size and accuracy for an overall +constant noise level 1 − x ∈ (0, 1]: +Theorem 1. Consider an M-mode Fock-state Boson Sampling with N single photons and a linear- +optical circuit with a global Haar-random unitary with M = ω(N 5). If there is an overall constant +circuit noise, we can classically simulate collision-free outcomes of the noisy Boson Sampling with +running time N O(log N,log ǫ−1,log δ−1) within total variation distance ǫ for 1−δ portion of Haar-random +unitary matrices. +The main reason that the running time is quasi-polynomial is that the noise rate is assumed +constant for the entire circuit instead a constant level of noise per gate as in [AGL+22], where noise +scales with the system size. To introduce a similar effect, we now consider the case where the total +noise rate scales as 1 − xγ +1 with γ = Ω(log N) and a constant x1 ∈ [0, 1) and show for this case that +the running time becomes polynomial: +Corollary 2. Consider an M-mode Fock-state Boson Sampling with N single photons and a linear- +optical circuit with a global Haar-random unitary with M = ω(N 5). If there is an overall circuit +4 + +noise 1 − xγ +1 with a constant x1 ∈ [0, 1) and γ = Ω(log N), we can classically simulate collision-free +outcomes of the noisy Boson Sampling with running time poly(N, ǫ−1, δ−1) within total variation +distance ǫ for 1 − δ portion of Haar-random unitary matrices. +Note that whereas [AGL+22] introduces noise for each gate, but also requires anticoncentra- +tion, we introduce the noise for the entire circuit at once with global Haar-random circuits but do +not explicitly require anticoncentration. It remains open to generalize our result as the setting in +[AGL+22]. +The key idea to channel the sparse low-degree polynomial approximation from [KK14] to sampling +is to employ the first quantization representation of Boson Sampling. We show that the marginals +of approximated quasi-probability distribution for the first quantization representation can also be +efficiently computed by sparse polynomials, and consequently the technique from [BMS17] can be +applied for sampling. Thus, it closes the gap between the approximate computation of probability +and sampling for circuit noise. Intriguingly, applying the same sparsity technique to physical noise +models such as partial distinguishability and photon loss hits barriers to finding a corresponding +classical sampler. First, for partial distinguishability noise, the barrier is that even after introducing +noise and approximating the probability with similar polynomials, computing the output probability +distribution still costs an exponential time. Thus, a naive approach does not successfully reduce the +complexity by exploiting the noise. Second, for photon loss, the barrier is that we need to choose +a large degree to suppress the approximation error, which implies that the algorithm might work +only for a large photon-loss regime. However, the large photon-loss regime can already be classically +simulated because lossy single-photon states are already sufficiently close to classical states (much +like the convergence of the output probability distribution to uniform at superlogarithmic depth for +qubit cases [ABOIN96]) [OB18, GPRS19, QBQGP20]. Thus, the sparsity technique does not provide +any benefits over the existing methods. +Our analysis of three different types of noise clearly reveals that the different behavior of output +distributions against different noise types poses difficulties in the generalization of the same technique +for more general noise models. Interestingly, both the output distribution of random circuits with +depolarizing noise and that of Boson Sampling with circuit noise converge to the uniform distribution, +while those of Boson Sampling with partial distinguishability and photon loss do not. This might +indicate that the current technique implicitly relies on a certain property of the noise model, which is +related to convergence to the uniform distribution, and that different noise models might require an +additional technique or perhaps even lead to a scalable demonstration of noisy quantum advantage. +We stress, however, that we do not prove such a formal connection to the uniform distribution in +this work, but leave this as an intriguing open direction for future research. +1.4 +Relation to previous results on Boson Sampling +As mentioned in the previous section, the low-degree polynomial approximation techniques for noisy +Boson Sampling have been discussed even before [GD18, AGL+22]. More specifically, Kalai and +Kindler showed that the output probabilities of noisy Boson Sampling can be approximated by +sparse low-degree polynomials under the assumption of Haar-randomness of the linear optical circuit +matrix (this seems analogous to the anticoncentration requirement of Aharonov et al. [AGL+22]) +[KK14]. Nevertheless, it is not obvious how to approximately sample from the output distribution +described by the sparse low-degree polynomials because the approximated distribution might not be +a proper probability distribution and it is not guaranteed that its marginal probabilities can also +be described by sparse polynomials. The latter is because it has to be shown that any probabilities +can be described by the same sparse low-degree polynomials. Our contribution is to channel the +low-degree polynomial approximation to a classical sampling algorithm using the first quantization +5 + +method and marginal-based sampler. +Several subsequent works studied more physical noise types such as partial distinguishability +[RMC+18, RSGP18, Shc19, MGPRT19] while their approaches also encounter the same obstacles +to finding a classical sampler 5. +In particular, [RMC+18] observed that the output probability +of partial distinguishable Boson Sampling can be approximated by low-degree polynomials, which +guarantees that the total variation distance can be made small by choosing an appropriate degree. It +was also claimed that each polynomial can be efficiently approximated (not exactly computed, unlike +[KK14, AGL+22]). Nevertheless, it did not analyze the effect of the approximation of polynomials and +did not provide a provable classical sampler; instead, it considered the Metropolis algorithm, which is +heuristic [NSC+17]. Thus, they did not provide a provable classical sampler for partial distinguishable +Boson Sampling. We show that indeed it is not immediately straightforward to construct a classical +sampler that exploits the low-degree polynomial approximation for partial distinguishable noise. +Finally, there have been extensive studies on the effect of photon loss on Boson Sampling [AB16, +OB18, RSGP18, GPRS19, Shc19, QBQGP20, ONFJ21], while a similar technique has not been +considered 6. Our analysis shows that the previous techniques that approximate lossy single photons +by classical states provide a better approximation error than a naive approach using the low-degree +polynomial approximation. +1.5 +Concluding remarks +We finally remark on several points that were not addressed in the present work and open questions. +• Efficient classical algorithms for physical noise models. As we claimed, the low-degree +polynomial approximation does not immediately lead to an efficient classical sampler for par- +tial distinguishability and photon loss, which are the most crucial noise models in practice +[ZWD+20, ZDQ+21, MLA+22]. It remains an open question to improve the technique to find +an efficient classical sampler for those noise models. For photon loss case in particular, when +the output photon number scales as Θ( +√ +N), the total variation distance of the classical algo- +rithms in [OB18, GPRS19, QBQGP20] to the lossy output probability distribution is fixed as a +constant, and it cannot be reduced by increasing the running time of the algorithms. Finding +a classical algorithm that can efficiently reduce the approximation error as [AGL+22] and our +result for Gaussian noise is another open question. +• Lifting the assumption of global Haar-randomness. In the present work, we have as- +sumed that the linear-optical circuits are constructed to be global Haar-random7, which is +a standard assumption for the hardness of Boson Sampling [AA11]. +On the other hand, +the recent Boson Sampling experiments have not implemented global Haar-random circuits +[ZWD+20, ZDQ+21, MLA+22, OLFJ22]. Also, the recent result for random circuits [AGL+22] +assumed anticoncentration with consideration of depth and noise effect per gate. Extending our +results further with a less stringent assumption is another future work, such as replacing the +global Haar-random assumption with anticoncentration. Note that whereas random circuits in +[AGL+22] with gate-set orthogonality enjoy the symmetry between different outcomes when +averaged over ensembles, Boson Sampling outcomes generally do not have such an apparent +5While [Shc19] claimed that there is an efficient classical sampler, this was not completely proved to the best of our +knowledge. +6[Shc19] has considered the combined effect of loss and dark count with assuming that the total photon number is +preserved by dark count effect, which is not satisfied solely by photon loss. +7Unlike random circuit sampling using qubits, the dimension of the unitary matrix for global Haar-random is +polynomial in the system size. Thus, it is not an unrealistic assumption in practice (see e.g., [RCOL17]). +6 + +symmetry, which hinders us from analyzing the upper bound of total variation distance except +for the global Haar-random case. +• Anticoncentration of Boson Sampling. Unlike random circuit sampling, we have less un- +derstanding of anticoncentration in Boson Sampling such as how much circuit depth is required +to attain anticoncentration property with what kinds of an ensemble of linear-optical circuits. +Even whether the anticoncentration property is achieved with global-Haar random remains a +conjecture to the best of our knowledge [AA11] despite interesting recent progress (see e.g., +[Nez21]). +• Gaussian Boson Sampling. We have considered Fock-state Boson Sampling only while the +quantum advantage experiments employed Gaussian Boson Sampling [HKS+17, DMV+22], +which is a variant of Fock-state Boson Sampling. While we expect a similar result to hold, we +leave it as an open question. +• Practical consideration. As emphasized before, the proposed algorithm assumes an asymp- +totic regime of noisy Boson Sampling and we do not expect the algorithm to spoof finite-size +near-term experiments. Specifically, for a small noise rate x1 ≈ 1, the degree of the polyno- +mial of the running time is given by 1/ log(1/x1) ≈ 1/(1 − x1), which makes the algorithm +impractical. In fact, the recent result in [AGL+22] observed the same issue, i.e., the degree of +the polynomial is a large constant 1/γ, where γ is the noise rate per gate in their notation. +An interesting future work is to improve the algorithm to be applicable to finite-size Boson +Sampling. +2 +Fock-state Boson Sampling in first quantization +Let us consider the standard Fock-state Boson Sampling [AA11]. The basic setup is to prepare N +single photons and to inject the photons into an M-mode linear-optical circuit ˆU, characterized by +an M × M unitary matrix, where M = poly(N). +We then measure the number of photons for +each output mode, which gives rise to a measurement outcome m ∈ ZM +≥0 with �M +i=1 mi = N, where +mi represents the number of photons at the ith output mode. We now describe the dynamics by +introducing the first quantization representation, which enables us to analyze marginal distributions +later easily. First, we write the input state as +1 +√ +N! +� +σ∈SN +|σ(1), . . . , σ(N)⟩, +(1) +where SN represents the permutation group for N elements, which accounts for the symmetrization +of N photons due to bosons’ indistinguishability nature. Thus, the density matrix of the input state +is written as +1 +N! +� +σ,ρ∈SN +|σ(1), . . . , σ(N)⟩⟨ρ(1), . . . , ρ(N)|. +(2) +After applying beam splitter network ˆU, we obtain the output state +1 +N! +� +σ,ρ∈SN +ˆU ⊗N|σ(1), . . . , σ(N)⟩⟨ρ(1), . . . , ρ(N)| ˆU †⊗N, +(3) +7 + +where the linear-optical operation, characterized by an M × M unitary matrix U, transforms the +state as +ˆU|i⟩ = +M +� +j=1 +Uij|j⟩. +(4) +Finally, we measure in each photon’s position r ∈ ZN +≥0, whose probability is written as +p(r) = 1 +N! +� +σ,ρ∈SN +⟨r| ˆU ⊗N|σ(1), . . . , σ(N)⟩⟨ρ(1), . . . , ρ(N)| ˆU †⊗N|r⟩ = 1 +N! +� +σ,ρ∈SN +� N +� +i=1 +Uσ(i),riU ∗ +ρ(i),ri +� +. +(5) +Especially for collision-free outcomes r, i.e. at most a single photon clicks for each output mode +(equivalently all ri’s are distinct), the probability reduces to +p(r) = |PerUN,r|2 +N! +, +(6) +where UN,r is the N × N submatrix of a unitary matrix U obtained by selecting the first N rows, +which accounts for the input photons, and r’s columns. +We first clarify the notation of outcomes m, r, and z and their relations, the latter of which +will be defined now. First, we will define z ∈ ZN +≥0 as the ordered vector of r in the nondecreasing +order, i.e., z1 ≤ z2 ≤ · · · ≤ zN. Notice that different r’s may reduce to the same vector z, which is +because we cannot distinguish which input photons correspond to which output photons in principle +due to the indistinguishability. Because of the symmetry, the different r’s that correspond to the +same z have the same probability. The photon number vector m’s elements mi’s can be obtained by +counting the number of i’s in z. Hence, we can write the probability by abusing the notation of p(·) +p(z) = +� +σ∈SN +p(σ(r)) = |PerUN,r|2. +(7) +We will often abuse the notation of the probability p(z), p(r) and p(m), which can be uniquely +identified by using different arguments z, r and m. +Especially when another distribution q(z) has the same property, namely q(z) = � +σ∈SN q(σ(r)), +the total variation distance between p(z) and q(z) with ordered outcomes and that between p(r) +and q(r) with unordered outcomes are equal: +∥p(z) − q(z)∥1 = ∥p(r) − q(r)∥1. +(8) +The property will play an important role for approximate sampling. +We will focus on the (strong) collision-free regime M = ω(N 5), where an N × N submatrix of an +M × M Haar-random unitary matrix U can be approximated by complex random Gaussian matrix +Z such that (UN,z)ij ≈ Zij/ +√ +M with Zij ∝ N(0, 1) with scaling factor 1/ +√ +M [AA11]. Also, we +will focus on simulating the probability distribution over collision-free outcomes, which suggests that +ri ̸= rj, or equivalently zi ̸= zj, for all i ̸= j ∈ [N], or mi ∈ {0, 1} for all i ∈ [M]. Since we do not +aim to simulate collision outcomes, we will set all the collision outcomes to be c, i.e., we treat them +as the same outcome c. +8 + +3 +Low-degree polynomial approximation with circuit noise +3.1 +Noise sensitivity and low-degree polynomial approximation +Let us consider the effect of noise on Boson Sampling output probability distributions. The first type +of Gaussian noise we consider is the noise on circuit unitary U. Although this type of noise might not +be physically or experimentally relevant, it provides profound insights into the noise sensitivity of +the output probability of Boson Sampling. More specifically, the introduced noise changes a unitary +matrix as U → √xU +√1 − xY , where Y is an M ×M complex random Gaussian matrix, x ∈ [0, 1], +and 1 − x is the noise rate [KK14]. We remark that the definition of the Gaussian noise seems to +be unphysical in the sense that for a single instance Y , √xU + √1 − xY is not necessarily unitary +and, furthermore, its spectral norm can be larger than 1. However, we show that the noisy output +probability distribution is a proper probability distribution (see Appendix A). +In this section, we will recall the result from [KK14] that a Boson Sampling probability dis- +tribution under this type of noise can be approximated in total variation distance by low-degree +polynomials. To this end, let us consider an output probability +p(z) = |PerUN,z|2 = |Per(Z)|2 +MN +, +(9) +where Z ≡ +√ +MUN,z is the rescaled N × N submatrix of unitary U corresponding to the outcome +z. We used the fact that a submatrix of a large Haar-random unitary matrix (M = ω(N 5)) can +be approximated by a complex random Gaussian matrix whose elements follow complex normal +distribution N(0, 1) [AA11]. Following [KK14], we can expand the absolute-squared permanent as +the sum of orthogonal polynomials: +|Per(Z)|2 = +N +� +k=0 +f =2(N−k), +(10) +where the degree 2(N − k) polynomials f =2(N−k) satisfy the following orthogonal relations +EZ[|f =2(N−k)|2] = (N!)2, +EZ[f =2(N−k1)f =2(N−k2)∗] = 0, +for k1 ̸= k2. +(11) +Here, the average EZ[·] is taken over complex Gaussian random matrices Z. To see this, let us expand +the absolute-squared permanent of a complex random Gaussian matrix as +|Per(Z)|2 = +� +σ,ρ∈SN +N +� +i=1 +zσ(i),iz∗ +ρ(i),i +(12) += +� +σ,ρ∈SN +� +i∈T +(zσ(i),iz∗ +σ(i),i) +� +i∈T c +(zσ(i),iz∗ +ρ(i),i) +(13) += +� +σ,ρ∈SN +� +i∈T +(1 + h2(zσ(i),i)) +� +i∈T c +(zσ(i),iz∗ +ρ(i),i) +(14) += +� +σ,ρ∈SN +� +R⊂T + + � +i∈T\R +h2(zσ(i),i) +� +i∈T c +zσ(i),iz∗ +ρ(i),i + + , +(15) +where we defined T ⊂ [N] as the set of indices such that σ(i) = ρ(i) for given permutations σ and +ρ and h2(z) ≡ zz∗ − 1. An important fact is that {1, z, z∗, h2(z)} forms an orthogonal basis, i.e., +EZ[f1f ∗ +2 ] = 0 if f1 and f2 are different functions out of the basis, and they are eigenvectors of the +9 + +noise operator Tx[f](z) ≡ Ey[f(√xz + √1 − xy)] with y being the complex random Gaussian noise +N(0, 1), namely, +1 → 1, +z → √xz, +z∗ → √xz∗, +and +h2(z) → xh2(z). +(16) +Here, we assign a degree for each by adding 1 for z or z∗ and 2 for h2. Thus, the degree of the term +in the parenthesis in Eq. (15) is 2(|T| − |R|) + 2(N − |T|) = 2(N − |R|). We further partition these +terms according to the image R′ of R under σ and ρ. Thus, we denote by σ′ and ρ′ the restriction of +σ and ρ on the complement of R, namely these are one-to-one functions from Rc and [N] \ R′. Let +S(σ′, ρ′) ⊂ Rc be the set of indices on which they agree. Using some algebra, one can show that the +degree 2(N − k) part is given by +f =2(N−k) = +� +R,R′⊂[N]: +|R|,|R′|=k +� +σ∈Sk: +R→R′ +� +σ′,ρ′∈SN−k: +Rc→R′c +� +i∈S(σ′,ρ′) +h2(zσ′(i),i) +� +i∈Rc\S(σ′,ρ′) +zσ′(i),iz∗ +ρ′(i),i +(17) += +� +R,R′⊂[N]: +|R|,|R′|=k +k! +� +σ′,ρ′∈SN−k: +Rc→R′c +� +i∈S(σ′,ρ′) +h2(zσ′(i),i) +� +i∈Rc\S(σ′,ρ′) +zσ′(i),iz∗ +ρ′(i),i. +(18) +Hence, we can rewrite the absolute-squared permanent as +|Per(Z)|2 = +N +� +k=0 +f =2(N−k), +(19) +as desired. +Let us introduce the noise. +As shown from Eq. (16), the noise operator Tx[f](z) introduces +additional prefactor xN−k for each 2(N − k)-degree polynomial, i.e., +f =2(N−k) → xN−kf =2(N−k). +(20) +Hence, the noisy output probability becomes +˜p(z) = +1 +MN +N +� +k=0 +xN−kf =2(N−k). +(21) +Also, the following relations can be easily checked from the orthogonality of basic elements [KK14]: +EZ[f =2(N−k1)f =2(N−k2)∗] = (N!)2δk1,k2, +(22) +where the average is over the complex random Gaussian matrix. Here, (N!)2 factor comes by counting +the number of orthogonal polynomials in f =2(N−k), +�N +k +�2 +(k!)2((N − k)!)2 = (N!)2, +(23) +where +�N +k +�2 are from the number of choices for R, R′ and (k!) from the coefficients, and ((N − k)!)2 +from the number of choices for σ′, ρ′. The noisy probability expression suggests that the high-degree +polynomials are more sensitive to the noise and they are suppressed exponentially in their degree. +Also, Eq. (22) shows that the contribution from high-degree polynomials does not scale as their +degree. Therefore, we will approximate the output probability by truncating the polynomials by +10 + +setting a cutoff of the degree. We will show in Sec. 3.3 that the complexity of computing f =2(N−k) +is determined by the degree 2(N − k). +More concretely, if we choose the maximum degree as 2l and truncate higher-degree contributions, +we obtain an approximated probability written by the sum of low-degree polynomials +¯q(z) ≡ +N +� +k=N−l +xN−kf =2(N−k). +(24) +Then the approximation error is written as +˜p(z) − ¯q(z) = +1 +MN +N−l−1 +� +k=0 +xN−kf =2(N−k). +(25) +3.2 +Bounds for the total variation distance +So far, we have focused on the approximation error of a single output probability. Using this, we +will derive the upper bound of the total variation distance of the full probability distribution, +∆ ≡ +� +z +|˜p(U, z) − ¯q(U, z)| = +� +z∈cf +|˜p(U, z) − ¯q(U, z)| + |˜p(U, c) − ¯q(U, c)|, +(26) +where cf represents the set of all collision-free outcomes and the first sum is over cf and collision +outcome c. Here, we explicitly expressed the dependency of U. We note that ˜p(U, z) is defined as +1−� +z∈cf ˜p(U, z) (see Appendix A). To find the upper bound of the total variation distance, we first +need to assign the value of ¯q(U, c). For this moment, let us assign this probability as +¯q(U, c) = 1 − +� +z∈cf +¯q(U, z). +(27) +We will show how to make the approximate distribution ¯q satisfy the assumption in Appendix B. +Such an assignment makes the analysis much easier because +|˜p(U, c) − ¯q(U, c)| ≤ +� +z∈cf +|˜p(U, z) − ¯q(U, z)|, +(28) +which is from the assumption and the triangular inequality. Then the average squared total variation +distance is upper bounded as +EU[∆2] ≤ 4EU + + + + � +z∈cf +|˜p(U, z) − ¯q(U, z)| + + +2 + +(29) +≤ 4 +�M +N +� +EU + + � +z∈cf +(˜p(U, z) − ¯q(U, z))2 + + +(30) +≤ 4 +�M +N +�2 +EU +� +(˜p(U, z) − ¯q(U, z))2� +, +(31) +where the average is taken over Haar-random unitaries U. We have used Jensen’s inequality for +the second inequality, and we have used the fact that the average over U gives rise to symmetry to +11 + +possible collision-free outcomes z ∈ cf, the number of which is +�M +N +� +, for the third equality. By using +the low-degree polynomial approximation, its upper bound can be written as +EU +� +(˜p(U, z) − ¯q(U, z))2� += +1 +M2N EU + + +�N−l−1 +� +k=0 +xN−kf =2(N−k) +�2 + +(32) += (N!)2 +M2N +N−l−1 +� +k=0 +x2(N−k) +(33) +≤ (N!)2 +M2N +N−l−1 +� +k=0 +x2(l+1) +(34) += (N − l + 1)x2(l+1)(N!)2 +M2N +, +(35) +where we have used the orthogonality, Eq. (22), replacing the Haar-random unitary average with +random Gaussian matrix average, for the second equality. Finally, +EU +� +∆2� +≤ 4 +�M +N +�2 (N − l + 1)x2(l+1)(N!)2 +M2N +(36) +≤ 4 +�MN +N! +�2 +(N!)2 (N − l + 1)x2(l+1) +M2N +(37) +≤ 4Nx2(l+1), +(38) +where we have used the inequality +�M +N +� +≤ MN/N! for the second inequality. Together with this, we +will use Markov’s inequality +PrU +� +∆ ≥ 1 +√ +δ +� +EU[∆2] +� += PrU +� +∆2 ≥ 1 +δEU[∆2] +� +≤ δ, +(39) +where the probability is over Haar-random unitary matrices. Thus for 1 − δ portion of Haar-random +unitary matrices, the approximation error of low-degree polynomial is upper-bounded by +� +z +|˜p(U, z) − ¯q(U, z)| ≤ 2 +√ +Nxl+1 +√ +δ +. +(40) +Therefore, to bound the error by ǫ > 0, it is sufficient to choose the cutoff of degree l such that +l ≥ log(2 +√ +N/ǫ +√ +δ) +log(1/x) +− 1 = O(log N, log(1/ǫ), log(1/δ)). +(41) +To introduce the noise effect that scales with the system size, we also consider the case where x scales +as x = xγ +1 with a constant x1. Then, the total variation distance bound becomes +2 +√ +Nxl+1 +√ +δ += 2 +√ +Nxγ(l+1) +1 +√ +δ +, +(42) +which implies that it is sufficient to choose the degree as +l ≥ +log 2 +√ +N +ǫ +√ +δ +γ log 1/x1 +− 1. +(43) +12 + +3.3 +Approximate sampling +In the previous section, we have shown that ¯q(z) with an appropriate cutoff of degree l approximates +the noisy distribution ˜p(z) with an error ǫ with high probability 1 − δ. It is worth emphasizing +again that a similar analysis was conducted in [KK14] while it focused on approximating a single +output probability only and did not provide the bound for total variation distance and a classi- +cal approximate sampler of low-degree approximated distribution ¯q(z). +The remaining challenge +from the previous section is to find a classical sampling algorithm from ¯q(z). A caveat is that the +approximated distribution ¯q(z) is not necessarily a proper probability distribution, i.e., it might +have a negative quantity. Nevertheless, the following lemma [BMS17] provides a recipe for dealing +with quasi-probability distribution, which can be straightforwardly generalized to M-level outcomes +instead of binary outcomes: +Lemma 3. (modified) Let ˜p be a probability distribution on MN. If there is an oracle that computes +a function ¯q : MN → R as well as its marginals satisfying � +x ¯q(x) = 1, such that ∥˜p − ¯q∥1 ≤ ǫ, then +there is an algorithm that samples from a probability distribution q using O(MN) calls to the oracle, +such that ∥˜p − q∥1 ≤ 2ǫ. +Here, the marginal is defined as ¯q(x1, . . . , xk) = �M +xk+1,...,xN=1 ¯q(x1, . . . , xN). We have added +an additional assumption � +x ¯q(x) = 1, which results in the approximation error by 2ǫ instead of +4ǫ/(1 − ǫ). Therefore, it suffices to find ¯q whose marginals can be efficiently computed and are close +to ˜p so that it can be used for the lemma for noisy Boson Sampling. The remaining section will show +that ¯q obtained by sparse low-degree polynomials satisfies such conditions. +One immediate difficulty of applying this lemma to Boson Sampling is that a restriction of +an outcome z such that z1 ≤ z2 ≤ · · · ≤ zN makes it difficult to compute its marginals. +To +circumvent such a difficulty, we will consider the unordered outcome vector r introduced with the +first quantization instead of the ordered vector z. While the output vector r without ordering has +a redundancy, it enables us to easily express the marginals since it does not have the restriction of +ordering. Thanks to the symmetry between r and z, we can rewrite it as +p(r) = p(z) +N! = 1 +N! +1 +MN +N +� +k=0 +f =2(N−k)(Z), +(44) +where Z corresponds to the submatrix of U by choosing the first N rows and r’s columns. Our +strategy was to set a cutoff on the degree, i.e., +¯q(r) = 1 +N! +1 +MN +N +� +k=N−l +xN−kf =2(N−k)(Z). +(45) +Note that changing the representation from z to r does not change the simulation error due to the +symmetry and Eq. (8). +We now show that marginals can also be computed using a similar method. +The marginal +13 + +probability of the noiseless distribution is +p(r1, . . . , rj) = 1 +N! +� +σ,ρ∈SN +� j� +i=1 +Uσ(i),riU ∗ +ρ(i),ri +�  + +N +� +i=j+1 +⟨ρ(i)|σ(i)⟩ + + +(46) += 1 +N! +� +J⊂[N]: +|J|=j +� +τ∈SN−j: +[j+1,N]→Jc +� +σ,ρ∈Sj: +[j]→J +� j� +i=1 +Uσ(i),riU ∗ +ρ(i),ri +�  + +N +� +i=j+1 +⟨τ(i)|τ(i)⟩ + + +(47) += (N − j)! +N! +� +J⊂[N]: +|J|=j +� +σ,ρ∈Sj: +[j]→J +� j� +i=1 +Uσ(i),riU ∗ +ρ(i),ri +� +(48) += +1 +Mj +(N − j)! +N! +� +J⊂[N]: +|J|=j +� +σ,ρ∈Sj: +[j]→J +� +R⊂T +�� +i∈T +h2(zσ(i),ri) +� +i∈T c +zσ(i),riz∗ +ρ(i),ri +� +. +(49) +Using the same procedure as in the probability case, we can rewrite the noiseless marginal probability +as +p(r1, . . . , rj) = (N − j)! +N! +1 +Mj +j +� +k=0 +g=2(j−k), +(50) +where +g=2(j−k) += +� +|R|,|R′|=k: +R⊂[j],R′⊂[N] +� +σ∈Sk: +R→R′ +� +K′⊂[N]\R′: +|K′|=j−k +� +σ′,ρ′∈Sj−k: +[j]\R→K′ +� +i∈S(σ′,ρ′) +h2(zσ′(i),ri) +� +i∈([j]\R)\S(σ′,ρ′) +zσ′(i),riz∗ +ρ′(i),ri +(51) += +� +|R|,|R′|=k: +R⊂[j],R′⊂[N] +k! +� +K′⊂[N]\R′ +:|K′|=j−k +� +σ′,ρ′∈Sj−k: +[j]\R→K′ +� +i∈S(σ′,ρ′) +h2(zσ′(i),ri) +� +i∈([j]\R)\S(σ′,ρ′) +zσ′(i),riz∗ +ρ′(i),ri +(52) += +� +|R|=k:R⊂[j] +k! +�N +k +� +� +K′⊂[N]: +|K′|=j−k +� +σ′,ρ′∈Sj−k: +[j]\R→K′ +� +i∈S(σ′,ρ′) +h2(zσ′(i),ri) +� +i∈([j]\R)\S(σ′,ρ′) +zσ′(i),riz∗ +ρ′(i),ri. +(53) +Here k! accounts for the permutations between R and R′ and +�N +k +� +accounts for the choice of R′. +Observe that when j = N, it reduces to f =2(N−k), which describes the full probability. Also, the +noisy marginal distribution is written as +˜p(r1, . . . , rj) = (N − j)! +N! +1 +Mj +j +� +k=0 +xj−kg=2(j−k). +(54) +Thus, the marginal of the approximate distribution is +¯q(r1, . . . , rj) = (N − j)! +N! +1 +Mj +j +� +k=j−l +xj−kg=2(j−k). +(55) +14 + +Here, the complexity of computing g=2(j−k) is given by +�j +k +�� N +j − k +� +((j − k)!)2 ≤ (Nj)j−k. +(56) +Recall that we set a cutoff of the degree as 2(j − k) ≤ 2l. When j ≤ l, since the maximum degree is +2j, we do not approximate and the complexity of computing ¯q(r1, . . . , rj) is upper-bounded by +j +� +k=0 +(Nj)j−k ≤ l(Nl)l ≤ N 2l+1 = O(N 2l+1), +(57) +where we have used j ≤ l ≤ N. +When j > l, we start to approximate and the complexity of +computing ¯q(r1, . . . , rj) is given by +j +� +k=j−l +�j +k +�� N +j − k +� +((j − k)!)2 ≤ (l + 1)(Nj)l ≤ (N + 1)(N 2)l = O(N 2l+1). +(58) +Therefore, we can compute any marginals of ¯q(r) in complexity O(N 2l+1) satisfying ∥˜p − ¯q∥1 ≤ ǫ. +Hence, we can simply apply Lemma 3 to sample from a proper probability distribution q such that +∥˜p − q∥1 ≤ 2ǫ. +From the previous section, for constant x we showed that the degree l can be chosen to be +l = O +� +log(2 +√ +N/ǫ +√ +δ) +log(1/x) +� +to bound the total variation distance and that the complexity of computing a +single probability (marginal is the same or less) is O(N 2l+1). Hence, by using the lemma, the total +complexity to generate a sample is then given by +N O(log N,log ǫ−1,log δ−1), +(59) +which proves Theorem 1. As mentioned before, the algorithm’s running time is quasi-polynomial +not polynomial as in [AGL+22]. The reason is that the noise rate does not scale as the system size +for our case. To properly introduce the noise that scales with the system size, we again consider the +case that x = xγ +1 with a constant x1. In this case, l can be chosen to be l = O +� +log 2 +√ +N +ǫ +√ +δ +γ log 1/x1 +� +. Hence, +for γ = Ω(log N), the complexity becomes polynomial: +O(poly(N, 1/ǫ, 1/δ)), +(60) +which proves Corollary 2. +We emphasize that the degree of the polynomial of the running time in the noise rate scales as +log(1/x1) ≈ 1/(1 − x1), where the approximation is valid for small noise rate x1 ≈ 1. Thus, the +running time of our algorithm can be very large due to the large degree of the polynomial, which +makes it impractical. Also, it is worthwhile to emphasize an extreme case where we only choose +the lowest degree polynomial, i.e., l = 0. Obviously, the lowest degree polynomial, in this case, is a +constant, i.e., the corresponding probability distribution is uniform. +4 +Low-degree approximation with partial distinguishability noise +4.1 +Noise sensitivity and low-degree polynomial approximation +In various optical experiments including Boson Sampling experiments, one of the most important +noise sources is partial distinguishability of particles, which is caused when the particles are not +15 + +fully indistinguishable because of other degrees of freedom. The effect of partial distinguishability +on Boson Sampling has been studied in [Tic15, RMC+18, RSGP18, MGPRT19]. Let us study the +effect of the noise and approximation method of noisy distribution. +Again, consider an output probability and expand it using an orthogonal polynomial basis +p(z) = |Per(UN,z)|2 = +1 +MN +� +σ,ρ∈SN +N +� +i=1 +zσ(i),iz∗ +ρ(i),i, +(61) +where Z corresponds to a rescaled submatrix of a unitary and is approximated by a random Gaussian +matrix. Then, after introducing the partial distinguishability of photons, the probability becomes +[Tic15] +|Per(Z)|2 = +� +σ,ρ∈SN +N +� +i=1 +zσ(i),iz∗ +ρ(i),i → +� +σ,ρ∈SN +xN−k +N +� +i=1 +zσ(i),iz∗ +ρ(i),i, +(62) +where k is the number of i’s such that σ(i) = ρ(i). In other words, whenever we have an interference +due to indistinguishability, i.e., i ∈ [M] such that σ(i) ̸= ρ(i), the partial distinguishability x is +multiplied as a noise factor (x = 1 for fully indistinguishable cases and x = 0 for fully distinguishable +cases.). Now, we expand the probability: +N +� +i=1 +zσ(i),iz∗ +ρ(i),i = +� +i∈T +(zσ(i),iz∗ +σ(i),i) +� +i∈T c +(zσ(i),iz∗ +ρ(i),i) = +� +i∈T +h1(zσ(i),i) +� +i∈T c +h2(zσ(i),i, zρ(i),i). +(63) +In this case, we have chosen a different basis of polynomials: +1, h1(z) ≡ zz∗, h2(z, z′) ≡ zz′∗, +for independent variables z and z′. +(64) +As we have seen, the effect of partial distinguishability is to transform each polynomial as +1 → 1, +h1(z) → h1(z), +h2(z) → xh2(z). +(65) +Here, we assign the degree by adding 0 for h1 and 1 for h2 based on the sensitivity to noise. Notice +a difference from the circuit noise in the previous section that h1(z) is not sensitive to the noise, and +thus it has degree 0. We rewrite the summation as +|Per(Z)|2 = +N +� +k=0 +� +T,T ′⊂[N] +|T|=|T ′|=k +� +σ∈Sk: +T→T ′ +� +σ′,ρ′∈SN−k: +σ′(i)̸=ρ′(i), +T c→T ′c +� +i∈T +h1(zσ(i),i) +� +i∈T c +h2(zσ′(i),i, zρ′(i),i) = +N +� +k=0 +f =(N−k), +(66) +where k is the number of i’s such that σ(i) = ρ(i) from the previous notation. For each k, we need +to decide k elements from [N] for input and output, which are represented by T and T ′. The new σ +is the permutation between these newly chosen sets. And σ′ and ρ′ are now permutations between +the remaining (N − k) indices and σ′(i) ̸= ρ′(i) for all i’s. Thus, the (N − k)th degree part is written +as +f =(N−k) = +� +T,T ′⊂[N] +|T|=|T ′|=k +� +σ∈Sk: +T→T ′ +� +σ′,ρ′∈SN−k: +σ′(i)̸=ρ′(i), +T c→T ′c +� +i∈T +h1(zσ(i),i) +� +i∈T c +h2(zσ′(i),i, zρ′(i),i). +(67) +16 + +After some algebra, we can show that (See Appendix C) +EZ[f =(N−k1)f =(N−k2)∗] = 0, +if k1 ̸= k2, +(68) +and that +EZ[|f =(N−k)|2] = +�N +k +�2 +(N − k)!(!(N − k)) +k +� +j=0 +�k +j +�2 +j!(k − j)!(!(k − j))2j, +(69) +where (!k) represents the number of derangements of k elements, i.e., the number of permutations σ +between k elements such that σ(i) ̸= i for any i ∈ [k]. When the photons in the system have partial +distinguishability x, the polynomial transforms as +f =(N−k) → xN−kf =(N−k). +(70) +Thus, our approximation strategy is to keep the polynomials up to degree l: +˜p(z) = +1 +MN +N +� +k=0 +xN−kf =(N−k) ≈ +1 +MN +N +� +k=N−l +xN−kf =(N−k) ≡ ¯q(z), +(71) +and the approximation error is +˜p(z) − ¯q(z) = +1 +MN +N−l−1 +� +k=0 +xN−kf =(N−k). +(72) +4.2 +Bounds for the total variation distance +Using the same method as the previous section, we can show that +EU[∆2] ≤ 4 +�M +N +�2 +EU [˜p(U, z) − ¯q(U, z)]2 = 4 +�M +N +�2 +1 +M2N +N +� +k=l+1 +x2kEZ[|f =k|2]. +(73) +In Appendix C, we show that +EZ[|f =k|2] ≤ e2(N!)2. +(74) +(Note that one can numerically check that e2 is generally not necessary [RMC+18] but we keep it +since it does not change our main result below.) Hence, the average squared total variation distance +is bounded as +EU[∆2] ≤ 4 +�M +N +�2 +1 +M2N +N +� +k=l+1 +x2ke2(N!)2 ≤ 4 +N +� +k=l+1 +x2(l+1)e2 ≤ 4e2Nx2(l+1). +(75) +By applying Markov’s inequality as the previous case, we can conclude that for 1−δ portion of Haar- +random linear-optical circuits, the approximation error of low-degree polynomial is upper-bounded +by +� +z +|˜p(U, z) − ¯q(U, z)| ≤ 2e +√ +Nxl+1 +√ +δ +. +(76) +To bound the error by ǫ, it is sufficient to choosse l to be +l = +log +� +2e +√ +N +ǫ +√ +δ +� +log(1/x) +− 1 = O(log N, log(1/ǫ), log(1/δ)). +(77) +17 + +4.3 +Barrier of approximate sampling +Now, we again try to find an analogous classical sampler to the previous case and show a barrier to +implementing it in an efficient way. First of all, the noisy distribution is written as +˜p(r) = +1 +MNN! +N +� +k=0 +xN−kf =(N−k). +(78) +Our strategy was to set a cutoff l on the degree, i.e., +¯q(r) = +1 +MNN! +N +� +k=N−l +xN−kf =(N−k). +(79) +One can easily check that the number of summands in f (N−k) is given by +�N +k +�2 +k!(N − k)!(!(N − k)), +(80) +which is larger than N! regardless of k. Thus, direct computation of f =(N−k) is inefficient to any +degrees. One might hope that there can still be a possibility of computing this quantity efficiently. +However, we can show that exact computation requires exponential time. To see this, consider the +lowest-degree polynomial l = 0, which is the fixed point of the noise: +� +σ∈SN +� N +� +i=1 +Uσ(i),riU ∗ +σ(i),ri +� += Per(|UN,r|2), +(81) +where |U|2 is the matrix obtained by taking absolute values on each matrix element. Therefore, it is +written as the permanent of a positive matrix, and its exact computation is known to be #P-hard +[Val79]. Meanwhile, [RMC+18] observed that the permanent of positive matrices can be efficiently +approximated in multiplicative error [JSV04]. Let us recall their method and present a caveat. We +can rewrite the polynomial as in [RMC+18]: +f =(N−k) = +� +T,T ′⊂[N] +|T|=|T ′|=k +� +σ∈Sk: +T→T ′ +� +σ′,ρ′∈SN−k: +σ′(i)̸=ρ′(i), +T c→T ′c +� +i∈T +h1(zσ(i),i) +� +i∈T c +h2(zσ′(i),i, zρ′(i),i) +(82) += +� +T,T ′⊂[N] +|T|=|T ′|=k +Per(|ZT ′,T |2) +� +τ ′∈SN−k: +τ ′(i)̸=i +T ′c→T ′c +� +σ′∈SN−k: +T c→T ′c +� +i∈T c +h2(zσ′(i),i, zτ ′(σ′(i)),i) +(83) += +� +T,T ′⊂[N] +|T|=|T ′|=k +Per(|ZT ′,T |2) +� +τ ′∈SN−k: +τ ′(i)̸=i +T ′c→T ′c +� +σ′∈SN−k: +T ′c→T c +� +i∈T ′c +h2(zi,σ′(i), zτ ′(i),σ′(i)) +(84) += +� +T,T ′⊂[N] +|T|=|T ′|=k +� +τ ′∈SN−k: +τ ′(i)̸=i +T c→T ′c +Per(|ZT ′,T|2)Per(ZT ′c,T c ∗ Zτ ′(T ′c),T c), +(85) +where ∗ represents the elementwise multiplication of two matrices and ZT ′c,T c is obtained by selecting +rows and columns corresponding to T ′c and T c, respectively, and Zτ ′(T ′c),T c is obtained similarly but +18 + +with permuting the rows by τ ′. One can notice that if we set N − k = l, the number of terms to sum +is +� N +N − l +�2 +(!l), +(86) +which is a polynomial in l. Also, the matrix size of ZT ′c,T c∗Zτ ′(T ′c),T c is given by l×l, whose permanent +can be exactly computed in ˜O(2l) [Rys63]. Meanwhile, the difficulty comes from computing the +permanent of |ZT ′,T|2, whose matrix size is (N − l) × (N − l). [RMC+18] claimed that since we can +efficiently approximate the permanent of positive matrices in multiplicative error [JSV04], it might +enable us to approximate f =l as well. However, this is not immediately obvious. To see this clearer, +we can simply write f =l as an inner product of two vectors a, b ∈ Cpoly(N), +f =l = a · b, +(87) +where all the elements of a can be exactly computed and those of b can be efficiently approximated in +multiplicative error, which corresponds to Per(ZT ′c,T c∗Zτ ′(T ′c),T c). The difficulty is the fact that even +though we have exact values of a, they can be negative (or even complex). Thus, the quantity f =l we +are approximating is the sum of many terms, which can be only be approximated, with different signs. +In general, it does not guarantee even a multiplicative error approximation for f =l. Therefore, the +difficulty of computing the probability becomes a barrier to applying the same technique to partial +distinguishability even though the approximation error using low-degree polynomials is sufficiently +small. Furthermore, even if we could approximate the probabilities in a multiplicative error, the +direct application of Lemma 3 still requires an exact computation of probabilities and marginals. +Our analysis reveals that channeling between the small approximation error (in total variation +distance) and constructing an efficient classical sampler is highly nontrivial. +More precisely, our +analysis implies that an additional condition is required for noise, which is that the low-degree +polynomials need to be composed only of polynomially many orthogonal basis polynomials. +We remark that even though the probability of the fixed point of partial distinguishability noise, +i.e., fully distinguishable Boson Sampling, is described by the permanent of a positive matrix and +computing the probability is hard, the corresponding sampling can be shown to be easy even exactly +[AA11, AA13]. This is because fully distinguishable particles do not interfere, so that we can sample +particle by particle, which does not require computing the probability of N particles. Therefore, it +remains open to adapt such a method without computing probabilities to circumvent the barrier and +construct an approximate sampler. +5 +Barriers to photon Loss +Finally, let us consider photon-loss which is one of the most detrimental noise models in Boson +Sampling experiments. We can assume that all the loss occurs at the beginning with total trans- +mission rate η = ηd +1, where d is the depth of the circuit and η1 is a constant loss rate per depth. +This simplification can be justified in many cases because uniform loss channel and beam splitters +commute. +In the second quantization representation, the density matrix of the state is written as +|1, . . . , 1, 0 . . . , 0⟩⟨1, . . . , 1, 0 . . . , 0|, +(88) +which represents the number of photons for each mode. The effect of photon loss is to transform a +single-photon state as +|1⟩⟨1| → η|1⟩⟨1| + (1 − η)|0⟩⟨0| +(89) +19 + +and the vacuum state |0⟩⟨0| does not change. +Therefore, if we introduce photon loss, the state +transforms +N +� +k=0 +�N +k +� +ηk(1 − η)N−k ˆρk, +(90) +where ˆρk is k-photon states with equal weight of selecting k photons out of the initial N photons. +One distinct feature of photon loss from other noise models is that the photon number changes and +that the output quantum state occupies lower than N photons. +If we exploit the same method as the previous cases, we will need to discard the terms having ηk +with k > l with a cutoff l. It implies that we discard +N +� +k=l+1 +�N +k +� +ηk(1 − η)N−k ˆρk, +(91) +which contains at least ηl+1 degrees, while there are other remaining terms that contain ηl+1 degrees; +thus, we will underestimate the approximation error. We note that by discarding the above term, +we do not obtain any outcomes which have larger than l photons because Boson Sampling circuit +does not change the number of photons. Even when underestimating the approximation error, one +can easily see that the probability of the discarded terms is given by +Tr +� +N +� +k=l+1 +�N +k +� +ηk(1 − η)1−k ˆρk +� += +N +� +k=l+1 +�N +k +� +ηk(1 − η)N−k. +(92) +Here, we emphasize that ˆρk’s for different k’s are orthogonal each other from the density matrix level, +which is a distinct property from the other noise models. Thus, regardless of a linear-optical circuit, +the probability that we have lost from discarding high-degree contributions of η is already large. To +be more precise, notice that the photon number distribution follows the binomial distribution with +mean ηN and standard deviation +� +Nη(1 − η). It suggests that we need to keep at least l ≥ ηN. As +a comparison, for circuit noise and partial distinguishability, the required degree was l = O(log N) +for a constant noise rate, which shows that the required degree for photon loss is much larger. +Now, let us now consider the output probability of obtaining r which has k clicks with N − k +photons lost and analyze the complexity. Without loss of generality, let us set ri = 0 for k+1 ≤ i ≤ N. +Then, the output probability of lossy Boson Sampling is written as +˜p(r) = ηk(1 − η)N−k +N! +�N +k +�−1 +� +T⊂[N]:|T|=k +|Per(UT,r)|2. +(93) +Approximating by low-degree in η only changes the prefactor as +¯q(r) = ηk +N! +�N +k +�−1 l−k +� +j=0 +�N − k +j +� +(−η)j +� +T⊂[N]:|T|=k +|Per(UT,r)|2. +(94) +Thus, the complexity of ¯q(r) by computing all the permanents and summing them is +˜O +��N +k +� +2k +� += ˜O +� +N k� +, +(95) +20 + +which is exponential in k. Therefore, to make the complexity at most quasi-polynomial as before, +l needs to be at most logarithmic in the system size N, l = O(log N), which requires the condition +ηN = O(log N). +However, it is known that when ηN = O( +√ +N), the corresponding noisy distribution can be +approximated by a separable state or thermal state input Boson Sampling [OB18, GPRS19], which +can be easily simulated using a classical computer. More specifically, the trace distance between lossy +single photons and a thermal state converges to 0 when ηN = o( +√ +N) in an asymptotic regime (it +converges to a constant when ηN = Θ( +√ +N)). Therefore, the regime in which the proposed technique +might work can already be classically simulated using different techniques with the approximation +error converging to zero in the asymptotic regime. +It is worth emphasizing that we assumed that the sum of permanents Eq. (94) can only be +obtained by computing individual permanents, which might not be the optimal method. For certain +cases, exponential sum of quantities that are hard to compute can be easily obtained [OLW+22]. +Acknowledgements +We thank Senrui Chen and Umesh Vazirani for interesting and fruitful discussions. LJ acknowledges +support from the ARO MURI (W911NF-21-1-0325), AFOSR MURI (FA9550-19-1-0399, FA9550- +21-1-0209), AFRL (FA8649-21-P-0781), DoE Q-NEXT, NSF (OMA-1936118, ERC-1941583, OMA- +2137642), NTT Research, and the Packard Foundation (2020-71479). +BF acknowledges support +from AFOSR (YIP number FA9550-18-1-0148 and FA9550-21-1-0008). This material is based upon +work partially supported by the National Science Foundation under Grant CCF-2044923 (CAREER) +and by the U.S. Department of Energy, Office of Science, National Quantum Information Science +Research Centers as well as by DOE QuantISED grant DE-SC0020360. +References +[AA11] +Scott Aaronson and Alex Arkhipov. The computational complexity of linear optics. In +Proceedings of the forty-third annual ACM symposium on Theory of computing, pages +333–342, 2011. +[AA13] +Scott Aaronson and Alex Arkhipov. Bosonsampling is far from uniform. arXiv preprint +arXiv:1309.7460, 2013. +[AAB+19] +Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami +Barends, Rupak Biswas, Sergio Boixo, Fernando GSL Brandao, David A Buell, +et al. Quantum supremacy using a programmable superconducting processor. Nature, +574(7779):505–510, 2019. +[Aar22] +Scott Aaronson. https://scottaaronson.blog/?p=6871, 2022. +[AB16] +Scott Aaronson and Daniel J Brod. Bosonsampling with lost photons. Physical Review +A, 93(1):012335, 2016. +[ABOIN96] +Dorit Aharonov, Michael Ben-Or, Russell Impagliazzo, and Noam Nisan. Limitations +of noisy reversible computation. arXiv preprint quant-ph/9611028, 1996. +[AC17] +Scott Aaronson and Lijie Chen. +Complexity-theoretic foundations of quantum +supremacy experiments. +In Ryan O’Donnell, editor, 32nd Computational Complex- +21 + +ity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages +22:1–22:67. Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2017. +[AG19] +Scott Aaronson and Sam Gunn. On the classical hardness of spoofing linear cross- +entropy benchmarking. CoRR, abs/1910.12085, 2019. +[AGL+22] +Dorit Aharonov, Xun Gao, Zeph Landau, Yunchao Liu, and Umesh Vazirani. +A +polynomial-time classical algorithm for noisy random circuit sampling. arXiv preprint +arXiv:2211.03999, 2022. +[BCG21] +Boaz Barak, Chi-Ning Chou, and Xun Gao. Spoofing linear cross-entropy benchmarking +in shallow quantum circuits. In James R. Lee, editor, 12th Innovations in Theoreti- +cal Computer Science Conference, ITCS 2021, January 6-8, 2021, Virtual Conference, +volume 185 of LIPIcs, pages 30:1–30:20. Schloss Dagstuhl - Leibniz-Zentrum f¨ur Infor- +matik, 2021. +[BFLL21] +Adam Bouland, Bill Fefferman, Zeph Landau, and Yunchao Liu. Noise and the frontier +of quantum supremacy. In 62nd IEEE Annual Symposium on Foundations of Computer +Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 1308–1317. IEEE, +2021. +[BFNV19] +Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani. On the complex- +ity and verification of quantum random circuit sampling. Nature Physics, 15(2):159–163, +2019. +[BHH16] +Fernando G. S. L. Brand˜ao, Aram W. Harrow, and Micha�l Horodecki. Local random +quantum circuits are approximate polynomial-designs. Communications in Mathemat- +ical Physics, 346(2):397–434, aug 2016. +[BJS10] +Michael J. Bremner, Richard Jozsa, and Dan J. Shepherd. Classical simulation of com- +muting quantum computations implies collapse of the polynomial hierarchy. Proceedings +of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, +467(2126):459–472, 2010. +[BMS17] +Michael J Bremner, Ashley Montanaro, and Dan J Shepherd. +Achieving quantum +supremacy with sparse and noisy commuting quantum computations. Quantum, 1:8, +2017. +[BSN17] +Sergio Boixo, Vadim N Smelyanskiy, and Hartmut Neven. Fourier analysis of sampling +from noisy chaotic quantum circuits. arXiv preprint arXiv:1708.01875, 2017. +[BV93] +Ethan Bernstein and Umesh Vazirani. Quantum complexity theory. In Proceedings of +the twenty-fifth annual ACM symposium on Theory of computing, pages 11–20, 1993. +[BVHS+18] +Juan Bermejo-Vega, Dominik Hangleiter, Martin Schwarz, Robert Raussendorf, and +Jens Eisert. Architectures for quantum simulation showing a quantum speedup. Phys- +ical Review X, 8(2), apr 2018. +[DHJB22] +Alexander M Dalzell, Nicholas Hunter-Jones, and Fernando GSL Brand˜ao. Random +quantum circuits anticoncentrate in log depth. PRX Quantum, 3(1):010333, 2022. +22 + +[DMV+22] +Abhinav Deshpande, Arthur Mehta, Trevor Vincent, Nicol´as Quesada, Marcel Hin- +sche, Marios Ioannou, Lars Madsen, Jonathan Lavoie, Haoyu Qi, Jens Eisert, Dominik +Hangleiter, Bill Fefferman, and Ish Dhand. +Quantum computational advantage via +high-dimensional Gaussian boson sampling. Science Advances, 8(1):eabi7894, 2022. +[DNS+22] +Abhinav Deshpande, Pradeep Niroula, Oles Shtanko, Alexey V Gorshkov, Bill Feffer- +man, and Michael J Gullans. Tight bounds on the convergence of noisy random circuits +to the uniform distribution. PRX Quantum, 3(4):040329, 2022. +[GD18] +Xun Gao and Luming Duan. Efficient classical simulation of noisy quantum computa- +tion. arXiv preprint arXiv:1810.03176, 2018. +[GPRS19] +Ra´ul Garc´ıa-Patr´on, Jelmer J Renema, and Valery Shchesnovich. Simulating boson +sampling in lossy architectures. Quantum, 3:169, 2019. +[HBVSE18] +Dominik Hangleiter, Juan Bermejo-Vega, Martin Schwarz, and Jens Eisert. Anticon- +centration theorems for schemes showing a quantum speedup. Quantum, 2:65, 2018. +[HE22] +Dominik Hangleiter and Jens Eisert. Computational advantage of quantum random +sampling. arXiv preprint arXiv:2206.04079, 2022. +[HHB+20] +Jonas Haferkamp, Dominik Hangleiter, Adam Bouland, Bill Fefferman, Jens Eisert, and +Juani Bermejo-Vega. Closing gaps of a quantum advantage with short-time hamiltonian +dynamics. Physical Review Letters, 125(25):250501, 2020. +[HKS+17] +Craig S Hamilton, Regina Kruse, Linda Sansoni, Sonja Barkhofen, Christine Silberhorn, +and Igor Jex. Gaussian boson sampling. Physical review letters, 119(17):170501, 2017. +[HM18] +Aram Harrow and Saeed Mehraban. +Approximate unitary t-designs by short ran- +dom quantum circuits using nearest-neighbor and long-range gates. +arXiv preprint +arXiv:1809.06957, 2018. +[JSV04] +Mark Jerrum, Alistair Sinclair, and Eric Vigoda. A polynomial-time approximation +algorithm for the permanent of a matrix with nonnegative entries. Journal of the ACM +(JACM), 51(4):671–697, 2004. +[KK14] +Gil Kalai and Guy Kindler. +Gaussian noise sensitivity and bosonsampling. +arXiv +preprint arXiv:1409.3093, 2014. +[KMM21] +Yasuhiro Kondo, Ryuhei Mori, and Ramis Movassagh. Quantum supremacy and hard- +ness of estimating output probabilities of quantum circuits. +In 62nd IEEE Annual +Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, +February 7-10, 2022, pages 1296–1307. IEEE, 2021. +[MGPRT19] Alexandra E Moylett, Ra´ul Garc´ıa-Patr´on, Jelmer J Renema, and Peter S Turner. +Classically simulating near-term partially-distinguishable and lossy boson sampling. +Quantum Science and Technology, 5(1):015001, 2019. +[MLA+22] +Lars S Madsen, Fabian Laudenbach, Mohsen Falamarzi Askarani, Fabien Rortais, +Trevor Vincent, Jacob FF Bulmer, Filippo M Miatto, Leonhard Neuhaus, Lukas G Helt, +Matthew J Collins, et al. Quantum computational advantage with a programmable +photonic processor. Nature, 606(7912):75–81, 2022. +23 + +[Nez21] +Sepehr Nezami. Permanent of random matrices from representation theory: moments, +numerics, concentration, and comments on hardness of boson-sampling. arXiv preprint +arXiv:2104.06423, 2021. +[NJF20] +Kyungjoo Noh, Liang Jiang, and Bill Fefferman. Efficient classical simulation of noisy +random quantum circuits in one dimension. Quantum, 4:318, 2020. +[NSC+17] +Alex Neville, Chris Sparrow, Rapha¨el Clifford, Eric Johnston, Patrick M Birchall, Ash- +ley Montanaro, and Anthony Laing. Classical boson sampling algorithms with superior +performance to near-term experiments. Nature Physics, 13(12):1153–1157, 2017. +[OB18] +Micha�l Oszmaniec and Daniel J Brod. Classical simulation of photonic linear optics +with lost particles. New Journal of Physics, 20(9):092002, 2018. +[OLFJ22] +Changhun Oh, Youngrong Lim, Bill Fefferman, and Liang Jiang. Classical simulation +of boson sampling based on graph structure. Physical Review Letters, 128(19):190501, +2022. +[OLW+22] +Changhun Oh, +Youngrong Lim, +Yat Wong, +Bill Fefferman, +and Liang Jiang. +Quantum-inspired classical algorithm for molecular vibronic spectra. arXiv preprint +arXiv:2202.01861, 2022. +[ONFJ21] +Changhun Oh, Kyungjoo Noh, Bill Fefferman, and Liang Jiang. +Classical simula- +tion of lossy boson sampling using matrix product operators. +Physical Review A, +104(2):022407, 2021. +[QBQGP20] Haoyu Qi, Daniel J Brod, Nicol´as Quesada, and Ra´ul Garc´ıa-Patr´on. +Regimes +of classical simulability for noisy gaussian boson sampling. +Physical review letters, +124(10):100502, 2020. +[RCOL17] +Nicholas J Russell, Levon Chakhmakhchyan, Jeremy L O’Brien, and Anthony Laing. +Direct dialling of haar random unitary matrices. New journal of physics, 19(3):033007, +2017. +[RMC+18] +Jelmer J Renema, Adrian Menssen, William R Clements, Gil Triginer, William S +Kolthammer, and Ian A Walmsley. +Efficient classical algorithm for boson sampling +with partially distinguishable photons. Physical review letters, 120(22):220502, 2018. +[RSGP18] +Jelmer Renema, Valery Shchesnovich, and Raul Garcia-Patron. Classical simulability +of noisy boson sampling. arXiv preprint arXiv:1809.01953, 2018. +[Rys63] +Herbert John Ryser. Combinatorial mathematics, volume 14. American Mathematical +Soc., 1963. +[Shc19] +Valery S Shchesnovich. Noise in boson sampling and the threshold of efficient classical +simulatability. Physical Review A, 100(1):012340, 2019. +[TD04] +Barbara M. Terhal and David P. DiVincenzo. +Adaptive quantum computation, +constant-depth quantum circuits and Arthur-Merlin games. Quantum Information and +Computation, 4(2):134–145, 2004. +[Tic15] +Malte C Tichy. Sampling of partially distinguishable bosons and the relation to the +multidimensional permanent. Physical Review A, 91(2):022316, 2015. +24 + +[TTT21] +Yasuhiro Takahashi, Yuki Takeuchi, and Seiichiro Tani. Classically simulating quantum +circuits with local depolarizing noise. Theoretical Computer Science, 893:117–132, 2021. +[Val79] +Leslie G Valiant. The complexity of computing the permanent. Theoretical computer +science, 8(2):189–201, 1979. +[VNL+21] +Benjamin Villalonga, Murphy Yuezhen Niu, Li Li, Hartmut Neven, John C Platt, +Vadim N Smelyanskiy, and Sergio Boixo. Efficient approximation of experimental gaus- +sian boson sampling. arXiv preprint arXiv:2109.11525, 2021. +[WBC+21] +Yulin Wu, Wan-Su Bao, Sirui Cao, Fusheng Chen, Ming-Cheng Chen, Xiawei Chen, +Tung-Hsun Chung, Hui Deng, Yajie Du, Daojin Fan, et al. Strong quantum computa- +tional advantage using a superconducting quantum processor. Physical review letters, +127(18):180501, 2021. +[ZDQ+21] +Han-Sen Zhong, Yu-Hao Deng, Jian Qin, Hui Wang, Ming-Cheng Chen, Li-Chao Peng, +Yi-Han Luo, Dian Wu, Si-Qiu Gong, Hao Su, et al. Phase-programmable Gaussian +boson sampling using stimulated squeezed light. Physical review letters, 127(18):180502, +2021. +[ZWD+20] +Han-Sen Zhong, Hui Wang, Yu-Hao Deng, Ming-Cheng Chen, Li-Chao Peng, Yi-Han +Luo, Jian Qin, Dian Wu, Xing Ding, Yi Hu, et al. Quantum computational advantage +using photons. Science, 370(6523):1460–1463, 2020. +A +Output distribution of Gaussian noise +In this Appendix, we show that the output probability distribution after Gaussian noise [KK14] +is a nontrivial and proper probability distribution. +Specifically, we show that the probability of +obtaining outcomes in the collision-free subspace is close to and smaller than one, as in the noiseless +case. Therefore, by defining the remaining probability to normalize the sum of probabilities, the +noise maps a noiseless output probability distribution into another proper probability distribution. +Recall that Gaussian noise transforms the unitary matrix of a boson sampling circuit as +U → √xU + +√ +1 − xY, +(96) +where Y is a random Gaussian matrix with variance 1/M. Consider a probability of detecting N +photons for the first N modes with N input photons from the first N modes: +p(U, z) = |Per(UN,N)|2 = +� +σ,ρ∈SN +N +� +i=1 +Ui,ρ(i)U ∗ +i,σ(i). +(97) +25 + +After Gaussian noise, it transforms to +˜p(U, z) = EY + + � +σ,ρ∈SN +N +� +i=1 +(√xUi,ρ(i) + +√ +1 − xYi,ρ(i))(√xUi,σ(i) + +√ +1 − xYi,σ(i))∗ + + +(98) += EY + + � +σ,ρ∈SN +N +� +i=1 +(xUi,σ(i)U ∗ +i,ρ(i) + (1 − x)Yi,σ(i)Y ∗ +i,ρ(i)) + + +(99) += +N +� +k=0 +xk(1 − x)N−k +MN−k +(N − k)! +� +K,K′⊂[N]: +|K|=|K′|=k +� +σ,ρ∈Sk:K→K′ +� +i∈K +Ui,σ(i)U ∗ +i,ρ(i) +(100) += +N +� +k=0 +xk(1 − x)N−k +MN−k +(N − k)! +� +K,K′⊂[N]: +|K|=|K′|=k +|Per(UK,K′)|2, +(101) +where for the second equality, we used the independence of matrix elements of Y , and for the third +equality, we split permutations into trivial permutations, from again independence of Y ’s elements, +and nontrivial permutations. Let us sum over all collision-free outcomes z: +� +z∈cf +˜p(U, z) = +� +z∈cf +N +� +k=0 +xk(1 − x)N−k +MN−k +(N − k)! +� +K⊂[N]: +|K|=k +� +K′⊂z: +|K′|=k +|Per(UK,K′)|2 +(102) += +N +� +k=0 +xk(1 − x)N−k +MN−k +�M − k +N − k +� +(N − k)! +� +K⊂[N]: +|K|=k +� +K′⊂[M]: +|K′|=k +|Per(UK,K′)|2 +(103) += +N +� +k=0 +xk(1 − x)N−k +MN−k +(M − k)! +(M − N)! +� +K⊂[N]: +|K|=k +(collision-free for K input boson sampling with U) +(104) += +N +� +k=0 +xk(1 − x)N−k +MN−k +(M − k)! +(M − N)! +� +K⊂[N]: +|K|=k +[1 − (collision for K input boson sampling with U)], +(105) +where (collision(-free) for K input boson sampling with U) represents the probability of obtaining +collision(-free) outcomes with |K| single photons in modes K with the circuit unitary U, and the +inclusion symbol from z is defined to be the subsets of the modes i’s such that zi = 1. First, we find +the upper bound: +N +� +k=0 +xk(1 − x)N−k +MN−k +(M − k)! +(M − N)! +� +K⊂[N]: +|K|=k +[1 − (collision for K input boson sampling with U)] +(106) +< +N +� +k=0 +xk(1 − x)N−k +�N +k +� += 1. +(107) +26 + +Now we find the lower bound of the average over Haar-random unitary U. Using the bosonic birthday +paradox [AA11], we can bound the total collision-free outcomes as +EU + + � +z∈cf +˜p(U, z) + + +(108) += EU + + +N +� +k=0 +xk(1 − x)N−k +MN−k +(M − k)! +(M − N)! +� +K⊂[N]: +|K|=k +[1 − (collision for K input boson sampling with U)] + + +(109) +> +N +� +k=0 +xk(1 − x)N−k +MN−k +(M − k)! +(M − N)! +� +K⊂[N]: +|K|=k +� +1 − 2k2 +M +� +(110) +≥ +N +� +k=0 +xk(1 − x)N−k +�N +k +� � +1 − N +M +�N � +1 − 2N 2 +M +� +(111) +→ 1, +(112) +where for the last expression, we used M = ω(N 2) for large N. Therefore, using the assumption of +the strong collision-free regime, i.e., M = ω(N 5), the noisy output probability distribution sums close +to one. Finally, we defined the collision case of the noisy distribution as the remaining probability, +so that the total probability is normalized to be one, +� +z∈cf +˜p(U, z) + ˜p(U, c) = 1. +(113) +B +Collision +In this Appendix, we will show how to make the distribution ¯q(r) to satisfy the sufficient condition +� +r∈[M]N ¯q(r) = 1 by assigning ¯q(r) for collision cases r properly. We will assume that we have +chosen the cutoff of degree as l ≥ 1 for simplicity. Then, the first-order marginal ¯q(r1) is exact, i.e., +¯q(r1) = ˜p(r1) for all r1 ∈ [M]. For the second-order marginals, we will define for each r1 ∈ [M] +¯q(r1, r2 = r1) = ¯q(r1) + +1 − +� +r2∈[M]\{r1} +¯q(r1, r2) + + , +(114) +which obviously guarantees that �M +r2=1 ¯q(r1, r2) = ¯q(r1). +Similarly, for given (r1, . . . , rk−1) with +distinct {ri}k−1 +i=1 , we define for each rk ∈ {ri}k−1 +i=1 +¯q(r1, . . . , rk) = ¯q(r1, . . . , rk−1) + +1 − +1 +k − 1 +� +rk∈[M]\{ri}k−1 +i=1 +¯q(r1, . . . , rk−1, rk) + + for each rk ∈ {ri}k−1 +i=1 , +(115) +which again guarantees that �M +rk=1 ¯q(r1, . . . , rk) = ¯q(r1, . . . , rk−1). For such rk’s and for all permu- +tations σ ∈ Sk, we also define +¯q(rσ(1), . . . , rσ(k)) ≡ ¯q(r1, . . . , rk). +(116) +27 + +We continue this procedure until k = N when we define all quantities of ¯q(r) of r ∈ [M]N. +Now, we have defined all relevant quantities of ¯q(r) and its marginals. Consequently, we can +easily show that the resultant distribution satisfies +� +r∈[M]N +¯q(r) = 1, +(117) +which can be easily shown by the marginal relation, +¯q(r1, . . . , rk−1) = +M +� +rk=1 +¯q(r1, . . . , rk). +(118) +As a remark, we argue why this procedure is necessary. Since the collision probability is inverse- +polynomially suppressed when M = ω(N 2) [AA11], one might be tempted to set it to be zero for ¯q(r). +However, one can immediately see that it might cause a large error. Suppose that a quasi-probability +distribution ¯q(r) is given for collision-free space, i.e., which is close to the target distribution +� +r∈cf +|˜p(r) − ¯q(r)| ≤ ǫ, +(119) +where cf accounts for the set of collision-free outcomes. We first show that a naive approach may +entail a large error. Let us denote the probability of collisions as ǫc. We will set ¯q(r) = 0 for collision +outcomes r. Then, for full distribution we have +� +r∈[M]N +|˜p(r) − ¯q(r)| = +� +r∈c +|˜p(r) − ¯q(r)| + +� +r∈cf +|˜p(r) − ¯q(r)| ≤ ǫc + ǫ ≡ ǫt, +(120) +where c accounts for the set of collision outcomes. Then after using the lemma from [BMS17], we +can sample from a proper probability distribution q(z) with the total variation distance given by +� +z +|˜p(z) − q(z)| ≤ +4ǫt +1 − ǫt +. +(121) +Since the collision probability ǫc is fixed for a given system, we cannot reduce the error as much as we +want. Thus, we need to assign appropriate quantities of ¯q(r) for collision outcomes before applying +the lemma. +C +Orthogonality of polynomials for partial distinguishability noise +In this Appendix, we show the orthogonality of polynomials introduced for partial distinguishability +noise. Consider +EZ[|f =(N−k)|2] = EZ +�  + +� +T,T ′⊂[N],|T|=|T ′|=k +� +σ,σ′,ρ′ +� +i∈T +h1(zσ(i),i) +� +i∈T c +h2(zσ′(i),i, zρ′(i),i) + + +× + + +� +T ∗,T ′∗⊂[N],|T ∗|=|T ′∗|=k +� +σ∗,σ′∗,ρ′∗ +� +i∈T ∗ +h∗ +1(zσ∗(i),i) +� +i∈T ∗c +h∗ +2(zσ′∗(i),i, zρ′∗(i),i) + + +� +. +(122) +28 + +Here if T ̸= T ∗ or T ′ ̸= T +′∗, one can easily check that the average over Z becomes zero. Thus, we +set T ∗ = T and T ′∗ = T ′. Now, we have, for |T| = k and a fixed T and T ′, +� +σ,σ∗ +� +i∈T +h1(zσ(i),i)h∗ +1(zσ∗(i),i) = +k +� +j=0 +�k +j +� +j!(k − j)!(!(k − j))2j, +(123) +which can be shown by splitting the factors |z|4 and |z|2 and counting each and using Ez[|z|4] = 2. +Here (!j) is the derangement, namely, the number of permutations σ ∈ Sj such that σ(i) ̸= i for all +i’s. Meanwhile, +� +σ′,ρ′ +� +σ′∗,ρ′∗ +� +i∈T c +h2(zσ′(i),i, zρ′(i),i)h∗ +2(zσ′∗(i),i, zρ′∗(i),i) = +� +σ′,ρ′ +� +σ′∗,ρ∗ +� +i∈T c +(zσ′(i),iz∗ +ρ′(i),iz∗ +σ′∗(i),izρ′∗(i),i) (124) += (N − k)!(!(N − k)), +(125) +where we used the fact that σ = σ′∗ and ρ = ρ′∗ is necessary to be nonzero. Thus, the number of +choices of T and T ′ has +�N +k +�2 and the number of choices of σ′ and ρ′ is (N − k)!(!(N − k)) and we +obtain +EZ[|f =(N−k)|2] = +�N +k +�2 +(N − k)!(!(N − k)) +k +� +j=0 +�k +j +�2 +j!(k − j)!(!(k − j))2j. +(126) +We now further upper bound the two-norm. Here, we first use +k +� +j=0 +�k +j +�2 +j!(k − j)!(!(k − j))2j ≤ +k +� +j=0 +�k +j +�2 +j!((k − j)!)22j +(127) += +k +� +j=0 +�k +j +� +k!(k − j)!2j +(128) += e2k!Γ(k + 1, 2) +(129) +≤ e2k!Γ(k + 1) +(130) += e2(k!)2. +(131) +Thus, +EZ[|f =(N−k)|2] = +�N +k +�2 +(N − k)!(!(N − k)) +k +� +j=0 +�k +j +�2 +j!(k − j)!(!(k − j))2j +(132) +≤ e2 +�N +k +�2 +(N − k)!(!(N − k))(k!)2 +(133) +≤ e2 +�N +k +�2 +((N − k)!)2(k!)2 +(134) +≤ e2(N!)2. +(135) +29 + diff --git a/l9FJT4oBgHgl3EQfZixR/content/tmp_files/load_file.txt b/l9FJT4oBgHgl3EQfZixR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2215f03d4fdf98da4fe397e7589de3d8c0d8bb64 --- /dev/null +++ b/l9FJT4oBgHgl3EQfZixR/content/tmp_files/load_file.txt @@ -0,0 +1,896 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf,len=895 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='11532v1 [quant-ph] 27 Jan 2023 On classical simulation algorithms for noisy Boson Sampling Changhun Oh ∗1, Liang Jiang †1, and Bill Fefferman ‡2 1Pritzker School of Molecular Engineering, University of Chicago, Chicago 2Department of Computer Science, University of Chicago, Chicago January 30, 2023 Abstract We present a classical algorithm that approximately samples from the output distribution of certain noisy Boson Sampling experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' This algorithm is inspired by a recent result of Aharonov, Gao, Landau, Liu and Vazirani and makes use of an observation originally due to Kalai and Kindler that the output probability of Boson Sampling experiments with a Gaussian noise model can be approximated by sparse low-degree polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' This observation alone does not suffice for classical sampling, because its marginal probabilities might not be approximated by sparse low-degree polynomials, and furthermore, the approximated probabilities might be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We solve this problem by employing the first quantization representation to give an algorithm for computing the marginal probabilities of these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We prove that when the overall noise rate is constant, the algorithm runs in time quasi- polynomial in the number of input photons N and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' When the overall noise rate scales as 1 − xγ 1 for constant x1 and γ = Ω(log N), the running time becomes polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Furthermore, we study noisy Boson Sampling with practically relevant noise models such as partial distinguishability and photon loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We show that the same technique does not immediately apply in these settings, leaving open the possibility of a scalable demonstration of noisy quantum advantage for these noise models in certain parameter regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1 Introduction We have recently seen the first claims of experimental quantum advantage using both the random circuit sampling proposal implemented with superconducting qubits [AAB+19, WBC+21] as well as the Gaussian Boson Sampling proposal implemented in a linear optical architecture [ZWD+20, ZDQ+21, MLA+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Such quantum advantage is a necessary step on the path toward building scal- able, fault-tolerant quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In addition quantum advantage is a fundamental milestone in its own right, where it can be interpreted as providing an experimental violation to the Extended Church-Turing thesis (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', [BV93, AA11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' With such an important milestone it is critical to analyze our evidence for believing that such experiments are classically intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Here much is still unknown, and to improve this situation we must both bolster the classical hardness arguments as well as develop new classical simulation algorithms to challenge our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In this work, inspired by a recent algorithm for simulating logarithmic depth noisy random quantum circuits due to Aharonov, Gao, Landau, Liu and Vazirani ∗changhun@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='edu †liangjiang@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='edu ‡wjf@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='edu [AGL+22] and earlier work due to Gao and Duan [GD18], we develop a classical algorithm to approx- imately sample from the output distribution of certain noisy Boson Sampling experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Much like the Aharonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' result, we do not expect that this algorithm is practical in its present form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' That is, it most likely will not “spoof” present day Boson Sampling experiments in a reasonable amount of classical running time, due mainly to the inefficient scaling of the algorithm’s running time with the noise rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Nonetheless we are able to prove that our algorithm works for a Gaussian noise model proposed in past work by Kalai and Kindler [KK14], which like depolarizing noise in the Aharonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' algorithm has the property that the noisy output distribution eventually converges to the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We then discuss the prospects for extending this algorithm to other noise models, including photon loss and partial distinguishability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='1 Putting recent simulation results in context After more than a decade of research in this area, there now is a body of work to support the classical intractability of these quantum advantage experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' This evidence comes primarily from complexity theoretic arguments proving that no efficient classical algorithm can simulate these experiments in the asymptotic regime as the system size increases under reasonable complexity theoretical assumptions (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', [TD04, BJS10, AA11, BFNV19, AC17, AG19, BFLL21, KMM21, DMV+22, HE22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' One potential challenge to these hardness arguments comes from uncorrected noise, which is perhaps the defining characteristic of near-term quantum computational experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' This noise degrades the quantum signal as the system size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Consequently it is reasonable to expect that classical algorithms could potentially take advantage of this weakness to simulate noisy experiments at a sufficiently large system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' While this has been an active subject of research with many results [KK14, BMS17, GD18, RSGP18, Shc19, GPRS19, NJF20, TTT21, QBQGP20, ONFJ21, VNL+21], we have arguably not yet seen a classical algorithm that simulates state-of-the-art quantum advantage experiments using a comparable amount of computational resources (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', [Aar22] for more discussion on this point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Indeed, at the moment there is hope that near-term quantum advantage experiments operate in a “Goldilocks” regime in which the system size is large enough to be classically intractable to simulate, but not so large that uncorrected noise overwhelms the quantum signal1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Short of spoofing fixed size near-term experiments, one can ask if classical algorithms can effi- ciently simulate noisy quantum advantage experiments in the asymptotic limit as the system size scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Such a classical algorithm would rule out a fully scalable demonstration of quantum advantage with uncorrected noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Indeed, such a scalable demonstration would be of great interest, but until very recently was thought to be infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' This pessimism was mainly due to two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The first major reason came from a foundational result due to Aharonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' from the late 1990’s [ABOIN96] showing that the total variation distance between the output distribution of a noisy quantum circuit with circuit depth d and the uniform distribution is upper bounded by 2−O(d) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' This early result already rules out scalable quantum advantage for any depth that is super-logarithmic in the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' To make things worse, there is numerical evidence that the output distribution of most noisy random quantum circuits converges to the uniform distribution at the even faster rate of 2−O(n·d) (see [BSN17] and the corresponding discussion in [BFLL21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' This rapid convergence would rule out scalable, noisy quantum advantage at any depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1This “Goldilocks” regime is also important to enable classical verification techniques such as the cross-entropy benchmark, which currently requires exponential time on a classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 2Strictly speaking this upper bound applies to any quantum circuit that is subject to depolarizing noise with constant noise rate, although more recent results have clarified that it is widely applicable to a variety of reasonable noise models (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', [GD18, DNS+22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 2 The second major reason for pessimism came from a statistical property of the output distribution of random quantum experiments known as “anticoncentration”, which is useful in the theoretical hardness analysis of these systems (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', [AA11] for more discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Anticoncentration is known to be a property of any ensemble of random quantum circuits that forms an approximate unitary two-design (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', [BHH16, BVHS+18, HBVSE18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' For D-dimensional local random quantum circuits with Haar random gates this property first arises at depth n1/D and this is believed to be optimal [BHH16, HM18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Consequently, if the spatial locality is constant, then combining this result together with the upper bound of Aharonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [ABOIN96] we find that the noisy output distribution of such circuits is inverse superpolynomially close to the uniform distribution, which again rules out noisy, scalable quantum advantage in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' However, in the last two years new results were proven which offered some brief hope that random quantum circuits might be able to achieve such a scalable noisy advantage at precisely logarithmic depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' First the results of Dalzell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' and Barak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' proved that random quantum circuits with Haar random two-qubit gates anticoncentrate at logarithmic depth3 [DHJB22, BCG21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Crucially, these papers directly analyze the anticoncentration property of the ensemble of circuits without relying on the approximate two-design property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Moreover, these results are optimal, in the sense that sublogarithmic depth random quantum circuits with two-qubit Haar random gates are known not to anticoncentrate [DHJB22, DNS+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In addition, a result of Deshpande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' proved that the total variation distance between the output distribution of most random quantum circuits and the uniform distribution is lower bounded by a quantity that scales as 2−O(d), matching the Aharonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' upper bound of 2−O(d) [DNS+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Putting these two results together gave rise to the (as it turns out, fleeting) hope that logarithmic depth random quantum circuits with Haar random gates could offer a “sweet-spot” regime in which the depth was both sufficient to have anticoncentration yet shallow enough so that uncorrected noise does not overwhelm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='2 The Aharonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' random circuit simulation algorithm This hope was very recently ruled out by a result of Aharonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [AGL+22] which presents an efficient algorithm for approximately sampling from the output distribution of noisy random circuit ensembles that anticoncentrate, modulo the “gate-set orthogonality” constraint which is satisfied e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', by two qubit Haar random gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' This algorithm follows up on earlier work of Gao and Duan, which achieved the same accuracy in quasi-polynomial time [GD18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Owing to the requirement of anticoncentration, these algorithms are useful for simulating random quantum circuits with depth that scales at least logarithmically in the system size 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In particular at logarithmic depth the earlier Aharonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' result implies that sampling from the uniform dis- tribution achieves total variation distance 1/2O(d) = 1/poly(n) [ABOIN96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' However, approximating the noisy output distribution by the uniform sampler cannot reduce the total variation distance by increasing the running time because the approximate sampler is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' By contrast this new result is stronger and gives a classical algorithm that can achieve any total variation distance parameter ǫ with a running time that scales as poly(1/ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The key observation behind this algorithm is that the output (or marginal) probabilities of 3Strictly speaking this is proven for 1D and all-to-all connectivities, but is believed to hold for intermediate regimes such as a 2D grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 4It still remains possible to prove hardness of sampling results for random quantum circuits with Haar random gates at sublogarithmic depths without needing anti-concentration, although it is likely that new ideas will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Additionally there exist ensembles of random circuits that anticoncentrate at constant depths [HHB+20] by using a distribution over gates that is very different from Haar random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' It remains unclear if the Aharonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' algorithm can be adapted to simulate such ensembles in the presence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 3 noisy random circuits with a constant rate of depolarizing noise per gate can be expressed as the sum of polynomially many dominant Fourier coefficients with exponentially many other Fourier coefficients that are highly suppressed due to the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In other words, the output probability of noisy random circuits can be approximately represented by sparse Fourier coefficients with a small error occurring by discarding other Fourier coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Using sparsity of the Fourier coefficients involved in the output (or marginal) probabilities, one can efficiently approximate the output (marginal) probabilities, which enables us to sample from the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We emphasize that it is crucial that any output probability of a given circuit has to be described by the same polynomially many Fourier coefficients to guarantee that all the marginals can also be efficiently computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The latter is not obvious because the marginal probabilities can be the sum of exponentially many probabilities, which may eventually require an exponential number of Fourier coefficients even though each probability has a sparse Fourier description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In addition, since the approximated distribution can be a quasi- probability distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', it can be negative, it was crucial to exploit a technique proposed in [BMS17], which enables us to approximately sample from a proper probability distribution when the quasi-probability distribution is sufficiently close to the noisy probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='3 Noisy Boson Sampling Let us turn our attention to Boson Sampling [AA11], which is our main focus in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The main question of the present work is whether the same type of Aharonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' classical algorithm [AGL+22] works to simulate noisy Boson Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Interestingly, even before studies on the sparsity of Fourier coefficients in noisy random circuit sampling [GD18, AGL+22], Kalai and Kindler already pointed out that low-degree polynomials can approximate the output probability of noisy Boson Sampling with a particular choice of noise type, which transforms a given linear-optical circuit U → √xU + √1 − xY , where Y is a random Gaussian matrix and 1 − x is the noise rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' To avoid confusion we emphasize that 1 − x is the noise rate not x, which is the case in [AGL+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' After Kalai and Kindler’s analysis on a mathematically appealing noise model, several subsequent works studied more physical noise types such as partial distinguishability using similar techniques [RMC+18, RSGP18, Shc19, MGPRT19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' However, the previous works did not provide a classical sampler to exploit the low-degree polynomial approximation (See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='4 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In this work, we present a classical algorithm that approximately simulates noisy Boson Sampling with noise studied in [KK14] using sparsity of low-degree polynomials and the method in [BMS17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In particular, assuming Haar-random linear-optical circuits (instead of anticoncentration), the classical algorithm’s running time is given by quasi-polynomial in the system size and accuracy for an overall constant noise level 1 − x ∈ (0, 1]: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Consider an M-mode Fock-state Boson Sampling with N single photons and a linear- optical circuit with a global Haar-random unitary with M = ω(N 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' If there is an overall constant circuit noise, we can classically simulate collision-free outcomes of the noisy Boson Sampling with running time N O(log N,log ǫ−1,log δ−1) within total variation distance ǫ for 1−δ portion of Haar-random unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The main reason that the running time is quasi-polynomial is that the noise rate is assumed constant for the entire circuit instead a constant level of noise per gate as in [AGL+22], where noise scales with the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' To introduce a similar effect, we now consider the case where the total noise rate scales as 1 − xγ 1 with γ = Ω(log N) and a constant x1 ∈ [0, 1) and show for this case that the running time becomes polynomial: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Consider an M-mode Fock-state Boson Sampling with N single photons and a linear- optical circuit with a global Haar-random unitary with M = ω(N 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' If there is an overall circuit 4 noise 1 − xγ 1 with a constant x1 ∈ [0, 1) and γ = Ω(log N), we can classically simulate collision-free outcomes of the noisy Boson Sampling with running time poly(N, ǫ−1, δ−1) within total variation distance ǫ for 1 − δ portion of Haar-random unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Note that whereas [AGL+22] introduces noise for each gate, but also requires anticoncentra- tion, we introduce the noise for the entire circuit at once with global Haar-random circuits but do not explicitly require anticoncentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' It remains open to generalize our result as the setting in [AGL+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The key idea to channel the sparse low-degree polynomial approximation from [KK14] to sampling is to employ the first quantization representation of Boson Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We show that the marginals of approximated quasi-probability distribution for the first quantization representation can also be efficiently computed by sparse polynomials, and consequently the technique from [BMS17] can be applied for sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, it closes the gap between the approximate computation of probability and sampling for circuit noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Intriguingly, applying the same sparsity technique to physical noise models such as partial distinguishability and photon loss hits barriers to finding a corresponding classical sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' First, for partial distinguishability noise, the barrier is that even after introducing noise and approximating the probability with similar polynomials, computing the output probability distribution still costs an exponential time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, a naive approach does not successfully reduce the complexity by exploiting the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Second, for photon loss, the barrier is that we need to choose a large degree to suppress the approximation error, which implies that the algorithm might work only for a large photon-loss regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' However, the large photon-loss regime can already be classically simulated because lossy single-photon states are already sufficiently close to classical states (much like the convergence of the output probability distribution to uniform at superlogarithmic depth for qubit cases [ABOIN96]) [OB18, GPRS19, QBQGP20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, the sparsity technique does not provide any benefits over the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Our analysis of three different types of noise clearly reveals that the different behavior of output distributions against different noise types poses difficulties in the generalization of the same technique for more general noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Interestingly, both the output distribution of random circuits with depolarizing noise and that of Boson Sampling with circuit noise converge to the uniform distribution, while those of Boson Sampling with partial distinguishability and photon loss do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' This might indicate that the current technique implicitly relies on a certain property of the noise model, which is related to convergence to the uniform distribution, and that different noise models might require an additional technique or perhaps even lead to a scalable demonstration of noisy quantum advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We stress, however, that we do not prove such a formal connection to the uniform distribution in this work, but leave this as an intriguing open direction for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='4 Relation to previous results on Boson Sampling As mentioned in the previous section, the low-degree polynomial approximation techniques for noisy Boson Sampling have been discussed even before [GD18, AGL+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' More specifically, Kalai and Kindler showed that the output probabilities of noisy Boson Sampling can be approximated by sparse low-degree polynomials under the assumption of Haar-randomness of the linear optical circuit matrix (this seems analogous to the anticoncentration requirement of Aharonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [AGL+22]) [KK14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Nevertheless, it is not obvious how to approximately sample from the output distribution described by the sparse low-degree polynomials because the approximated distribution might not be a proper probability distribution and it is not guaranteed that its marginal probabilities can also be described by sparse polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The latter is because it has to be shown that any probabilities can be described by the same sparse low-degree polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Our contribution is to channel the low-degree polynomial approximation to a classical sampling algorithm using the first quantization 5 method and marginal-based sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Several subsequent works studied more physical noise types such as partial distinguishability [RMC+18, RSGP18, Shc19, MGPRT19] while their approaches also encounter the same obstacles to finding a classical sampler 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In particular, [RMC+18] observed that the output probability of partial distinguishable Boson Sampling can be approximated by low-degree polynomials, which guarantees that the total variation distance can be made small by choosing an appropriate degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' It was also claimed that each polynomial can be efficiently approximated (not exactly computed, unlike [KK14, AGL+22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Nevertheless, it did not analyze the effect of the approximation of polynomials and did not provide a provable classical sampler;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' instead, it considered the Metropolis algorithm, which is heuristic [NSC+17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, they did not provide a provable classical sampler for partial distinguishable Boson Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We show that indeed it is not immediately straightforward to construct a classical sampler that exploits the low-degree polynomial approximation for partial distinguishable noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Finally, there have been extensive studies on the effect of photon loss on Boson Sampling [AB16, OB18, RSGP18, GPRS19, Shc19, QBQGP20, ONFJ21], while a similar technique has not been considered 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Our analysis shows that the previous techniques that approximate lossy single photons by classical states provide a better approximation error than a naive approach using the low-degree polynomial approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='5 Concluding remarks We finally remark on several points that were not addressed in the present work and open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Efficient classical algorithms for physical noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' As we claimed, the low-degree polynomial approximation does not immediately lead to an efficient classical sampler for par- tial distinguishability and photon loss, which are the most crucial noise models in practice [ZWD+20, ZDQ+21, MLA+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' It remains an open question to improve the technique to find an efficient classical sampler for those noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' For photon loss case in particular, when the output photon number scales as Θ( √ N), the total variation distance of the classical algo- rithms in [OB18, GPRS19, QBQGP20] to the lossy output probability distribution is fixed as a constant, and it cannot be reduced by increasing the running time of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Finding a classical algorithm that can efficiently reduce the approximation error as [AGL+22] and our result for Gaussian noise is another open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Lifting the assumption of global Haar-randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In the present work, we have as- sumed that the linear-optical circuits are constructed to be global Haar-random7, which is a standard assumption for the hardness of Boson Sampling [AA11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' On the other hand, the recent Boson Sampling experiments have not implemented global Haar-random circuits [ZWD+20, ZDQ+21, MLA+22, OLFJ22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Also, the recent result for random circuits [AGL+22] assumed anticoncentration with consideration of depth and noise effect per gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Extending our results further with a less stringent assumption is another future work, such as replacing the global Haar-random assumption with anticoncentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Note that whereas random circuits in [AGL+22] with gate-set orthogonality enjoy the symmetry between different outcomes when averaged over ensembles, Boson Sampling outcomes generally do not have such an apparent 5While [Shc19] claimed that there is an efficient classical sampler, this was not completely proved to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 6[Shc19] has considered the combined effect of loss and dark count with assuming that the total photon number is preserved by dark count effect, which is not satisfied solely by photon loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 7Unlike random circuit sampling using qubits, the dimension of the unitary matrix for global Haar-random is polynomial in the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, it is not an unrealistic assumption in practice (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', [RCOL17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 6 symmetry, which hinders us from analyzing the upper bound of total variation distance except for the global Haar-random case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Anticoncentration of Boson Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Unlike random circuit sampling, we have less un- derstanding of anticoncentration in Boson Sampling such as how much circuit depth is required to attain anticoncentration property with what kinds of an ensemble of linear-optical circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Even whether the anticoncentration property is achieved with global-Haar random remains a conjecture to the best of our knowledge [AA11] despite interesting recent progress (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', [Nez21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Gaussian Boson Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We have considered Fock-state Boson Sampling only while the quantum advantage experiments employed Gaussian Boson Sampling [HKS+17, DMV+22], which is a variant of Fock-state Boson Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' While we expect a similar result to hold, we leave it as an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Practical consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' As emphasized before, the proposed algorithm assumes an asymp- totic regime of noisy Boson Sampling and we do not expect the algorithm to spoof finite-size near-term experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Specifically, for a small noise rate x1 ≈ 1, the degree of the polyno- mial of the running time is given by 1/ log(1/x1) ≈ 1/(1 − x1), which makes the algorithm impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In fact, the recent result in [AGL+22] observed the same issue, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', the degree of the polynomial is a large constant 1/γ, where γ is the noise rate per gate in their notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' An interesting future work is to improve the algorithm to be applicable to finite-size Boson Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 2 Fock-state Boson Sampling in first quantization Let us consider the standard Fock-state Boson Sampling [AA11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The basic setup is to prepare N single photons and to inject the photons into an M-mode linear-optical circuit ˆU, characterized by an M × M unitary matrix, where M = poly(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We then measure the number of photons for each output mode, which gives rise to a measurement outcome m ∈ ZM ≥0 with �M i=1 mi = N, where mi represents the number of photons at the ith output mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We now describe the dynamics by introducing the first quantization representation, which enables us to analyze marginal distributions later easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' First, we write the input state as 1 √ N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � σ∈SN |σ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , σ(N)⟩, (1) where SN represents the permutation group for N elements, which accounts for the symmetrization of N photons due to bosons’ indistinguishability nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, the density matrix of the input state is written as 1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � σ,ρ∈SN |σ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , σ(N)⟩⟨ρ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , ρ(N)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (2) After applying beam splitter network ˆU, we obtain the output state 1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � σ,ρ∈SN ˆU ⊗N|σ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , σ(N)⟩⟨ρ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , ρ(N)| ˆU †⊗N, (3) 7 where the linear-optical operation, characterized by an M × M unitary matrix U, transforms the state as ˆU|i⟩ = M � j=1 Uij|j⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (4) Finally, we measure in each photon’s position r ∈ ZN ≥0, whose probability is written as p(r) = 1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � σ,ρ∈SN ⟨r| ˆU ⊗N|σ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , σ(N)⟩⟨ρ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , ρ(N)| ˆU †⊗N|r⟩ = 1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � σ,ρ∈SN � N � i=1 Uσ(i),riU ∗ ρ(i),ri � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (5) Especially for collision-free outcomes r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' at most a single photon clicks for each output mode (equivalently all ri’s are distinct), the probability reduces to p(r) = |PerUN,r|2 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , (6) where UN,r is the N × N submatrix of a unitary matrix U obtained by selecting the first N rows, which accounts for the input photons, and r’s columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We first clarify the notation of outcomes m, r, and z and their relations, the latter of which will be defined now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' First, we will define z ∈ ZN ≥0 as the ordered vector of r in the nondecreasing order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', z1 ≤ z2 ≤ · · · ≤ zN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Notice that different r’s may reduce to the same vector z, which is because we cannot distinguish which input photons correspond to which output photons in principle due to the indistinguishability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Because of the symmetry, the different r’s that correspond to the same z have the same probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The photon number vector m’s elements mi’s can be obtained by counting the number of i’s in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Hence, we can write the probability by abusing the notation of p(·) p(z) = � σ∈SN p(σ(r)) = |PerUN,r|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (7) We will often abuse the notation of the probability p(z), p(r) and p(m), which can be uniquely identified by using different arguments z, r and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Especially when another distribution q(z) has the same property, namely q(z) = � σ∈SN q(σ(r)), the total variation distance between p(z) and q(z) with ordered outcomes and that between p(r) and q(r) with unordered outcomes are equal: ∥p(z) − q(z)∥1 = ∥p(r) − q(r)∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (8) The property will play an important role for approximate sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We will focus on the (strong) collision-free regime M = ω(N 5), where an N × N submatrix of an M × M Haar-random unitary matrix U can be approximated by complex random Gaussian matrix Z such that (UN,z)ij ≈ Zij/ √ M with Zij ∝ N(0, 1) with scaling factor 1/ √ M [AA11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Also, we will focus on simulating the probability distribution over collision-free outcomes, which suggests that ri ̸= rj, or equivalently zi ̸= zj, for all i ̸= j ∈ [N], or mi ∈ {0, 1} for all i ∈ [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Since we do not aim to simulate collision outcomes, we will set all the collision outcomes to be c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', we treat them as the same outcome c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 8 3 Low-degree polynomial approximation with circuit noise 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='1 Noise sensitivity and low-degree polynomial approximation Let us consider the effect of noise on Boson Sampling output probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The first type of Gaussian noise we consider is the noise on circuit unitary U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Although this type of noise might not be physically or experimentally relevant, it provides profound insights into the noise sensitivity of the output probability of Boson Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' More specifically, the introduced noise changes a unitary matrix as U → √xU +√1 − xY , where Y is an M ×M complex random Gaussian matrix, x ∈ [0, 1], and 1 − x is the noise rate [KK14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We remark that the definition of the Gaussian noise seems to be unphysical in the sense that for a single instance Y , √xU + √1 − xY is not necessarily unitary and, furthermore, its spectral norm can be larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' However, we show that the noisy output probability distribution is a proper probability distribution (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In this section, we will recall the result from [KK14] that a Boson Sampling probability dis- tribution under this type of noise can be approximated in total variation distance by low-degree polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' To this end, let us consider an output probability p(z) = |PerUN,z|2 = |Per(Z)|2 MN , (9) where Z ≡ √ MUN,z is the rescaled N × N submatrix of unitary U corresponding to the outcome z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We used the fact that a submatrix of a large Haar-random unitary matrix (M = ω(N 5)) can be approximated by a complex random Gaussian matrix whose elements follow complex normal distribution N(0, 1) [AA11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Following [KK14], we can expand the absolute-squared permanent as the sum of orthogonal polynomials: |Per(Z)|2 = N � k=0 f =2(N−k), (10) where the degree 2(N − k) polynomials f =2(N−k) satisfy the following orthogonal relations EZ[|f =2(N−k)|2] = (N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2, EZ[f =2(N−k1)f =2(N−k2)∗] = 0, for k1 ̸= k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (11) Here, the average EZ[·] is taken over complex Gaussian random matrices Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' To see this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' let us expand the absolute-squared permanent of a complex random Gaussian matrix as |Per(Z)|2 = � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='ρ∈SN N � i=1 zσ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='iz∗ ρ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='i (12) = � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='ρ∈SN � i∈T (zσ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='iz∗ σ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='i) � i∈T c (zσ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='iz∗ ρ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='i) (13) = � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='ρ∈SN � i∈T (1 + h2(zσ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='i)) � i∈T c (zσ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='iz∗ ρ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='i) (14) = � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='ρ∈SN � R⊂T \uf8ee \uf8f0 � i∈T\\R h2(zσ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='i) � i∈T c zσ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='iz∗ ρ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='i \uf8f9 \uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (15) where we defined T ⊂ [N] as the set of indices such that σ(i) = ρ(i) for given permutations σ and ρ and h2(z) ≡ zz∗ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' An important fact is that {1, z, z∗, h2(z)} forms an orthogonal basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', EZ[f1f ∗ 2 ] = 0 if f1 and f2 are different functions out of the basis, and they are eigenvectors of the 9 noise operator Tx[f](z) ≡ Ey[f(√xz + √1 − xy)] with y being the complex random Gaussian noise N(0, 1), namely, 1 → 1, z → √xz, z∗ → √xz∗, and h2(z) → xh2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (16) Here, we assign a degree for each by adding 1 for z or z∗ and 2 for h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, the degree of the term in the parenthesis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (15) is 2(|T| − |R|) + 2(N − |T|) = 2(N − |R|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We further partition these terms according to the image R′ of R under σ and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, we denote by σ′ and ρ′ the restriction of σ and ρ on the complement of R, namely these are one-to-one functions from Rc and [N] \\ R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Let S(σ′, ρ′) ⊂ Rc be the set of indices on which they agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Using some algebra, one can show that the degree 2(N − k) part is given by f =2(N−k) = � R,R′⊂[N]: |R|,|R′|=k � σ∈Sk: R→R′ � σ′,ρ′∈SN−k: Rc→R′c � i∈S(σ′,ρ′) h2(zσ′(i),i) � i∈Rc\\S(σ′,ρ′) zσ′(i),iz∗ ρ′(i),i (17) = � R,R′⊂[N]: |R|,|R′|=k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � σ′,ρ′∈SN−k: Rc→R′c � i∈S(σ′,ρ′) h2(zσ′(i),i) � i∈Rc\\S(σ′,ρ′) zσ′(i),iz∗ ρ′(i),i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (18) Hence, we can rewrite the absolute-squared permanent as |Per(Z)|2 = N � k=0 f =2(N−k), (19) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Let us introduce the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' As shown from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (16), the noise operator Tx[f](z) introduces additional prefactor xN−k for each 2(N − k)-degree polynomial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', f =2(N−k) → xN−kf =2(N−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (20) Hence, the noisy output probability becomes ˜p(z) = 1 MN N � k=0 xN−kf =2(N−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (21) Also, the following relations can be easily checked from the orthogonality of basic elements [KK14]: EZ[f =2(N−k1)f =2(N−k2)∗] = (N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2δk1,k2, (22) where the average is over the complex random Gaussian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Here, (N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 factor comes by counting the number of orthogonal polynomials in f =2(N−k), �N k �2 (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2((N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 = (N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2, (23) where �N k �2 are from the number of choices for R, R′ and (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=') from the coefficients, and ((N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 from the number of choices for σ′, ρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The noisy probability expression suggests that the high-degree polynomials are more sensitive to the noise and they are suppressed exponentially in their degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Also, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (22) shows that the contribution from high-degree polynomials does not scale as their degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Therefore, we will approximate the output probability by truncating the polynomials by 10 setting a cutoff of the degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We will show in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='3 that the complexity of computing f =2(N−k) is determined by the degree 2(N − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' More concretely, if we choose the maximum degree as 2l and truncate higher-degree contributions, we obtain an approximated probability written by the sum of low-degree polynomials ¯q(z) ≡ N � k=N−l xN−kf =2(N−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (24) Then the approximation error is written as ˜p(z) − ¯q(z) = 1 MN N−l−1 � k=0 xN−kf =2(N−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (25) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='2 Bounds for the total variation distance So far, we have focused on the approximation error of a single output probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Using this, we will derive the upper bound of the total variation distance of the full probability distribution, ∆ ≡ � z |˜p(U, z) − ¯q(U, z)| = � z∈cf |˜p(U, z) − ¯q(U, z)| + |˜p(U, c) − ¯q(U, c)|, (26) where cf represents the set of all collision-free outcomes and the first sum is over cf and collision outcome c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Here, we explicitly expressed the dependency of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We note that ˜p(U, z) is defined as 1−� z∈cf ˜p(U, z) (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' To find the upper bound of the total variation distance, we first need to assign the value of ¯q(U, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' For this moment, let us assign this probability as ¯q(U, c) = 1 − � z∈cf ¯q(U, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (27) We will show how to make the approximate distribution ¯q satisfy the assumption in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Such an assignment makes the analysis much easier because |˜p(U, c) − ¯q(U, c)| ≤ � z∈cf |˜p(U, z) − ¯q(U, z)|, (28) which is from the assumption and the triangular inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Then the average squared total variation distance is upper bounded as EU[∆2] ≤ 4EU \uf8ee \uf8f0 \uf8eb \uf8ed � z∈cf |˜p(U, z) − ¯q(U, z)| \uf8f6 \uf8f8 2\uf8f9 \uf8fb (29) ≤ 4 �M N � EU \uf8ee \uf8f0 � z∈cf (˜p(U, z) − ¯q(U, z))2 \uf8f9 \uf8fb (30) ≤ 4 �M N �2 EU � (˜p(U, z) − ¯q(U, z))2� , (31) where the average is taken over Haar-random unitaries U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We have used Jensen’s inequality for the second inequality, and we have used the fact that the average over U gives rise to symmetry to 11 possible collision-free outcomes z ∈ cf, the number of which is �M N � , for the third equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' By using the low-degree polynomial approximation, its upper bound can be written as EU � (˜p(U, z) − ¯q(U, z))2� = 1 M2N EU \uf8ee \uf8f0 �N−l−1 � k=0 xN−kf =2(N−k) �2\uf8f9 \uf8fb (32) = (N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 M2N N−l−1 � k=0 x2(N−k) (33) ≤ (N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 M2N N−l−1 � k=0 x2(l+1) (34) = (N − l + 1)x2(l+1)(N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 M2N , (35) where we have used the orthogonality, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (22), replacing the Haar-random unitary average with random Gaussian matrix average, for the second equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Finally, EU � ∆2� ≤ 4 �M N �2 (N − l + 1)x2(l+1)(N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 M2N (36) ≤ 4 �MN N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' �2 (N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 (N − l + 1)x2(l+1) M2N (37) ≤ 4Nx2(l+1), (38) where we have used the inequality �M N � ≤ MN/N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' for the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Together with this, we will use Markov’s inequality PrU � ∆ ≥ 1 √ δ � EU[∆2] � = PrU � ∆2 ≥ 1 δEU[∆2] � ≤ δ, (39) where the probability is over Haar-random unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus for 1 − δ portion of Haar-random unitary matrices, the approximation error of low-degree polynomial is upper-bounded by � z |˜p(U, z) − ¯q(U, z)| ≤ 2 √ Nxl+1 √ δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (40) Therefore, to bound the error by ǫ > 0, it is sufficient to choose the cutoff of degree l such that l ≥ log(2 √ N/ǫ √ δ) log(1/x) − 1 = O(log N, log(1/ǫ), log(1/δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (41) To introduce the noise effect that scales with the system size, we also consider the case where x scales as x = xγ 1 with a constant x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Then, the total variation distance bound becomes 2 √ Nxl+1 √ δ = 2 √ Nxγ(l+1) 1 √ δ , (42) which implies that it is sufficient to choose the degree as l ≥ log 2 √ N ǫ √ δ γ log 1/x1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (43) 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='3 Approximate sampling In the previous section, we have shown that ¯q(z) with an appropriate cutoff of degree l approximates the noisy distribution ˜p(z) with an error ǫ with high probability 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' It is worth emphasizing again that a similar analysis was conducted in [KK14] while it focused on approximating a single output probability only and did not provide the bound for total variation distance and a classi- cal approximate sampler of low-degree approximated distribution ¯q(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The remaining challenge from the previous section is to find a classical sampling algorithm from ¯q(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' A caveat is that the approximated distribution ¯q(z) is not necessarily a proper probability distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', it might have a negative quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Nevertheless, the following lemma [BMS17] provides a recipe for dealing with quasi-probability distribution, which can be straightforwardly generalized to M-level outcomes instead of binary outcomes: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (modified) Let ˜p be a probability distribution on MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' If there is an oracle that computes a function ¯q : MN → R as well as its marginals satisfying � x ¯q(x) = 1, such that ∥˜p − ¯q∥1 ≤ ǫ, then there is an algorithm that samples from a probability distribution q using O(MN) calls to the oracle, such that ∥˜p − q∥1 ≤ 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Here, the marginal is defined as ¯q(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , xk) = �M xk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=',xN=1 ¯q(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , xN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We have added an additional assumption � x ¯q(x) = 1, which results in the approximation error by 2ǫ instead of 4ǫ/(1 − ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Therefore, it suffices to find ¯q whose marginals can be efficiently computed and are close to ˜p so that it can be used for the lemma for noisy Boson Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The remaining section will show that ¯q obtained by sparse low-degree polynomials satisfies such conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' One immediate difficulty of applying this lemma to Boson Sampling is that a restriction of an outcome z such that z1 ≤ z2 ≤ · · · ≤ zN makes it difficult to compute its marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' To circumvent such a difficulty, we will consider the unordered outcome vector r introduced with the first quantization instead of the ordered vector z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' While the output vector r without ordering has a redundancy, it enables us to easily express the marginals since it does not have the restriction of ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thanks to the symmetry between r and z, we can rewrite it as p(r) = p(z) N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' = 1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1 MN N � k=0 f =2(N−k)(Z), (44) where Z corresponds to the submatrix of U by choosing the first N rows and r’s columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Our strategy was to set a cutoff on the degree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', ¯q(r) = 1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1 MN N � k=N−l xN−kf =2(N−k)(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (45) Note that changing the representation from z to r does not change the simulation error due to the symmetry and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We now show that marginals can also be computed using a similar method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The marginal 13 probability of the noiseless distribution is p(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rj) = 1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � σ,ρ∈SN � j� i=1 Uσ(i),riU ∗ ρ(i),ri � \uf8eb \uf8ed N � i=j+1 ⟨ρ(i)|σ(i)⟩ \uf8f6 \uf8f8 (46) = 1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � J⊂[N]: |J|=j � τ∈SN−j: [j+1,N]→Jc � σ,ρ∈Sj: [j]→J � j� i=1 Uσ(i),riU ∗ ρ(i),ri � \uf8eb \uf8ed N � i=j+1 ⟨τ(i)|τ(i)⟩ \uf8f6 \uf8f8 (47) = (N − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � J⊂[N]: |J|=j � σ,ρ∈Sj: [j]→J � j� i=1 Uσ(i),riU ∗ ρ(i),ri � (48) = 1 Mj (N − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � J⊂[N]: |J|=j � σ,ρ∈Sj: [j]→J � R⊂T �� i∈T h2(zσ(i),ri) � i∈T c zσ(i),riz∗ ρ(i),ri � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (49) Using the same procedure as in the probability case, we can rewrite the noiseless marginal probability as p(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rj) = (N − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1 Mj j � k=0 g=2(j−k), (50) where g=2(j−k) = � |R|,|R′|=k: R⊂[j],R′⊂[N] � σ∈Sk: R→R′ � K′⊂[N]\\R′: |K′|=j−k � σ′,ρ′∈Sj−k: [j]\\R→K′ � i∈S(σ′,ρ′) h2(zσ′(i),ri) � i∈([j]\\R)\\S(σ′,ρ′) zσ′(i),riz∗ ρ′(i),ri (51) = � |R|,|R′|=k: R⊂[j],R′⊂[N] k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � K′⊂[N]\\R′ :|K′|=j−k � σ′,ρ′∈Sj−k: [j]\\R→K′ � i∈S(σ′,ρ′) h2(zσ′(i),ri) � i∈([j]\\R)\\S(σ′,ρ′) zσ′(i),riz∗ ρ′(i),ri (52) = � |R|=k:R⊂[j] k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' �N k � � K′⊂[N]: |K′|=j−k � σ′,ρ′∈Sj−k: [j]\\R→K′ � i∈S(σ′,ρ′) h2(zσ′(i),ri) � i∈([j]\\R)\\S(σ′,ρ′) zσ′(i),riz∗ ρ′(i),ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (53) Here k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' accounts for the permutations between R and R′ and �N k � accounts for the choice of R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Observe that when j = N, it reduces to f =2(N−k), which describes the full probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Also, the noisy marginal distribution is written as ˜p(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rj) = (N − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1 Mj j � k=0 xj−kg=2(j−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (54) Thus, the marginal of the approximate distribution is ¯q(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rj) = (N − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 1 Mj j � k=j−l xj−kg=2(j−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (55) 14 Here, the complexity of computing g=2(j−k) is given by �j k �� N j − k � ((j − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 ≤ (Nj)j−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (56) Recall that we set a cutoff of the degree as 2(j − k) ≤ 2l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' When j ≤ l, since the maximum degree is 2j, we do not approximate and the complexity of computing ¯q(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rj) is upper-bounded by j � k=0 (Nj)j−k ≤ l(Nl)l ≤ N 2l+1 = O(N 2l+1), (57) where we have used j ≤ l ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' When j > l, we start to approximate and the complexity of computing ¯q(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rj) is given by j � k=j−l �j k �� N j − k � ((j − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 ≤ (l + 1)(Nj)l ≤ (N + 1)(N 2)l = O(N 2l+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (58) Therefore, we can compute any marginals of ¯q(r) in complexity O(N 2l+1) satisfying ∥˜p − ¯q∥1 ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Hence, we can simply apply Lemma 3 to sample from a proper probability distribution q such that ∥˜p − q∥1 ≤ 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' From the previous section, for constant x we showed that the degree l can be chosen to be l = O � log(2 √ N/ǫ √ δ) log(1/x) � to bound the total variation distance and that the complexity of computing a single probability (marginal is the same or less) is O(N 2l+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Hence, by using the lemma, the total complexity to generate a sample is then given by N O(log N,log ǫ−1,log δ−1), (59) which proves Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' As mentioned before, the algorithm’s running time is quasi-polynomial not polynomial as in [AGL+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The reason is that the noise rate does not scale as the system size for our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' To properly introduce the noise that scales with the system size, we again consider the case that x = xγ 1 with a constant x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In this case, l can be chosen to be l = O � log 2 √ N ǫ √ δ γ log 1/x1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Hence, for γ = Ω(log N), the complexity becomes polynomial: O(poly(N, 1/ǫ, 1/δ)), (60) which proves Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We emphasize that the degree of the polynomial of the running time in the noise rate scales as log(1/x1) ≈ 1/(1 − x1), where the approximation is valid for small noise rate x1 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, the running time of our algorithm can be very large due to the large degree of the polynomial, which makes it impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Also, it is worthwhile to emphasize an extreme case where we only choose the lowest degree polynomial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Obviously, the lowest degree polynomial, in this case, is a constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', the corresponding probability distribution is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 4 Low-degree approximation with partial distinguishability noise 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='1 Noise sensitivity and low-degree polynomial approximation In various optical experiments including Boson Sampling experiments, one of the most important noise sources is partial distinguishability of particles, which is caused when the particles are not 15 fully indistinguishable because of other degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The effect of partial distinguishability on Boson Sampling has been studied in [Tic15, RMC+18, RSGP18, MGPRT19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Let us study the effect of the noise and approximation method of noisy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Again, consider an output probability and expand it using an orthogonal polynomial basis p(z) = |Per(UN,z)|2 = 1 MN � σ,ρ∈SN N � i=1 zσ(i),iz∗ ρ(i),i, (61) where Z corresponds to a rescaled submatrix of a unitary and is approximated by a random Gaussian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Then, after introducing the partial distinguishability of photons, the probability becomes [Tic15] |Per(Z)|2 = � σ,ρ∈SN N � i=1 zσ(i),iz∗ ρ(i),i → � σ,ρ∈SN xN−k N � i=1 zσ(i),iz∗ ρ(i),i, (62) where k is the number of i’s such that σ(i) = ρ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In other words, whenever we have an interference due to indistinguishability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', i ∈ [M] such that σ(i) ̸= ρ(i), the partial distinguishability x is multiplied as a noise factor (x = 1 for fully indistinguishable cases and x = 0 for fully distinguishable cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Now, we expand the probability: N � i=1 zσ(i),iz∗ ρ(i),i = � i∈T (zσ(i),iz∗ σ(i),i) � i∈T c (zσ(i),iz∗ ρ(i),i) = � i∈T h1(zσ(i),i) � i∈T c h2(zσ(i),i, zρ(i),i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (63) In this case, we have chosen a different basis of polynomials: 1, h1(z) ≡ zz∗, h2(z, z′) ≡ zz′∗, for independent variables z and z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (64) As we have seen, the effect of partial distinguishability is to transform each polynomial as 1 → 1, h1(z) → h1(z), h2(z) → xh2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (65) Here, we assign the degree by adding 0 for h1 and 1 for h2 based on the sensitivity to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Notice a difference from the circuit noise in the previous section that h1(z) is not sensitive to the noise, and thus it has degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We rewrite the summation as |Per(Z)|2 = N � k=0 � T,T ′⊂[N] |T|=|T ′|=k � σ∈Sk: T→T ′ � σ′,ρ′∈SN−k: σ′(i)̸=ρ′(i), T c→T ′c � i∈T h1(zσ(i),i) � i∈T c h2(zσ′(i),i, zρ′(i),i) = N � k=0 f =(N−k), (66) where k is the number of i’s such that σ(i) = ρ(i) from the previous notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' For each k, we need to decide k elements from [N] for input and output, which are represented by T and T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The new σ is the permutation between these newly chosen sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' And σ′ and ρ′ are now permutations between the remaining (N − k) indices and σ′(i) ̸= ρ′(i) for all i’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, the (N − k)th degree part is written as f =(N−k) = � T,T ′⊂[N] |T|=|T ′|=k � σ∈Sk: T→T ′ � σ′,ρ′∈SN−k: σ′(i)̸=ρ′(i), T c→T ′c � i∈T h1(zσ(i),i) � i∈T c h2(zσ′(i),i, zρ′(i),i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (67) 16 After some algebra, we can show that (See Appendix C) EZ[f =(N−k1)f =(N−k2)∗] = 0, if k1 ̸= k2, (68) and that EZ[|f =(N−k)|2] = �N k �2 (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (N − k)) k � j=0 �k j �2 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (k − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (k − j))2j, (69) where (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='k) represents the number of derangements of k elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', the number of permutations σ between k elements such that σ(i) ̸= i for any i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' When the photons in the system have partial distinguishability x, the polynomial transforms as f =(N−k) → xN−kf =(N−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (70) Thus, our approximation strategy is to keep the polynomials up to degree l: ˜p(z) = 1 MN N � k=0 xN−kf =(N−k) ≈ 1 MN N � k=N−l xN−kf =(N−k) ≡ ¯q(z), (71) and the approximation error is ˜p(z) − ¯q(z) = 1 MN N−l−1 � k=0 xN−kf =(N−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (72) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='2 Bounds for the total variation distance Using the same method as the previous section, we can show that EU[∆2] ≤ 4 �M N �2 EU [˜p(U, z) − ¯q(U, z)]2 = 4 �M N �2 1 M2N N � k=l+1 x2kEZ[|f =k|2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (73) In Appendix C, we show that EZ[|f =k|2] ≤ e2(N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (74) (Note that one can numerically check that e2 is generally not necessary [RMC+18] but we keep it since it does not change our main result below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=') Hence, the average squared total variation distance is bounded as EU[∆2] ≤ 4 �M N �2 1 M2N N � k=l+1 x2ke2(N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 ≤ 4 N � k=l+1 x2(l+1)e2 ≤ 4e2Nx2(l+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (75) By applying Markov’s inequality as the previous case, we can conclude that for 1−δ portion of Haar- random linear-optical circuits, the approximation error of low-degree polynomial is upper-bounded by � z |˜p(U, z) − ¯q(U, z)| ≤ 2e √ Nxl+1 √ δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (76) To bound the error by ǫ, it is sufficient to choosse l to be l = log � 2e √ N ǫ √ δ � log(1/x) − 1 = O(log N, log(1/ǫ), log(1/δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (77) 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='3 Barrier of approximate sampling Now, we again try to find an analogous classical sampler to the previous case and show a barrier to implementing it in an efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' First of all, the noisy distribution is written as ˜p(r) = 1 MNN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' N � k=0 xN−kf =(N−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (78) Our strategy was to set a cutoff l on the degree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', ¯q(r) = 1 MNN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' N � k=N−l xN−kf =(N−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (79) One can easily check that the number of summands in f (N−k) is given by �N k �2 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (N − k)), (80) which is larger than N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' regardless of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, direct computation of f =(N−k) is inefficient to any degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' One might hope that there can still be a possibility of computing this quantity efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' However, we can show that exact computation requires exponential time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' To see this, consider the lowest-degree polynomial l = 0, which is the fixed point of the noise: � σ∈SN � N � i=1 Uσ(i),riU ∗ σ(i),ri � = Per(|UN,r|2), (81) where |U|2 is the matrix obtained by taking absolute values on each matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Therefore, it is written as the permanent of a positive matrix, and its exact computation is known to be #P-hard [Val79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Meanwhile, [RMC+18] observed that the permanent of positive matrices can be efficiently approximated in multiplicative error [JSV04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Let us recall their method and present a caveat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We can rewrite the polynomial as in [RMC+18]: f =(N−k) = � T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='T ′⊂[N] |T|=|T ′|=k � σ∈Sk: T→T ′ � σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='ρ′∈SN−k: σ′(i)̸=ρ′(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' T c→T ′c � i∈T h1(zσ(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='i) � i∈T c h2(zσ′(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' zρ′(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='i) (82) = � T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='T ′⊂[N] |T|=|T ′|=k Per(|ZT ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='T |2) � τ ′∈SN−k: τ ′(i)̸=i T ′c→T ′c � σ′∈SN−k: T c→T ′c � i∈T c h2(zσ′(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' zτ ′(σ′(i)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='i) (83) = � T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='T ′⊂[N] |T|=|T ′|=k Per(|ZT ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='T |2) � τ ′∈SN−k: τ ′(i)̸=i T ′c→T ′c � σ′∈SN−k: T ′c→T c � i∈T ′c h2(zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='σ′(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' zτ ′(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='σ′(i)) (84) = � T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='T ′⊂[N] |T|=|T ′|=k � τ ′∈SN−k: τ ′(i)̸=i T c→T ′c Per(|ZT ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='T|2)Per(ZT ′c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='T c ∗ Zτ ′(T ′c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='T c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (85) where ∗ represents the elementwise multiplication of two matrices and ZT ′c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='T c is obtained by selecting rows and columns corresponding to T ′c and T c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' and Zτ ′(T ′c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='T c is obtained similarly but 18 with permuting the rows by τ ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' One can notice that if we set N − k = l, the number of terms to sum is � N N − l �2 (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='l), (86) which is a polynomial in l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Also, the matrix size of ZT ′c,T c∗Zτ ′(T ′c),T c is given by l×l, whose permanent can be exactly computed in ˜O(2l) [Rys63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Meanwhile, the difficulty comes from computing the permanent of |ZT ′,T|2, whose matrix size is (N − l) × (N − l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [RMC+18] claimed that since we can efficiently approximate the permanent of positive matrices in multiplicative error [JSV04], it might enable us to approximate f =l as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' However, this is not immediately obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' To see this clearer, we can simply write f =l as an inner product of two vectors a, b ∈ Cpoly(N), f =l = a · b, (87) where all the elements of a can be exactly computed and those of b can be efficiently approximated in multiplicative error, which corresponds to Per(ZT ′c,T c∗Zτ ′(T ′c),T c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The difficulty is the fact that even though we have exact values of a, they can be negative (or even complex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, the quantity f =l we are approximating is the sum of many terms, which can be only be approximated, with different signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In general, it does not guarantee even a multiplicative error approximation for f =l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Therefore, the difficulty of computing the probability becomes a barrier to applying the same technique to partial distinguishability even though the approximation error using low-degree polynomials is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Furthermore, even if we could approximate the probabilities in a multiplicative error, the direct application of Lemma 3 still requires an exact computation of probabilities and marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Our analysis reveals that channeling between the small approximation error (in total variation distance) and constructing an efficient classical sampler is highly nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' More precisely, our analysis implies that an additional condition is required for noise, which is that the low-degree polynomials need to be composed only of polynomially many orthogonal basis polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We remark that even though the probability of the fixed point of partial distinguishability noise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', fully distinguishable Boson Sampling, is described by the permanent of a positive matrix and computing the probability is hard, the corresponding sampling can be shown to be easy even exactly [AA11, AA13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' This is because fully distinguishable particles do not interfere, so that we can sample particle by particle, which does not require computing the probability of N particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Therefore, it remains open to adapt such a method without computing probabilities to circumvent the barrier and construct an approximate sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 5 Barriers to photon Loss Finally, let us consider photon-loss which is one of the most detrimental noise models in Boson Sampling experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We can assume that all the loss occurs at the beginning with total trans- mission rate η = ηd 1, where d is the depth of the circuit and η1 is a constant loss rate per depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' This simplification can be justified in many cases because uniform loss channel and beam splitters commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In the second quantization representation, the density matrix of the state is written as |1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , 1, 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , 0⟩⟨1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , 1, 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , 0|, (88) which represents the number of photons for each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The effect of photon loss is to transform a single-photon state as |1⟩⟨1| → η|1⟩⟨1| + (1 − η)|0⟩⟨0| (89) 19 and the vacuum state |0⟩⟨0| does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Therefore, if we introduce photon loss, the state transforms N � k=0 �N k � ηk(1 − η)N−k ˆρk, (90) where ˆρk is k-photon states with equal weight of selecting k photons out of the initial N photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' One distinct feature of photon loss from other noise models is that the photon number changes and that the output quantum state occupies lower than N photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' If we exploit the same method as the previous cases, we will need to discard the terms having ηk with k > l with a cutoff l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' It implies that we discard N � k=l+1 �N k � ηk(1 − η)N−k ˆρk, (91) which contains at least ηl+1 degrees, while there are other remaining terms that contain ηl+1 degrees;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' thus, we will underestimate the approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We note that by discarding the above term, we do not obtain any outcomes which have larger than l photons because Boson Sampling circuit does not change the number of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Even when underestimating the approximation error, one can easily see that the probability of the discarded terms is given by Tr � N � k=l+1 �N k � ηk(1 − η)1−k ˆρk � = N � k=l+1 �N k � ηk(1 − η)N−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (92) Here, we emphasize that ˆρk’s for different k’s are orthogonal each other from the density matrix level, which is a distinct property from the other noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, regardless of a linear-optical circuit, the probability that we have lost from discarding high-degree contributions of η is already large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' To be more precise, notice that the photon number distribution follows the binomial distribution with mean ηN and standard deviation � Nη(1 − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' It suggests that we need to keep at least l ≥ ηN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' As a comparison, for circuit noise and partial distinguishability, the required degree was l = O(log N) for a constant noise rate, which shows that the required degree for photon loss is much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Now, let us now consider the output probability of obtaining r which has k clicks with N − k photons lost and analyze the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Without loss of generality, let us set ri = 0 for k+1 ≤ i ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Then, the output probability of lossy Boson Sampling is written as ˜p(r) = ηk(1 − η)N−k N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' �N k �−1 � T⊂[N]:|T|=k |Per(UT,r)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (93) Approximating by low-degree in η only changes the prefactor as ¯q(r) = ηk N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' �N k �−1 l−k � j=0 �N − k j � (−η)j � T⊂[N]:|T|=k |Per(UT,r)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (94) Thus, the complexity of ¯q(r) by computing all the permanents and summing them is ˜O ��N k � 2k � = ˜O � N k� , (95) 20 which is exponential in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Therefore, to make the complexity at most quasi-polynomial as before, l needs to be at most logarithmic in the system size N, l = O(log N), which requires the condition ηN = O(log N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' However, it is known that when ηN = O( √ N), the corresponding noisy distribution can be approximated by a separable state or thermal state input Boson Sampling [OB18, GPRS19], which can be easily simulated using a classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' More specifically, the trace distance between lossy single photons and a thermal state converges to 0 when ηN = o( √ N) in an asymptotic regime (it converges to a constant when ηN = Θ( √ N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Therefore, the regime in which the proposed technique might work can already be classically simulated using different techniques with the approximation error converging to zero in the asymptotic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' It is worth emphasizing that we assumed that the sum of permanents Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (94) can only be obtained by computing individual permanents, which might not be the optimal method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' For certain cases, exponential sum of quantities that are hard to compute can be easily obtained [OLW+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Acknowledgements We thank Senrui Chen and Umesh Vazirani for interesting and fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' LJ acknowledges support from the ARO MURI (W911NF-21-1-0325), AFOSR MURI (FA9550-19-1-0399, FA9550- 21-1-0209), AFRL (FA8649-21-P-0781), DoE Q-NEXT, NSF (OMA-1936118, ERC-1941583, OMA- 2137642), NTT Research, and the Packard Foundation (2020-71479).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' BF acknowledges support from AFOSR (YIP number FA9550-18-1-0148 and FA9550-21-1-0008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' This material is based upon work partially supported by the National Science Foundation under Grant CCF-2044923 (CAREER) and by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Department of Energy, Office of Science, National Quantum Information Science Research Centers as well as by DOE QuantISED grant DE-SC0020360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' References [AA11] Scott Aaronson and Alex Arkhipov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The computational complexity of linear optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 333–342, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [AA13] Scott Aaronson and Alex Arkhipov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Bosonsampling is far from uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' arXiv preprint arXiv:1309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='7460, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [AAB+19] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando GSL Brandao, David A Buell, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum supremacy using a programmable superconducting processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Nature, 574(7779):505–510, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [Aar22] Scott Aaronson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' https://scottaaronson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='blog/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='p=6871, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [AB16] Scott Aaronson and Daniel J Brod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Bosonsampling with lost photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Physical Review A, 93(1):012335, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [ABOIN96] Dorit Aharonov, Michael Ben-Or, Russell Impagliazzo, and Noam Nisan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Limitations of noisy reversible computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' arXiv preprint quant-ph/9611028, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [AC17] Scott Aaronson and Lijie Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Complexity-theoretic foundations of quantum supremacy experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In Ryan O’Donnell, editor, 32nd Computational Complex- 21 ity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 22:1–22:67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [AG19] Scott Aaronson and Sam Gunn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' On the classical hardness of spoofing linear cross- entropy benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' CoRR, abs/1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='12085, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [AGL+22] Dorit Aharonov, Xun Gao, Zeph Landau, Yunchao Liu, and Umesh Vazirani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' A polynomial-time classical algorithm for noisy random circuit sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='03999, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [BCG21] Boaz Barak, Chi-Ning Chou, and Xun Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Spoofing linear cross-entropy benchmarking in shallow quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In James R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Lee, editor, 12th Innovations in Theoreti- cal Computer Science Conference, ITCS 2021, January 6-8, 2021, Virtual Conference, volume 185 of LIPIcs, pages 30:1–30:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum f¨ur Infor- matik, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [BFLL21] Adam Bouland, Bill Fefferman, Zeph Landau, and Yunchao Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Noise and the frontier of quantum supremacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 1308–1317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [BFNV19] Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' On the complex- ity and verification of quantum random circuit sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Nature Physics, 15(2):159–163, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [BHH16] Fernando G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Brand˜ao, Aram W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Harrow, and Micha�l Horodecki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Local random quantum circuits are approximate polynomial-designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Communications in Mathemat- ical Physics, 346(2):397–434, aug 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [BJS10] Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Bremner, Richard Jozsa, and Dan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Shepherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Classical simulation of com- muting quantum computations implies collapse of the polynomial hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 467(2126):459–472, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [BMS17] Michael J Bremner, Ashley Montanaro, and Dan J Shepherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Achieving quantum supremacy with sparse and noisy commuting quantum computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum, 1:8, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [BSN17] Sergio Boixo, Vadim N Smelyanskiy, and Hartmut Neven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Fourier analysis of sampling from noisy chaotic quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' arXiv preprint arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='01875, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [BV93] Ethan Bernstein and Umesh Vazirani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum complexity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, pages 11–20, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [BVHS+18] Juan Bermejo-Vega, Dominik Hangleiter, Martin Schwarz, Robert Raussendorf, and Jens Eisert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Architectures for quantum simulation showing a quantum speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Phys- ical Review X, 8(2), apr 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [DHJB22] Alexander M Dalzell, Nicholas Hunter-Jones, and Fernando GSL Brand˜ao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Random quantum circuits anticoncentrate in log depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' PRX Quantum, 3(1):010333, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 22 [DMV+22] Abhinav Deshpande, Arthur Mehta, Trevor Vincent, Nicol´as Quesada, Marcel Hin- sche, Marios Ioannou, Lars Madsen, Jonathan Lavoie, Haoyu Qi, Jens Eisert, Dominik Hangleiter, Bill Fefferman, and Ish Dhand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum computational advantage via high-dimensional Gaussian boson sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Science Advances, 8(1):eabi7894, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [DNS+22] Abhinav Deshpande, Pradeep Niroula, Oles Shtanko, Alexey V Gorshkov, Bill Feffer- man, and Michael J Gullans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Tight bounds on the convergence of noisy random circuits to the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' PRX Quantum, 3(4):040329, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [GD18] Xun Gao and Luming Duan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Efficient classical simulation of noisy quantum computa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='03176, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [GPRS19] Ra´ul Garc´ıa-Patr´on, Jelmer J Renema, and Valery Shchesnovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Simulating boson sampling in lossy architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum, 3:169, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [HBVSE18] Dominik Hangleiter, Juan Bermejo-Vega, Martin Schwarz, and Jens Eisert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Anticon- centration theorems for schemes showing a quantum speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum, 2:65, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [HE22] Dominik Hangleiter and Jens Eisert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Computational advantage of quantum random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='04079, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [HHB+20] Jonas Haferkamp, Dominik Hangleiter, Adam Bouland, Bill Fefferman, Jens Eisert, and Juani Bermejo-Vega.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Closing gaps of a quantum advantage with short-time hamiltonian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Physical Review Letters, 125(25):250501, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [HKS+17] Craig S Hamilton, Regina Kruse, Linda Sansoni, Sonja Barkhofen, Christine Silberhorn, and Igor Jex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Gaussian boson sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Physical review letters, 119(17):170501, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [HM18] Aram Harrow and Saeed Mehraban.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Approximate unitary t-designs by short ran- dom quantum circuits using nearest-neighbor and long-range gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' arXiv preprint arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='06957, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [JSV04] Mark Jerrum, Alistair Sinclair, and Eric Vigoda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Journal of the ACM (JACM), 51(4):671–697, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [KK14] Gil Kalai and Guy Kindler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Gaussian noise sensitivity and bosonsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' arXiv preprint arXiv:1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='3093, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [KMM21] Yasuhiro Kondo, Ryuhei Mori, and Ramis Movassagh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum supremacy and hard- ness of estimating output probabilities of quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 1296–1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [MGPRT19] Alexandra E Moylett, Ra´ul Garc´ıa-Patr´on, Jelmer J Renema, and Peter S Turner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Classically simulating near-term partially-distinguishable and lossy boson sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum Science and Technology, 5(1):015001, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [MLA+22] Lars S Madsen, Fabian Laudenbach, Mohsen Falamarzi Askarani, Fabien Rortais, Trevor Vincent, Jacob FF Bulmer, Filippo M Miatto, Leonhard Neuhaus, Lukas G Helt, Matthew J Collins, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum computational advantage with a programmable photonic processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Nature, 606(7912):75–81, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 23 [Nez21] Sepehr Nezami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Permanent of random matrices from representation theory: moments, numerics, concentration, and comments on hardness of boson-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='06423, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [NJF20] Kyungjoo Noh, Liang Jiang, and Bill Fefferman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Efficient classical simulation of noisy random quantum circuits in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum, 4:318, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [NSC+17] Alex Neville, Chris Sparrow, Rapha¨el Clifford, Eric Johnston, Patrick M Birchall, Ash- ley Montanaro, and Anthony Laing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Classical boson sampling algorithms with superior performance to near-term experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Nature Physics, 13(12):1153–1157, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [OB18] Micha�l Oszmaniec and Daniel J Brod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Classical simulation of photonic linear optics with lost particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' New Journal of Physics, 20(9):092002, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [OLFJ22] Changhun Oh, Youngrong Lim, Bill Fefferman, and Liang Jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Classical simulation of boson sampling based on graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Physical Review Letters, 128(19):190501, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [OLW+22] Changhun Oh, Youngrong Lim, Yat Wong, Bill Fefferman, and Liang Jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum-inspired classical algorithm for molecular vibronic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='01861, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [ONFJ21] Changhun Oh, Kyungjoo Noh, Bill Fefferman, and Liang Jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Classical simula- tion of lossy boson sampling using matrix product operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Physical Review A, 104(2):022407, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [QBQGP20] Haoyu Qi, Daniel J Brod, Nicol´as Quesada, and Ra´ul Garc´ıa-Patr´on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Regimes of classical simulability for noisy gaussian boson sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Physical review letters, 124(10):100502, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [RCOL17] Nicholas J Russell, Levon Chakhmakhchyan, Jeremy L O’Brien, and Anthony Laing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Direct dialling of haar random unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' New journal of physics, 19(3):033007, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [RMC+18] Jelmer J Renema, Adrian Menssen, William R Clements, Gil Triginer, William S Kolthammer, and Ian A Walmsley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Efficient classical algorithm for boson sampling with partially distinguishable photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Physical review letters, 120(22):220502, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [RSGP18] Jelmer Renema, Valery Shchesnovich, and Raul Garcia-Patron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Classical simulability of noisy boson sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' arXiv preprint arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='01953, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [Rys63] Herbert John Ryser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Combinatorial mathematics, volume 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' American Mathematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [Shc19] Valery S Shchesnovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Noise in boson sampling and the threshold of efficient classical simulatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Physical Review A, 100(1):012340, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [TD04] Barbara M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Terhal and David P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' DiVincenzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Adaptive quantum computation, constant-depth quantum circuits and Arthur-Merlin games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum Information and Computation, 4(2):134–145, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [Tic15] Malte C Tichy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Sampling of partially distinguishable bosons and the relation to the multidimensional permanent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Physical Review A, 91(2):022316, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' 24 [TTT21] Yasuhiro Takahashi, Yuki Takeuchi, and Seiichiro Tani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Classically simulating quantum circuits with local depolarizing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Theoretical Computer Science, 893:117–132, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [Val79] Leslie G Valiant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' The complexity of computing the permanent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Theoretical computer science, 8(2):189–201, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [VNL+21] Benjamin Villalonga, Murphy Yuezhen Niu, Li Li, Hartmut Neven, John C Platt, Vadim N Smelyanskiy, and Sergio Boixo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Efficient approximation of experimental gaus- sian boson sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='11525, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [WBC+21] Yulin Wu, Wan-Su Bao, Sirui Cao, Fusheng Chen, Ming-Cheng Chen, Xiawei Chen, Tung-Hsun Chung, Hui Deng, Yajie Du, Daojin Fan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Strong quantum computa- tional advantage using a superconducting quantum processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Physical review letters, 127(18):180501, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [ZDQ+21] Han-Sen Zhong, Yu-Hao Deng, Jian Qin, Hui Wang, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Dian Wu, Si-Qiu Gong, Hao Su, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Phase-programmable Gaussian boson sampling using stimulated squeezed light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Physical review letters, 127(18):180502, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' [ZWD+20] Han-Sen Zhong, Hui Wang, Yu-Hao Deng, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Jian Qin, Dian Wu, Xing Ding, Yi Hu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Quantum computational advantage using photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Science, 370(6523):1460–1463, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' A Output distribution of Gaussian noise In this Appendix, we show that the output probability distribution after Gaussian noise [KK14] is a nontrivial and proper probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Specifically, we show that the probability of obtaining outcomes in the collision-free subspace is close to and smaller than one, as in the noiseless case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Therefore, by defining the remaining probability to normalize the sum of probabilities, the noise maps a noiseless output probability distribution into another proper probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Recall that Gaussian noise transforms the unitary matrix of a boson sampling circuit as U → √xU + √ 1 − xY, (96) where Y is a random Gaussian matrix with variance 1/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Consider a probability of detecting N photons for the first N modes with N input photons from the first N modes: p(U, z) = |Per(UN,N)|2 = � σ,ρ∈SN N � i=1 Ui,ρ(i)U ∗ i,σ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (97) 25 After Gaussian noise, it transforms to ˜p(U, z) = EY \uf8ee \uf8f0 � σ,ρ∈SN N � i=1 (√xUi,ρ(i) + √ 1 − xYi,ρ(i))(√xUi,σ(i) + √ 1 − xYi,σ(i))∗ \uf8f9 \uf8fb (98) = EY \uf8ee \uf8f0 � σ,ρ∈SN N � i=1 (xUi,σ(i)U ∗ i,ρ(i) + (1 − x)Yi,σ(i)Y ∗ i,ρ(i)) \uf8f9 \uf8fb (99) = N � k=0 xk(1 − x)N−k MN−k (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � K,K′⊂[N]: |K|=|K′|=k � σ,ρ∈Sk:K→K′ � i∈K Ui,σ(i)U ∗ i,ρ(i) (100) = N � k=0 xk(1 − x)N−k MN−k (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � K,K′⊂[N]: |K|=|K′|=k |Per(UK,K′)|2, (101) where for the second equality, we used the independence of matrix elements of Y , and for the third equality, we split permutations into trivial permutations, from again independence of Y ’s elements, and nontrivial permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Let us sum over all collision-free outcomes z: � z∈cf ˜p(U, z) = � z∈cf N � k=0 xk(1 − x)N−k MN−k (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � K⊂[N]: |K|=k � K′⊂z: |K′|=k |Per(UK,K′)|2 (102) = N � k=0 xk(1 − x)N−k MN−k �M − k N − k � (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � K⊂[N]: |K|=k � K′⊂[M]: |K′|=k |Per(UK,K′)|2 (103) = N � k=0 xk(1 − x)N−k MN−k (M − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (M − N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � K⊂[N]: |K|=k (collision-free for K input boson sampling with U) (104) = N � k=0 xk(1 − x)N−k MN−k (M − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (M − N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � K⊂[N]: |K|=k [1 − (collision for K input boson sampling with U)], (105) where (collision(-free) for K input boson sampling with U) represents the probability of obtaining collision(-free) outcomes with |K| single photons in modes K with the circuit unitary U, and the inclusion symbol from z is defined to be the subsets of the modes i’s such that zi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' First, we find the upper bound: N � k=0 xk(1 − x)N−k MN−k (M − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (M − N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � K⊂[N]: |K|=k [1 − (collision for K input boson sampling with U)] (106) < N � k=0 xk(1 − x)N−k �N k � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (107) 26 Now we find the lower bound of the average over Haar-random unitary U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Using the bosonic birthday paradox [AA11], we can bound the total collision-free outcomes as EU \uf8ee \uf8f0 � z∈cf ˜p(U, z) \uf8f9 \uf8fb (108) = EU \uf8ee \uf8ef\uf8ef\uf8f0 N � k=0 xk(1 − x)N−k MN−k (M − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (M − N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � K⊂[N]: |K|=k [1 − (collision for K input boson sampling with U)] \uf8f9 \uf8fa\uf8fa\uf8fb (109) > N � k=0 xk(1 − x)N−k MN−k (M − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (M − N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' � K⊂[N]: |K|=k � 1 − 2k2 M � (110) ≥ N � k=0 xk(1 − x)N−k �N k � � 1 − N M �N � 1 − 2N 2 M � (111) → 1, (112) where for the last expression, we used M = ω(N 2) for large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Therefore, using the assumption of the strong collision-free regime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', M = ω(N 5), the noisy output probability distribution sums close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Finally, we defined the collision case of the noisy distribution as the remaining probability, so that the total probability is normalized to be one, � z∈cf ˜p(U, z) + ˜p(U, c) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (113) B Collision In this Appendix, we will show how to make the distribution ¯q(r) to satisfy the sufficient condition � r∈[M]N ¯q(r) = 1 by assigning ¯q(r) for collision cases r properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We will assume that we have chosen the cutoff of degree as l ≥ 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Then, the first-order marginal ¯q(r1) is exact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', ¯q(r1) = ˜p(r1) for all r1 ∈ [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' For the second-order marginals, we will define for each r1 ∈ [M] ¯q(r1, r2 = r1) = ¯q(r1) \uf8ee \uf8f01 − � r2∈[M]\\{r1} ¯q(r1, r2) \uf8f9 \uf8fb , (114) which obviously guarantees that �M r2=1 ¯q(r1, r2) = ¯q(r1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Similarly, for given (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rk−1) with distinct {ri}k−1 i=1 , we define for each rk ∈ {ri}k−1 i=1 ¯q(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rk) = ¯q(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rk−1) \uf8ee \uf8ef\uf8f01 − 1 k − 1 � rk∈[M]\\{ri}k−1 i=1 ¯q(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rk−1, rk) \uf8f9 \uf8fa\uf8fb for each rk ∈ {ri}k−1 i=1 , (115) which again guarantees that �M rk=1 ¯q(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rk) = ¯q(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' For such rk’s and for all permu- tations σ ∈ Sk, we also define ¯q(rσ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rσ(k)) ≡ ¯q(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (116) 27 We continue this procedure until k = N when we define all quantities of ¯q(r) of r ∈ [M]N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Now, we have defined all relevant quantities of ¯q(r) and its marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Consequently, we can easily show that the resultant distribution satisfies � r∈[M]N ¯q(r) = 1, (117) which can be easily shown by the marginal relation, ¯q(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rk−1) = M � rk=1 ¯q(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' , rk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (118) As a remark, we argue why this procedure is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Since the collision probability is inverse- polynomially suppressed when M = ω(N 2) [AA11], one might be tempted to set it to be zero for ¯q(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' However, one can immediately see that it might cause a large error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Suppose that a quasi-probability distribution ¯q(r) is given for collision-free space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=', which is close to the target distribution � r∈cf |˜p(r) − ¯q(r)| ≤ ǫ, (119) where cf accounts for the set of collision-free outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We first show that a naive approach may entail a large error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Let us denote the probability of collisions as ǫc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' We will set ¯q(r) = 0 for collision outcomes r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Then, for full distribution we have � r∈[M]N |˜p(r) − ¯q(r)| = � r∈c |˜p(r) − ¯q(r)| + � r∈cf |˜p(r) − ¯q(r)| ≤ ǫc + ǫ ≡ ǫt, (120) where c accounts for the set of collision outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Then after using the lemma from [BMS17], we can sample from a proper probability distribution q(z) with the total variation distance given by � z |˜p(z) − q(z)| ≤ 4ǫt 1 − ǫt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (121) Since the collision probability ǫc is fixed for a given system, we cannot reduce the error as much as we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, we need to assign appropriate quantities of ¯q(r) for collision outcomes before applying the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' C Orthogonality of polynomials for partial distinguishability noise In this Appendix, we show the orthogonality of polynomials introduced for partial distinguishability noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Consider EZ[|f =(N−k)|2] = EZ � \uf8eb \uf8ed � T,T ′⊂[N],|T|=|T ′|=k � σ,σ′,ρ′ � i∈T h1(zσ(i),i) � i∈T c h2(zσ′(i),i, zρ′(i),i) \uf8f6 \uf8f8 × \uf8eb \uf8ed � T ∗,T ′∗⊂[N],|T ∗|=|T ′∗|=k � σ∗,σ′∗,ρ′∗ � i∈T ∗ h∗ 1(zσ∗(i),i) � i∈T ∗c h∗ 2(zσ′∗(i),i, zρ′∗(i),i) \uf8f6 \uf8f8 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (122) 28 Here if T ̸= T ∗ or T ′ ̸= T ′∗, one can easily check that the average over Z becomes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, we set T ∗ = T and T ′∗ = T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Now, we have, for |T| = k and a fixed T and T ′, � σ,σ∗ � i∈T h1(zσ(i),i)h∗ 1(zσ∗(i),i) = k � j=0 �k j � j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (k − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (k − j))2j, (123) which can be shown by splitting the factors |z|4 and |z|2 and counting each and using Ez[|z|4] = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Here (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='j) is the derangement, namely, the number of permutations σ ∈ Sj such that σ(i) ̸= i for all i’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Meanwhile, � σ′,ρ′ � σ′∗,ρ′∗ � i∈T c h2(zσ′(i),i, zρ′(i),i)h∗ 2(zσ′∗(i),i, zρ′∗(i),i) = � σ′,ρ′ � σ′∗,ρ∗ � i∈T c (zσ′(i),iz∗ ρ′(i),iz∗ σ′∗(i),izρ′∗(i),i) (124) = (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (N − k)), (125) where we used the fact that σ = σ′∗ and ρ = ρ′∗ is necessary to be nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Thus, the number of choices of T and T ′ has �N k �2 and the number of choices of σ′ and ρ′ is (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (N − k)) and we obtain EZ[|f =(N−k)|2] = �N k �2 (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (N − k)) k � j=0 �k j �2 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (k − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (k − j))2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (126) We now further upper bound the two-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' Here, we first use k � j=0 �k j �2 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (k − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (k − j))2j ≤ k � j=0 �k j �2 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' ((k − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )22j (127) = k � j=0 �k j � k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (k − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='2j (128) = e2k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='Γ(k + 1, 2) (129) ≤ e2k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='Γ(k + 1) (130) = e2(k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (131) Thus, EZ[|f =(N−k)|2] = �N k �2 (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (N − k)) k � j=0 �k j �2 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (k − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (k − j))2j (132) ≤ e2 �N k �2 (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content='(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (N − k))(k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 (133) ≤ e2 �N k �2 ((N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=')2(k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2 (134) ≤ e2(N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} +page_content=' (135) 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FJT4oBgHgl3EQfZixR/content/2301.11532v1.pdf'} diff --git a/nNE0T4oBgHgl3EQfqAE4/content/tmp_files/2301.02546v1.pdf.txt b/nNE0T4oBgHgl3EQfqAE4/content/tmp_files/2301.02546v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..db88531f57e47f431fa96657289e91248e39dd6a --- /dev/null +++ b/nNE0T4oBgHgl3EQfqAE4/content/tmp_files/2301.02546v1.pdf.txt @@ -0,0 +1,339 @@ + AHFE 2023: Vol. XX, 2023 +doi: 10.54941/ahfeXXXX +© 2022. Published by AHFE Open Access. All rights reserved. +1 +A new conversational interaction concept +for document creation and editing on mobile +devices for visually impaired users +Alireza Darvishy1, Hans-Peter Hutter1, Edin Beljulji1, Zeno Heeb1 +1Zurich University of Applied Sciences, 8400 Winterthur, Switzerland +ABSTRACT +This paper describes the ongoing development of a conversational interaction concept that +allows visually impaired users to easily create and edit text documents on mobile devices +using mainly voice input. In order to verify the concept, a prototype app was developed and +tested for both iOS and Android systems, based on the natural-language understanding +(NLU) platform Google Dialogflow. The app and interaction concept were repeatedly tested +by users with and without visual impairments. Based on their feedback, the concept was +continuously refined, adapted and improved on both mobile platforms. In an iterative user- +centred design approach, the following research questions were investigated: Can a visually +impaired user rely mainly on speech commands to efficiently create and edit a document on +mobile devices? User testing found that an interaction concept based on conversational +speech commands was easy and intuitive for visually impaired users. However, it was also +found that relying on speech commands alone created its own obstacles, and that a +combination of gestures and voice interaction would be more robust. Future research and +more extensive useability tests should be carried out among visually impaired users in order +to optimize the interaction concept. +Keywords: Visual impairment, mobile devices, non-visual interaction, NLP, speech input, speech +output, document creation + +AHFE +InternationaDarvishy et al. +INTRODUCTION AND RELATED WORKS +According to WHO, there are more than 250 million people living with a visual +impairment worldwide (WHO, 2012) Many people with visual impairments use +assistive technologies such as screen readers to read digital content aloud. A screen +reader is a software application that allows individuals who are blind or have low +vision to access and use the features and functions of a computer or smartphone. +Screen readers work by providing synthesized speech output of the text and other +information displayed on the screen, as well as allowing the user to interact with +the device using keyboard commands or gestures. There are different screen +readers available for mobile devices, such as “Voice Over” in iOS and “TalkBack” +in Android. +In addition to speech output, recent advances in speech technologies have also +enabled users to harness speech input to interact with their mobile devices. For +example, users are able to control their mobile devices solely with voice +commands, e.g. using “Voice Control” in iOS or “Voice Access” in Android +devices. As a use case, users can use voice commands to edit a text on their mobile +devices. These are useful tools for many people with visual impairments, when +compared to other text entry methods such as onscreen keyboards, gesture-based +text entry (wherein the user draws the desired letter/character on the screen using +a finger or stylus), or braille-based digital keyboards (e.g. Mattheiss et. al, 2015; +Oliveria et. al, 2011). However, despite the usefulness of this new software, current +mainstream speech input programs are not designed with visually impaired users +in mind. As a result, many accessibility issues remain. For example: In iOS, the +Voice Control command needed to delete a certain section of text is the phrase +“delete that” – a command which assumes that the user can see which section of +text is currently highlighted. +Speech input has been found to be one of the most efficient text entry methods +for users with visual impairments. A 2013 user study by Azenkot and Lee found +that speech input was almost five times faster for users with visual impairments +than using a standard touch keyboard. However, the same study noted that, despite +the relative ease of entering text, errors in speech recognition remained a major +problem: on average, when composing a text, participants spent 80% of their time +finding and correcting errors (Azenkot & Lee, 2013). +Text editing and error correction can present significant barriers to visually +impaired users, particularly with regards to unstructured and inaccessible +document formats. Structural elements such as headings, tables, and lists, are not +only more difficult to create (requiring additional commands beyond simple +dictation), but they are also important elements in document accessibility, +necessary for efficient navigation via screen readers (see e.g. Rajkumar et al. 2020, +Darvishy et al. 2011, Munteanu et al. 1995). + + + A new conversational interaction concept for document creation and editing on mobile devices for +visually impaired users +3 +In late 2015, Google rolled out a feature called “Voice Typing”, a speech-to- +text dictation feature that can be used in conjunction with Google Docs on the +Chrome browser (Moynihan, 2016). This technology could hold great promise for +visually impaired users because, unlike many other dictation programs and apps, +it includes features such as inserting structural elements (headings, tables, lists, +etc.), jumping to a desired word/sentence in the document, and adjusting +formatting using speech commands. However, Google Docs creation using Voice +Typing is currently only available for desktop use via Chrome. It is also not +designed with visually impaired users in mind – for example, users must click the +microphone icon to start dictation. + +CONCEPT DEVELOPMENT +The major goal of this innovative interaction concept is to allow users to easily +create documents in a conversational manner via speech input, including +commonly used formatting features such as headings and subheadings, +indentation, bullet lists, enumerations, etc. The interaction concept also allows +users to edit the document in a conversational manner, e.g. adding, removing or +changing text by directly referring to them. Consequently, speech output was also +implemented to help users navigate and orient themselves within the document +without relying on a cursor or other visual cues. +In order to develop the conversational interaction model for that in a user- +centered manner, a first user research and analysis of user needs was carried out. +Subsequently, based on the user needs identified, a first design and prototype +iteration was implemented. With this prototype, extensive user tests were carried +out. +In order to define the user needs, initial informal interviews were conducted with +visually impaired potential users. These discussions revealed that the major tasks +needed for mobile document creation, such as text dictation, editing, and basic +structuring, are particularly cumbersome and time-consuming for visually +impaired users. Based on this feedback, an initial concept was developed by +defining context scenarios and sample dialogues for a prototype app called +“Spectra”. +Context Scenarios +A series of context scenarios were defined to demonstrate some envisioned +interactions with the prototype app. In each context scenario, fictional personas +were described performing a task using the app. For example, one such context +scenario, “proofreading a document,” reads as follows: +Angelica works as a technical assistant at a university. It's Monday afternoon +and she has been asked to proofread a few documents for technical accuracy before +they are submitted next week. As Angelica has an optic nerve disease, she cannot +do this visually. Instead, she launches the Spectra app on her smartphone, opening +her first document for today. She can sit back and relax in her chair while the app +reads her a brief overview of the headings in this document. Since Angelica is only +interested in the technically important chapters, she instructs the app to jump + +Darvishy et al. +directly to a specific chapter and read it aloud. With the stop command, she +interrupts the reading, “Please repeat the last sentence”. Spectra reads her the last +sentence again. Because the content of the sentence is incorrect, she uses a voice +command to insert a comment into the document: “Insert comment: This sentence +should be reformulated”. Spectra replies: “Comment inserted: This sentence +should be reformulated”. After Angelica’s reply, “go on,” Spectra continues with +reading on. + +The following context scenarios were considered: +• +Create a shopping list +• +Write a letter +• +Write a report +• +Review and edit a document +Sample dialogues +A number of sample dialogues were created to conceptualize concrete interactions +between the user and the system, showing both the user’s verbal input and the +system’s speech output. By default, the app is programmed to confirm/read back +each command or dictated sentence, so that the user can detect and correct any +errors or misunderstandings. A snippet of such a sample dialogue is shown in Table +1. +Table 1. Snippet of sample dialogue, in which the user writes and edits a document about a +new product (an anti-theft system) +Turn +Utterance +user +Title «anti theft system» +system +Document title “Anti Theft System” +user +Heading one «introduction» +system +Heading 1 «instructions» +user +Replace «instructions» with «instruction» +system +Heading 1 «Introduction» +user +Dictation mode +system +Dictation mode started +user +This new system should achieve protection against burglary comma +both in the absence and presence of residents period +system +This new system should achieve protection against burglary, both in +the absence and presence of residents. +user +Insert “control” before “system” +system +This new control system should achieve protection against burglary, +both in the absence and presence of residents. +Interaction Concept +The interaction concept envisages two modes: the dictation mode and the +command mode. In the dictation mode, the running text of the document is dictated +by voice. Hence, this mode mainly provides speech-to-text (STT) where the voice +input is interpreted as text. The text is entered utterance by utterance. The dictation +mode also allows punctuation information within the dictated text, e.g., commas, +semicolon, etc (see Table 1). The system automatically reads back the recognised +utterance as confirmation, including formatting and punctuation information + + A new conversational interaction concept for document creation and editing on mobile devices for +visually impaired users +5 +(except normal periods and question marks at the end of a sentence, which are +indicated by prosodic means) +After readback of the recognised utterance the command mode becomes +automatically active where the user can immediately edit the text just entered by +replacing, moving, inserting or deleting words. The user can thereby directly refer +to the words entered before, e.g. “Insert ’control’ before ‘system’ in Table 1. If the +user just goes on with dictating, the system automatically switches back to dictation +mode. The user can explicitly switch between dictation and command mode with +the corresponding commands. +With the command mode the dictated text can be reviewed, edited, formatted +and structured (headings, paragraphs, lists etc.) Erroneous entries or commands +can be easily undone with the corresponding commands. There are also various +jump commands to navigate quickly in the document. To get a better overview of +the content of a document, users can also input various read commands, i.e. to +request that certain words, sentences, paragraphs or the entire document be read +aloud. Commands related to a relative position, e.g., “delete last word” are +interpreted with respect to the current focus (virtual cursor). This is set at the end +of the last word read back or at the position of the last editing command. +Alternatively to the pure voice interaction, the app also supports screen readers, +which can be used to read out elements on the touchscreen using tapping gestures. +An export function is also available to share the dictated documents with other +people. +Prototype Implementation +In order to efficiently implement a prototype of the interaction concept, a +distributed speech recognition approach is used based on the local STT +components on the smartphone (iOS or Android). The recognised text is then sent +to the remote intent-based Dialogflow NLU service (Sabharwal & Agrawal, 2020) +for the recognition and interpretation of the commands, after having set the right +context for the intent-based interpretation of the commands. The recognised +commands are finally sent back to the mobile device, where they are executed by +the app, or the recognized sentence is read back to the user. Immediately after that, +the next recognition loop starts. +User Testing +To test the usability of the initial prototype, a group of test users was established +which included both visually impaired and non-visually impaired users. Each test +user was asked to download the Spectra app and was then sent a sample text. The +test task was to re-create the sample text using only the app (alongside their +device’s built-in screen reader, if needed). Test users then submitted the resulting +text document to the authors. All test users also submitted unstructured written +feedback describing their overall experience with the app and noting any obstacles +they encountered. The current iteration of the Spectra app implements changes and +improvements based on this feedback. + + +Darvishy et al. +RESULTS AND DISCUSSION +This ongoing work has delivered two major findings so far: +1) +Using content-based commands, as described below, are most efficient +for visually impaired users in many cases, and is often more practical +than cursor-based commands. +2) +Relying only on speech commands and voice output functions is +insufficient for visually impaired users. Rather, a combination of gestures +and voice interaction is ideal. +In the planning and the first prototype implementation, the app primarily took +“cursor-based” commands, such as “select word” or “delete sentence” which were +interpreted relative to the current cursor position. In this case, if a word is to be +deleted, the cursor must be first moved to the correct position in the text, as in +normal text editors. However, based on user feedback, we have learned that such +commands are not suitable for non-visual text editing. In order to find the desired +word or sentence, users either had to have large sections of content read aloud and +then stop at the right place, or had to use navigation commands to move around. +The alternative to cursor-based commands are more “content-based commands”, +which more closely mimic how people would communicate with one another. For +example, the command “Replace x with y” is more reminiscent of conversational +phrases like “sorry, I meant to say x, not y”. These commands do not require the +position to be specified and are less cumbersome than navigating to find “x” and +giving the commands “delete word” and “dictation mode: y”. In the newest +prototype, we have thus implemented content-based commands in which the old +and new content is specified directly within the command, so that no navigation +commands must be executed first. However, because content-based commands are +not always easier to use or can be misinterpreted by the system, standard cursor- +based commands have also been kept in the prototype, giving users the choice of +using either kind. +The current version of the app includes commands to insert text or structured +content such as paragraphs, lists or headings, as well as commands to undo or +amend incorrect entries. There are also commands to have parts of the document +or the entire document read aloud. The reading can be interrupted using a single- +tap gesture in order to make changes at the desired point. Navigation commands +such as “start of paragraph”, “end of paragraph” etc., are also available to change +the current position in the document. +User testing also identified screen reader support, particularly iOS VoiceOver +support, as a desirable feature. This support means that changes to the text can also +be made using the device’s built-in screen reader functions, whenever Spectra and +the screen reader are running simultaneously. Although iOS and Android both +feature built-in screen readers, the overall accessibility support on iOS is +significantly higher and our target group is therefore increasingly using the iOS +operating system; as such, the focus for the current iteration of Spectra is on +VoiceOver support. + + A new conversational interaction concept for document creation and editing on mobile devices for +visually impaired users +7 +Delegating the NLU to Dialogflow brought several advantages, since the +analysis takes place independently of the native mobile system, and without the +need for a local agent. As a result, the Dialogflow agent has significantly more +resources available than would be possible on a single mobile device. However, of +disadvantage is that there is a delay before the data is sent to and evaluated by +Dialogflow, and the result is received. While this process is not significantly slower +on Android, it takes longer on the iOS operating system. As a result, the speed of +use is lower than would be the case with a local agent. +One major research question remains unanswered: What is the optimal +combination of speech commands, gestures and screen reader output to allow +visually impaired users to efficiently create and edit text documents on mobile +devices? +This question will be addressed in the future development of Spectra, and with +more extensive user testing. +REFERENCES +1. Azenkot, S. and Lee, N. (2013). Exploring the use of speech input by blind people on +mobile devices. Proceedings of the 15th International ACM SIGACCESS Conference +on Computers and Accessibility, ASSETS 2013. +2. Azenkot, S. Wobbrock, J.O. Prasain, S. Ladner, R.E. (2012). Input finger detection for +nonvisual touch screen text entry in Perkinput. Proceedings of graphics interface 2012, +pp. 121-129. +3. Darvishy, A., Hutter, HP., Mannhart, O. (2011). Web Application for Analysis, +Manipulation and Generation of Accessible PDF Documents. In: Stephanidis, C. (eds) +Universal Access in Human-Computer Interaction. Applications and Services. UAHCI +2011. Lecture Notes in Computer Science, vol 6768. Springer, Berlin, Heidelberg. +4. Mattheiss, E. Regal, G. Schrammel, J. Garschall, M. Tscheligi, M. (2015). +EdgeBraille: Braille-based text input for touch devices. Journal of Assistive +Technologies. +5. Darvishy, A., Leemann, T., Hutter, HP. (2012). Two Software Plugins for the Creation +of Fully Accessible PDF Documents Based on a Flexible Software Architecture. In: +Miesenberger, K., Karshmer, A., Penaz, P., Zagler, W. (eds) Computers Helping +People with Special Needs. ICCHP 2012. Lecture Notes in Computer Science, vol +7382. Springer, Berlin, Heidelberg. +6. Darvishy, A., Hutter, HP. (2013). Comparison of the Effectiveness of Different +Accessibility Plugins Based on Important Accessibility Criteria. In: Stephanidis, C., +Antona, M. (eds) Universal Access in Human-Computer Interaction. Applications and +Services for Quality of Life. UAHCI 2013. Lecture Notes in Computer Science, vol +8011. Springer, Berlin, Heidelberg. +7. Moynihan, Tim. (2016) “Now You Can Edit Google Docs by Speaking.” Wired, 29 +February. https://www.wired.com/2016/02/now-can-type-google-docs-speaking/ +8. Munteanu, E., Guggiana, V., Darvishi, A., Schauer, H., Rauterberg, G. W. M., & +Motavalli, M. (1995). Physical modelling of environmental sounds. In F. Pedrielli +(Ed.), Proceedings of the 2nd international conference on acoustic and musical +research, CIARM '95 (pp. 107-112). Universita di Ferrara. +9. Oliveira, J. Guerreiro, T. Nicolau, H. Jorge, J. Gonçalves, D. (2011). BrailleType: +unleashing braille over touch screen mobile phones. IFIP Conference on Human- +Computer Interaction, Springer, Berlin, Heidelberg, pp. 100-107. + +Darvishy et al. +10. Rajkumar, A., J. Lazar, J. B. Jordan, A. Darvishy, and H. Hutter. (2020). PDF +Accessibility of Research Papers: What Tools are Needed for Assessment and +Remediation?. In HICSS. +11. Sabharwal, N., Agrawal, A. (2020). Introduction to Google Dialogflow. In: Cognitive +Virtual Assistants Using Google Dialogflow. Apress, Berkeley, CA. +12. World Health Organization (WHO). (2012). Global Data on Visual Impairments 2010. +Geneva, World Health Organization. + diff --git a/nNE0T4oBgHgl3EQfqAE4/content/tmp_files/load_file.txt b/nNE0T4oBgHgl3EQfqAE4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..17a36c19ff445b6ecddf629715162c8b88332812 --- /dev/null +++ b/nNE0T4oBgHgl3EQfqAE4/content/tmp_files/load_file.txt @@ -0,0 +1,292 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf,len=291 +page_content='AHFE 2023: Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' XX, 2023 doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='54941/ahfeXXXX © 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Published by AHFE Open Access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 1 A new conversational interaction concept for document creation and editing on mobile devices for visually impaired users Alireza Darvishy1, Hans-Peter Hutter1, Edin Beljulji1, Zeno Heeb1 1Zurich University of Applied Sciences, 8400 Winterthur, Switzerland ABSTRACT This paper describes the ongoing development of a conversational interaction concept that allows visually impaired users to easily create and edit text documents on mobile devices using mainly voice input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In order to verify the concept, a prototype app was developed and tested for both iOS and Android systems, based on the natural-language understanding (NLU) platform Google Dialogflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The app and interaction concept were repeatedly tested by users with and without visual impairments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Based on their feedback, the concept was continuously refined, adapted and improved on both mobile platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In an iterative user- centred design approach, the following research questions were investigated: Can a visually impaired user rely mainly on speech commands to efficiently create and edit a document on mobile devices?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' User testing found that an interaction concept based on conversational speech commands was easy and intuitive for visually impaired users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' However, it was also found that relying on speech commands alone created its own obstacles, and that a combination of gestures and voice interaction would be more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Future research and more extensive useability tests should be carried out among visually impaired users in order to optimize the interaction concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Keywords: Visual impairment, mobile devices, non-visual interaction, NLP, speech input, speech output, document creation AHFE InternationaDarvishy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' INTRODUCTION AND RELATED WORKS According to WHO, there are more than 250 million people living with a visual impairment worldwide (WHO, 2012) Many people with visual impairments use assistive technologies such as screen readers to read digital content aloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' A screen reader is a software application that allows individuals who are blind or have low vision to access and use the features and functions of a computer or smartphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Screen readers work by providing synthesized speech output of the text and other information displayed on the screen, as well as allowing the user to interact with the device using keyboard commands or gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' There are different screen readers available for mobile devices, such as “Voice Over” in iOS and “TalkBack” in Android.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In addition to speech output, recent advances in speech technologies have also enabled users to harness speech input to interact with their mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' For example, users are able to control their mobile devices solely with voice commands, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' using “Voice Control” in iOS or “Voice Access” in Android devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' As a use case, users can use voice commands to edit a text on their mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' These are useful tools for many people with visual impairments, when compared to other text entry methods such as onscreen keyboards, gesture-based text entry (wherein the user draws the desired letter/character on the screen using a finger or stylus), or braille-based digital keyboards (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Mattheiss et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' al, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Oliveria et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' al, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' However, despite the usefulness of this new software, current mainstream speech input programs are not designed with visually impaired users in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' As a result, many accessibility issues remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' For example: In iOS, the Voice Control command needed to delete a certain section of text is the phrase “delete that” – a command which assumes that the user can see which section of text is currently highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Speech input has been found to be one of the most efficient text entry methods for users with visual impairments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' A 2013 user study by Azenkot and Lee found that speech input was almost five times faster for users with visual impairments than using a standard touch keyboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' However, the same study noted that, despite the relative ease of entering text, errors in speech recognition remained a major problem: on average, when composing a text, participants spent 80% of their time finding and correcting errors (Azenkot & Lee, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Text editing and error correction can present significant barriers to visually impaired users, particularly with regards to unstructured and inaccessible document formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Structural elements such as headings, tables, and lists, are not only more difficult to create (requiring additional commands beyond simple dictation), but they are also important elements in document accessibility, necessary for efficient navigation via screen readers (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Rajkumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 2020, Darvishy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 2011, Munteanu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' A new conversational interaction concept for document creation and editing on mobile devices for visually impaired users 3 In late 2015, Google rolled out a feature called “Voice Typing”, a speech-to- text dictation feature that can be used in conjunction with Google Docs on the Chrome browser (Moynihan, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' This technology could hold great promise for visually impaired users because, unlike many other dictation programs and apps, it includes features such as inserting structural elements (headings, tables, lists, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' ), jumping to a desired word/sentence in the document, and adjusting formatting using speech commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' However, Google Docs creation using Voice Typing is currently only available for desktop use via Chrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' It is also not designed with visually impaired users in mind – for example, users must click the microphone icon to start dictation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' CONCEPT DEVELOPMENT The major goal of this innovative interaction concept is to allow users to easily create documents in a conversational manner via speech input, including commonly used formatting features such as headings and subheadings, indentation, bullet lists, enumerations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The interaction concept also allows users to edit the document in a conversational manner, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' adding, removing or changing text by directly referring to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Consequently, speech output was also implemented to help users navigate and orient themselves within the document without relying on a cursor or other visual cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In order to develop the conversational interaction model for that in a user- centered manner, a first user research and analysis of user needs was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Subsequently, based on the user needs identified, a first design and prototype iteration was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' With this prototype, extensive user tests were carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In order to define the user needs, initial informal interviews were conducted with visually impaired potential users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' These discussions revealed that the major tasks needed for mobile document creation, such as text dictation, editing, and basic structuring, are particularly cumbersome and time-consuming for visually impaired users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Based on this feedback, an initial concept was developed by defining context scenarios and sample dialogues for a prototype app called “Spectra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Context Scenarios A series of context scenarios were defined to demonstrate some envisioned interactions with the prototype app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In each context scenario, fictional personas were described performing a task using the app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' For example, one such context scenario, “proofreading a document,” reads as follows: Angelica works as a technical assistant at a university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=" It's Monday afternoon and she has been asked to proofread a few documents for technical accuracy before they are submitted next week." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' As Angelica has an optic nerve disease, she cannot do this visually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Instead, she launches the Spectra app on her smartphone, opening her first document for today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' She can sit back and relax in her chair while the app reads her a brief overview of the headings in this document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Since Angelica is only interested in the technically important chapters, she instructs the app to jump Darvishy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' directly to a specific chapter and read it aloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' With the stop command, she interrupts the reading, “Please repeat the last sentence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Spectra reads her the last sentence again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Because the content of the sentence is incorrect, she uses a voice command to insert a comment into the document: “Insert comment: This sentence should be reformulated”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Spectra replies: “Comment inserted: This sentence should be reformulated”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' After Angelica’s reply, “go on,” Spectra continues with reading on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The following context scenarios were considered: Create a shopping list Write a letter Write a report Review and edit a document Sample dialogues A number of sample dialogues were created to conceptualize concrete interactions between the user and the system, showing both the user’s verbal input and the system’s speech output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' By default, the app is programmed to confirm/read back each command or dictated sentence, so that the user can detect and correct any errors or misunderstandings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' A snippet of such a sample dialogue is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Snippet of sample dialogue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' in which the user writes and edits a document about a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='new product (an anti-theft system) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='Turn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='Utterance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='Title «anti theft system» ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='Document title “Anti Theft System” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='Heading one «introduction» ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='Heading 1 «instructions» ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='Replace «instructions» with «instruction» ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='Heading 1 «Introduction» ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='Dictation mode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='Dictation mode started ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='This new system should achieve protection against burglary comma ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='both in the absence and presence of residents period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='This new system should achieve protection against burglary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' both in the absence and presence of residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' user Insert “control” before “system” system This new control system should achieve protection against burglary, both in the absence and presence of residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Interaction Concept The interaction concept envisages two modes: the dictation mode and the command mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In the dictation mode, the running text of the document is dictated by voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Hence, this mode mainly provides speech-to-text (STT) where the voice input is interpreted as text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The text is entered utterance by utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The dictation mode also allows punctuation information within the dictated text, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', commas, semicolon, etc (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The system automatically reads back the recognised utterance as confirmation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' including formatting and punctuation information A new conversational interaction concept for document creation and editing on mobile devices for visually impaired users 5 (except normal periods and question marks at the end of a sentence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' which are indicated by prosodic means) After readback of the recognised utterance the command mode becomes automatically active where the user can immediately edit the text just entered by replacing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' moving,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' inserting or deleting words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The user can thereby directly refer to the words entered before, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' “Insert ’control’ before ‘system’ in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' If the user just goes on with dictating, the system automatically switches back to dictation mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The user can explicitly switch between dictation and command mode with the corresponding commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' With the command mode the dictated text can be reviewed, edited, formatted and structured (headings, paragraphs, lists etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=') Erroneous entries or commands can be easily undone with the corresponding commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' There are also various jump commands to navigate quickly in the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' To get a better overview of the content of a document, users can also input various read commands, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' to request that certain words, sentences, paragraphs or the entire document be read aloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Commands related to a relative position, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', “delete last word” are interpreted with respect to the current focus (virtual cursor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' This is set at the end of the last word read back or at the position of the last editing command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Alternatively to the pure voice interaction, the app also supports screen readers, which can be used to read out elements on the touchscreen using tapping gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' An export function is also available to share the dictated documents with other people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Prototype Implementation In order to efficiently implement a prototype of the interaction concept, a distributed speech recognition approach is used based on the local STT components on the smartphone (iOS or Android).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The recognised text is then sent to the remote intent-based Dialogflow NLU service (Sabharwal & Agrawal, 2020) for the recognition and interpretation of the commands, after having set the right context for the intent-based interpretation of the commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The recognised commands are finally sent back to the mobile device, where they are executed by the app, or the recognized sentence is read back to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Immediately after that, the next recognition loop starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' User Testing To test the usability of the initial prototype, a group of test users was established which included both visually impaired and non-visually impaired users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Each test user was asked to download the Spectra app and was then sent a sample text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The test task was to re-create the sample text using only the app (alongside their device’s built-in screen reader, if needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Test users then submitted the resulting text document to the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' All test users also submitted unstructured written feedback describing their overall experience with the app and noting any obstacles they encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The current iteration of the Spectra app implements changes and improvements based on this feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Darvishy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' RESULTS AND DISCUSSION This ongoing work has delivered two major findings so far: 1) Using content-based commands, as described below, are most efficient for visually impaired users in many cases, and is often more practical than cursor-based commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 2) Relying only on speech commands and voice output functions is insufficient for visually impaired users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Rather, a combination of gestures and voice interaction is ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In the planning and the first prototype implementation, the app primarily took “cursor-based” commands, such as “select word” or “delete sentence” which were interpreted relative to the current cursor position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In this case, if a word is to be deleted, the cursor must be first moved to the correct position in the text, as in normal text editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' However, based on user feedback, we have learned that such commands are not suitable for non-visual text editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In order to find the desired word or sentence, users either had to have large sections of content read aloud and then stop at the right place, or had to use navigation commands to move around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The alternative to cursor-based commands are more “content-based commands”, which more closely mimic how people would communicate with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' For example, the command “Replace x with y” is more reminiscent of conversational phrases like “sorry, I meant to say x, not y”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' These commands do not require the position to be specified and are less cumbersome than navigating to find “x” and giving the commands “delete word” and “dictation mode: y”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In the newest prototype, we have thus implemented content-based commands in which the old and new content is specified directly within the command, so that no navigation commands must be executed first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' However, because content-based commands are not always easier to use or can be misinterpreted by the system, standard cursor- based commands have also been kept in the prototype, giving users the choice of using either kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The current version of the app includes commands to insert text or structured content such as paragraphs, lists or headings, as well as commands to undo or amend incorrect entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' There are also commands to have parts of the document or the entire document read aloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' The reading can be interrupted using a single- tap gesture in order to make changes at the desired point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Navigation commands such as “start of paragraph”, “end of paragraph” etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', are also available to change the current position in the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' User testing also identified screen reader support, particularly iOS VoiceOver support, as a desirable feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' This support means that changes to the text can also be made using the device’s built-in screen reader functions, whenever Spectra and the screen reader are running simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Although iOS and Android both feature built-in screen readers, the overall accessibility support on iOS is significantly higher and our target group is therefore increasingly using the iOS operating system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' as such, the focus for the current iteration of Spectra is on VoiceOver support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' A new conversational interaction concept for document creation and editing on mobile devices for visually impaired users 7 Delegating the NLU to Dialogflow brought several advantages, since the analysis takes place independently of the native mobile system, and without the need for a local agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' As a result, the Dialogflow agent has significantly more resources available than would be possible on a single mobile device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' However, of disadvantage is that there is a delay before the data is sent to and evaluated by Dialogflow, and the result is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' While this process is not significantly slower on Android, it takes longer on the iOS operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' As a result, the speed of use is lower than would be the case with a local agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' One major research question remains unanswered: What is the optimal combination of speech commands, gestures and screen reader output to allow visually impaired users to efficiently create and edit text documents on mobile devices?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' This question will be addressed in the future development of Spectra, and with more extensive user testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Azenkot, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' and Lee, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Exploring the use of speech input by blind people on mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Proceedings of the 15th International ACM SIGACCESS Conference on Computers and Accessibility, ASSETS 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Azenkot, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Wobbrock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Prasain, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Ladner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Input finger detection for nonvisual touch screen text entry in Perkinput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Proceedings of graphics interface 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 121-129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Darvishy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Hutter, HP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Mannhart, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Web Application for Analysis, Manipulation and Generation of Accessible PDF Documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In: Stephanidis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (eds) Universal Access in Human-Computer Interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Applications and Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' UAHCI 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Lecture Notes in Computer Science, vol 6768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Springer, Berlin, Heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Mattheiss, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Regal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Schrammel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Garschall, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Tscheligi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' EdgeBraille: Braille-based text input for touch devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Journal of Assistive Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Darvishy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Leemann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Hutter, HP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Two Software Plugins for the Creation of Fully Accessible PDF Documents Based on a Flexible Software Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In: Miesenberger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Karshmer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Penaz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Zagler, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (eds) Computers Helping People with Special Needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' ICCHP 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Lecture Notes in Computer Science, vol 7382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Springer, Berlin, Heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Darvishy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Hutter, HP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Comparison of the Effectiveness of Different Accessibility Plugins Based on Important Accessibility Criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In: Stephanidis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Antona, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (eds) Universal Access in Human-Computer Interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Applications and Services for Quality of Life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' UAHCI 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Lecture Notes in Computer Science, vol 8011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Springer, Berlin, Heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Moynihan, Tim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (2016) “Now You Can Edit Google Docs by Speaking.” Wired, 29 February.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='wired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='com/2016/02/now-can-type-google-docs-speaking/ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Munteanu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Guggiana, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Darvishi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Schauer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Rauterberg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', & Motavalli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Physical modelling of environmental sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Pedrielli (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=" ), Proceedings of the 2nd international conference on acoustic and musical research, CIARM '95 (pp." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 107-112).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Universita di Ferrara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Oliveira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Guerreiro, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Nicolau, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Jorge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Gonçalves, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' BrailleType: unleashing braille over touch screen mobile phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' IFIP Conference on Human- Computer Interaction, Springer, Berlin, Heidelberg, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 100-107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Darvishy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Rajkumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Lazar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Jordan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Darvishy, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' PDF Accessibility of Research Papers: What Tools are Needed for Assessment and Remediation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content='. In HICSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Sabharwal, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=', Agrawal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Introduction to Google Dialogflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' In: Cognitive Virtual Assistants Using Google Dialogflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Apress, Berkeley, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' World Health Organization (WHO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Global Data on Visual Impairments 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} +page_content=' Geneva, World Health Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE0T4oBgHgl3EQfqAE4/content/2301.02546v1.pdf'} diff --git a/nNFAT4oBgHgl3EQfcx2g/content/tmp_files/2301.08566v1.pdf.txt b/nNFAT4oBgHgl3EQfcx2g/content/tmp_files/2301.08566v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f4db519c11765c9f5e690944ae76e1893a96f47 --- /dev/null +++ b/nNFAT4oBgHgl3EQfcx2g/content/tmp_files/2301.08566v1.pdf.txt @@ -0,0 +1,1936 @@ +arXiv:2301.08566v1 [math.AG] 20 Jan 2023 +The higher direct images of locally constant group schemes +from the Kummer log flat topology to the classical flat +topology +Heer Zhao +Abstract. Let S be an fs log scheme, and let F be a group scheme over the +underlying scheme which is ´etale locally representable by (1) a finite dimen- +sional Q-vector space, or (2) a finite rank free abelian group, or (3) a finite +abelian group. We give a full description of all the higher direct images of F +from the Kummer log flat site to the classical flat site. In particular, we show +that: in case (1) the higher direct images of F vanish; and in case (2) the first +higher direct image of F vanishes and the n-th (n > 1) higher direct image +of F is isomorphic to the (n − 1)-th higher direct image of F ⊗Z Q/Z. In the +end, we make some computations when the base is a standard log trait or a +Dedekind scheme endowed with the log structure associated to a finite set of +closed points. +1. Introduction +Let S be an fs log scheme, (fs/S) the category of fs log schemes over S. We +endow (fs/S) with the Kummer log flat topology (resp. the classical flat topol- +ogy, resp. +the classical ´etale topology), see [KAT21, §2] (resp. +[KAT21, §4], +resp. [Ill02, §2.4]), and denote the resulting site by (fs/S)kfl (resp. (fs/S)fl, resp. +(fs/S)´et)1. We have a canonical map +εfl : (fs/S)kfl → (fs/S)fl +of sites. To understand the cohomology of a sheaf of abelian groups F on the site +(fs/S)kfl, one needs to understand the higher direct images Riεfl∗F. +The first higher direct image R1εfl∗F has been determined by Kato when S is +locally noetherian and F is either a finite flat (commutative) group scheme or a +smooth affine (commutative) group scheme over the underlying scheme of S, see +[KAT21, Thm. 4.1] or [Niz08, Thm. 3.12]. Kato’s theorem about R1εfl∗F has +2020 Mathematics Subject Classification. 14F20 (primary), 14A21 (secondary). +Key words and phrases. log schemes, Kummer flat topology, comparison of cohomology. +1In [KAT21] the sites (fs/S)kfl and (fs/S)fl are denoted as Slog +fl +and Scl +fl respectively. Our +notation here is analogous to that of the Kummer log ´etale site from [KKN15, §5.3] +1 + +2 +HEER ZHAO +been generalized to quasi-projective smooth (commutative) group schemes by the +author, see [Zha21a, Thm. 3.14]. +Assume that the underlying scheme of S is locally noetherian. The second +higher direct image R2εfl∗F has been described in [Zha21a, Thm. 3.23] (resp. +[Zha21b, Thm. 1.2], resp. [Zha21b, Thm. 1.3], resp. [Zha21b, Thm. 1.4]) when +F is representable by a torus (resp. a smooth affine commutative group scheme, +resp. a finite flat commutative group scheme, resp. an extension of an abelian +scheme by a torus) over the underlying scheme of S. When F is representable by a +smooth quasi-projective commutative group scheme, the higher direct images are +always torsion by [Zha21b, Thm. 1.1]. +In this article, we investigate Riεfl∗F for all i > 0 in Section 2 when F is +representable by a group scheme which is ´etale locally isomorphic either to a finite +dimension Q-vector space, or a finite rank free abelian group, or a finite abelian +group. The main results are the following three theorems. +Theorem 1.1. (See also Theorem 2.9) Let S be a locally noetherian fs log +scheme. Let l be a prime number, U the open locus on S where l is invertible, +and j : U ֒→ S the corresponding strict open immersion. Let F be a finite ´etale +group scheme over the underlying scheme of S, and we endow it with the induced +log structure from S. Assume that F is killed by an l-power, then we have +Riεfl∗F ∼= jfl!((j−1 +kfl F)(−i) ⊗Z +i� +(Gm,log/Gm)Ukfl) +for i ≥ 1, where jkfl : (fs/U)kfl → (fs/S)kfl (resp. jfl : (fs/U)fl → (fs/S)fl) is the +morphism on the Kummer log flat sites (resp. the classical flat sites) induced by j. +Theorem 1.2. (See also Theorem 2.10) Let S be an fs log scheme. Let F be a +group scheme over the underlying scheme of S which is ´etale locally representable +by a finite dimensional Q-vector space. Then we have Riεfl∗F = 0 for i ≥ 1. +Theorem 1.3. (See also Theorem 2.11) Let S be a locally noetherian fs log +scheme, and F a group scheme over the underlying scheme of S which is ´etale +locally isomorphic to a finite rank free abelian group. Then we have the following. +(1) R1εfl∗F = 0. +(2) Let i > 1. For each prime number l, let Ul be the locus on S on which +l is invertible and lj : Ul ֒→ S the corresponding strict open immersion. +Then +Riεfl∗F ∼= +� +l prime +Ri−1εfl∗(F ⊗Z Ql/Zl) +∼= +� +l prime +ljfl!(lj−1F ⊗Z Ql/Zl(−i + 1) ⊗Z +i−1 +� +(Gm,log/Gm)(Ul)fl). +The proof of Theorem 1.3 is reduced to Theorem 1.1 and Theorem 1.2 via the +short exact sequence +0 → F → F ⊗Z Q → F ⊗Z Q/Z → 0. + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +3 +Apparently the vanishing of Riεfl∗F in Theorem 1.2 is reduced to the vanishing of +Hi +kfl(X, F) for any X ∈ (fs/S) such that the underlying scheme of X is Spec R with +R a strictly henselian local ring. For the proof of Theorem 1.1, we first construct +a canonical map +Φ : jfl!Riεfl∗j−1 +kfl F → Riεfl∗F, +see (B.2), then determine Riεfl∗j−1 +kfl F (note that the order of F is invertible on +U), and finally prove that Φ is an isomorphism. The main computation tools are +ˇCech cohomology, ˇCech-to-derived functor spectral sequence, and Leray spectral +sequence. +In section 3, we apply the results from Section 2 to make some computations on +Kummer log flat cohomology when the base is a standard log trait or a Dedekind +scheme endowed with the log structure associated to a finite set of closed points. +An application. Theorem 1.3 (1) can be used to prove that log abelian va- +rieties with constant degeneration are sheaves for the Kummer log flat topology. +According to [KKN08, Def. 3.3], a log abelian variety with constant degeneration +A over an fs log scheme S is a sheaf on (fs/S)´et which is isomorphic to the quotient +sheaf G(Y ) +log /Y for a pointwise polarizable log 1-motive [Y +u−→ Glog] over S. One +can consider G(Y ) +log as a uniformization of A, and Y as the corresponding periods +lattice. It is not surprising that Y is important for understanding A. Indeed we +have R1εfl∗Y = 0 by Theorem 1.3 (1), and this vanishing is used in [Zha17, Thm. +2.1 (1)] to prove that A is a sheaf for the Kummer log flat topology. +In fact Theorem 1.3 (1) is just [Zha17, Lem. 2.4]. However the proof loc. cit. +makes use of fpqc descent of schemes which probably does not always hold and +deserves a precise reference (see [Sta21, Lemma 0APK] for the situation loc. cit.). +The original motivation of this article is to present a new proof to [Zha17, Lem. +2.4]. +2. The higher direct images +We make a few lemmas first. +Lemma 2.1. Let X be an fs log scheme. We make the following assumption on +X, as well as some constructions associated to X. +⋆ The underlying scheme is Spec R with R a strictly henselian local ring. +Let x denote the closed point, k the residue field of R, p the characteristic +of k, and PX → MX a chart of the log structure of X with P an fs monoid, +such that P +∼ += +−→ MX,x/O× +X,x. Let P 1/n denote the monoid P regarded as +a monoid above P via the homomorphism P +n−→ P. Let +Xn := X ×Spec Z[P ] Spec Z[P 1/n] +endowed with the canonical log structure associated to P 1/n and let Hn +denote the group scheme Spec Z[(P 1/n)gp/P gp] over Spec Z, then Xn is a +Kummer log flat cover of X such that +Xn ×X Xn ∼= Xn ×Spec Z Hn, + +4 +HEER ZHAO +and it is even a Kummer log ´etale cover in case (p, n) = 1. +Let F be a constant group scheme over Spec R associated to an abelian group, and +we write n = m · pt with (m, p) = 1. Then we have an isomorphism +ˇHi +kfl(Xn/X, F) ∼= Hi(Hm(X), F), +where the first term is the i-th ˇCech cohomology of F with respect to the cover +Xn/X, and the second term is the abstract group cohomology of the abstract group +Hm(X) with coefficients in F. +In particular, we have the following. +(1) ˇHi +kfl(Xn/X, Q) = 0 for i > 0. +(2) ˇH1 +kfl(Xn/X, Z) = 0 and ˇHi +kfl(Xn/X, Z) is torsion and p-torsion-free for +i > 0. +(3) The canonical map +ˇHi +kfl(Xm/X, Z) +∼ += +−→ ˇHi +kfl(Xn/X, Z) +is an isomorphism for i > 0. +(4) Suppose that F is killed by a power of p. Then +lim +−→ +n +ˇHi +kfl(Xn/X, F) = 0. +(5) Suppose that F is torsion and p-torsion-free. Then +lim +−→ +n +ˇHi +kfl(Xn/X, F) ∼= lim +−→ +n +Hi(Hm(X), F) ∼= F(−i) ⊗Z +i� +P gp +for i > 0. +Proof. Since R is a strictly henselian local ring, X ×Spec Z Hr +m is the constant +group scheme over X associated to the abstract group Hm(X)r and X ×Spec Z Hr +pt +is a connected group scheme over X, therefore we have +Γ(Xn ×X · · · ×X Xn +� +�� +� +r + 1 times +, F) =Γ(Xn ×Spec Z Hr +n, F) +=Γ((X ×Spec Z Hr +pt) ×X (Xn ×Spec Z Hr +m), F) += +� +h∈Hm(X)r +F +=Map(Hm(X)r, F). +We consider the ˇCech complex +(2.1) +Γ(Xn, F) +d0 +−→ Γ(Xn ×X Xn, F) +d1 +−→ Γ(Xn ×X Xn ×X Xn, F) +d2 +−→ · · · +for F with respect to the cover Xn/X. Let Γn := (P +1 +n )gp/P gp. By [Mil80, Chap. +III, Example 2.6], the ˇCech nerve of the Kummer log flat cover Xn/X can be + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +5 +identified with the sequence +Xn +Xn × Hn +d1,0 +� +d1,1 +� +Xn × H2 +n +d2,0 +�� +d2,2 +� +Xn × H3 +n · · · , +d3,0 +��� +d3,3 +� +where the map dr,i on the ring level is given by the R-linear ring homomorphism +R ⊗Z[P ] Z[P +1 +n ⊕ Γr−1 +n +] → R ⊗Z[P ] Z[P +1 +n ⊕ Γr +n] +(a, ¯a1, · · · , ¯ar−1) �→ + + + + + +(a, ¯a, ¯a1, · · · , ¯ar−1), +if i = 0; +(a, ¯a1, · · · , ¯ai, ¯ai, · · · , ¯ar−1), +if 0 < i < r; +(a, ¯a1, · · · , ¯ar−1, 0), +if i = r. +for any (a, ¯a1, · · · , ¯ar−1) ∈ P +1 +n ⊕ Γr−1 +n +. If m = 1, i.e. n = pt, we have +Γ(Xpt ×Spec Z Hr +pt, F) = F. +The map +d∗ +r,i : Γ(Xpt ×Spec Z Hr−1 +pt +, F) → Γ(Xpt ×Spec Z Hr +pt, F) +is clearly just the identity map Id : F → F. In general, the map +d∗ +r,i : Γ(Xn ×Spec Z Hr−1 +n +, F) → Γ(Xn ×Spec Z Hr +n, F) +can be identified with the map +Map(Hm(X)r−1, F) +∂r,i +−−→ Map(Hm(X)r, F) +which maps f ∈ Map(Hm(X)r−1, F) to +∂r,i(f) : (h1, · · · , hr) �→ + + + + + +f(h2, · · · , · · · , hr), +if i = 0; +f(h1, · · · , hi + hi+1, · · · , hr), +if 0 < i < r; +f(h1, · · · · · · , hr−1), +if i = r. +Therefore the complex (2.1) can be identified with the standard complex +F → Map(Hm(X), F) → Map(Hm(X)2, F) → · · · +which computes the abstract group cohomology of the trivial Hm(X)-module F. +It follows that ˇHi +kfl(Xn/X, F) ∼= Hi(Hm(X), F). +In particular, we have part (1), part (2), and part (4) by the corresponding +results for the abstract group cohomology Hi(Hm(X), F). Part (3) is also clear by +the construction of the isomorphism ˇHi +kfl(Xn/X, F) ∼= Hi(Hm(X), F). + +6 +HEER ZHAO +We are left with part (5). We have +lim +−→ +n +ˇHi +kfl(Xn/X, F) ∼= +lim +−→ +n = mpt with (p, m) = 1 +Hi(Hm(X), F) +(3) += +lim +−→ +(p,m)=1 +Hi(Hm(X), F) +=Hi( +lim +←− +(p,m)=1 +Hm(X), F) +=Hi(Hom(P gp, ˆZ′(1)), F). +By the description of the profinite group cohomology of (ˆZ′)r from Lemma C.1, we +further have +lim +−→ +n +ˇHi +kfl(Xn/X, F) ∼=Hi(Hom(P gp, ˆZ′(1)), F) += +lim +−→ +(n,p)=1 +Hi(Hom(P gp, ˆZ′(1)), F[n]) += +lim +−→ +(n,p)=1 +F[n] ⊗Z/nZ Hi(Hom(P gp, ˆZ′(1)), Z/nZ) += +lim +−→ +(n,p)=1 +F[n] ⊗Z/nZ +i� +H1(Hom(P gp, ˆZ′(1)), Z/nZ) += +lim +−→ +(n,p)=1 +F[n] ⊗Z/nZ +i� +Hom(Hom(P gp, ˆZ′(1)), Z/nZ) += +lim +−→ +(n,p)=1 +F[n] ⊗Z/nZ ( +i� +P gp ⊗Z Z/nZ(−i)) +=F(−i) ⊗Z +i� +P gp +for i > 0. This finishes the proof of part (5). +□ +Remark 2.1. In the proof of Lemma 2.1 (5), one can use Lemma C.1 to get +Hi(Hom(P gp, ˆZ′(1)), F) ∼= F ⊗Z +�i P gp directly. However we repeat the proof of +Lemma C.1 here in order to keep track of the Tate twist. +Lemma 2.2. Let X and Xn be as in Lemma 2.1. Let F be a sheaf on (fs/X)kfl. +Then the family XN := {Xn → X}n≥1 of Kummer log flat covers of X satisfies +the condition (L3) from [Art62, §2], whence a ˇCech-to-derived functor spectral +sequence +(2.2) +Ei,j +2 += ˇHi +kfl(XN, Hj +kfl(F)) ⇒ Hi+j +kfl (X, F), +where +ˇHi +kfl(XN, −) := lim +−→ +n≥1 +ˇHi +kfl(Xn/X, −). + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +7 +Proof. This follows from [Art62, Chap. II, Sec. 3, (3.3)]. +□ +A nice thing about the ˇCech cohomology with respect to all coverings, is that +the zero-th cohomology vanishes, see [Zha21a, Prop. +3.1]. +This is often very +useful for computations. This is of course not the case for the ˇCech cohomology +with respect to an arbitrary family. The following lemma is about the vanishing of +the zero-th ˇCech cohomology for the covering family XN from Lemma 2.2 under a +suitable assumption. +Lemma 2.3. Let X and Xn be as in Lemma 2.1. Let F be a sheaf on (fs/X)kfl, +and N a non-negative integer. For any U ∈ (fs/X), let (st/U) be the full subcategory +of (fs/U) consisting of strict objects over U. Assume that for any U ∈ (fs/X) and +any 0 ≤ i ≤ N, the restriction of the flat sheaf Riεfl∗F to (st/U) is the inverse +image of some sheaf on the small ´etale site of U, i.e. it lies in the image of the +functor a−1 +U +from [Sta21, Lemma 0DDU]. Then we have +lim +−→ +n≥1 +Hi +kfl(Xn, F) = 0 +for any 0 < i ≤ N + 1, in particular lim +−→n≥1 ˇH0 +kfl(Xn/X, Hi +kfl(F)) = 0 for any +0 < i ≤ N + 1. +Proof. Clearly we only need to prove the vanishing for i = N + 1. It suffices +to show that for any γ ∈ HN+1 +kfl +(Xr, F) there exists s such that γ goes to zero in +HN+1 +kfl +(Xrs, F). +By [Zha21a, Prop. 3.1], we can find a Kummer log flat cover T → Xr such +that γ dies in HN+1 +kfl +(T, F). By [Niz08, Cor. 2.16], we may assume that for some +s, we have a factorization T → Xrs → Xr with T → Xrs a classical flat cover. It +follows that the pull-back γs of γ to Xrs is trivialized by a classical flat cover, i.e. +γs ∈ ker(HN+1 +kfl +(Xrs, F) → H0 +fl(Xrs, RN+1εfl∗F)). +By our assumption on the flat sheaf Riεfl∗F for any 0 ≤ i ≤ N, we get +Ha +fl(Xrs, Riεfl∗F) = Ha +´et(Xrs, Riεfl∗F) = 0 +for any a > 0 and any 0 ≤ i ≤ N by [Sta21, Lemma 0DDU]. Therefore the Leray +spectral sequence +Eu,v +2 += Hu +fl (Xrs, Rvεfl∗F) ⇒ Hv+v +kfl (Xrs, F), +implies that γs = 0. This finishes the proof. +□ +In the setting-up of Lemma 2.3, the following lemma shows that Ha +kfl(Xn,d, F) +is determined only by Raεfl∗F for 0 < a < N + 1, where Xn,d denotes the fiber +product of d + 1 copies of Xn over X. Apparently this is useful for computations +on the spectral sequence (2.2). +Lemma 2.4. Let the setting-up be as in Lemma 2.3, and let Xn,d denote the +fiber product of d + 1 copies of Xn over X. Then we have +Ha +kfl(Xn,d, F) = Γ(Xn,d, Raεfl∗F) +for 0 < a < N + 1. + +8 +HEER ZHAO +Proof. We have Xn,d = Hd +n ×Spec Z Xn = (Hn)d +X ×X Xn, where Hn := +Spec Z[(P 1/n)gp/P gp] and (Hn)X := Hn ×Spec Z X. Note that (Hn)X is the con- +stant group scheme associated to the abstract group Hn(X) over X if (n, p) = 1. +We write n = pr · n′ with (p, n′) = 1, then we have +Xn,d = (Hn′)d +X ×X ((Hpr)d +X ×X Xn). +Thus +Hu +fl (Xn,d, Rvεfl∗F) = Hu +´et(Xn,d, Rvεfl∗F) += +� +x∈Hn′(X)d +Hu +´et((Hpr)d +X ×X Xn, Rvεfl∗F) += 0 +for 1 ≤ u and 0 ≤ v ≤ N. Therefore the result follows from the Leray spectral +sequence Eu,v +2 += Hu +fl (Xn,d, Rvεfl∗F) ⇒ Hv+v +kfl (Xn,d, F). +□ +2.1. Case of finite abelian l-groups. +Lemma 2.5. Let l be a prime number, and S a locally noetherian fs log scheme +on which l is invertible. Let F be a finite ´etale group scheme over the underlying +scheme of S, we endow it with the induced log structure from S. Assume that F is +killed by lr for some positive integer r, then we have +Riεfl∗F ∼= F(−i) ⊗Z +i� +(Gm,log/Gm)Sfl +for i > 0. +Proof. The proof is analogous to that of [KN99, Thm. +2.4]. +By Kato’s +theorem, see [KAT21, Thm. 4.1], we have an isomorphism +R1εfl∗F ∼= Hom(Z/lrZ(1), F) ⊗Z (Gm,log/Gm)Sfl = F(−1) ⊗Z (Gm,log/Gm)Sfl. +Using cup-product, this isomorphism induces a homomorphism +ϕi : F(−i) ⊗Z +i� +(Gm,log/Gm)Sfl → F ⊗Z Riεfl∗Z/lrZ → Riεfl∗F. +To finish the proof, it suffices to prove that ϕi is an isomorphism for i ≥ 1 +which we will proceed by induction. +Base case. The case i = 1 is just Kato’s theorem. +Inductive step. Let N be a positive integer, and we assume that ϕi is an +isomorphism for any 1 ≤ i ≤ N. +We are going to show that ϕN+1 is also an +isomorphism. The map ϕN+1 induces a map +φN+1 : Γ(X, F(−N − 1)) ⊗Z +N+1 +� +P gp → HN+1 +kfl +(X, F) +for any X ∈ (fs/S) satisfying the assumption ⋆ in Lemma 2.1. It suffices to show +that φN+1 is an isomorphism. +In the spectral sequence (2.2) +Eu,v +2 += lim +−→ +n +ˇHu +kfl(Xn/X, Hv +kfl(F)) ⇒ Hu+v +kfl (X, F), + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +9 +the terms Eu,v +2 +with u + v ≤ N + 1 and v ̸= 0 (see the picture below) vanish (see +the picture below) by Lemma 2.6 below. +× +× +× +× +× +× +× +× +× +× +• +• +• +• +• +u +N + 1 +1 +v +N + 1 +1 +0 +Therefore the canonical map lim +−→n ˇHN+1 +kfl +(Xn/X, F) +∼ += +−→ HN+1 +kfl +(X, F) is an isomor- +phism. Thus we can identify φN+1 as a map +Γ(X, F(−N − 1)) ⊗Z +N+1 +� +P gp → lim +−→ +n +ˇHN+1 +kfl +(Xn/X, F), +and still call the resulting map φN+1. Since both φN+1 and the identification of +Lemma 2.1 (5) are constructed via cup-product, they agree. In particular φN+1 is +an isomorphism. This finishes the induction step. +□ +Lemma 2.6. Let the setting-up be as in the inductive step of the proof of Lemma +2.5. Then the terms Eu,v +2 +with u + v ≤ N + 1 and v ̸= 0 vanish in the spectral +sequence (2.2). +Proof. We have E0,v +2 += 0 for 0 < v ≤ N + 1 by Lemma 2.3. We are left with +the case that uv ̸= 0 and u + v ≤ N + 1. +We denote by Xn,d the fiber product of d + 1 copies of Xn over X. We have +Xn,d = Hd +n ×Spec Z Xn = (Hn)d +X ×X Xn, where Hn := Spec Z[(P 1/n)gp/P gp] and +(Hn)X := Hn ×Spec Z X. Note that (Hn)X is the constant group scheme associated +to the abstract group Hn(X) over X if (n, p) = 1. +We compute the ˇCech cohomology group ˇHu +kfl(Xn/X, Hv +kfl(F)) for 0 < v < +N + 1. Let n = pr · n′ with (p, n′) = 1, then we have +Xn,d = (Hn′)d +X ×X ((Hpr)d +X ×X Xn) +and +Hv +kfl(Xn,d, F) = +� +a∈Hn′(X)d +Hv +kfl((Hpr)d +X ×X Xn, F) += Map(Hn′(X)d, Hv +kfl((Hpr)d +X ×X Xn, F)). + +10 +HEER ZHAO +By the inductive hypothesis from the proof of Lemma 2.5, the assumption of Lemma +2.3 is satisfied. Thus by Lemma 2.4, we have +Hv +kfl((Hpr)d +X ×X Xn, F) =Γ((Hpr)d +X ×X Xn, Rvεfl∗F) +=Γ((Hpr)d +X ×X Xn, F(−v) ⊗Z +v� +(Gm,log/Gm)Sfl) +=Γ(X, F(−v)) ⊗Z +v� +(P +1 +n )gp +=Hv +kfl(Xn, F). +It follows that Hv +kfl(Xn,d, F) = Map(Hn′(X)d, Hv +kfl(Xn, F)). +The ˇCech complex +for Hv +kfl(F) with respect to the cover Xn/X can be identified with the standard +complex that computes the group cohomology of Hv +kfl(Xn, F) regarded as a trivial +Hn′(X)-module, hence we get ˇHu +kfl(Xn/X, Hv +kfl(F)) = Hu(Hn′(X), Hv +kfl(Xn, F)). +Therefore +lim +−→ +n +ˇHu +kfl(Xn/X, Hv +kfl(F)) = +lim +−→ +n=pr·n′ +Hu(Hn′(X), Hv +kfl(Xn, F)) +=Hu(lim +←− +n′ +Hn′(X), +lim +−→ +n=pr·n′ +Hv +kfl(Xn, F)), +where the second identification follows from [Ser02, §2, Prop. 8]. By Lemma 2.3 +we have lim +−→n Hv +kfl(Xn, F) = 0, and thus Eu,v +2 += lim +−→n ˇHu +kfl(Xn/X, Hv +kfl(F)) = 0 for +and 0 < v < N + 1. This finishes the proof of the lemma. +□ +Lemma 2.7. Let S be a locally noetherian fs log scheme. Let l be a prime num- +ber, U the open locus on S where l is invertible, and j : U ֒→ S the corresponding +strict open immersion. Let F be a finite ´etale group scheme over the underlying +scheme of S such that it is killed by a power of l. We endow F with the induced +log structure from S. Assume that the sheaf Riεfl∗F is supported on U for i > 0 +(which we will show later), then we have the following. +(1) Riεfl∗F ∼= jfl!((j−1 +kfl F)(−i) ⊗Z +�i(Gm,log/Gm)Ufl). +(2) Let X ∈ (fs/S) satisfying the assumption ⋆ in Lemma 2.1 with p = l, +then Ha +fl(X, Riεfl∗F) = 0. +Proof. By Proposition B.4, we get Riεfl∗F ∼= jfl!Riεfl∗j−1 +kfl F. By Lemma 2.5, +we have Riεfl∗j−1 +kfl F ∼= (j−1 +kfl F)(−i) ⊗Z +�i(Gm,log/Gm)Ufl. Then part (1) follows. +By the description of from part (1), the restriction of Riεfl∗F to (st/X)fl is the +inverse image of some sheaf on the small ´etale site of X. Thus Ha +fl(X, Riεfl∗F) = +Ha +´et(X, Riεfl∗F) by [Sta21, Lemma 0DDU]. Let x denote the closed point of X. +By Gabber’s theorem, see [Sta21, Theorem 09ZI], we get +Ha +´et(X, Riεfl∗F) = Ha +´et(x, Riεfl∗F). +Since Riεfl∗F is supported on U and x /∈ U, part (2) follows. +□ +Lemma 2.8. Let S be a locally noetherian fs log scheme. +Let l be a prime +number, and U the open locus on S where l is invertible. Let F be a finite ´etale +group scheme over the underlying scheme of S, and we endow it with the induced + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +11 +log structure from S. Assume that F is killed by a power of l, then the sheaf Riεfl∗F +is supported on U for i > 0. +Proof. We use induction on i. +Base case. First of all we consider the case i = 1. It suffices to show that +H1 +kfl(X, F) = 0 for any X ∈ (fs/S) satisfying the assumption ⋆ in Lemma 2.1 with +p = l. +The spectral sequence (2.2) gives rise to an exact sequence +0 → lim +−→ +n≥1 +ˇH1 +kfl(Xn/X, F) → H1 +kfl(X, F) → lim +−→ +n≥1 +ˇH0 +kfl(Xn/X, H1 +kfl(F)). +By Lemma 2.1 (4), we have lim +−→n≥1 ˇH1 +kfl(Xn/X, F) = 0. By Lemma 2.3, we have +lim +−→n≥1 ˇH0 +kfl(Xn/X, H1 +kfl(F)) = 0. Hence H1 +kfl(X, F) = 0. +Inductive step. Let N be a positive integer, and we assume that Riεfl∗F is +supported on U for any 1 ≤ i ≤ N. We are going to prove that the same is true for +i = N + 1. It suffices to show that HN+1 +kfl +(X, F) = 0 for any X ∈ (fs/S) satisfying +the assumption ⋆ in Lemma 2.1 with p = l. +Let γ ∈ HN+1 +kfl +(X, F). By [Zha21a, Prop. 3.1], we can find a Kummer log flat +cover T → X such that γ dies in HN+1 +kfl +(T, F). By [Niz08, Cor. 2.16], we may +assume that for some m, we have a factorization T → Xm → X with T → Xm a +classical flat cover. It follows that the pull-back γm of γ to Xm is trivialized by a +classical flat cover, i.e. +γm ∈ ker(HN+1 +kfl +(Xm, F) → H0 +fl(Xm, RN+1εfl∗F)). +Note that Xm also satisfies the assumption ⋆ in Lemma 2.1 with p = l. Since +Riεfl∗F is supported on U for 0 < i < N + 1 by induction hypothesis, we have +Ha +fl(Xm, Riεfl∗F) = 0 +for any a ≥ 0 and any 0 < i < N + 1 by Lemma 2.7 (2). We also have +HN+1 +fl +(Xm, F) = HN+1 +´et +(Xm, F) = 0. +Thus the Leray spectral sequence +Ei,j +2 += Hi +fl(Xm, Rjεfl∗F) ⇒ Hi+j +kfl (Xm, F) +implies that γm = 0. +It follows that +γ ∈ Ker(HN+1 +kfl +(X, F) → lim +−→ +n +ˇH0 +kfl(Xn/X, HN+1 +kfl +(F))). +Therefore to show that γ is itself zero, it suffices to show that +lim +−→ +n +ˇHi +kfl(Xn/X, HN+1−i +kfl +(F)) = 0 +for 0 < i ≤ N + 1. By Lemma 2.1 (4), we are left with the case 0 < i < N + 1. +Further it suffices to show that ˇHi +kfl(Xn/X, HN+1−i +kfl +(F)) = 0 for any 0 < i < N + 1 +and any n. + +12 +HEER ZHAO +Let n = lrn′ with (l, n′) = 1, then we have Hn′ ×Spec Z Xn is a constant group +over Xn and +Xn ×X · · · ×X Xn +� +�� +� +d + 1 folded += Xn ×Spec Z Hd +n = +� +a∈Hn′(X) +Xn ×Spec Z Hd +lr +with Xn ×Spec Z Hd +lr satisfying the assumption ⋆ in Lemma 2.1 with p = l. By the +same argument showing γm = 0, one can also show that +HN+1−i +kfl +(Xn ×Spec Z Hd +lr, F) = 0 +for any 0 < i < N + 1 and any d ≥ 0. Thus +HN+1−i +kfl +(Xn ×X · · · ×X Xn +� +�� +� +d + 1 folded +, F) = 0 +for any 0 < i < N + 1 and any d ≥ 0. It follows that ˇHi +kfl(Xn/X, HN+1−i +kfl +(F)) = 0 +for any 0 < i < N + 1 and any n. Therefore γ = 0, and thus HN+1 +kfl +(X, F) = 0. +This shows that RN+1εfl∗F is supported on U. +Conclusion. The lemma is proven. +□ +Theorem 2.9. Let S be a locally noetherian fs log scheme. Let l be a prime +number, U the open locus on S where l is invertible, and j : U ֒→ S the correspond- +ing strict open immersion. Let F be a finite ´etale group scheme over the underlying +scheme of S, and we endow it with the induced log structure from S. Assume that +F is killed by a power of l, then we have +Riεfl∗F ∼= jfl!((j−1 +kfl F)(−i) ⊗Z +i� +(Gm,log/Gm)Ufl). +Proof. This follows from Lemma 2.7 and Lemma 2.8. +□ +2.2. Case of rational vector spaces. In this subsection, we consider the +case that F is ´etale locally isomorphic to a finite dimensional Q-vector space. +Theorem 2.10. Let S be an fs log scheme. Let F be a group scheme over the +underlying scheme of S which is ´etale locally representable by a finite dimensional +Q-vector space. Then we have Riεfl∗F = 0 for i ≥ 1. +Proof. It suffices to consider the case F = Q. We use induction on i. +Base case. First of all we consider the case i = 1. It suffices to show that +H1 +kfl(X, Q) = 0 for any X ∈ (fs/S) which satisfies the assumption ⋆ in Lemma 2.1. +The spectral sequence (2.2) gives rise to an exact sequence +0 → lim +−→ +n≥1 +ˇH1 +kfl(Xn/X, Q) → H1 +kfl(X, Q) → lim +−→ +n≥1 +ˇH0 +kfl(Xn/X, H1 +kfl(Q)). +By Lemma 2.1 (1), we have lim +−→n≥1 ˇH1 +kfl(Xn/X, Q) = 0. By Lemma 2.3, we have +lim +−→n≥1 ˇH0 +kfl(Xn/X, H1 +kfl(Q)) = 0. Hence H1 +kfl(X, Q) = 0, and thus R1εfl∗Q = 0. +Inductive step. Let N be a positive integer, and assume that Riεfl∗Q = 0 for +any 1 ≤ i ≤ N. We are going to prove that RN+1εfl∗Q = 0. It suffices to show that + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +13 +HN+1 +kfl +(X, Q) = 0 for any X ∈ (fs/S) which satisfies the assumption ⋆ in Lemma +2.1. +By the inductive hypothesis and Lemma 2.3, we have +lim +−→ +n≥1 +ˇH0 +kfl(Xn/X, HN+1 +kfl +(Q))) = 0. +By the spectral sequence (2.2), to show that HN+1 +kfl +(X, Q) = 0, it suffices to show +that +lim +−→ +n≥1 +ˇHi +kfl(Xn/X, HN+1−i +kfl +(Q)) = 0 +for 0 < i ≤ N + 1. By Lemma 2.1 (1), we are left with the case 0 < i < N + 1. +Further it suffices to show that ˇHi +kfl(Xn/X, HN+1−i +kfl +(Q)) = 0 for any 0 < i < N + 1 +and any n. Since Rtεfl∗Q = 0 for 1 ≤ t ≤ N and +Hr +fl(Xn ×X · · · ×X Xn +� +�� +� +d + 1 folded +, Q) = Hr +fl(Xn ×Spec Z Hd +n, Q) = Hr +´et(Xn ×Spec Z Hd +n, Q) = 0 +for any r > 0 and any d ≥ 0, the Leray spectral sequence +Eu,v +2 += Hu +fl (Xn ×X · · · ×X Xn +� +�� +� +d + 1 folded +, Rvεfl∗Q) ⇒ Hu+v +kfl (Xn ×X · · · ×X Xn +� +�� +� +d + 1 folded +, Q) +implies that HN+1−i +kfl +(Xn ×X · · · ×X Xn +� +�� +� +d + 1 folded +, Q) = 0 for any 0 < i < N + 1 and any +d ≥ 0. It follows that ˇHi +kfl(Xn/X, HN+1−i +kfl +(Q)) = 0 for any 0 < i < N + 1 and any +n. Therefore HN+1 +kfl +(X, Q) = 0. This finishes the proof of RN+1εfl∗Q = 0. +Conclusion. The theorem is proven. +□ +2.3. Case of finite rank torsion-free abelian groups. +Theorem 2.11. Let S be a locally noetherian fs log scheme, and F a group +scheme over the underlying scheme of S which is ´etale locally isomorphic to a finite +rank free abelian group. Then we have the following. +(1) R1εfl∗F = 0. +(2) Let i > 1. For each prime number l, let Ul be the locus on S on which +l is invertible and lj : Ul ֒→ S the corresponding strict open immersion. +Then +Riεfl∗F ∼= +� +l prime +Ri−1εfl∗(F ⊗Z Ql/Zl) +∼= +� +l prime +ljfl!(lj−1F ⊗Z Ql/Zl(−i + 1) ⊗Z +i−1 +� +(Gm,log/Gm)(Ul)fl). +Proof. Applying the functor εfl∗ to the short exact sequence +0 → F → F ⊗Z Q → +� +l prime +F ⊗Z Ql/Zl → 0 + +14 +HEER ZHAO +of sheaves of abelian groups on (fs/S)kfl, we get a long exact sequence +0 →F → F ⊗Z Q → +� +l prime +F ⊗Z Ql/Zl → R1εfl∗F → R1εfl∗(F ⊗Z Q) → · · · +→Ri−1εfl∗(F ⊗Z Q) → +� +l prime +Ri−1εfl∗(F ⊗Z Ql/Zl) → Riεfl∗F +→Riεfl∗(F ⊗Z Q) → · · · . +By Theorem 2.10, we get an exact sequence +0 → F → F ⊗Z Q → +� +l prime +F ⊗Z Ql/Zl → R1εfl∗F → 0 +and +(2.3) +� +l prime +Ri−1εfl∗(F ⊗Z Ql/Zl) +∼ += +−→ Riεfl∗F +for i > 1. +Since the map F ⊗Z Q → � +l prime F ⊗Z Ql/Zl remains surjective on (fs/S)fl, +we get R1εfl∗F = 0. +Part (2) follows from the isomorphism (2.3) and Theorem 2.9. +□ +3. Examples +3.1. Some general results. +Lemma 3.1. Let X be an fs log scheme whose underlying scheme is locally +noetherian, and let F be a group scheme over the underlying scheme of X which is +´etale locally isomorphic to a finite rank free abelian group. We endow F with the +induced log structure from X. Then we have +(1) an isomorphism H1 +fl(X, F) +∼ += +−→ H1 +kfl(X, F), +(2) and an exact sequence +(3.1) +0 → H2 +fl(X, F) → H2 +kfl(X, F) → H0 +fl(X, R2εfl∗F) → H3 +fl(X, F) → H3 +kfl(X, F). +Proof. The results follow from the Leray spectral sequence +Ei,j +2 += Hi +fl(X, Rjεfl∗F) ⇒ Hi+j +kfl (X, F) +and the vanishing of R1εfl∗F from Theorem 2.11 (1). +□ +Proposition 3.2. Let X be an fs log scheme whose underlying scheme is noe- +therian and normal. Then we have +H1 +kfl(X, Z) = 0. +Proof. This follows from the isomorphism from Lemma 3.1 (1) and the van- +ishing of H1 +fl(X, Z) from [CTS21, Chapter 2, Prop. 2.4.2]. +□ + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +15 +Lemma 3.3. Let X, F be as in Lemma 3.1, and we further assume that the +ranks of the stalks of the ´etale sheaf M gp +X /O× +X are at most one. Then the exact +sequence (3.1) can be extended further as +0 → H2 +fl(X, F) → H2 +kfl(X, F) → H0 +fl(X, R2εfl∗F) +→ H3 +fl(X, F) → H3 +kfl(X, F) → H1 +fl(X, R2εfl∗F) → · · · +→ Hi +fl(X, F) → Hi +kfl(X, F) → Hi−2 +fl +(X, R2εfl∗F) → · · · . +Proof. By Theorem 2.11 (1), we have R1εfl∗F = 0. By Theorem 2.11 (2) +and our assumption, the restriction of Riεfl∗F to (st/X) vanishes for i > 2, +where (st/X) denotes the full subcategory of (fs/X) consisting of strict fs log +schemes over X. +Then the result is an easy exercise on the spectral sequence +Ei,j +2 += Hi +fl(X, Rjεfl∗F) ⇒ Hi+j +kfl (X, F). +□ +Lemma 3.4. Let X be as in Lemma 3.3. Let F be a finite ´etale group scheme +over the underlying scheme of X, and we endow it with the induced log structure +from X. Then we have a long exact sequence +0 → H1 +fl(X, F) → H1 +kfl(X, F) → H0 +fl(X, R1εfl∗F) +→ H2 +fl(X, F) → H2 +kfl(X, F) → H1 +fl(X, R1εfl∗F) → · · · +→ Hi +fl(X, F) → Hi +kfl(X, F) → Hi−1 +fl +(X, R1εfl∗F) → · · · . +Proof. By Theorem 2.9 and our assumption, the restriction of Riεfl∗F to +(st/X) vanishes for i > 1. +Then the result is an easy exercise on the spectral +sequence Ei,j +2 += Hi +fl(X, Rjεfl∗F) ⇒ Hi+j +kfl (X, F). +□ +3.2. Discrete valuation rings. Throughout this subsection, let R be a dis- +crete valuation ring with fraction field K and residue field k. Assume that k is of +positive characteristic p. Let π be a uniformizer of R, and we endow X = Spec R +with the log structure associated to the homomorphism N → R, 1 �→ π. Let x be +the closed point of X and i the closed immersion x ֒→ X, and we endow x with +the induced log structure from X. Let η be the generic point of X and j the open +immersion η ֒→ X. +Example 3.1. Let F be a group scheme over the underlying scheme of X which +is ´etale locally isomorphic to a finite rank free abelian group. We have +H1 +fl(X, F) +∼ += +−→ H1 +kfl(X, F) +by Lemma 3.1 (1). Clearly X satisfies the assumption on log structure from Lemma +3.3, thus we get a long exact sequence +0 → H2 +fl(X, F) → H2 +kfl(X, F) → H0 +fl(X, R2εfl∗F) +→ H3 +fl(X, F) → H3 +kfl(X, F) → H1 +fl(X, R2εfl∗F) → · · · +→ Hi +fl(X, F) → Hi +kfl(X, F) → Hi−2 +fl +(X, R2εfl∗F) → · · · . + +16 +HEER ZHAO +Apparently the log structure of X is supported on the closed point x and the +restriction of (Gm,log/Gm)Xfl to (st/X) is isomorphic to i∗Z. Since the only non- +invertible prime on X is p, the restriction of R2εfl∗F to (st/X) is isomorphic to +i∗(F ⊗Z (Q/Z)′(−1)), +where (Q/Z)′ denotes the prime to p part of Q/Z. Thus we have +Hu +fl (X, R2εfl∗F) = Hu +fl (x, F ⊗Z (Q/Z)′(−1)) += Hu +´et(x, F ⊗Z (Q/Z)′(−1)) += Hu(Gal(ks/k), F ⊗Z (Q/Z)′(−1)), +where ks is a separable closure of k and the second identification follows from +[Sta21, Lemma 0DDU]. We also have +Hu +fl (X, F) ∼= Hu +´et(X, F) ∼= Hu +´et(x, F) ∼= Hu(Gal(ks/k), F), +where the first identification also follows from [Sta21, Lemma 0DDU]. Thus we +have +H1 +kfl(X, F) ∼= H1(Gal(ks/k), F), +and the above exact sequence can be rewritten as +0 → H2(Gal(ks/k), F) → H2 +kfl(X, F) → H0(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) +→ H3(Gal(ks/k), F) → H3 +kfl(X, F) → H1(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) → · · · +→ Hi(Gal(ks/k), F) → Hi +kfl(X, F) → Hi−2(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) → · · · +(3.2) +Assume that the cohomological dimension of k is d, then we have +Hu(Gal(ks/k), F) = 0 +for u > d + 1 and +Hu(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) = 0 +for u > d. Thus we have an exact sequence +0 → H2(Gal(ks/k), F) → H2 +kfl(X, F) → H0(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) +→ H3(Gal(ks/k), F) → H3 +kfl(X, F) → H1(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) → · · · +→ Hd+1(Gal(ks/k), F) → Hd+1 +kfl (X, F) → Hd−1(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) +→ 0, +an isomorphism +Hd+2 +kfl (X, F) +∼ += +−→ Hd(Gal(ks/k), F ⊗Z (Q/Z)′(−1)), +and +Hi +kfl(X, F) = 0 +for i ≥ d + 3 by the exact sequence (3.2). +Moreover we assume that k is finite, then Gal(ks/k) ∼= ˆZ and d = 1. Thus we +have an exact sequence +0 → H2(ˆZ, F) → H2 +kfl(X, F) → H0(ˆZ, F ⊗Z (Q/Z)′(−1)) → 0, + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +17 +an isomorphism +H3 +kfl(X, F) ∼= H1(ˆZ, F ⊗Z (Q/Z)′(−1)), +and +Hi +kfl(X, F) = 0 +for i ≥ 4. Now we claim that +H0(ˆZ, F ⊗Z (Q/Z)′(−1)) = H1(ˆZ, F ⊗Z (Q/Z)′(−1)) = 0. +This is clear if F = Z. In general, take a finite extension k′ of k such that F ×X +Spec k′ ∼= Zr, then the claim follows from the Hochschild-Serre spectral sequence +and the case of F = Z. If follows that +Hi +kfl(X, F) = +� +Hi(ˆZ, F), +if i = 0, 1, 2; +0, +if i > 2. +In particular we have +Hi +kfl(X, Z) = + + + + + + + + + +Z, +if i = 0; +0, +if i = 1; +Q/Z, +if i = 2; +0, +if i > 2. +Example 3.2. Let F be a finite ´etale group scheme over the underlying scheme +of X. Then F = � +l prime F(l), where F(l) is the l-primary subgroup of F and +also finite ´etale. To compute the Kummer log flat cohomology of F, it suffices to +compute that of F(l) for each prime l. Without loss of generality, we assume that +F = F(l) for some prime l. Since X satisfies the assumption on log structure from +Lemma 3.3, we get a long exact sequence +0 → H1 +fl(X, F) → H1 +kfl(X, F) → H0 +fl(X, R1εfl∗F) +→ H2 +fl(X, F) → H2 +kfl(X, F) → H1 +fl(X, R1εfl∗F) → · · · +→ Hi +fl(X, F) → Hi +kfl(X, F) → Hi−1 +fl +(X, R1εfl∗F) → · · · . +by Lemma 3.4. +Now we proceed by considering the two cases l = p and l ̸= p. +Case (1): l = p. In this case, we have R1εfl∗F = 0. Thus +Hi +kfl(X, F) ∼= Hi +fl(X, F) ∼= Hi +´et(X, F) = Hi(Gal(ks/k), F) +for any i ≥ 0. +Case (2): l ̸= p. Since l ̸= p, we have R1εfl∗F ∼= i∗F(−1). Similar to Example +3.1, we have +Hu +fl (X, R1εfl∗F) = Hu +fl (x, F(−1)) = Hu +´et(x, F(−1)) = Hu(Gal(ks/k), F(−1)) +and +Hu +fl (X, F) ∼= Hu +´et(X, F) ∼= Hu +´et(x, F) ∼= Hu(Gal(ks/k), F). + +18 +HEER ZHAO +Thus the above exact sequence can be rewritten as +0 → H1(Gal(ks/k), F) → H1 +kfl(X, F) → H0(Gal(ks/k), F(−1)) +→ H2(Gal(ks/k), F) → H2 +kfl(X, F) → H1(Gal(ks/k), F(−1)) → · · · +→ Hi(Gal(ks/k), F) → Hi +kfl(X, F) → Hi−1(Gal(ks/k), F(−1)) → · · · . +(3.3) +Assume that the cohomological dimension of k is d, then we have +Hu(Gal(ks/k), F) = Hu(Gal(ks/k), F(−1)) = 0 +for u > d. Thus we have an exact sequence +0 → H1(Gal(ks/k), F) → H1 +kfl(X, F) → H0(Gal(ks/k), F(−1)) +→ H2(Gal(ks/k), F) → H2 +kfl(X, F) → H1(Gal(ks/k), F(−1)) → · · · +→ Hd(Gal(ks/k), F) → Hd +kfl(X, F) → Hd−1(Gal(ks/k), F(−1)) → 0, +an isomorphism +Hd+1 +kfl (X, F) ∼= Hd(Gal(ks/k), F(−1)), +and +Hi +kfl(X, F) = 0 +for i ≥ d + 2 by the exact sequence (3.3). +Moreover we assume that k is finite, then Gal(ks/k) ∼= ˆZ and d = 1. Thus we +have an exact sequence +0 → H1(ˆZ, F) → H1 +kfl(X, F) → H0(ˆZ, F(−1)) → 0, +an isomorphism +H2 +kfl(X, F) ∼= H1(ˆZ, F(−1)), +and +Hi +kfl(X, F) = 0 +for i ≥ 3. In particular, we have +H0(ˆZ, Z/lrZ(−1)) = H1(ˆZ, Z/lrZ(−1)) = 0, +and thus +Hi +kfl(X, Z/lrZ) = + + + + + +Z/lrZ, +if i = 0; +Z/lrZ, +if i = 1; +0, +if i > 1. +3.3. Global Dedekind domains. Throughout this subsection, let K be a +global field. When K is a number field, X denotes the spectrum of the ring of +integers in K, and when K is a function field, k denotes the field of constants of +K and X denotes the unique connected smooth projective curve over k having K +as its function field. Let S be a finite set of closed points of X, U := X − S, +j : U ֒→ X, and ix : x ֒→ X for each closed point x ∈ X. We endow X with the +log structure j∗O× +U ∩ OX → OX. +Apparently X satisfies the assumption on log structure from Lemma 3.3. + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +19 +Example 3.3. Let F be a group scheme over the underlying scheme of X which +is ´etale locally isomorphic to a finite rank free abelian group. We have +H1 +fl(X, F) +∼ += +−→ H1 +kfl(X, F) +by Lemma 3.1 (1). Since X satisfies the assumption on log structure from Lemma +3.3, we have a long exact sequence as in Lemma 3.3. +The sheaf MX/O× +X is supported on the closed subset S, and thus the restriction +of (Gm,log/Gm)Xfl to (st/X) is isomorphic to � +x∈S ix∗Z. For any prime number l, +let +Sl := {x ∈ S | the characteristic of the residue field of x is not l}. +Thus by Theorem 2.11, we have +R2εfl∗F ∼= +� +l prime +R1εfl∗(F ⊗Z Ql/Zl) +and +R1εfl∗(F ⊗Z Ql/Zl) ∼= +� +x∈Sl +ix∗(F ⊗Z Ql/Zl(−1)). +Therefore we get +Hu +fl (X, R2εfl∗F) = +� +l prime +� +x∈Sl +Hu +fl (X, ix∗(F ⊗Z Ql/Zl(−1))) += +� +l prime +� +x∈Sl +Hu +fl (x, F ⊗Z Ql/Zl(−1)) += +� +l prime +� +x∈Sl +Hu +´et(x, F ⊗Z Ql/Zl(−1)) += +� +l prime +� +x∈Sl +Hu(Γx, F ⊗Z Ql/Zl(−1)), +where Γx := Gal(κ(x)s/κ(x)). We also have H2 +fl(X, F) ∼= H2 +´et(X, F) by [Sta21, +Lemma 0DDU]. So we can rewrite the exact sequence from Lemma 3.3 as +0 → H2 +´et(X, F) → H2 +kfl(X, F) → +� +l prime +� +x∈Sl +H0(Γx, F ⊗Z Ql/Zl(−1)) +→ H3 +´et(X, F) → H3 +kfl(X, F) → +� +l prime +� +x∈Sl +H1(Γx, F ⊗Z Ql/Zl(−1)) → · · · +→ Hi +´et(X, F) → Hi +kfl(X, F) → +� +l prime +� +x∈Sl +Hi−2(Γx, F ⊗Z Ql/Zl(−1)) → · · · . +Now we assume that the residue fields of X at its closed points are finite, then +we have +Hu(Γx, F ⊗Z (Ql/Zl)(−1)) = 0 +for u > 1 due to cohomological dimension reason. Moreover, by the same argument +as in Example 3.1, we even have +H0(Γx, F ⊗Z (Ql/Zl)(−1)) = H1(Γx, F ⊗Z (Ql/Zl)(−1)) = 0. + +20 +HEER ZHAO +It follows that +Hi +kfl(X, F) ∼= Hi +´et(X, F) +for any i ≥ 0. +Example 3.4. Let F be a finite ´etale group scheme over the underlying scheme +of X. Then F = � +l prime F(l), where F(l) is the l-primary subgroup of F and +also finite ´etale. To compute the Kummer log flat cohomology of F, it suffices to +compute that of F(l) for each prime l. Without loss of generality, we assume that +F = F(l) for some prime l. Similar to the situation of Example 3.3, we have +R1εfl∗F ∼= +� +x∈Sl +ix∗(F(−1)). +Therefore by Lemma 3.4 and similar arguments as in Example 3.3, we have a long +exact sequence +0 → H1 +´et(X, F) → H1 +kfl(X, F) → +� +x∈Sl +H0(Γx, F(−1)) +→ H2 +´et(X, F) → H2 +kfl(X, F) → +� +x∈Sl +H1(Γx, F(−1)) → · · · +→ Hi +´et(X, F) → Hi +kfl(X, F) → +� +x∈Sl +Hi−1(Γx, F(−1)) → · · · . +Now we assume that the residue fields of X at the closed points are finite, then we +have +Hu(Γx, F(−1)) = 0 +for u > 1 due to cohomological dimension reason. Therefore we have an exact +sequence +0 → H1 +´et(X, F) → H1 +kfl(X, F) → +� +x∈Sl +H0(Γx, F(−1)) +→ H2 +´et(X, F) → H2 +kfl(X, F) → +� +x∈Sl +H1(Γx, F(−1)) +→ H3 +´et(X, F) → H3 +kfl(X, F) → 0, +and isomorphisms +Hi +´et(X, F) +∼ += +−→ Hi +kfl(X, F) +for i ≥ 4. +For F = Z/lrZ, we further have +H0(Γx, Z/lrZ(−1)) = H1(Γx, Z/lrZ(−1)) = 0 +for x ∈ Sl. Therefore +Hi +´et(X, Z/lrZ) +∼ += +−→ Hi +kfl(X, Z/lrZ) +for i ≥ 0. + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +21 +Appendix A. Sites and sheaves +In this appendix, we collect some general results about sites from [Sta21, +Chapter 00UZ]. +A.1. Sites. +Definition A.1. [Sta21, Definition 00VH] A site is given by a category C and +a set Cov(C) = � +U∈C Cov(U) with Cov(U) being a set of families of morphisms +with target U, such that the following conditions hold. +(1) If V → U is an isomorphism, then {V → U} ∈ Cov(U). +(2) If {Ui → U}i∈I ∈ Cov(U) and for each i we have {Vij → Ui}j∈Ji ∈ +Cov(Ui), then {Vij → U}i∈I,j∈Ji ∈ Cov(U). +(3) If {Ui → U}i∈I ∈ Cov(U) and V → U is a morphism of C, then Ui ×U V +exists for each i ∈ I and {Ui ×U V → V }i∈I ∈ Cov(V ). +A.2. Continuous functors. +Definition A.2. [Sta21, Tag 00WV] A functor u : C → D of sites is called +continuous, if for every {Vi → V }i∈I ∈ Cov(C) we have the following +(1) {u(Vi) → u(V )} ∈ Cov(D), and +(2) for any morphism T → V in C the morphism +u(T ×V Vi) → u(T ) ×u(V ) u(Vi) +is an isomorphism. +Recall that given a functor u as above, and a presheaf of sets F on D we can +define upF to be simply the presheaf F ◦ u, in other words +upF(V ) = F(u(V )) +for every object V of C (see [Sta21, Tag 00VC] for up as well as for up). +Suppose that the functor u : C → D is continuous, then F ∈ Sh(D) ⇒ upF ∈ +Sh(D). We denote +us : Sh(D) → Sh(C) +the functor up restricted to the subcategory of sheaves of sets. +Recall that up +admits a left adjoint up, see [Sta21, Tag 00VE]. This is also the case for us. +Lemma A.1. [Sta21, Tag 00WX] Let u : C → D be a continuous functor of +sites. Then the functor +us : Sh(C) → Sh(D), +G �→ (upG)♯ +is left adjoint to us. +Definition A.3. [Sta21, Tag 00X1] A morphism of sites f : D → C is given +by a continuous functor u : C → D such that us is exact. +Notice how the functor u goes in the direct opposite the morphism f. If f ↔ u +is a morphism of sites, then we use the notation f −1 = us and f∗ = us. The functor +f −1 is called the pullback functor, and the functor f∗ is called the pushforward +functor. As in topology we have the adjunction (f −1, f∗). + +22 +HEER ZHAO +See [Sta21, Tag 00X2] (examples associated to maps between two topological +spaces) and [Sta21, Tag 0EWI] (examples for different topologies on the same +space, comparison of topologies) for examples of morphisms of sites. +A.3. Cocontinuous functors. +Definition A.4. A functor u : C → D of sites is called cocontinuous, if for +every U ∈ C and every {Vj → u(U)}j∈J ∈ Cov(D), there exists {Ui → U}i∈I ∈ +Cov(C) such that {u(Ui) → u(U)}i∈I refines {Vj → u(U)}j∈J. +Warning: In general {u(Ui) → u(U)}i∈I is not a covering of D. +Example A.1. For an fs log scheme S, we denote by (Sch/S) the category of +fs log schemes over S. Let (Sch/S)k´et and (Sch/S)´et be the Kummer log ´etale +site and the classical ´etale site for (Sch/S) respectively, see [Ill02, §2.5]. Let +(Sch/S)kfl be the Kummer log flat site for (Sch/S), see [KAT21, Def. 2.3], and +let (Sch/S)fl be the classical flat site for (Sch/S), which is an obvious analogue +of (Sch/S)´et. +Now let j : U ֒→ X be a strict open immersion of fs log schemes, and let +τ ∈ {k´et, ´et, kfl, fl}. Then the functor of sites +u : (fs/X)τ → (fs/U)τ, Y �→ Y ×X U +is continuous, and the functor of sites +v : (fs/U)τ → (fs/X)τ, V �→ V +is continuous and cocontinuous +Appendix B. The Kummer log flat site +In this appendix, we focus on the Kummer log flat site. +B.1. Morphisms of sites associated to a strict open immersion. +Warning: We are going to follow [Sta21, Chapter 00UZ] and [Sta21, +Chapter 03A4] to construct a homomorphism of sheaves of abelian groups. +Morphisms of sites and topoi in [Sta21, Chapter 00UZ] are often formulated for +sheaves of sets, while the map to be constructed is for sheaves of abelian groups. +Often in order to use results from [Sta21, Chapter 00UZ], we have to use [Sta21, +Tag 00YV]. Below whenever we refer to a result about sheaves of sets from +[Sta21, Chapter 00UZ] for sheaves of abelian groups, we are referring to [Sta21, +Tag 00YV] without mention at the same time. +Let X be an fs log scheme, and j : U ֒→ X a strict open immersion of fs log +schemes. Let τ ∈ {k´et, ´et, kfl, fl}. The functor +u : (fs/U)τ → (fs/X)τ, (V → U) �→ (V → U → X) +of sites is continuous and cocontinuous, hence it gives rise to a morphism of topoi +g = (g−1, g∗) : Ab((fs/U)τ) → Ab((fs/X)τ) + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +23 +with g−1 exact by [Sta21, Tag 00XO] and [Sta21, Tag 00XL]. For any presheaf +F ∈ PAb((fs/U)τ), we define gp!F ∈ PAb((fs/X)τ) as the presheaf +Y �→ +lim +−→ +Y →u(V ) +F(V ) +with colimits over (Iv +Y )opp (see [Sta21, Equation 053L] for this index category) +taken in the category of abelian groups. For F ∈ Ab((fs/U)τ), we set g!F to be +the sheafification of gp!F, see [Sta21, Tag 04BF], and called it the extension by +zero of F. +Lemma B.1. Let X, U, u, g = (g−1, g∗), and g! be as above. Then we have the +following. +(1) The functor g! is left adjoint to g−1 and exact. +(2) The functor g−1 is exact and preserves injective objects. +(3) For any F ∈ Ab((fs/U)τ), the canonical maps +F → g−1g!F and g−1g∗F → F +are isomorphisms. +Proof. Part (1) follows from [Sta21, Tag 04BG] and [Sta21, Tag 04BH]. +Since g−1 admits both a left adjoint g! and a right adjoint g∗, it is exact by +[Sta21, Tag 0039]. Since the left adjoint g! of g−1 is exact, it preserves injective +objects by [Sta21, Tag 015Z]. This finishes the proof of part (2). +Since j is a strict open immersion, the functor u is fully faithful. The functor u is +continuous and cocontinuous. Thus part (3) follows from [Sta21, Lemma 077I]. +□ +The functor +v : (fs/X)τ → (fs/U)τ, Y �→ Y ×X U +of sites is continuous with vs exact, hence gives rise to a morphism +jτ : (fs/U)τ → (fs/X)τ +of sites, and further a morphism +jτ = (j−1 +τ += vs, jτ∗ = vs) : Ab((fs/U)τ) → Ab((fs/X)τ) +of topoi. The cocontinuous functor u is left adjoint to the continuous functor v. By +[Sta21, Tag 00XY], the two morphisms g and jτ of topoi agree. We set jτ! := g!, +and call it the functor of extension by zero. To sum up, we have the following +lemma. +Lemma B.2. Let j : U → X be a strict open immersion of fs log schemes. +Then we have a morphism of sites +jτ : (fs/U)τ → (fs/X)τ, +and a sequence of functors +jτ!, j−1 +τ , jτ∗ +where in each consecutive pair the first is exact and left adjoint to the second. +Moreover we have the following. +(1) The functor jτ! is exact. + +24 +HEER ZHAO +(2) The functor j−1 +τ +is exact and preserves injective objects. +(3) For any F ∈ Ab((fs/U)τ), the canonical maps +F → j−1 +τ jτ!F and j−1 +τ jτ∗F → F +are isomorphisms. +B.2. Comparison morphism from the Kummer log flat site to the +classical flat site. Let j : U → X be a strict open immersion of fs log schemes. +We have canonical forgetful morphisms of sites +(fs/X)kfl → (fs/X)fl +and +(fs/U)kfl → (fs/U)fl. +By abuse of notation, we denote both of them by εfl. It is easy to see that we have +the following commutative diagram +(fs/U)kfl +jkfl � +εfl +� +(fs/X)kfl +εfl +� +(fs/U)fl +jfl +� (fs/X)fl +of morphisms of sites. +For F ∈ Ab((fs/X)kfl), the adjunction (jfl!, j−1 +fl ) gives a canonical map +(B.1) +jfl!j−1 +fl Riεfl∗F → Riεfl∗F. +Lemma B.3. We have +j−1 +fl Riεfl∗F = Riεfl∗j−1 +kfl F. +Proof. Let F → I• be an injective resolution of F on (fs/X)kfl. Since the +functor j−1 +kfl is exact and preserves injective objects by Lemma B.2 (2), we get an +injective resolution j−1 +kfl F → j−1 +kfl I• of j−1 +kfl F. Thus +Riεfl∗j−1 +kfl F = Hi(εfl∗j−1 +kfl I•) = Hi(j−1 +fl εfl∗I•) = j−1 +fl Hi(εfl∗I•) = j−1 +fl Riεfl∗F. +□ +By the identification from Lemma B.3, we get a map +(B.2) +Φ : jfl!Riεfl∗j−1 +kfl F → Riεfl∗F. +Proposition B.4. Let X be an fs log scheme and U an open subscheme of +the underlying scheme of X. We endow U with the induced log structure, and let +j : U ֒→ X be the corresponding strict open immersion. Let F be a sheaf of abelian +groups on (fs/X)kfl such that Riεfl∗F is supported over U, then the map +Φ : jfl!Riεfl∗j−1 +kfl F → Riεfl∗F +is an isomorphism. + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +25 +Proof. Applying the functor j−1 +fl +to the canonical map (B.1), we get a map +(B.3) +j−1 +fl jfl!j−1 +fl Riεfl∗F → j−1 +fl Riεfl∗F. +Since j−1 +fl jfl!j−1 +fl Riεfl∗F is identified to j−1 +fl Riεfl∗F by Lemma B.2 (3), the map +(B.3) is an isomorphism. Since Riεfl∗F is supported on U and the extension by +zero sheaf jfl!j−1 +fl Riεfl∗F is clearly supported on U, the map (B.1) is actually an +isomorphism. Therefore Φ is also an isomorphism by construction. +□ +Appendix C. A lemma on profinite group cohomology +Let p be a fixed prime number, and let ˆZ′ := lim +←−(p,m)=1 Z/mZ. Let Gr = (ˆZ′)r +and M a torsion abelian group, we regard M as a Gr-module with respect to +the trivial action. In this appendix, we compute the profinite group cohomology +Hi(Gr, M). The result should be well-known to the experts, but we are not able +to find a reference so present a computation here. +According to [Ols09, A.2], the cohomology groups Hi(Gr, M) are computed +by the cohomology groups of the standard homogeneous cochain complex of Gr +with coefficients in M +RΓ(Gr, M) : Mapcts +Gr(Gr, M) → Mapcts +Gr(G2 +r, M) → · · · → Mapcts +Gr(Gi +r, M) → · · · , +where Mapcts +Gr(Gi +r, M) denotes the set of equivariant continuous functions φ : Gi +r → +M (where M is endowed with the discrete topology). Note that Mapcts +Gr(Gi +r, M) is +denoted as Homcts +Gr(G[i−1] +r +, M) in [Ols09, App. A]. +First we consider the case that M = Z/nZ with (n, p) = 1. +The complex +RΓ(Gr, Z/nZ) of abelian groups is also naturally a complex of modules over the +ring Z/nZ. Since Z/nZ as a module over itself is flat and RΓ(Gr, Z/nZ) lies in +Db(Z/nZ) (the bounded derived category of complexes of Z/nZ-modules), we have +(C.1) +RΓ(Gr, Z/nZ) ⊗L +Z/nZ RΓ(Gs, Z/nZ) +∼ += +−→ RΓ(Gr+s, Z/nZ) +by K¨unneth formula, see [Ols09, Thm. A.6]. We have +Hi(G1, Z/nZ) = +� +Z/nZ, +if i = 0, 1; +0, +if i > 0, +and RΓ(G1, Z/nZ) is isomorphic to Z/nZ +0−→ Z/nZ in Db(Z/nZ). By [Ols09, Cor. +A.7], the natural map of graded Z/nZ-modules +H∗(G1, Z/nZ) ⊗Z/nZ H∗(G1, Z/nZ) → H∗(G2, Z/nZ) +is an isomorphism, and thus Hi(G2, Z/nZ) are free Z/nZ-modules for all i. Ap- +plying [Ols09, Cor. +A.7] inductively, we have that the natural map of graded +Z/nZ-modules +H∗(G1, Z/nZ)⊗r → H∗(Gr, Z/nZ) +is an isomorphism. It follows that the graded cohomology ring H∗(Gr, Z/nZ) is +isomorphic to the exterior algebra of the module +H1(Gr, Z/nZ) = Hom(Gr, Z/nZ) ∼= (Z/nZ)r + +26 +HEER ZHAO +over Z/nZ. +Now let M be a torsion abelian group which is killed by n with (n, p) = 1. We +regard M as a module over Z/nZ. Clearly we have +RΓ(Gr, M) ∼= RΓ(Gr, Z/nZ) ⊗Z/nZ M. +Since Hi(Gr, Z/nZ) are free Z/nZ-modules for all i, we get +Hi(Gr, M) ∼= Hi(Gr, Z/nZ) ⊗Z/nZ M +canonically. +In general for a torsion abelian group M, we have +M = M[p∞] ⊕ M ′ = M[p∞] ⊕ +lim +−→ +(n,p)=1 +M[n], +where M[p∞] (resp. M ′, resp. M[n]) denotes the p-primary part (resp. prime to p +part, resp. n-torsion part) of M. Thus +Hi(Gr, M) = +lim +−→ +(n,p)=1 +Hi(Gr, M[n]) +∼= +lim +−→ +(n,p)=1 +Hi(Gr, Z/nZ) ⊗Z/nZ M[n] +∼= +lim +−→ +(n,p)=1 +( +i� +H1(Gr, Z/nZ)) ⊗Z/nZ M[n]. +(C.2) +We fix an isomorphism Gr = Hom(Xr, ˆZ′) with Xr := Zr, and thus +H1(Gr, Z/nZ) = Hom(Gr, Z/nZ) = Xr ⊗Z Z/nZ. +Therefore we further have +Hi(Gr, M) ∼= +lim +−→ +(n,p)=1 +( +i� +(Xr ⊗Z Z/nZ)) ⊗Z/nZ M[n] += +lim +−→ +(n,p)=1 +( +i� +Xr) ⊗Z M[n] +=( +i� +Xr) ⊗Z M ′, +where the last two wedges are for Z-module structure. Apparently the identification +(C.2) is induced by cup-product. +To sum up, we have the following lemma. +Lemma C.1. Let p be a fixed prime number, and ˆZ′ := lim +←−(p,m)=1 Z/mZ. Let +M be a torsion abelian group, and we regard it as a (ˆZ′)r-module with respect +to the trivial action. Let M[p∞] (resp. M ′, resp. M[n]) denote the p-primary +part (resp. prime to p part, resp. n-torsion part) of M. We fix an isomorphism +(ˆZ′)r = Hom(Xr, ˆZ′) with Xr := Zr. +Then + +THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES +27 +(1) for any positive integer n with (n, p) = 1, we have +H1((ˆZ′)r, M[n]) = Hom((ˆZ′)r, M[n]) ∼= M[n] ⊗Z Xr +and the cup product induces an isomorphism +Hi((ˆZ′)r, M[n]) ∼= M[n] ⊗Z/nZ +i� +H1((ˆZ′)r, Z/nZ), +and the latter can be further identified with M[n] ⊗Z (�i Xr); +(2) for the profinite group cohomology of (ˆZ′)r with coefficients in M, we have +Hi((ˆZ′)r, M) = +lim +−→ +(n,p)=1 +Hi((ˆZ′)r, M[n]) = +lim +−→ +(n,p)=1 +M[n] ⊗Z ( +i� +Xr) +=M ′ ⊗Z ( +i� +Xr). +Remark C.1. Clearly the above computation works also for the profinite +groups ˆZr and Zr +l for any prime number l. Such results are the profinite group +cohomology analogues of the description of the singular cohomology of topological +tori (see [Hat02, Chap. 3, Exa. 3.16]). +Acknowledgement +The author thanks Professor Chicara Nakayama for very helpful discussions. +This work was partially supported by the Research Training Group 2553 of the +German Research Foundation DFG. +References +[Art62] Michael Artin. Grothendieck topologies: notes on a seminar. Harvard University, Dept. +of Mathematics, 1962. +[CTS21] Jean-Louis Colliot-Th´elene and Alexei N Skorobogatov. The Brauer–Grothendieck group, +volume 71. Springer Nature, 2021. +[Hat02] Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. +[Ill02] +Luc Illusie. An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on +logarithmic ´etale cohomology. Ast´erisque, (279):271–322, 2002. Cohomologies p-adiques +et applications arithm´etiques, II. +[KAT21] Kazuya KATO. Logarithmic Structures of Fontaine-Illusie. II —Logarithmic Flat Topol- +ogy. Tokyo Journal of Mathematics, 44(1):125 – 155, 2021. +[KKN08] Takeshi Kajiwara, Kazuya Kato, and Chikara Nakayama. Logarithmic abelian varieties. +Nagoya Math. J., 189:63–138, 2008. +[KKN15] Takeshi Kajiwara, Kazuya Kato, and Chikara Nakayama. Logarithmic abelian varieties, +Part IV: Proper models. Nagoya Math. J., 219:9–63, 2015. +[KN99] Kazuya Kato and Chikara Nakayama. Log Betti cohomology, log ´etale cohomology, and +log de Rham cohomology of log schemes over C. Kodai Math. J., 22(2):161–186, 1999. +[Mil80] James S. Milne. ´Etale cohomology, volume 33 of Princeton Mathematical Series. Princeton +University Press, Princeton, N.J., 1980. +[Niz08] Wies�lawa Nizio�l. K-theory of log-schemes. I. Doc. Math., 13:505–551, 2008. +[Ols09] Martin C. Olsson. On Faltings’ method of almost ´etale extensions. In Algebraic geometry— +Seattle 2005. Part 2, volume 80 of Proc. Sympos. Pure Math., pages 811–936. Amer. Math. +Soc., Providence, RI, 2009. + +28 +HEER ZHAO +[Ser02] Jean-Pierre Serre. Galois cohomology. Springer Monographs in Mathematics. Springer- +Verlag, Berlin, english edition, 2002. Translated from the French by Patrick Ion and +revised by the author. +[Sta21] The Stacks Project Authors. Stacks Project. http://stacks.math.columbia.edu, 2021. +[Zha17] Heer Zhao. Log abelian varieties over a log point. Doc. Math., 22:505–550, 2017. +[Zha21a] Heer Zhao. Comparison of Kummer logarithmic topologies with classical topologies. +Journal of the Institute of Mathematics of Jussieu, pages 1–31, 2021. +[Zha21b] Heer Zhao. Comparison of kummer logarithmic topologies with classical topologies ii, +2021. +Heer Zhao, Fakult¨at f¨ur Mathematik, Universit¨at Duisburg-Essen, Essen 45117, +Germany, heer.zhao@uni-due.de + diff --git a/nNFAT4oBgHgl3EQfcx2g/content/tmp_files/load_file.txt b/nNFAT4oBgHgl3EQfcx2g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7148d9349e1d4d61b90695e5eff27884d9b136f3 --- /dev/null +++ b/nNFAT4oBgHgl3EQfcx2g/content/tmp_files/load_file.txt @@ -0,0 +1,886 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf,len=885 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='08566v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='AG] 20 Jan 2023 The higher direct images of locally constant group schemes from the Kummer log flat topology to the classical flat topology Heer Zhao Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let S be an fs log scheme, and let F be a group scheme over the underlying scheme which is ´etale locally representable by (1) a finite dimen- sional Q-vector space, or (2) a finite rank free abelian group, or (3) a finite abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We give a full description of all the higher direct images of F from the Kummer log flat site to the classical flat site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In particular, we show that: in case (1) the higher direct images of F vanish;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' and in case (2) the first higher direct image of F vanishes and the n-th (n > 1) higher direct image of F is isomorphic to the (n − 1)-th higher direct image of F ⊗Z Q/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In the end, we make some computations when the base is a standard log trait or a Dedekind scheme endowed with the log structure associated to a finite set of closed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Introduction Let S be an fs log scheme, (fs/S) the category of fs log schemes over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We endow (fs/S) with the Kummer log flat topology (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' the classical flat topol- ogy, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' the classical ´etale topology), see [KAT21, §2] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [KAT21, §4], resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Ill02, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4]), and denote the resulting site by (fs/S)kfl (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (fs/S)fl, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (fs/S)´et)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We have a canonical map εfl : (fs/S)kfl → (fs/S)fl of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' To understand the cohomology of a sheaf of abelian groups F on the site (fs/S)kfl, one needs to understand the higher direct images Riεfl∗F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The first higher direct image R1εfl∗F has been determined by Kato when S is locally noetherian and F is either a finite flat (commutative) group scheme or a smooth affine (commutative) group scheme over the underlying scheme of S, see [KAT21, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1] or [Niz08, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Kato’s theorem about R1εfl∗F has 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 14F20 (primary), 14A21 (secondary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' log schemes, Kummer flat topology, comparison of cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 1In [KAT21] the sites (fs/S)kfl and (fs/S)fl are denoted as Slog fl and Scl fl respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Our notation here is analogous to that of the Kummer log ´etale site from [KKN15, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3] 1 2 HEER ZHAO been generalized to quasi-projective smooth (commutative) group schemes by the author, see [Zha21a, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Assume that the underlying scheme of S is locally noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The second higher direct image R2εfl∗F has been described in [Zha21a, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='23] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Zha21b, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2], resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Zha21b, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3], resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Zha21b, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4]) when F is representable by a torus (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' a smooth affine commutative group scheme, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' a finite flat commutative group scheme, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' an extension of an abelian scheme by a torus) over the underlying scheme of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' When F is representable by a smooth quasi-projective commutative group scheme, the higher direct images are always torsion by [Zha21b, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In this article, we investigate Riεfl∗F for all i > 0 in Section 2 when F is representable by a group scheme which is ´etale locally isomorphic either to a finite dimension Q-vector space, or a finite rank free abelian group, or a finite abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The main results are the following three theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (See also Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='9) Let S be a locally noetherian fs log scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let l be a prime number, U the open locus on S where l is invertible, and j : U ֒→ S the corresponding strict open immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a finite ´etale group scheme over the underlying scheme of S, and we endow it with the induced log structure from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Assume that F is killed by an l-power, then we have Riεfl∗F ∼= jfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' ((j−1 kfl F)(−i) ⊗Z i� (Gm,log/Gm)Ukfl) for i ≥ 1, where jkfl : (fs/U)kfl → (fs/S)kfl (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' jfl : (fs/U)fl → (fs/S)fl) is the morphism on the Kummer log flat sites (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' the classical flat sites) induced by j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (See also Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='10) Let S be an fs log scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a group scheme over the underlying scheme of S which is ´etale locally representable by a finite dimensional Q-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then we have Riεfl∗F = 0 for i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (See also Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='11) Let S be a locally noetherian fs log scheme, and F a group scheme over the underlying scheme of S which is ´etale locally isomorphic to a finite rank free abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (1) R1εfl∗F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (2) Let i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' For each prime number l, let Ul be the locus on S on which l is invertible and lj : Ul ֒→ S the corresponding strict open immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then Riεfl∗F ∼= � l prime Ri−1εfl∗(F ⊗Z Ql/Zl) ∼= � l prime ljfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (lj−1F ⊗Z Ql/Zl(−i + 1) ⊗Z i−1 � (Gm,log/Gm)(Ul)fl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3 is reduced to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2 via the short exact sequence 0 → F → F ⊗Z Q → F ⊗Z Q/Z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 3 Apparently the vanishing of Riεfl∗F in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2 is reduced to the vanishing of Hi kfl(X, F) for any X ∈ (fs/S) such that the underlying scheme of X is Spec R with R a strictly henselian local ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' For the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1, we first construct a canonical map Φ : jfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='Riεfl∗j−1 kfl F → Riεfl∗F, see (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2), then determine Riεfl∗j−1 kfl F (note that the order of F is invertible on U), and finally prove that Φ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The main computation tools are ˇCech cohomology, ˇCech-to-derived functor spectral sequence, and Leray spectral sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In section 3, we apply the results from Section 2 to make some computations on Kummer log flat cohomology when the base is a standard log trait or a Dedekind scheme endowed with the log structure associated to a finite set of closed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' An application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3 (1) can be used to prove that log abelian va- rieties with constant degeneration are sheaves for the Kummer log flat topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' According to [KKN08, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3], a log abelian variety with constant degeneration A over an fs log scheme S is a sheaf on (fs/S)´et which is isomorphic to the quotient sheaf G(Y ) log /Y for a pointwise polarizable log 1-motive [Y u−→ Glog] over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' One can consider G(Y ) log as a uniformization of A, and Y as the corresponding periods lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It is not surprising that Y is important for understanding A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Indeed we have R1εfl∗Y = 0 by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3 (1), and this vanishing is used in [Zha17, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 (1)] to prove that A is a sheaf for the Kummer log flat topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In fact Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3 (1) is just [Zha17, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' However the proof loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' makes use of fpqc descent of schemes which probably does not always hold and deserves a precise reference (see [Sta21, Lemma 0APK] for the situation loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The original motivation of this article is to present a new proof to [Zha17, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The higher direct images We make a few lemmas first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let X be an fs log scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We make the following assumption on X, as well as some constructions associated to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' ⋆ The underlying scheme is Spec R with R a strictly henselian local ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let x denote the closed point, k the residue field of R, p the characteristic of k, and PX → MX a chart of the log structure of X with P an fs monoid, such that P ∼ = −→ MX,x/O× X,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let P 1/n denote the monoid P regarded as a monoid above P via the homomorphism P n−→ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let Xn := X ×Spec Z[P ] Spec Z[P 1/n] endowed with the canonical log structure associated to P 1/n and let Hn denote the group scheme Spec Z[(P 1/n)gp/P gp] over Spec Z, then Xn is a Kummer log flat cover of X such that Xn ×X Xn ∼= Xn ×Spec Z Hn, 4 HEER ZHAO and it is even a Kummer log ´etale cover in case (p, n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a constant group scheme over Spec R associated to an abelian group, and we write n = m · pt with (m, p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then we have an isomorphism ˇHi kfl(Xn/X, F) ∼= Hi(Hm(X), F), where the first term is the i-th ˇCech cohomology of F with respect to the cover Xn/X, and the second term is the abstract group cohomology of the abstract group Hm(X) with coefficients in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In particular, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (1) ˇHi kfl(Xn/X, Q) = 0 for i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (2) ˇH1 kfl(Xn/X, Z) = 0 and ˇHi kfl(Xn/X, Z) is torsion and p-torsion-free for i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (3) The canonical map ˇHi kfl(Xm/X, Z) ∼ = −→ ˇHi kfl(Xn/X, Z) is an isomorphism for i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (4) Suppose that F is killed by a power of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then lim −→ n ˇHi kfl(Xn/X, F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (5) Suppose that F is torsion and p-torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then lim −→ n ˇHi kfl(Xn/X, F) ∼= lim −→ n Hi(Hm(X), F) ∼= F(−i) ⊗Z i� P gp for i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since R is a strictly henselian local ring, X ×Spec Z Hr m is the constant group scheme over X associated to the abstract group Hm(X)r and X ×Spec Z Hr pt is a connected group scheme over X, therefore we have Γ(Xn ×X · · · ×X Xn � �� � r + 1 times , F) =Γ(Xn ×Spec Z Hr n, F) =Γ((X ×Spec Z Hr pt) ×X (Xn ×Spec Z Hr m), F) = � h∈Hm(X)r F =Map(Hm(X)r, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We consider the ˇCech complex (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1) Γ(Xn, F) d0 −→ Γ(Xn ×X Xn, F) d1 −→ Γ(Xn ×X Xn ×X Xn, F) d2 −→ · · · for F with respect to the cover Xn/X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let Γn := (P 1 n )gp/P gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By [Mil80, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' III, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='6], the ˇCech nerve of the Kummer log flat cover Xn/X can be THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 5 identified with the sequence Xn Xn × Hn d1,0 � d1,1 � Xn × H2 n d2,0 �� d2,2 � Xn × H3 n · · · , d3,0 ��� d3,3 � where the map dr,i on the ring level is given by the R-linear ring homomorphism R ⊗Z[P ] Z[P 1 n ⊕ Γr−1 n ] → R ⊗Z[P ] Z[P 1 n ⊕ Γr n] (a, ¯a1, · · · , ¯ar−1) �→ \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (a, ¯a, ¯a1, · · · , ¯ar−1), if i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (a, ¯a1, · · · , ¯ai, ¯ai, · · · , ¯ar−1), if 0 < i < r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (a, ¯a1, · · · , ¯ar−1, 0), if i = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' for any (a, ¯a1, · · · , ¯ar−1) ∈ P 1 n ⊕ Γr−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' If m = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' n = pt, we have Γ(Xpt ×Spec Z Hr pt, F) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The map d∗ r,i : Γ(Xpt ×Spec Z Hr−1 pt , F) → Γ(Xpt ×Spec Z Hr pt, F) is clearly just the identity map Id : F → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In general, the map d∗ r,i : Γ(Xn ×Spec Z Hr−1 n , F) → Γ(Xn ×Spec Z Hr n, F) can be identified with the map Map(Hm(X)r−1, F) ∂r,i −−→ Map(Hm(X)r, F) which maps f ∈ Map(Hm(X)r−1, F) to ∂r,i(f) : (h1, · · · , hr) �→ \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 f(h2, · · · , · · · , hr), if i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' f(h1, · · · , hi + hi+1, · · · , hr), if 0 < i < r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' f(h1, · · · · · · , hr−1), if i = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore the complex (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1) can be identified with the standard complex F → Map(Hm(X), F) → Map(Hm(X)2, F) → · · · which computes the abstract group cohomology of the trivial Hm(X)-module F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It follows that ˇHi kfl(Xn/X, F) ∼= Hi(Hm(X), F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In particular, we have part (1), part (2), and part (4) by the corresponding results for the abstract group cohomology Hi(Hm(X), F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Part (3) is also clear by the construction of the isomorphism ˇHi kfl(Xn/X, F) ∼= Hi(Hm(X), F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 6 HEER ZHAO We are left with part (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We have lim −→ n ˇHi kfl(Xn/X, F) ∼= lim −→ n = mpt with (p, m) = 1 Hi(Hm(X), F) (3) = lim −→ (p,m)=1 Hi(Hm(X), F) =Hi( lim ←− (p,m)=1 Hm(X), F) =Hi(Hom(P gp, ˆZ′(1)), F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By the description of the profinite group cohomology of (ˆZ′)r from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1, we further have lim −→ n ˇHi kfl(Xn/X, F) ∼=Hi(Hom(P gp, ˆZ′(1)), F) = lim −→ (n,p)=1 Hi(Hom(P gp, ˆZ′(1)), F[n]) = lim −→ (n,p)=1 F[n] ⊗Z/nZ Hi(Hom(P gp, ˆZ′(1)), Z/nZ) = lim −→ (n,p)=1 F[n] ⊗Z/nZ i� H1(Hom(P gp, ˆZ′(1)), Z/nZ) = lim −→ (n,p)=1 F[n] ⊗Z/nZ i� Hom(Hom(P gp, ˆZ′(1)), Z/nZ) = lim −→ (n,p)=1 F[n] ⊗Z/nZ ( i� P gp ⊗Z Z/nZ(−i)) =F(−i) ⊗Z i� P gp for i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This finishes the proof of part (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 (5), one can use Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 to get Hi(Hom(P gp, ˆZ′(1)), F) ∼= F ⊗Z �i P gp directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' However we repeat the proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 here in order to keep track of the Tate twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let X and Xn be as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a sheaf on (fs/X)kfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then the family XN := {Xn → X}n≥1 of Kummer log flat covers of X satisfies the condition (L3) from [Art62, §2], whence a ˇCech-to-derived functor spectral sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2) Ei,j 2 = ˇHi kfl(XN, Hj kfl(F)) ⇒ Hi+j kfl (X, F), where ˇHi kfl(XN, −) := lim −→ n≥1 ˇHi kfl(Xn/X, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This follows from [Art62, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' II, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ A nice thing about the ˇCech cohomology with respect to all coverings, is that the zero-th cohomology vanishes, see [Zha21a, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This is often very useful for computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This is of course not the case for the ˇCech cohomology with respect to an arbitrary family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The following lemma is about the vanishing of the zero-th ˇCech cohomology for the covering family XN from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2 under a suitable assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let X and Xn be as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a sheaf on (fs/X)kfl, and N a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' For any U ∈ (fs/X), let (st/U) be the full subcategory of (fs/U) consisting of strict objects over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Assume that for any U ∈ (fs/X) and any 0 ≤ i ≤ N, the restriction of the flat sheaf Riεfl∗F to (st/U) is the inverse image of some sheaf on the small ´etale site of U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' it lies in the image of the functor a−1 U from [Sta21, Lemma 0DDU].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then we have lim −→ n≥1 Hi kfl(Xn, F) = 0 for any 0 < i ≤ N + 1, in particular lim −→n≥1 ˇH0 kfl(Xn/X, Hi kfl(F)) = 0 for any 0 < i ≤ N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Clearly we only need to prove the vanishing for i = N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It suffices to show that for any γ ∈ HN+1 kfl (Xr, F) there exists s such that γ goes to zero in HN+1 kfl (Xrs, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By [Zha21a, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1], we can find a Kummer log flat cover T → Xr such that γ dies in HN+1 kfl (T, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By [Niz08, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='16], we may assume that for some s, we have a factorization T → Xrs → Xr with T → Xrs a classical flat cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It follows that the pull-back γs of γ to Xrs is trivialized by a classical flat cover, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' γs ∈ ker(HN+1 kfl (Xrs, F) → H0 fl(Xrs, RN+1εfl∗F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By our assumption on the flat sheaf Riεfl∗F for any 0 ≤ i ≤ N, we get Ha fl(Xrs, Riεfl∗F) = Ha ´et(Xrs, Riεfl∗F) = 0 for any a > 0 and any 0 ≤ i ≤ N by [Sta21, Lemma 0DDU].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore the Leray spectral sequence Eu,v 2 = Hu fl (Xrs, Rvεfl∗F) ⇒ Hv+v kfl (Xrs, F), implies that γs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ In the setting-up of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3, the following lemma shows that Ha kfl(Xn,d, F) is determined only by Raεfl∗F for 0 < a < N + 1, where Xn,d denotes the fiber product of d + 1 copies of Xn over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Apparently this is useful for computations on the spectral sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let the setting-up be as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3, and let Xn,d denote the fiber product of d + 1 copies of Xn over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then we have Ha kfl(Xn,d, F) = Γ(Xn,d, Raεfl∗F) for 0 < a < N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 8 HEER ZHAO Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We have Xn,d = Hd n ×Spec Z Xn = (Hn)d X ×X Xn, where Hn := Spec Z[(P 1/n)gp/P gp] and (Hn)X := Hn ×Spec Z X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Note that (Hn)X is the con- stant group scheme associated to the abstract group Hn(X) over X if (n, p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We write n = pr · n′ with (p, n′) = 1, then we have Xn,d = (Hn′)d X ×X ((Hpr)d X ×X Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus Hu fl (Xn,d, Rvεfl∗F) = Hu ´et(Xn,d, Rvεfl∗F) = � x∈Hn′(X)d Hu ´et((Hpr)d X ×X Xn, Rvεfl∗F) = 0 for 1 ≤ u and 0 ≤ v ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore the result follows from the Leray spectral sequence Eu,v 2 = Hu fl (Xn,d, Rvεfl∗F) ⇒ Hv+v kfl (Xn,d, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Case of finite abelian l-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let l be a prime number, and S a locally noetherian fs log scheme on which l is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a finite ´etale group scheme over the underlying scheme of S, we endow it with the induced log structure from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Assume that F is killed by lr for some positive integer r, then we have Riεfl∗F ∼= F(−i) ⊗Z i� (Gm,log/Gm)Sfl for i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The proof is analogous to that of [KN99, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Kato’s theorem, see [KAT21, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1], we have an isomorphism R1εfl∗F ∼= Hom(Z/lrZ(1), F) ⊗Z (Gm,log/Gm)Sfl = F(−1) ⊗Z (Gm,log/Gm)Sfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Using cup-product, this isomorphism induces a homomorphism ϕi : F(−i) ⊗Z i� (Gm,log/Gm)Sfl → F ⊗Z Riεfl∗Z/lrZ → Riεfl∗F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' To finish the proof, it suffices to prove that ϕi is an isomorphism for i ≥ 1 which we will proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The case i = 1 is just Kato’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Inductive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let N be a positive integer, and we assume that ϕi is an isomorphism for any 1 ≤ i ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We are going to show that ϕN+1 is also an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The map ϕN+1 induces a map φN+1 : Γ(X, F(−N − 1)) ⊗Z N+1 � P gp → HN+1 kfl (X, F) for any X ∈ (fs/S) satisfying the assumption ⋆ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It suffices to show that φN+1 is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In the spectral sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2) Eu,v 2 = lim −→ n ˇHu kfl(Xn/X, Hv kfl(F)) ⇒ Hu+v kfl (X, F), THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 9 the terms Eu,v 2 with u + v ≤ N + 1 and v ̸= 0 (see the picture below) vanish (see the picture below) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' × × × × × × × × × × u N + 1 1 v N + 1 1 0 Therefore the canonical map lim −→n ˇHN+1 kfl (Xn/X, F) ∼ = −→ HN+1 kfl (X, F) is an isomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus we can identify φN+1 as a map Γ(X, F(−N − 1)) ⊗Z N+1 � P gp → lim −→ n ˇHN+1 kfl (Xn/X, F), and still call the resulting map φN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since both φN+1 and the identification of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 (5) are constructed via cup-product, they agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In particular φN+1 is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This finishes the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let the setting-up be as in the inductive step of the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then the terms Eu,v 2 with u + v ≤ N + 1 and v ̸= 0 vanish in the spectral sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We have E0,v 2 = 0 for 0 < v ≤ N + 1 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We are left with the case that uv ̸= 0 and u + v ≤ N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We denote by Xn,d the fiber product of d + 1 copies of Xn over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We have Xn,d = Hd n ×Spec Z Xn = (Hn)d X ×X Xn, where Hn := Spec Z[(P 1/n)gp/P gp] and (Hn)X := Hn ×Spec Z X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Note that (Hn)X is the constant group scheme associated to the abstract group Hn(X) over X if (n, p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We compute the ˇCech cohomology group ˇHu kfl(Xn/X, Hv kfl(F)) for 0 < v < N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let n = pr · n′ with (p, n′) = 1, then we have Xn,d = (Hn′)d X ×X ((Hpr)d X ×X Xn) and Hv kfl(Xn,d, F) = � a∈Hn′(X)d Hv kfl((Hpr)d X ×X Xn, F) = Map(Hn′(X)d, Hv kfl((Hpr)d X ×X Xn, F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 10 HEER ZHAO By the inductive hypothesis from the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='5, the assumption of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4, we have Hv kfl((Hpr)d X ×X Xn, F) =Γ((Hpr)d X ×X Xn, Rvεfl∗F) =Γ((Hpr)d X ×X Xn, F(−v) ⊗Z v� (Gm,log/Gm)Sfl) =Γ(X, F(−v)) ⊗Z v� (P 1 n )gp =Hv kfl(Xn, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It follows that Hv kfl(Xn,d, F) = Map(Hn′(X)d, Hv kfl(Xn, F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The ˇCech complex for Hv kfl(F) with respect to the cover Xn/X can be identified with the standard complex that computes the group cohomology of Hv kfl(Xn, F) regarded as a trivial Hn′(X)-module, hence we get ˇHu kfl(Xn/X, Hv kfl(F)) = Hu(Hn′(X), Hv kfl(Xn, F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore lim −→ n ˇHu kfl(Xn/X, Hv kfl(F)) = lim −→ n=pr·n′ Hu(Hn′(X), Hv kfl(Xn, F)) =Hu(lim ←− n′ Hn′(X), lim −→ n=pr·n′ Hv kfl(Xn, F)), where the second identification follows from [Ser02, §2, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3 we have lim −→n Hv kfl(Xn, F) = 0, and thus Eu,v 2 = lim −→n ˇHu kfl(Xn/X, Hv kfl(F)) = 0 for and 0 < v < N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This finishes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let S be a locally noetherian fs log scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let l be a prime num- ber, U the open locus on S where l is invertible, and j : U ֒→ S the corresponding strict open immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a finite ´etale group scheme over the underlying scheme of S such that it is killed by a power of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We endow F with the induced log structure from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Assume that the sheaf Riεfl∗F is supported on U for i > 0 (which we will show later), then we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (1) Riεfl∗F ∼= jfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' ((j−1 kfl F)(−i) ⊗Z �i(Gm,log/Gm)Ufl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (2) Let X ∈ (fs/S) satisfying the assumption ⋆ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 with p = l, then Ha fl(X, Riεfl∗F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4, we get Riεfl∗F ∼= jfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='Riεfl∗j−1 kfl F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='5, we have Riεfl∗j−1 kfl F ∼= (j−1 kfl F)(−i) ⊗Z �i(Gm,log/Gm)Ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then part (1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By the description of from part (1), the restriction of Riεfl∗F to (st/X)fl is the inverse image of some sheaf on the small ´etale site of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus Ha fl(X, Riεfl∗F) = Ha ´et(X, Riεfl∗F) by [Sta21, Lemma 0DDU].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let x denote the closed point of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Gabber’s theorem, see [Sta21, Theorem 09ZI], we get Ha ´et(X, Riεfl∗F) = Ha ´et(x, Riεfl∗F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since Riεfl∗F is supported on U and x /∈ U, part (2) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let S be a locally noetherian fs log scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let l be a prime number, and U the open locus on S where l is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a finite ´etale group scheme over the underlying scheme of S, and we endow it with the induced THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 11 log structure from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Assume that F is killed by a power of l, then the sheaf Riεfl∗F is supported on U for i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We use induction on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' First of all we consider the case i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It suffices to show that H1 kfl(X, F) = 0 for any X ∈ (fs/S) satisfying the assumption ⋆ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 with p = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The spectral sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2) gives rise to an exact sequence 0 → lim −→ n≥1 ˇH1 kfl(Xn/X, F) → H1 kfl(X, F) → lim −→ n≥1 ˇH0 kfl(Xn/X, H1 kfl(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 (4), we have lim −→n≥1 ˇH1 kfl(Xn/X, F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3, we have lim −→n≥1 ˇH0 kfl(Xn/X, H1 kfl(F)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Hence H1 kfl(X, F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Inductive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let N be a positive integer, and we assume that Riεfl∗F is supported on U for any 1 ≤ i ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We are going to prove that the same is true for i = N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It suffices to show that HN+1 kfl (X, F) = 0 for any X ∈ (fs/S) satisfying the assumption ⋆ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 with p = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let γ ∈ HN+1 kfl (X, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By [Zha21a, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1], we can find a Kummer log flat cover T → X such that γ dies in HN+1 kfl (T, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By [Niz08, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='16], we may assume that for some m, we have a factorization T → Xm → X with T → Xm a classical flat cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It follows that the pull-back γm of γ to Xm is trivialized by a classical flat cover, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' γm ∈ ker(HN+1 kfl (Xm, F) → H0 fl(Xm, RN+1εfl∗F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Note that Xm also satisfies the assumption ⋆ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 with p = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since Riεfl∗F is supported on U for 0 < i < N + 1 by induction hypothesis, we have Ha fl(Xm, Riεfl∗F) = 0 for any a ≥ 0 and any 0 < i < N + 1 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='7 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We also have HN+1 fl (Xm, F) = HN+1 ´et (Xm, F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus the Leray spectral sequence Ei,j 2 = Hi fl(Xm, Rjεfl∗F) ⇒ Hi+j kfl (Xm, F) implies that γm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It follows that γ ∈ Ker(HN+1 kfl (X, F) → lim −→ n ˇH0 kfl(Xn/X, HN+1 kfl (F))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore to show that γ is itself zero, it suffices to show that lim −→ n ˇHi kfl(Xn/X, HN+1−i kfl (F)) = 0 for 0 < i ≤ N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 (4), we are left with the case 0 < i < N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Further it suffices to show that ˇHi kfl(Xn/X, HN+1−i kfl (F)) = 0 for any 0 < i < N + 1 and any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 12 HEER ZHAO Let n = lrn′ with (l, n′) = 1, then we have Hn′ ×Spec Z Xn is a constant group over Xn and Xn ×X · · · ×X Xn � �� � d + 1 folded = Xn ×Spec Z Hd n = � a∈Hn′(X) Xn ×Spec Z Hd lr with Xn ×Spec Z Hd lr satisfying the assumption ⋆ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 with p = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By the same argument showing γm = 0, one can also show that HN+1−i kfl (Xn ×Spec Z Hd lr, F) = 0 for any 0 < i < N + 1 and any d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus HN+1−i kfl (Xn ×X · · · ×X Xn � �� � d + 1 folded , F) = 0 for any 0 < i < N + 1 and any d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It follows that ˇHi kfl(Xn/X, HN+1−i kfl (F)) = 0 for any 0 < i < N + 1 and any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore γ = 0, and thus HN+1 kfl (X, F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This shows that RN+1εfl∗F is supported on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The lemma is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let S be a locally noetherian fs log scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let l be a prime number, U the open locus on S where l is invertible, and j : U ֒→ S the correspond- ing strict open immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a finite ´etale group scheme over the underlying scheme of S, and we endow it with the induced log structure from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Assume that F is killed by a power of l, then we have Riεfl∗F ∼= jfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' ((j−1 kfl F)(−i) ⊗Z i� (Gm,log/Gm)Ufl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='7 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Case of rational vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In this subsection, we consider the case that F is ´etale locally isomorphic to a finite dimensional Q-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let S be an fs log scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a group scheme over the underlying scheme of S which is ´etale locally representable by a finite dimensional Q-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then we have Riεfl∗F = 0 for i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It suffices to consider the case F = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We use induction on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' First of all we consider the case i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It suffices to show that H1 kfl(X, Q) = 0 for any X ∈ (fs/S) which satisfies the assumption ⋆ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The spectral sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2) gives rise to an exact sequence 0 → lim −→ n≥1 ˇH1 kfl(Xn/X, Q) → H1 kfl(X, Q) → lim −→ n≥1 ˇH0 kfl(Xn/X, H1 kfl(Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 (1), we have lim −→n≥1 ˇH1 kfl(Xn/X, Q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3, we have lim −→n≥1 ˇH0 kfl(Xn/X, H1 kfl(Q)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Hence H1 kfl(X, Q) = 0, and thus R1εfl∗Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Inductive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let N be a positive integer, and assume that Riεfl∗Q = 0 for any 1 ≤ i ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We are going to prove that RN+1εfl∗Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It suffices to show that THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 13 HN+1 kfl (X, Q) = 0 for any X ∈ (fs/S) which satisfies the assumption ⋆ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By the inductive hypothesis and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3, we have lim −→ n≥1 ˇH0 kfl(Xn/X, HN+1 kfl (Q))) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By the spectral sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2), to show that HN+1 kfl (X, Q) = 0, it suffices to show that lim −→ n≥1 ˇHi kfl(Xn/X, HN+1−i kfl (Q)) = 0 for 0 < i ≤ N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 (1), we are left with the case 0 < i < N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Further it suffices to show that ˇHi kfl(Xn/X, HN+1−i kfl (Q)) = 0 for any 0 < i < N + 1 and any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since Rtεfl∗Q = 0 for 1 ≤ t ≤ N and Hr fl(Xn ×X · · · ×X Xn � �� � d + 1 folded , Q) = Hr fl(Xn ×Spec Z Hd n, Q) = Hr ´et(Xn ×Spec Z Hd n, Q) = 0 for any r > 0 and any d ≥ 0, the Leray spectral sequence Eu,v 2 = Hu fl (Xn ×X · · · ×X Xn � �� � d + 1 folded , Rvεfl∗Q) ⇒ Hu+v kfl (Xn ×X · · · ×X Xn � �� � d + 1 folded , Q) implies that HN+1−i kfl (Xn ×X · · · ×X Xn � �� � d + 1 folded , Q) = 0 for any 0 < i < N + 1 and any d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It follows that ˇHi kfl(Xn/X, HN+1−i kfl (Q)) = 0 for any 0 < i < N + 1 and any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore HN+1 kfl (X, Q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This finishes the proof of RN+1εfl∗Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The theorem is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Case of finite rank torsion-free abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let S be a locally noetherian fs log scheme, and F a group scheme over the underlying scheme of S which is ´etale locally isomorphic to a finite rank free abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (1) R1εfl∗F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (2) Let i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' For each prime number l, let Ul be the locus on S on which l is invertible and lj : Ul ֒→ S the corresponding strict open immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then Riεfl∗F ∼= � l prime Ri−1εfl∗(F ⊗Z Ql/Zl) ∼= � l prime ljfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (lj−1F ⊗Z Ql/Zl(−i + 1) ⊗Z i−1 � (Gm,log/Gm)(Ul)fl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Applying the functor εfl∗ to the short exact sequence 0 → F → F ⊗Z Q → � l prime F ⊗Z Ql/Zl → 0 14 HEER ZHAO of sheaves of abelian groups on (fs/S)kfl, we get a long exact sequence 0 →F → F ⊗Z Q → � l prime F ⊗Z Ql/Zl → R1εfl∗F → R1εfl∗(F ⊗Z Q) → · · · →Ri−1εfl∗(F ⊗Z Q) → � l prime Ri−1εfl∗(F ⊗Z Ql/Zl) → Riεfl∗F →Riεfl∗(F ⊗Z Q) → · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='10, we get an exact sequence 0 → F → F ⊗Z Q → � l prime F ⊗Z Ql/Zl → R1εfl∗F → 0 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3) � l prime Ri−1εfl∗(F ⊗Z Ql/Zl) ∼ = −→ Riεfl∗F for i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since the map F ⊗Z Q → � l prime F ⊗Z Ql/Zl remains surjective on (fs/S)fl, we get R1εfl∗F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Part (2) follows from the isomorphism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3) and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Some general results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let X be an fs log scheme whose underlying scheme is locally noetherian, and let F be a group scheme over the underlying scheme of X which is ´etale locally isomorphic to a finite rank free abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We endow F with the induced log structure from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then we have (1) an isomorphism H1 fl(X, F) ∼ = −→ H1 kfl(X, F), (2) and an exact sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1) 0 → H2 fl(X, F) → H2 kfl(X, F) → H0 fl(X, R2εfl∗F) → H3 fl(X, F) → H3 kfl(X, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The results follow from the Leray spectral sequence Ei,j 2 = Hi fl(X, Rjεfl∗F) ⇒ Hi+j kfl (X, F) and the vanishing of R1εfl∗F from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='11 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let X be an fs log scheme whose underlying scheme is noe- therian and normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then we have H1 kfl(X, Z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This follows from the isomorphism from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 (1) and the van- ishing of H1 fl(X, Z) from [CTS21, Chapter 2, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 15 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let X, F be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1, and we further assume that the ranks of the stalks of the ´etale sheaf M gp X /O× X are at most one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then the exact sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1) can be extended further as 0 → H2 fl(X, F) → H2 kfl(X, F) → H0 fl(X, R2εfl∗F) → H3 fl(X, F) → H3 kfl(X, F) → H1 fl(X, R2εfl∗F) → · · · → Hi fl(X, F) → Hi kfl(X, F) → Hi−2 fl (X, R2εfl∗F) → · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='11 (1), we have R1εfl∗F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='11 (2) and our assumption, the restriction of Riεfl∗F to (st/X) vanishes for i > 2, where (st/X) denotes the full subcategory of (fs/X) consisting of strict fs log schemes over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then the result is an easy exercise on the spectral sequence Ei,j 2 = Hi fl(X, Rjεfl∗F) ⇒ Hi+j kfl (X, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let X be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a finite ´etale group scheme over the underlying scheme of X, and we endow it with the induced log structure from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then we have a long exact sequence 0 → H1 fl(X, F) → H1 kfl(X, F) → H0 fl(X, R1εfl∗F) → H2 fl(X, F) → H2 kfl(X, F) → H1 fl(X, R1εfl∗F) → · · · → Hi fl(X, F) → Hi kfl(X, F) → Hi−1 fl (X, R1εfl∗F) → · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='9 and our assumption, the restriction of Riεfl∗F to (st/X) vanishes for i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then the result is an easy exercise on the spectral sequence Ei,j 2 = Hi fl(X, Rjεfl∗F) ⇒ Hi+j kfl (X, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Discrete valuation rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Throughout this subsection, let R be a dis- crete valuation ring with fraction field K and residue field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Assume that k is of positive characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let π be a uniformizer of R, and we endow X = Spec R with the log structure associated to the homomorphism N → R, 1 �→ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let x be the closed point of X and i the closed immersion x ֒→ X, and we endow x with the induced log structure from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let η be the generic point of X and j the open immersion η ֒→ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a group scheme over the underlying scheme of X which is ´etale locally isomorphic to a finite rank free abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We have H1 fl(X, F) ∼ = −→ H1 kfl(X, F) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Clearly X satisfies the assumption on log structure from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3, thus we get a long exact sequence 0 → H2 fl(X, F) → H2 kfl(X, F) → H0 fl(X, R2εfl∗F) → H3 fl(X, F) → H3 kfl(X, F) → H1 fl(X, R2εfl∗F) → · · · → Hi fl(X, F) → Hi kfl(X, F) → Hi−2 fl (X, R2εfl∗F) → · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 16 HEER ZHAO Apparently the log structure of X is supported on the closed point x and the restriction of (Gm,log/Gm)Xfl to (st/X) is isomorphic to i∗Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since the only non- invertible prime on X is p, the restriction of R2εfl∗F to (st/X) is isomorphic to i∗(F ⊗Z (Q/Z)′(−1)), where (Q/Z)′ denotes the prime to p part of Q/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus we have Hu fl (X, R2εfl∗F) = Hu fl (x, F ⊗Z (Q/Z)′(−1)) = Hu ´et(x, F ⊗Z (Q/Z)′(−1)) = Hu(Gal(ks/k), F ⊗Z (Q/Z)′(−1)), where ks is a separable closure of k and the second identification follows from [Sta21, Lemma 0DDU].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We also have Hu fl (X, F) ∼= Hu ´et(X, F) ∼= Hu ´et(x, F) ∼= Hu(Gal(ks/k), F), where the first identification also follows from [Sta21, Lemma 0DDU].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus we have H1 kfl(X, F) ∼= H1(Gal(ks/k), F), and the above exact sequence can be rewritten as 0 → H2(Gal(ks/k), F) → H2 kfl(X, F) → H0(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) → H3(Gal(ks/k), F) → H3 kfl(X, F) → H1(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) → · · · → Hi(Gal(ks/k), F) → Hi kfl(X, F) → Hi−2(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) → · · · (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2) Assume that the cohomological dimension of k is d, then we have Hu(Gal(ks/k), F) = 0 for u > d + 1 and Hu(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) = 0 for u > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus we have an exact sequence 0 → H2(Gal(ks/k), F) → H2 kfl(X, F) → H0(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) → H3(Gal(ks/k), F) → H3 kfl(X, F) → H1(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) → · · · → Hd+1(Gal(ks/k), F) → Hd+1 kfl (X, F) → Hd−1(Gal(ks/k), F ⊗Z (Q/Z)′(−1)) → 0, an isomorphism Hd+2 kfl (X, F) ∼ = −→ Hd(Gal(ks/k), F ⊗Z (Q/Z)′(−1)), and Hi kfl(X, F) = 0 for i ≥ d + 3 by the exact sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Moreover we assume that k is finite, then Gal(ks/k) ∼= ˆZ and d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus we have an exact sequence 0 → H2(ˆZ, F) → H2 kfl(X, F) → H0(ˆZ, F ⊗Z (Q/Z)′(−1)) → 0, THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 17 an isomorphism H3 kfl(X, F) ∼= H1(ˆZ, F ⊗Z (Q/Z)′(−1)), and Hi kfl(X, F) = 0 for i ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Now we claim that H0(ˆZ, F ⊗Z (Q/Z)′(−1)) = H1(ˆZ, F ⊗Z (Q/Z)′(−1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This is clear if F = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In general, take a finite extension k′ of k such that F ×X Spec k′ ∼= Zr, then the claim follows from the Hochschild-Serre spectral sequence and the case of F = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' If follows that Hi kfl(X, F) = � Hi(ˆZ, F), if i = 0, 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 0, if i > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In particular we have Hi kfl(X, Z) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 Z, if i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 0, if i = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Q/Z, if i = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 0, if i > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a finite ´etale group scheme over the underlying scheme of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then F = � l prime F(l), where F(l) is the l-primary subgroup of F and also finite ´etale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' To compute the Kummer log flat cohomology of F, it suffices to compute that of F(l) for each prime l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Without loss of generality, we assume that F = F(l) for some prime l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since X satisfies the assumption on log structure from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3, we get a long exact sequence 0 → H1 fl(X, F) → H1 kfl(X, F) → H0 fl(X, R1εfl∗F) → H2 fl(X, F) → H2 kfl(X, F) → H1 fl(X, R1εfl∗F) → · · · → Hi fl(X, F) → Hi kfl(X, F) → Hi−1 fl (X, R1εfl∗F) → · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Now we proceed by considering the two cases l = p and l ̸= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Case (1): l = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In this case, we have R1εfl∗F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus Hi kfl(X, F) ∼= Hi fl(X, F) ∼= Hi ´et(X, F) = Hi(Gal(ks/k), F) for any i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Case (2): l ̸= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since l ̸= p, we have R1εfl∗F ∼= i∗F(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Similar to Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1, we have Hu fl (X, R1εfl∗F) = Hu fl (x, F(−1)) = Hu ´et(x, F(−1)) = Hu(Gal(ks/k), F(−1)) and Hu fl (X, F) ∼= Hu ´et(X, F) ∼= Hu ´et(x, F) ∼= Hu(Gal(ks/k), F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 18 HEER ZHAO Thus the above exact sequence can be rewritten as 0 → H1(Gal(ks/k), F) → H1 kfl(X, F) → H0(Gal(ks/k), F(−1)) → H2(Gal(ks/k), F) → H2 kfl(X, F) → H1(Gal(ks/k), F(−1)) → · · · → Hi(Gal(ks/k), F) → Hi kfl(X, F) → Hi−1(Gal(ks/k), F(−1)) → · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3) Assume that the cohomological dimension of k is d, then we have Hu(Gal(ks/k), F) = Hu(Gal(ks/k), F(−1)) = 0 for u > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus we have an exact sequence 0 → H1(Gal(ks/k), F) → H1 kfl(X, F) → H0(Gal(ks/k), F(−1)) → H2(Gal(ks/k), F) → H2 kfl(X, F) → H1(Gal(ks/k), F(−1)) → · · · → Hd(Gal(ks/k), F) → Hd kfl(X, F) → Hd−1(Gal(ks/k), F(−1)) → 0, an isomorphism Hd+1 kfl (X, F) ∼= Hd(Gal(ks/k), F(−1)), and Hi kfl(X, F) = 0 for i ≥ d + 2 by the exact sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Moreover we assume that k is finite, then Gal(ks/k) ∼= ˆZ and d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus we have an exact sequence 0 → H1(ˆZ, F) → H1 kfl(X, F) → H0(ˆZ, F(−1)) → 0, an isomorphism H2 kfl(X, F) ∼= H1(ˆZ, F(−1)), and Hi kfl(X, F) = 0 for i ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In particular, we have H0(ˆZ, Z/lrZ(−1)) = H1(ˆZ, Z/lrZ(−1)) = 0, and thus Hi kfl(X, Z/lrZ) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Z/lrZ, if i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Z/lrZ, if i = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 0, if i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Global Dedekind domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Throughout this subsection, let K be a global field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' When K is a number field, X denotes the spectrum of the ring of integers in K, and when K is a function field, k denotes the field of constants of K and X denotes the unique connected smooth projective curve over k having K as its function field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let S be a finite set of closed points of X, U := X − S, j : U ֒→ X, and ix : x ֒→ X for each closed point x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We endow X with the log structure j∗O× U ∩ OX → OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Apparently X satisfies the assumption on log structure from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 19 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a group scheme over the underlying scheme of X which is ´etale locally isomorphic to a finite rank free abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We have H1 fl(X, F) ∼ = −→ H1 kfl(X, F) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since X satisfies the assumption on log structure from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3, we have a long exact sequence as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The sheaf MX/O× X is supported on the closed subset S, and thus the restriction of (Gm,log/Gm)Xfl to (st/X) is isomorphic to � x∈S ix∗Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' For any prime number l, let Sl := {x ∈ S | the characteristic of the residue field of x is not l}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='11, we have R2εfl∗F ∼= � l prime R1εfl∗(F ⊗Z Ql/Zl) and R1εfl∗(F ⊗Z Ql/Zl) ∼= � x∈Sl ix∗(F ⊗Z Ql/Zl(−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore we get Hu fl (X, R2εfl∗F) = � l prime � x∈Sl Hu fl (X, ix∗(F ⊗Z Ql/Zl(−1))) = � l prime � x∈Sl Hu fl (x, F ⊗Z Ql/Zl(−1)) = � l prime � x∈Sl Hu ´et(x, F ⊗Z Ql/Zl(−1)) = � l prime � x∈Sl Hu(Γx, F ⊗Z Ql/Zl(−1)), where Γx := Gal(κ(x)s/κ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We also have H2 fl(X, F) ∼= H2 ´et(X, F) by [Sta21, Lemma 0DDU].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' So we can rewrite the exact sequence from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3 as 0 → H2 ´et(X, F) → H2 kfl(X, F) → � l prime � x∈Sl H0(Γx, F ⊗Z Ql/Zl(−1)) → H3 ´et(X, F) → H3 kfl(X, F) → � l prime � x∈Sl H1(Γx, F ⊗Z Ql/Zl(−1)) → · · · → Hi ´et(X, F) → Hi kfl(X, F) → � l prime � x∈Sl Hi−2(Γx, F ⊗Z Ql/Zl(−1)) → · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Now we assume that the residue fields of X at its closed points are finite, then we have Hu(Γx, F ⊗Z (Ql/Zl)(−1)) = 0 for u > 1 due to cohomological dimension reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Moreover, by the same argument as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1, we even have H0(Γx, F ⊗Z (Ql/Zl)(−1)) = H1(Γx, F ⊗Z (Ql/Zl)(−1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 20 HEER ZHAO It follows that Hi kfl(X, F) ∼= Hi ´et(X, F) for any i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a finite ´etale group scheme over the underlying scheme of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then F = � l prime F(l), where F(l) is the l-primary subgroup of F and also finite ´etale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' To compute the Kummer log flat cohomology of F, it suffices to compute that of F(l) for each prime l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Without loss of generality, we assume that F = F(l) for some prime l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Similar to the situation of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3, we have R1εfl∗F ∼= � x∈Sl ix∗(F(−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4 and similar arguments as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3, we have a long exact sequence 0 → H1 ´et(X, F) → H1 kfl(X, F) → � x∈Sl H0(Γx, F(−1)) → H2 ´et(X, F) → H2 kfl(X, F) → � x∈Sl H1(Γx, F(−1)) → · · · → Hi ´et(X, F) → Hi kfl(X, F) → � x∈Sl Hi−1(Γx, F(−1)) → · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Now we assume that the residue fields of X at the closed points are finite, then we have Hu(Γx, F(−1)) = 0 for u > 1 due to cohomological dimension reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore we have an exact sequence 0 → H1 ´et(X, F) → H1 kfl(X, F) → � x∈Sl H0(Γx, F(−1)) → H2 ´et(X, F) → H2 kfl(X, F) → � x∈Sl H1(Γx, F(−1)) → H3 ´et(X, F) → H3 kfl(X, F) → 0, and isomorphisms Hi ´et(X, F) ∼ = −→ Hi kfl(X, F) for i ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' For F = Z/lrZ, we further have H0(Γx, Z/lrZ(−1)) = H1(Γx, Z/lrZ(−1)) = 0 for x ∈ Sl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore Hi ´et(X, Z/lrZ) ∼ = −→ Hi kfl(X, Z/lrZ) for i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 21 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Sites and sheaves In this appendix, we collect some general results about sites from [Sta21, Chapter 00UZ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Sta21, Definition 00VH] A site is given by a category C and a set Cov(C) = � U∈C Cov(U) with Cov(U) being a set of families of morphisms with target U, such that the following conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (1) If V → U is an isomorphism, then {V → U} ∈ Cov(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (2) If {Ui → U}i∈I ∈ Cov(U) and for each i we have {Vij → Ui}j∈Ji ∈ Cov(Ui), then {Vij → U}i∈I,j∈Ji ∈ Cov(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (3) If {Ui → U}i∈I ∈ Cov(U) and V → U is a morphism of C, then Ui ×U V exists for each i ∈ I and {Ui ×U V → V }i∈I ∈ Cov(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Continuous functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Sta21, Tag 00WV] A functor u : C → D of sites is called continuous, if for every {Vi → V }i∈I ∈ Cov(C) we have the following (1) {u(Vi) → u(V )} ∈ Cov(D), and (2) for any morphism T → V in C the morphism u(T ×V Vi) → u(T ) ×u(V ) u(Vi) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Recall that given a functor u as above, and a presheaf of sets F on D we can define upF to be simply the presheaf F ◦ u, in other words upF(V ) = F(u(V )) for every object V of C (see [Sta21, Tag 00VC] for up as well as for up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Suppose that the functor u : C → D is continuous, then F ∈ Sh(D) ⇒ upF ∈ Sh(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We denote us : Sh(D) → Sh(C) the functor up restricted to the subcategory of sheaves of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Recall that up admits a left adjoint up, see [Sta21, Tag 00VE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This is also the case for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Sta21, Tag 00WX] Let u : C → D be a continuous functor of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then the functor us : Sh(C) → Sh(D), G �→ (upG)♯ is left adjoint to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Sta21, Tag 00X1] A morphism of sites f : D → C is given by a continuous functor u : C → D such that us is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Notice how the functor u goes in the direct opposite the morphism f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' If f ↔ u is a morphism of sites, then we use the notation f −1 = us and f∗ = us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The functor f −1 is called the pullback functor, and the functor f∗ is called the pushforward functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' As in topology we have the adjunction (f −1, f∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 22 HEER ZHAO See [Sta21, Tag 00X2] (examples associated to maps between two topological spaces) and [Sta21, Tag 0EWI] (examples for different topologies on the same space, comparison of topologies) for examples of morphisms of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Cocontinuous functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' A functor u : C → D of sites is called cocontinuous, if for every U ∈ C and every {Vj → u(U)}j∈J ∈ Cov(D), there exists {Ui → U}i∈I ∈ Cov(C) such that {u(Ui) → u(U)}i∈I refines {Vj → u(U)}j∈J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Warning: In general {u(Ui) → u(U)}i∈I is not a covering of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' For an fs log scheme S, we denote by (Sch/S) the category of fs log schemes over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let (Sch/S)k´et and (Sch/S)´et be the Kummer log ´etale site and the classical ´etale site for (Sch/S) respectively, see [Ill02, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let (Sch/S)kfl be the Kummer log flat site for (Sch/S), see [KAT21, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3], and let (Sch/S)fl be the classical flat site for (Sch/S), which is an obvious analogue of (Sch/S)´et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Now let j : U ֒→ X be a strict open immersion of fs log schemes, and let τ ∈ {k´et, ´et, kfl, fl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then the functor of sites u : (fs/X)τ → (fs/U)τ, Y �→ Y ×X U is continuous, and the functor of sites v : (fs/U)τ → (fs/X)τ, V �→ V is continuous and cocontinuous Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The Kummer log flat site In this appendix, we focus on the Kummer log flat site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Morphisms of sites associated to a strict open immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Warning: We are going to follow [Sta21, Chapter 00UZ] and [Sta21, Chapter 03A4] to construct a homomorphism of sheaves of abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Morphisms of sites and topoi in [Sta21, Chapter 00UZ] are often formulated for sheaves of sets, while the map to be constructed is for sheaves of abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Often in order to use results from [Sta21, Chapter 00UZ], we have to use [Sta21, Tag 00YV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Below whenever we refer to a result about sheaves of sets from [Sta21, Chapter 00UZ] for sheaves of abelian groups, we are referring to [Sta21, Tag 00YV] without mention at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let X be an fs log scheme, and j : U ֒→ X a strict open immersion of fs log schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let τ ∈ {k´et, ´et, kfl, fl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The functor u : (fs/U)τ → (fs/X)τ, (V → U) �→ (V → U → X) of sites is continuous and cocontinuous, hence it gives rise to a morphism of topoi g = (g−1, g∗) : Ab((fs/U)τ) → Ab((fs/X)τ) THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 23 with g−1 exact by [Sta21, Tag 00XO] and [Sta21, Tag 00XL].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' For any presheaf F ∈ PAb((fs/U)τ), we define gp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='F ∈ PAb((fs/X)τ) as the presheaf Y �→ lim −→ Y →u(V ) F(V ) with colimits over (Iv Y )opp (see [Sta21, Equation 053L] for this index category) taken in the category of abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' For F ∈ Ab((fs/U)τ), we set g!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='F to be the sheafification of gp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='F, see [Sta21, Tag 04BF], and called it the extension by zero of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let X, U, u, g = (g−1, g∗), and g!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (1) The functor g!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' is left adjoint to g−1 and exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (2) The functor g−1 is exact and preserves injective objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (3) For any F ∈ Ab((fs/U)τ), the canonical maps F → g−1g!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='F and g−1g∗F → F are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Part (1) follows from [Sta21, Tag 04BG] and [Sta21, Tag 04BH].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since g−1 admits both a left adjoint g!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' and a right adjoint g∗, it is exact by [Sta21, Tag 0039].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since the left adjoint g!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' of g−1 is exact, it preserves injective objects by [Sta21, Tag 015Z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This finishes the proof of part (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since j is a strict open immersion, the functor u is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The functor u is continuous and cocontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus part (3) follows from [Sta21, Lemma 077I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ The functor v : (fs/X)τ → (fs/U)τ, Y �→ Y ×X U of sites is continuous with vs exact, hence gives rise to a morphism jτ : (fs/U)τ → (fs/X)τ of sites, and further a morphism jτ = (j−1 τ = vs, jτ∗ = vs) : Ab((fs/U)τ) → Ab((fs/X)τ) of topoi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The cocontinuous functor u is left adjoint to the continuous functor v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By [Sta21, Tag 00XY], the two morphisms g and jτ of topoi agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We set jτ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' := g!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=', and call it the functor of extension by zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' To sum up, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let j : U → X be a strict open immersion of fs log schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then we have a morphism of sites jτ : (fs/U)τ → (fs/X)τ, and a sequence of functors jτ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=', j−1 τ , jτ∗ where in each consecutive pair the first is exact and left adjoint to the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Moreover we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (1) The functor jτ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 24 HEER ZHAO (2) The functor j−1 τ is exact and preserves injective objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (3) For any F ∈ Ab((fs/U)τ), the canonical maps F → j−1 τ jτ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='F and j−1 τ jτ∗F → F are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Comparison morphism from the Kummer log flat site to the classical flat site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let j : U → X be a strict open immersion of fs log schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We have canonical forgetful morphisms of sites (fs/X)kfl → (fs/X)fl and (fs/U)kfl → (fs/U)fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By abuse of notation, we denote both of them by εfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It is easy to see that we have the following commutative diagram (fs/U)kfl jkfl � εfl � (fs/X)kfl εfl � (fs/U)fl jfl � (fs/X)fl of morphisms of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' For F ∈ Ab((fs/X)kfl), the adjunction (jfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=', j−1 fl ) gives a canonical map (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1) jfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='j−1 fl Riεfl∗F → Riεfl∗F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We have j−1 fl Riεfl∗F = Riεfl∗j−1 kfl F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F → I• be an injective resolution of F on (fs/X)kfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since the functor j−1 kfl is exact and preserves injective objects by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2 (2), we get an injective resolution j−1 kfl F → j−1 kfl I• of j−1 kfl F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus Riεfl∗j−1 kfl F = Hi(εfl∗j−1 kfl I•) = Hi(j−1 fl εfl∗I•) = j−1 fl Hi(εfl∗I•) = j−1 fl Riεfl∗F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ By the identification from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3, we get a map (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2) Φ : jfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='Riεfl∗j−1 kfl F → Riεfl∗F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let X be an fs log scheme and U an open subscheme of the underlying scheme of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We endow U with the induced log structure, and let j : U ֒→ X be the corresponding strict open immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let F be a sheaf of abelian groups on (fs/X)kfl such that Riεfl∗F is supported over U, then the map Φ : jfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='Riεfl∗j−1 kfl F → Riεfl∗F is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Applying the functor j−1 fl to the canonical map (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1), we get a map (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3) j−1 fl jfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='j−1 fl Riεfl∗F → j−1 fl Riεfl∗F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since j−1 fl jfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='j−1 fl Riεfl∗F is identified to j−1 fl Riεfl∗F by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2 (3), the map (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='3) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since Riεfl∗F is supported on U and the extension by zero sheaf jfl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='j−1 fl Riεfl∗F is clearly supported on U, the map (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1) is actually an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore Φ is also an isomorphism by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' □ Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' A lemma on profinite group cohomology Let p be a fixed prime number, and let ˆZ′ := lim ←−(p,m)=1 Z/mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let Gr = (ˆZ′)r and M a torsion abelian group, we regard M as a Gr-module with respect to the trivial action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In this appendix, we compute the profinite group cohomology Hi(Gr, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The result should be well-known to the experts, but we are not able to find a reference so present a computation here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' According to [Ols09, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2], the cohomology groups Hi(Gr, M) are computed by the cohomology groups of the standard homogeneous cochain complex of Gr with coefficients in M RΓ(Gr, M) : Mapcts Gr(Gr, M) → Mapcts Gr(G2 r, M) → · · · → Mapcts Gr(Gi r, M) → · · · , where Mapcts Gr(Gi r, M) denotes the set of equivariant continuous functions φ : Gi r → M (where M is endowed with the discrete topology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Note that Mapcts Gr(Gi r, M) is denoted as Homcts Gr(G[i−1] r , M) in [Ols09, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' First we consider the case that M = Z/nZ with (n, p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The complex RΓ(Gr, Z/nZ) of abelian groups is also naturally a complex of modules over the ring Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since Z/nZ as a module over itself is flat and RΓ(Gr, Z/nZ) lies in Db(Z/nZ) (the bounded derived category of complexes of Z/nZ-modules), we have (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1) RΓ(Gr, Z/nZ) ⊗L Z/nZ RΓ(Gs, Z/nZ) ∼ = −→ RΓ(Gr+s, Z/nZ) by K¨unneth formula, see [Ols09, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We have Hi(G1, Z/nZ) = � Z/nZ, if i = 0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 0, if i > 0, and RΓ(G1, Z/nZ) is isomorphic to Z/nZ 0−→ Z/nZ in Db(Z/nZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' By [Ols09, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='7], the natural map of graded Z/nZ-modules H∗(G1, Z/nZ) ⊗Z/nZ H∗(G1, Z/nZ) → H∗(G2, Z/nZ) is an isomorphism, and thus Hi(G2, Z/nZ) are free Z/nZ-modules for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Ap- plying [Ols09, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='7] inductively, we have that the natural map of graded Z/nZ-modules H∗(G1, Z/nZ)⊗r → H∗(Gr, Z/nZ) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' It follows that the graded cohomology ring H∗(Gr, Z/nZ) is isomorphic to the exterior algebra of the module H1(Gr, Z/nZ) = Hom(Gr, Z/nZ) ∼= (Z/nZ)r 26 HEER ZHAO over Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Now let M be a torsion abelian group which is killed by n with (n, p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We regard M as a module over Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Clearly we have RΓ(Gr, M) ∼= RΓ(Gr, Z/nZ) ⊗Z/nZ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Since Hi(Gr, Z/nZ) are free Z/nZ-modules for all i, we get Hi(Gr, M) ∼= Hi(Gr, Z/nZ) ⊗Z/nZ M canonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In general for a torsion abelian group M, we have M = M[p∞] ⊕ M ′ = M[p∞] ⊕ lim −→ (n,p)=1 M[n], where M[p∞] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' M ′, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' M[n]) denotes the p-primary part (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' prime to p part, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' n-torsion part) of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Thus Hi(Gr, M) = lim −→ (n,p)=1 Hi(Gr, M[n]) ∼= lim −→ (n,p)=1 Hi(Gr, Z/nZ) ⊗Z/nZ M[n] ∼= lim −→ (n,p)=1 ( i� H1(Gr, Z/nZ)) ⊗Z/nZ M[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2) We fix an isomorphism Gr = Hom(Xr, ˆZ′) with Xr := Zr, and thus H1(Gr, Z/nZ) = Hom(Gr, Z/nZ) = Xr ⊗Z Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Therefore we further have Hi(Gr, M) ∼= lim −→ (n,p)=1 ( i� (Xr ⊗Z Z/nZ)) ⊗Z/nZ M[n] = lim −→ (n,p)=1 ( i� Xr) ⊗Z M[n] =( i� Xr) ⊗Z M ′, where the last two wedges are for Z-module structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Apparently the identification (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='2) is induced by cup-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' To sum up, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let p be a fixed prime number, and ˆZ′ := lim ←−(p,m)=1 Z/mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let M be a torsion abelian group, and we regard it as a (ˆZ′)r-module with respect to the trivial action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Let M[p∞] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' M ′, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' M[n]) denote the p-primary part (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' prime to p part, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' n-torsion part) of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' We fix an isomorphism (ˆZ′)r = Hom(Xr, ˆZ′) with Xr := Zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Then THE HIGHER DIRECT IMAGES OF LOCALLY CONSTANT GROUP SCHEMES 27 (1) for any positive integer n with (n, p) = 1, we have H1((ˆZ′)r, M[n]) = Hom((ˆZ′)r, M[n]) ∼= M[n] ⊗Z Xr and the cup product induces an isomorphism Hi((ˆZ′)r, M[n]) ∼= M[n] ⊗Z/nZ i� H1((ˆZ′)r, Z/nZ), and the latter can be further identified with M[n] ⊗Z (�i Xr);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' (2) for the profinite group cohomology of (ˆZ′)r with coefficients in M, we have Hi((ˆZ′)r, M) = lim −→ (n,p)=1 Hi((ˆZ′)r, M[n]) = lim −→ (n,p)=1 M[n] ⊗Z ( i� Xr) =M ′ ⊗Z ( i� Xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Clearly the above computation works also for the profinite groups ˆZr and Zr l for any prime number l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Such results are the profinite group cohomology analogues of the description of the singular cohomology of topological tori (see [Hat02, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 3, Exa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Acknowledgement The author thanks Professor Chicara Nakayama for very helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' This work was partially supported by the Research Training Group 2553 of the German Research Foundation DFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' References [Art62] Michael Artin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Grothendieck topologies: notes on a seminar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Harvard University, Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' of Mathematics, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [CTS21] Jean-Louis Colliot-Th´elene and Alexei N Skorobogatov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' The Brauer–Grothendieck group, volume 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Springer Nature, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Hat02] Allen Hatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Algebraic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Ill02] Luc Illusie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' An overview of the work of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Fujiwara, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Kato, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Nakayama on logarithmic ´etale cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Ast´erisque, (279):271–322, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Cohomologies p-adiques et applications arithm´etiques, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [KAT21] Kazuya KATO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Logarithmic Structures of Fontaine-Illusie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' II —Logarithmic Flat Topol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Tokyo Journal of Mathematics, 44(1):125 – 155, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [KKN08] Takeshi Kajiwara, Kazuya Kato, and Chikara Nakayama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Logarithmic abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Nagoya Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=', 189:63–138, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [KKN15] Takeshi Kajiwara, Kazuya Kato, and Chikara Nakayama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Logarithmic abelian varieties, Part IV: Proper models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Nagoya Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=', 219:9–63, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [KN99] Kazuya Kato and Chikara Nakayama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Log Betti cohomology, log ´etale cohomology, and log de Rham cohomology of log schemes over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Kodai Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=', 22(2):161–186, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Mil80] James S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Milne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' ´Etale cohomology, volume 33 of Princeton Mathematical Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Princeton University Press, Princeton, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=', 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Niz08] Wies�lawa Nizio�l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' K-theory of log-schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=', 13:505–551, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Ols09] Martin C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' On Faltings’ method of almost ´etale extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' In Algebraic geometry— Seattle 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Part 2, volume 80 of Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=', pages 811–936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=', Providence, RI, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' 28 HEER ZHAO [Ser02] Jean-Pierre Serre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Galois cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Springer Monographs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Springer- Verlag, Berlin, english edition, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Translated from the French by Patrick Ion and revised by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Sta21] The Stacks Project Authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Stacks Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' http://stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='edu, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Zha17] Heer Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Log abelian varieties over a log point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=', 22:505–550, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Zha21a] Heer Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Comparison of Kummer logarithmic topologies with classical topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Journal of the Institute of Mathematics of Jussieu, pages 1–31, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' [Zha21b] Heer Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Comparison of kummer logarithmic topologies with classical topologies ii, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content=' Heer Zhao, Fakult¨at f¨ur Mathematik, Universit¨at Duisburg-Essen, Essen 45117, Germany, heer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='zhao@uni-due.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} +page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFAT4oBgHgl3EQfcx2g/content/2301.08566v1.pdf'} diff --git a/ndE1T4oBgHgl3EQf1QXp/content/tmp_files/2301.03467v1.pdf.txt b/ndE1T4oBgHgl3EQf1QXp/content/tmp_files/2301.03467v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..980dba65fbebf46cf2f14004e7bbf3206c10ccda --- /dev/null +++ b/ndE1T4oBgHgl3EQf1QXp/content/tmp_files/2301.03467v1.pdf.txt @@ -0,0 +1,1871 @@ +arXiv:2301.03467v1 [eess.SP] 8 Dec 2022 +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +1 +ORKA: Accelerated Kaczmarz Algorithms for +Signal Recovery from One-Bit Samples +Arian Eamaz, Student Member, IEEE, Farhang Yeganegi, Deanna Needell, Member, IEEE, and +Mojtaba Soltanalian, Senior Member, IEEE +Abstract +One-bit quantization with time-varying sampling thresholds has recently found significant utilization +potential in statistical signal processing applications due to its relatively low power consumption and +low implementation cost. In addition to such advantages, an attractive feature of one-bit analog-to- +digital converters (ADCs) is their superior sampling rates as compared to their conventional multi- +bit counterparts. This characteristic endows one-bit signal processing frameworks with what we refer +to as sample abundance. On the other hand, many signal recovery and optimization problems are +formulated as (possibly non-convex) quadratic programs with linear feasibility constraints in the one- +bit sampling regime. We demonstrate, with a particular focus on the nuclear norm minimization, that +the sample abundance paradigm allows for the transformation of such quadratic problems to merely +a linear feasibility problem by forming a large-scale overdetermined linear system; thus removing the +need for costly optimization constraints and objectives. To make this achievable, we propose enhanced +randomized Kaczmarz algorithms to tackle these highly overdetermined feasibility problems. Several +numerical results are presented to illustrate the effectiveness of the proposed methodologies. +Index Terms +Convex-relaxed problems, nuclear norm minimization, one-bit quantization, one-bit ADCs, random- +ized Kaczmarz algorithm, statistical signal processing, time-varying sampling thresholds. +This work was supported in part by National Science Foundation Grant CCF-1704401. The first two authors contributed +equally to this work. +A. Eamaz, F. Yeganegi and M. Soltanalian are with the Department of Electrical and Computer Engineering, University of +Illinois Chicago, Chicago, IL 60607, USA (Corresponding author: Arian Eamaz). +D. Needell is with the Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095 USA. +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +2 +I. INTRODUCTION +We consider an optimization problem of the form +min +X +f(X) +s.t. +A (X) = y, +X ∈ Ωc, +(1) +where f(.) is a cost function, X ∈ Cn1×n2 is the matrix of unknowns, y ∈ Rn is the measurement vector, +and A is a linear transformation mapping Cn1×n2 into Rn. +This problem has been used as a relaxed version of some well-known NP-hard problems, and emerging +in wide variety of statistical signal processing applications. Although many problems can be expressed +in the form in (1), the applications we will focus on in this paper include some specific problems of +interest in statistical signal processing, which can take advantage of low-resolution (and particularly one +bit) sampling and processing: +• Low-rank matrix recovery: The task of recovering a low-rank matrix from its linear measurements +plays a central role in computational science. The problem occurs in many areas of applied math- +ematics such as signal processing [1]–[7], machine learning [8]–[13], and computer vision [14]. In +this scenario, the cost function of (1), f(.), is typically to be the nuclear norm or the Frobenius +norm, and the constraint set Ωc would be a amplitude restriction on the elements of matrix X; see +[2], [15]. +• Phase retrieval: Phase retrieval has received a great deal of interest as it aims to recover an unknown +signal solely from phaseless measurements that depend on the signal through a linear observation, +commanding numerous applications in applied physics and statistical signal processing communities +over the past decades [16]–[28]. To have a convex formulation, the phase retrieval has been relaxed +into semi-definite programs where the problem boils down to a trace minimization while considering +the positive semi-definite constraint [1], [16]. +• Compressed sensing: Compressed sensing (CS) offers a framework for simultaneous sensing and +compression of finite dimensional vectors, that relies on linear dimensionality reduction. Through a +CS formulation, sparse signals may be recovered from highly incomplete measurements [29]. The +problem (1) can be adopted in the CS content when f (X) = ∥vec (X)∥1. +• Magnetic resonance imaging: Reconstructing magnetic resonance images commonly involves col- +lecting a series of frames of data in which a radio frequency excitation produces new transverse +magnetization, which is then sampled along a particular trajectory in k-sparse representation. Due +to meet various physical and physiological constraints, most MRI methods utilize a sequence of +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +3 +acquisitions, each of which partially samples the representation. Let the acquired sequence of +measurements be represented by yi, where i is the sequence index, and {Ai (X)} denote a linear +transformation, chosen in a manner that promotes sparsity in the range space. In this example, the +cost function can be considered to be the ℓ1-norm, and the sequence of acquisitions are used as +linear constraints in (1) [30], [31]. +Sampling the signals of interest at high data rates with high-resolution ADCs would dramatically +increase the overall manufacturing cost and power consumption of such ADCs. In multi-bit sampling +scenarios, a very large number of quantization levels is necessary in order to represent the original +continuous signal in with high accuracy, which in turn leads to a considerable reduction in sampling rate +[32]. This attribute of multi-bit sampling is the key reason for the general emergence of underdetermined +systems n1n2 ≥ n in (1) [1], [16], [27]. An alternative solution to such challenges is to deploy one-bit +quantization which is an extreme sampling scenario, where the signals are merely compared with given +threshold levels at the ADCs, producing sign data (±1). This enables signal processing equipments to +sample at a very high rate, with a considerably lower cost and energy consumption, compared to their +counterparts which employ multi-bit ADCs [32]–[35]. +In traditional one-bit sampling schemes, the signal recovery is accomplished by comparing the signal +with a fixed threshold, usually zero. This creates some difficulties in estimating signal parameters. In +contrast, recent works have employed time-varying sampling thresholds, which exhibit enhanced recovery +performance for the signal parameters [16], [32], [36]–[40]. +In this paper, we consider the deployment of one-bit sampling with time-varying thresholds,leading to +an increased sample size and a highly overdetermined system as a result. The proposed One-bit aided +Randomized Kaczmarz Algorithm, which we refer to as ORKA, can find the desired signal X⋆ in (1) +by (i) generating abundant one-bit measurements, in order to define a large scale overdetermined system +where a finite volume feasible set is created for (1), and (ii) solving this obtained linear feasibility problem +by leveraging one of the efficient solver families of overdetermined systems, Kaczmarz algorithms. The +Kaczmarz method [41] is an iterative projection algorithm for solving linear systems of equations and +inequalities. It is usually applied to highly overdetermined systems because of its simplicity. Each iteration +projects onto the solution space corresponding to one row in the linear system, in a sequential regimen. +The method has been applied to various applications in image reconstruction, digital signal processing, +and computer tomography [16], [42], [43]. Many variants of this iterative method and their convergence +rates have been proposed and studied in recent decades for both consistent and inconsistent systems +including the randomized Kaczmarz algorithm, the randomized block Kaczmarz algorithm and most +recently, the sampling Kaczmarz-Motzkin method [44]–[48]. +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +4 +A. Contributions of the Paper +In [16], we showed that the sheer number of measurements acquired in one bit sampling facilitates +recovering the signal of interest in a less costly manner by making costly constraints such as semidef- +initeness and rank redundant. Then, a simple randomized Kaczmarz algorithm (RKA) was utilized to +solve the obtained linear feasibility problem. This idea is generalized in this paper to (1) where we +generate the abundant samples and eventually introduce a one-bit linear feasibility region named the +one-bit polyhedron. In other words, by using this technique, we make (1) a large-scale overdetermined +system which is the desired application setting for Kaczmarz algorithms. +To solve our highly overdetermined system, we propose two novel variants of RKA which will be +compared with the existing RKA variants. Furthermore, an algorithm is proposed based on our model to +adaptively evaluate the time-varying sampling thresholds. The convergence rate of the proposed algorithm +is investigated based on the moments generating function of recovery errors and the scaled condition +number of the constraint matrix. Finally, the performance of the proposed method is examined in nuclear +norm minimization-based problems. +B. Organization of the Paper +Section II is dedicated to a review of proximal methods which have been utilized to tackle (1) by +projecting the final solution on the desired feasible set. In Section III, we will introduce our algorithm to +solve (1), ORKA, which tackles the problem as a large-scale overdetermined system and finds the optimal +point in the one-bit polyhedron by an accelerated Kaczmarz approach. Moreover, two new variants of the +Kaczmarz algorithms are proposed that enhance the convergence rate and the computational complexity of +these solvers. To investigate the convergence rate of ORKA, at first, we will introduce a penalty function in +Section IV based on the Chernoff bound. Section V discusses an iterative algorithm to achieve optimized +time-varying sampling threshold sequences which benefit the signal recovery process with enhanced +accuracy. As a representative application, in Section VI, ORKA and other proposed algorithms will be +applied in the context of low-rank matrix recovery in the form of a nuclear norm minimization problem. +Finally, Section IX concludes the paper. +Notation: We use bold lowercase letters for vectors and bold uppercase letters for matrices. C and +R represent the set of complex and real numbers, respectively. (·)⊤ and (·)H denote the vector/matrix +transpose, and the Hermitian transpose, respectively. IN ∈ RN×N is the identity matrix of size N. Tr(.) +denotes the trace of the matrix argument. ⟨B1, B2⟩ = Tr(BH +1 B2) is the standard inner product between +two matrices. The nuclear norm of a matrix B ∈ CN1×N2 is denoted ∥B∥⋆ = �M +i=1 σi where M and +{σi} are the rank and singular values of B, respectively. The Frobenius norm of a matrix B is defined +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +5 +as ∥B∥F= +��N1 +r=1 +�N2 +s=1 |brs|2 where {brs} are elements of B. The ℓk-norm of a vector b is defined as +∥b∥k +k= � +i|b|k +i . The Hadamard (element-wise) product of two matrices B1 and B2 is denoted as B1⊙B2. +Additionally, the Kronecker product is denoted as B1 ⊗B2. The vectorized form of a matrix B is written +as vec(B). 1s is the s-dimensional all-one vector. Given a scalar x, we define (x)+ as max {x, 0}. f ≍ g +means f and g are asymptotically equal. Diag {b} denotes a diagonal matrix with {bi} as its diagonal +elements. +II. PROJECTIONS ON CONVEX SETS: DEALING WITH COSTLY CONSTRAINTS +To tackle (1), many non-convex and local optimization algorithms have been developed over the years. +Nevertheless, in recent decades, convex programming formulations via relaxation have come to the fore to +approximate global solutions. In the convex framework, various iterative methods have been proposed to +tackle the problem with a Lagrangian formulation such as Uzawa’s algorithm and the proximal forward- +backward splitting method (PFBS) [2], [49], [50]. Moreover, to keep the problem solution inside the +constraint set Ωc, the orthogonal projection PΩc is applied to solutions in each iteration. This process is +briefly explained below. +The Lagrangian for (1) is written as [2], +L (X, λ) = f (X) + ⟨λ, y − A (X)⟩ , +(2) +where λ ∈ Rn. Uzawa’s algorithm aims to find a saddle point (X⋆, λ⋆), where supλ infX L (X, λ) = +infX supλ L (X, λ), with the iterative procedure: + + + + + +L +� +Xk, λk−1� += minX L +� +X, λk−1� +, +λk = PΩc +� +λk−1 + αk +� +y − A +� +Xk��� +, +(3) +where αk is the step size. This iterative steps can be rewritten as + + + + + +Xk = Proxf +� +A⋆ � +λk−1�� +, +λk = PΩc +� +λk−1 + αk +� +y − A � +Xk��� +, +(4) +where Proxf is the proximal operator minimizing the Lagrangian function, and A⋆ is the adjoint of A. +Since every linear equation can be reformulated in standard form, we recast A (X) = y as Ax = y, +where A ∈ Cn×n1n2 is a matrix version of the operator A, and x = vec (X) [15]. The optimization +problem (1) is equivalently given by [1], [2] +min +X +g (X) = 1 +2 ∥y − A vec (X)∥2 +2 + λf(X) +s.t. +X ∈ Ωc. +(5) +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +6 +To solve this problem, instead of using proximal methods, a projected gradient method such as Nesterov +iterative approach may be utilized, i.e., Xk = PΩc +� +Xk−1 − αk∇g � +Xk−1��. +Famous examples for Proxf and PΩc, are the singular value thresholding operator (SVT) and the semi- +definite orthogonal projector, respectively. SVT is useful when f (X) = ∥X∥⋆, mathematically defined +as [2]: +Dδ = U Diag +� +(σk − δ)+� +V⊤, +(6) +where U and V are unitary matrices from singular value decomposition (SVD), and {σk} are the singular +values. Furthermore, the semi-definite projector emerges in semi-definite programming where the convex +constraint set is a positive semi-definite (PSD) matrix. It compares eigenvalues of the solution in each +iteration with zero or a fixed threshold [1], i.e., +PΩc = U Diag +� +(λk − δ)+� +U⊤, +(7) +where U is the unitary matrix coming from the Schur decomposition. In the case of both operators, the +approximate solution should be projected onto a feasible convex set at each iteration via recovering all +singular values and eigenvalues and comparing their smaller elements with a threshold, which is quite +expensive [1]. +An interesting alternative to enforcing the feasible set FX = {Proxf ∩ Ωc} in (1) emerges when +one increases the number of samples n, and solves the overdetermined linear system of equations with +n ≥ n1n2. In this sample abundance regimen, the linear constraint A (X) = y may actually yield the +optimum inside FX. As a result of increasing the number of samples, it is possible that the intersection of +these hyperplanes will achieve the optimal point without the need to consider costly constraints. However, +this idea may face practical limitations in the case of multi-bit quantization systems since ADCs capable +of ultra-high rate sampling are difficult and expensive to produce. Moreover, one cannot necessarily +expect these constraints to intersect with FX in such a way to form a finite-volume space before the +optimum is obtained [16], [27]. +In the next section, by deploying the idea of one-bit sampling with time-varying thresholds, linear +equality constraints are superseded by a massive array of linear inequalities in forming the feasible +polyhedron. Therefore, by increasing the number of samples, a finite-volume space may be created inside +FX with shrinking size; making projections on Ωc redundant. From a practical point of view, one-bit +sampling is done efficiently at a very high rate with a significantly lower cost compared to its high- +resolution counterpart. It has been examined in [16] that even though only partial information is made +available to one-bit signal processing algorithms, they can achieve acceptable recovery performance with +less complexity compared to the high-resolution scenario. Thus, it is both practical and necessary to study +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +7 +the ground-breaking opportunities that emerge in the context of the wide array of problems formulated +as (1) due to the availability of a large number of one-bit samples. +III. PROPOSED ALGORITHM +In this section, at first we begin by presenting a summarized review of randomized Kaczmarz algo- +rithms. Then, we propose a novel Kaczmarz method variant formulated based on the sampling Kaczmarz- +Motzkin algorithm (SKM) and a preconditioning approach. One-bit sampling via time-varying thresholds +will be combined with the proposed randomized Kaczmarz method to create highly overdetermined +linear inequalities. This paves the way for the recovery of the desired signal X⋆ in (1) without solving +the original optimization problem; merely by tacking accounts of its linear constraints. We name our +algorithm One-bit aided Randomized Kaczmarz Algorithm (ORKA). Due to the block structure of the +linear feasibility in ORKA, we will propose a block-based Kaczmarz algorithm accordingly. +A. Randomized Kaczmarz Algorithm (RKA) +The randomized Kaczmarz algorithm (RKA) is a sub-conjugate gradient method to solve a linear +feasibility problem, i.e, Cx ⪯ b where C is a m × n matrix with m > n [44], [45]. Conjugate-gradient +methods immediately turn the mentioned inequality to an equality in the following form: +(Cx − b)+ = 0, +(8) +and then, approach the solution by the same process as used for systems of equations. Without any loss +of generality, consider (8) to be a polyhedron: + + + + + +cjx ≤ bj +(j ∈ I≤) , +cjx = bj +(j ∈ I=) , +(9) +where the disjoint index sets I≤ and I= partition our sample index set J , and {cj} denote the rows of +C. Based on this problem, the projection coefficient βi of the RKA is defined as [45], [47], [51]: +βi = + + + + + +(cjxi − bj)+ +(j ∈ I≤) , +cjxi − bj +(j ∈ I=) . +(10) +Also, the unknown column vector x is iteratively updated as +xi+1 = xi − +βi +∥cj∥2 +2 +cH +j , +(11) +where, at each iteration i, the index j is chosen independently at random from the set J , following the +distribution +P{j = k} = ∥ck∥2 +2 +∥C∥2 +F +. +(12) +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +8 +If the system (9) is consistent and its feasible region is nonempty, RKA converges linearly in expectation +[44], [45]: +E {¯h (xi, x⋆)} ≤ qi ¯h (x0, x⋆) , +(13) +where ¯h (xi, x⋆) = ∥xi − x⋆∥2 +2 is the distance function between two points in the space, x⋆ is a desired +point, i is the number of required iterations for RKA, and q ∈ (0, 1) is given as +q = 1 − +1 +κ2 (C), +(14) +with κ (C) = ∥C∥F∥C†∥2 denoting the scaled condition number. +B. Sampling Kaczmarz-Motzkin Algorithm (SKM) +The SKM combines the ideas of both the RKA and the Motzkin method. Its generalized convergence +theorem, and a validation of feasibility, which has been formulated based on the convergence analysis of +RKA and sampling Motzkin method for solving linear feasibility problem have been fully explored in +[48]. +The central contribution of SKM lies in its innovative way of projection plane selection. The hyperplane +selection is done as follows. At iteration i the SKM algorithm selects a collection of γ (denoted by the +set τi), uniformly at random out of m rows of the constraint matrix C. Then, out of these γ rows, the +row with maximum positive residual is selected. Finally, the solution is updated as [48], [52]: +xi+1 = xi − λi +βi +∥cj⋆∥2 +2 +cH +j⋆, +(15) +where j⋆ = argmax +� +(cjxi − bj)+� +, j ∈ τi, and λi is a relaxation parameter which for consistent +systems must satisfy [44], +0 ≤ lim +i→∞ inf λi ≤ lim +i→∞ sup λi < 2, +(16) +to ensure convergence. The convergence bound for SKM is given by +E {¯h (xi, x⋆)} ≤ +� +1 − 2λi − λ2 +i +κ2 (C) +�i +¯h (x0, x⋆) . +(17) +In the case where the constraint matrix is normalized, i.e. ∥cj∥2 +2= 1, si is the number of satisfied +constraints after iteration i, and Li = max {m − si, m − γ}, for the (i + 1)th iteration we have [48], +E {¯h (xi, x⋆)} ≤ +� +1 − σ2 +min +� +2λi − λ2 +i +� +Vi +�i +¯h (x0, x⋆) . +(18) +This recovery error bound is tighter than (17). +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +9 +C. Our Contribution: Preconditioned SKM (PrSKM) +According to the convergence rate formula of RKA, if we can reduce the value of the scaled condition +number, the convergence is accelerated, and the upper bound of the recovery error E +� +∥xi − x⋆∥2 +2 +� +decreases. Moreover, by having a lower value of q, a lower number of iterations is required to achieve a +specific recovery error bound, usually considered to be the algorithm’s termination criterion. Consequently, +let I is the number of iterations, the computational cost of RKA which behaves as O (In), is diminished +as well. To make this happen, one can start from reducing q = 1 − +1 +κ2(C) which occurs when the scaled +condition number κ (C) is diminished; a condition that can be satisfied by considering the following +theorem. +Theorem 1. The infimum scaled condition number of a matrix C ∈ Rm×n is given by +inf +C κ (C) = √n, +(19) +which is achieved if and only if C is of the form C = αU, where U is an orthonormal matrix and α ∈ R +is a scalar. +Proof. The condition number of the matrix C is defined as ̺ (C) = σmax +σmin , where σmax and σmin are its +minimum and maximum singular values, respectively [53]. The scaled condition number can be written +as κ (C) = ∥C∥F +σmin . Therefore, the scaled condition number has the following relation with ̺ (C): +κ (C) = ∥C∥F +σmax +̺ (C) . +(20) +Furthermore, the condition number can be considered to be an upper bound for the scaled condition +number as well based on the readily-known inequality relation between norm-2 and the Frobenius norm +[53]: +∥C∥F ≤ √n∥C∥2, +∥C∥F +σmin +≤ √n∥C∥2 +σmin +, +(21) +or equivalently, +κ (C) ≤ √n̺ (C) . +(22) +Thus, lowering ̺ (C) generally leads to a decreasing scaled condition number. Additionally, the lowest +possible value for ̺ is 1 which is achieved for scaled unitary matrices U as if we let S = αU, and +O = S⊤S = α2In, then, σmin = σmax = α, and ̺ = 1. Vice versa, if ̺ = 1, it means σmin = σmax +which leads to a diagonal matrix O = α2In. It is straightforward to verify that the decomposition of +O results in an S that is a scaled-version of an orthonormal matrix. As a result, the lowest achievable +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +10 +upper bound for the scaled condition number is obtained as κ (C) ≤ √n, and according to (20), κ (C) = +α∥U∥F +α += √n. +Accordingly, it would be enough to make our matrix C unitary by a process which is referred to as +the preconditioning method. In preconditioning, the linear feasibility is rewritten as +CMz ⪯ b, +(23) +where M is the preconditioner and x = Mz. The straightforward way to approach this task is to use +QR decomposition where the constraint matrix is decomposed as C = QcRc, with unitary Qc ∈ Rm×n, +and Rc ∈ Rn×n is an upper triangular matrix, leading to +Qc = CR−1. +(24) +Thus, the good choice for the preconditioner is M = R−1 +c . To find the desired point z⋆, the SKM +is selected in order to apply to the linear feasibility (23), then the desired signal x⋆ is obtained from +x⋆ = R−1 +c z⋆. We refer to this method Preconditioned SKM (PrSKM). +Proposition 1 (PrSKM). The proposed algorithm, PrSKM, can be summarized as follows: +1) Calculate the QR decomposition of the constraint matrix C to obtain the preconditioner M. +2) Using the change of variables, x = Mz, obtain Hz ⪯ b, where H = CM and M = R−1 +c . +3) Choose a sample set of γ constraints (denoted as τi) uniformly at random from the rows of H. +4) From these γ constraints, choose j⋆ = argmax +� +(hjzi − bj)+� +, j ∈ τi where hj is the jth row of +H. +5) Update the solution via the iterations zi+1 = zi − λi +(hj⋆zi−bj⋆)+ +∥hj⋆∥2 +2 +hH +j⋆ until convergence. +6) Recover the desired signal from the final solution of SKM as x⋆ = Mz⋆. +The scaled condition number of PrSKM is obtained as κ (H) = √n, which implies q = n−1 +n . +D. One-Bit Polyhedron +Consider a bandlimited signal y ∈ L2 to be represented by its samples via the standard sampling +formula [54], +0 < T ⩽ π +Ω, +y(t) = +k=+∞ +� +k=−∞ +y(kT) sinc +� t +T − k +� +, +(25) +where 1/T is the sampling rate and sinc(t) = sin(πt) +(πt) +is an ideal low-pass filter. Suppose yk = y(kT) +denotes the uniform samples of y(t) with the sampling rate T. Let rk denote the quantized version of +y[k] with the formulation +rk = Q(yk), +(26) +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +11 +Algorithm 1 Block SKM Algorithm +Input: Matrix B ∈ Rmn×d where B = +� +B⊤ +1 +· · · +B⊤ +m +�⊤ +, right-hand side b with dimension mn, +initial value of x0 with dimension n, convergence tolerance ǫ > 0, and relaxation parameter λi. +Output: An estimate x for the solution to the linear feasibility problem Bx ⪯ b. +1: Initiate the following loop by setting i = 0. +2: while +��(Bxi − b)+�� +2 ≤ ǫ do +3: +Choose a block Bj uniformly at random with the probability P{j = k} = ∥Bk∥2 +F +∥B∥2 +F . +4: +Let e′ denote the sorted version of e from emax (the maximum element of e) to emin (the minimum +element of e). +5: +Select the first k′ < d element of e′ and construct the problem B′ +jx ⪯ b′ +j, where B′ +j ∈ Rk′×d +and b′ +j ∈ Rk′×1. +6: +Compute the Moore-Penrose of B′ +j, i.e., +B′† +j ← B′⊤ +j +� +B′ +jB′⊤ +j +�−1 +. +7: +Update the solution xi+1 as: +xi+1 ← xi − λiB′† +j +� +B′ +jx − b′ +j +�+ . +8: +Increase i by one. +9: end while +where Q denotes the quantization effect. Consider a non-zero time-varying Gaussian threshold τ = [τk] +with the distribution τ ∼ N (d = 1d, Σ). In the case of one-bit quantization with such time-varying +sampling thresholds, (26) is simply written as +rk = sgn (yk − τk) , +(27) +where sgn(.) is the sign function. The information gathered through the one-bit sampling with time- +varying thresholds presented here may be formulated in terms of an overdetermined linear system of +inequalities. In Eq. (27), +rk = + + + + + ++1 +yk > τk, +−1 +yk < τk. +(28) +Therefore, one can formulate the geometric location of the signal as +rk (yk − τk) ≥ 0. +(29) +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +12 +Let y = [yk] and r = [rk]. Then, the vectorized representation of (29) is +r ⊙ (y − τ) ≥ 0, +(30) +or equivalently +Ωy ⪰ r ⊙ τ, +(31) +where Ω ≜ diag {r}. Suppose y, τ ∈ Rn, and that τ(ℓ) denotes the time-varying sampling threshold +sequence in ℓ-th experiment where ℓ ∈ L = {1, · · · , m}. According to (31), we have + + + + + + + + + + + + + + + + + + + +Ω(1)y ⪰ r(1) ⊙ τ(1) +r(1) = sgn � +y − τ(1)� +, +Ω(2)y ⪰ r(2) ⊙ τ(2) +r(2) = sgn � +y − τ(2)� +, +... +... +Ω(m)y ⪰ r(m) ⊙ τ(m) +r(m) = sgn � +y − τ(m)� +, +(32) +where Ω(ℓ) = diag +� +r(ℓ)� +. In Eq. (32), we have m linear system of inequalities which can also be merged +in one inequality as +˜Ωy ⪰ vec (R) ⊙ vec (Γ) , +(33) +where R and Γ are matrices with +� +r(ℓ)� +and +� +τ(ℓ)� +representing their columns, respectively, and ˜Ω is +˜Ω = +� +Ω(1) +· · · +Ω(m) +�⊤ +, +˜Ω ∈ Rmn×n. +(34) +Assuming a large number of samples which is a common situation in one-bit sampling scenarios, hereafter, +we consider (33) as an overdetermined linear system of inequalities associated with the sampling scheme +presented in (27). The inequality (33) can be recast by a polyhedron as +Py = +� +y | ˜Ωy ⪰ vec (R) ⊙ vec (Γ) +� +, +(35) +which we refer to as the one-bit polyhedron. +E. ORKA: Towards Circumventing Costly Constraints +If one applies one-bit sampling with time-varying sampling thresholds to the measurement y ∈ Rn +from (1) following the process defined in Subsection III-D, the arising inequality system is simply given +by +˜ΩAx ⪰ vec (R) ⊙ vec (Γ) , +(36) +where Ax = y, and x = vec (X). Consequently, the one-bit polyhedron for this problem is obtained as +Px = {x | Px ⪰ vec (R) ⊙ vec (Γ)} , +(37) +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +13 +Algorithm 2 Architecture of ORKA. +Input: The measurement vector y obtained as A (X) = y from (1), m sequences of time-varying +sampling thresholds generated as +� +τ(ℓ) ∼ N (0, I) ; ℓ ∈ L +� +. +Output: Recovered optimal signal x⋆ = vec (X⋆). +1: Apply one-bit sampling on y and generate sequences of one-bit measurements from: +r(ℓ) ← sgn +� +x − τ(ℓ)� +, +ℓ ∈ L. +2: Construct a linear feasibility region from the one-bit sampled data as: +Ω(ℓ)y ⪰ r(ℓ) ⊙ τ(ℓ), +ℓ ∈ L. +3: Define a highly-overdetermined system, the one-bit polyhedron, based on obtained inequalities: +Px = {x | Px ⪰ vec (R) ⊙ vec (Γ)} , +where R and Γ are matrices with +� +r(ℓ)� +and +� +τ(ℓ)� +representing their columns, respectively. +4: Employ RKA variants (PrSKM or Block SKM) to recover X⋆ within the one-bit polyhedron. +where P = ˜ΩA. +By taking advantage of one-bit sampling, in the asymptotic scenario of with sample abundance, the +space restricted by the one-bit polyhedron Px, shrinks to become contained inside the feasible set FX. +Note that this shrinking space always contains the global minima, with a volume that is diminished with +an increased sample size. As a result, instead of using proximal operators and orthogonal projectors, it +is enough to find the desired signal x⋆ in (37). To do so, one can use the PrSKM algorithm proposed in +Section III-C. It is worth noting that the PrSKM is a row-based algorithm, where at each iteration, the +row index is chosen independently at random. However, the matrix P in (37) has a block structure with +the following formulation +P = +� +A⊤Ω(1) +· · · +A⊤Ω(m) +�⊤ +, +P ∈ Rmn×d, +(38) +where d = n1n2. Therefore, it is useful to investigate the block-based RKA methods to find the desired +signal in Px for further efficiency enhancement. Our proposed algorithm, Block SKM, is described as +follows. +Proposition 2 (Block SKM). We have a linear feasibility problem Bx ⪯ b where B = +� +B⊤ +1 +· · · +B⊤ +m +�⊤ +, +and b = +� +b⊤ +1 +· · · +b⊤ +m +�⊤ +. The proposed algorithm for feasible signal recovery, Block SKM, can be +summarized as follows: +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +14 +0 +100 +200 +300 +Iteration +-30 +-25 +-20 +-15 +-10 +-5 +0 +RKA +SKM +PrSKM +Block SKM +(a) +0 +50 +100 +150 +200 +Iteration +-3 +-2.5 +-2 +-1.5 +-1 +-0.5 +0 +RKA +SKM +PrSKM +Block SKM +(b) +Figure 1: Comparing the NMSE recovery performance of the two proposed Kaczmarz algorithms, namely +the PrSKM and the block SKM, with that of SKM and RKA for: (a) a linear equation system, (b) a +linear inequality system. +1) Choose a block Bj uniformly at random with the probability P{j = k} = ∥Bk∥2 +F +∥B∥2 +F . +2) Compute e = Bjx − bj. +3) Let e′ denote the sorted version of e from emax (the maximum element of e) to emin (the minimum +element of e). This step is inspired by the idea of the Motzkin sampling, presented in Section III-B, +to have an accelerated convergence. +4) Select the first k′ < d element of e′ and construct the sub-problem B′ +jx ⪯ b′ +j, where B′ +j ∈ Rk′×d +and b′ +j ∈ Rk′×1. The reason behind choosing k′ < d is due to the computation of +� +B′ +jB′⊤ +j +�−1 +in the +next step (Step 5). For k′ > d, the matrix B′ +jB′⊤ +j +is rank-deficient and its inverse is not available. +5) Compute the Moore-Penrose of B′ +j, B′† +j = B′⊤ +j +� +B′ +jB′⊤ +j +�−1 +. +6) Update the solution xi+1 = xi − λiB′† +j +� +B′ +jx − b′ +j +�+ +. This update process is inspired from the +randomized block Kaczmarz method [46], [55] which takes advantage of the efficient matrix-vector +multiplication, thus giving the method a significant reduction in computational cost [47]. +The steps of the block SKM and ORKA are summarized in Algorithm 1 and Algorithm 2, respectively. +To examine the performance of the block SKM, we compare it to the PrSKM, SKM and RKA. +F. Comparing RKA, SKM, PrSKM and Block SKM +In this section, we numerically compare the RKA, SKM, PrSKM, and Block SKM in linear systems +of equalities as well as those formed by inequalities. +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +15 +Linear feasibility of equalities: Herein, we consider a block linear system of equalities Ax = b, +where A = +� +A⊤ +1 +· · · +A⊤ +100 +�⊤ +, Ai ∈ R10×10, x ∈ R10, and b ∈ R1000. Each row of Ai is generated +as ai +j ∼ N (0, I10). Also, the unknown signal x is generated as x ∼ N (0, I10). The normalized mean +square error (NMSE) is defined as +NMSE ≜ ∥x⋆ − ¯x∥2 +2 +∥x⋆∥2 +2 +, +(39) +where x⋆ and ¯x denote the true discretized signal and its recovered version, respectively. +Fig. 1 illustrates the performance of RKA, SKM, PrSKM, and Block SKM in the recovery of x from +the system Ax = b with NMSE results. As can be observed, the Block SKM outperforms the other three +approaches in the recovery task. Also, it can be seen that the PrSKM has a better recovery performance +compared to that of the RKA and the SKM. +Linear feasibility of inequalities: We utilize ORKA to make a linear equation Bx = y linear inequal- +ities system, where the number of time-varying sampling threshold sequences is m = 40, B ∈ R100×10, +x ∈ R10, and y ∈ R100. Each row of B is generated as bj ∼ N (0, I10). Also, the desired signal x +is generated as x ∈∼ N (0, I10). Each time-varying sampling threshold sequence τ(ℓ) is considered to +have the distribution τ(ℓ) ∼ N (0, I10). The performance of the RKA, SKM, PrSKM, and Block SKM +is illustrated in Fig. 1. Similar to the linear feasibility of equalities, it can be seen that the Block SKM +has a better accuracy in the recovery of the desired signal x in the one-bit polyhedron (37) compared to +the other three approaches. The NMSE results in Fig. 1 are averaged over 15 experiments. +IV. PROBABILISTIC EFFECT OF SAMPLE ABUNDANCE IN ORKA +An integral part of our proposed recovery algorithm is RKA, whose recovery error was readily given +in (17). As shown in [44], the convergence rate of RKA does not depend on the number of equations in +the system. We will show that the convergence rate of ORKA for linear feasibility is the same as RKA. +Nevertheless, when we face a non-linear constraint in our problem, as is generally the case in (1), it is +desirable are made redundant by using the opportunity of having a large number of samples; as typically +provided via one-bit sampling. In such a case, The offered convergence rate appears to be insufficient +since we must have enough number of samples to fulfill costly constraints. So, an extra term as a penalty +must be considered to present the importance of sample size in our algorithm [16]. +By adding more inequality constraints in (37) as a result of extra one-bit samples, the shrinkage of +the said polyhedron will put a downward pressure on the distance between the desired signal x⋆ and +its surrounding hyperplanes, each presenting an informative measurement that will shrink the feasibility +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +16 +space. We will show that by judicious sampling, the average of these distances will be bounded, which +may be considered to be a finite-volume space created around x⋆. Moreover, as a result of using an +overdetermined linear system of inequalities, the convergence of the RKA is guaranteed [16], [45], [46], +[48]. +A. Recovery Error Upper Bound for ORKA +As the scaled condition number is the central parameter governing the recovery error of Kaczmarz +algorithms, we will evaluate it for ORKA-created matrix P in the following, starting with σmin. The +singular values of P may be determined based on the following theorem, which thus unveils the value +of σmin. +Theorem 2. In ORKA, the rank of P is equal to that of the constraint matrix A, and its singular values +are given by +{σi} = √m {σiA} , +(40) +where {σiA} are singular values of A, and m is the number of time-varying sampling threshold sequences. +Moreover, the scaled condition number of the ORKA-created matrix P is equal to that of the constraint +matrix A: +κ (P) = κ (A) . +(41) +Proof. To obtain the singular values of P, the matrix W = P⊤P is computed as +W = P⊤P, += +� +A⊤Ω(1)...A⊤Ω(2)... · · · ...A⊤Ω(m) +� + + +Ω(1)A +· · · +Ω(2)A +... +· · · +Ω(m)A + + +, += A⊤Ω(1)Ω(1)A + · · · + A⊤Ω(m)Ω(m)A, += mA⊤IA = mA⊤A, +(42) +which means the singular values of P are {σi} = √m {σiA}. +Also, the Frobenius norm of P is obtained as +∥P∥2 +F = Tr +� +P⊤P +� +, += Tr +� +mA⊤A +� += m∥A∥2 +F. +(43) +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +17 +Consequently, the scaled condition number is independent of the number of time-varying thresholds +sequences. It follows that κ (P) = κ (A). +Corollary 1. For A = I, corresponding to the one-bit sampled signal recovery problem, the scaled +condition number of Ω = +� +Ω(1)...Ω(2)... · · · ...Ω(m) +� +is κ (Ω) = √n, which is the infimum of the scaled +condition number as shown in Theorem 1. +Note that the convergence bound (17) for ORKA is independent of the number of time-varying sampling +threshold sequences m, and it cannot take into account the effect of an increasing number of time-varying +threshold sequences. Inspired by [16], we augment (17) with a sample size-dependent penalty function +to make it useful in a sample abundance scenario: +Proposition 3 (Convergence rate of ORKA). In the proposed recovery approach, it is deemed necessary +to have a sufficient number of samples (inequalities) in order to guarantee a finite-volume feasible region +and a bounded recovery error. Once our search area is located inside FX, we may effectively deploy +(17) for the convergence rate. The convergence rate of the Kaczmarz variants is useful when we have a +linear feasibility problem. On the other hand, in (1), the main constraints are non-linear and they may +be considered to be redundant by deploying enough samples [16]. A sample size-aware convergence rate +for ORKA may be formulated as: +E {¯h (xi, x⋆)} ≤ +� +1 − 2λi − λ2 +i +κ2 (A) +�i +¯h (x0, x⋆) + Υ (m) , +(44) +where Υ(.) is an asymptotically decreasing function, such that if the number of time-varying threshold +sequences is enough for the one-bit polyhedron to fit inside FX, the sample size-dependent penalty +function Υ (m) approaches zero. +To propose an appropriate sample size-dependent penalty function, we will utilize the first theorem in +[16], which studies the possibility of creating a finite-volume space around the optimal signal. +B. Sample Size-Dependent Penalty Function via Moment Generating Functions +We investigate the convergence of ORKA through a probabilistic lens. To do so, define the distance +between the optimal point x⋆ and the j-th hyperplane presented in (37) as +dj +� +x⋆, τ(ℓ)� += +���rj ⊙ +� +ajx⋆ − τ (ℓ) +j +���� +2 +, +j ∈ +� +1, · · · , m′� +, +(45) +where rj ⊙ aj is the j-th row of P, aj is the j-th row of A, and aj = aj+n. It is easy to observe that by +generally reducing the distances between x⋆ and the constraint-associated hyperplanes, the possibility of +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +18 +capturing the optimal point is increased. For a specific sample size m′ = mn, when the volume of the +finite space around the optimal point is reduced, the average of � +dj +�˜x⋆, τ(ℓ)�� is diminished as well. +This average of distance can be written as [45]: +Tave = 1 +m′ +m′ +� +j=1 +dj +� +˜x⋆, τ(ℓ)� +. +(46) +The possibility of creating a finite-volume, and the importance of the number of samples in the recovery +performance of RKA, can be captured by the Chernoff bound as illustrated below. +Theorem 3 (See [16]). Assume the distances +� +dj +� +˜x⋆, τ(ℓ)�� +between the desired point ˜x⋆ and the +hyperplanes of the polyhedron defined in (12) are i.i.d. random variables. Then: +• The Chernoff bound of Tave is given by +Pr + + 1 +m′ +m′ +� +j=1 +dj +� +˜x⋆, τ(ℓ)� +≤ a + + ≥ 1 − inf +t≥0 +ΨT +eta , +(47) +where a is an average distance point in space at which the finite-volume space around the desired +signal is created, and ΨT is the moment generating function (MGF) of the error recovery, given as +ΨT = + +1 + t +µ(1) +dj +m′ + · · · + tκ µ(κ) +dj +κ! m′κ + R +� +m′� + + +m′ +, +(48) +with µ(κ) +dj = E +� +dκ +j +� +, and R denoting a bounded remainder associated with truncating the Taylor +series expansion of ΨT . +• ΨT is decreasing with an increasing sample size in the sample abundance scenario, leading to an +increasing lower bound in (47). +The MGF ΨT contains two parts. The first part has an increasing trend until a specific sample size +m⋆, which indicates the existence of an abundant number of samples. After that, the function has a +decreasing behavior. Therefore, ΨT with m ≥ m⋆ can be a good choice for a sample size-dependent +penalty function. Particularly, the penalty function can be chosen as ΨT −Ψ∞, where Ψ∞ = limm→∞ ΨT , +to ensure Υ(m) → 0 as m → ∞. +Since, we do not have access to the probability density function of {dj}, thus, the MGF must be +evaluated by the truncated Taylor series expansion, which may be accurately approximated by a rational +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +19 +function such as a Pad´e approximation (PA). The decreasing part of ΨT in m > m⋆ with PA is modeled +as follows [32], [38]: +Υ(m) ≍ + +1 + · · · + tκ µ(κ) +dj +κ! m′κ + + +m′ +− ΨT, += a0 + a1 +m +b0 + b1 +m +− a0 +b0 +, +(49) +where {a0, a1, b0, b1} are the PA coefficients as given by +a0 = eu � +12u2 − 24v +� +, +a1 = eu � +−3u4 + 8u3 + 12u2v − 24uv − 12v2� +, +b0 = 12u2 − 24v, +b1 = 3u4 + 8u3 − 12u2v − 24uv + 12v2, +(50) +where u = µ(1) +dj t and v = +µ(2) +dj t2 +2 +. +V. JUDICIOUS SAMPLING WITH ADAPTIVE THRESHOLDING FOR ORKA +By the spirit of using the iterative RKA, a suitable time-varying threshold can be selected in order to +enhance the recovery performance. In ORKA, we face a highly overdetermined linear feasibility problem +creating a finite-volume space. To capture the desired signal x⋆ more efficiently, the right-hand side of the +inequalities in (37), i.e. vec (R) ⊙ vec (Γ), must be determined in a way that each associated hyperplane +passes through the desired feasible region within FX. Therefore, an algorithm is proposed to ensure that +this occurs in practice. +We propose an iterative algorithm generating an adaptive sampling threshold to accurately obtain the +desired solution. To have the smaller area of the finite-volume space around the desired signal x⋆, one +can somehow choose thresholds to reduce distances between them and the desired point. To do so, we +update the time-varying threshold for ℓ ∈ {1, · · · , m} as +Axk − r(ℓ) +k ⊙ ǫ(ℓ) +k += τ(ℓ) +k+1, +(51) +where ǫ(ℓ) +k +are positive vectors in the k-th iteration of the algorithm. This updating process is based on +rj = + + + + + ++1 +ajx > τj, +−1 +ajx < τj, +(52) +where {aj} are the rows of A. The one-bit measurements +� +r(ℓ) +k +� +are updated in the way to satisfy (37) +in iteration k, i.e. the inequalities Ω(ℓ) +k Ax⋆ ≥ r(ℓ) +k +⊙ τ(ℓ) +k . The reason behind this updating is to ensure +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +20 +that each halfspace associated with a threshold in iteration k is getting closer to the optimal point in the +correct direction which means the main side of the halfspace is forced to cover the optimal solution. +Proposition 4 (ORKA with Adaptive Thresholding). Consider applying ORKA to a linear feasibility +problem Ax = y as part of the linear constraints of (1). Suppose the initial time-varying threshold +sequences are +� +τ(ℓ) +0 +� +∼ N (0, 1) (with the same length as r(ℓ)), and +� +δ(ℓ)� +are positive numbers. Also, +xk, τ(ℓ) +k , r(ℓ) +k +and ǫk denote their associated values at iteration k. The proposed sampling algorithm is +summarized as follows: +1) Find a point inside the following polyhedron with proposed accelerated Kaczmarz algorithms, i.e. +the PrSKM or the block SKM for τ = +� +τ(ℓ) +k +� +: +Pk = +� +xk | ˜ΩkAxk ⪰ bk +� +, +(53) +where bk = vec (Rk) ⊙ vec (Γk), Rk and Γk are matrices with +� +r(ℓ) +k +� +and +� +τ(ℓ) +k +� +representing +their columns, respectively. +2) Update Γk+1 as: +vec (Rk) ⊙ vec (Γk+1) = ˜ΩkAxk − ǫk +2 . +(54) +3) Compute ǫk, a block vector containing +� +ǫ(ℓ) +k +� +, as: +ǫk = ˜ΩkAxk − vec (Rk) ⊙ vec (Γk) . +(55) +4) Update Rk+1 based on (52): +r(ℓ) +k+1 = sgn +� +y − τ(ℓ) +k+1 +� +, +ℓ ∈ {1, · · · , m} . +(56) +5) Increase k by one. +6) Stop when +���τ(ℓ) +k+1 − τ(ℓ) +k +��� +2 ≤ δ(ℓ). +One can observe that by deploying this adaptive thresholding algorithm, smaller values of {dj} will +emerge which leads to their moments +� +µ(κ) +dj +� +to further diminish. Therefore, ΨT is smaller in this +scenario and a smaller number of time-varying sampling threshold sequences can be utilized in ORKA +with similar recovery performance. Additionally, non-informative sampling thresholds, which appear as +extra inequality constraints in the random time-varying sampling thresholds scenario, may be efficiently +removed by choosing the adaptive thresholds with closer hyperplanes to the desired point. +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +21 +(a) m = 2 +(b) m = 6 +(c) m = 60 +(d) m = 2 +(e) m = 6 +(f) m = 60 +Figure 2: Shrinkage of the one-bit polyhedron (37) in blue, ultimately placed within the unit ball of the +nuclear norm ∥X∥⋆ ≤ 1 shown with black cylindrical region and its red contours, when the number +constraints (samples) grows large. The arrows point to the half-space associated with each inequality +constraint. The evolution of the feasible regime is depicted with increasing samples in three cases: +(a) and (d) small sample-size regime, constraints not forming a finite-value polyhedron; (b) and (e) +medium sample-size regime, constraints forming a finite-volume polyhedron, parts of which are outside +the cylinder; (c) and (f) large sample-size regime, constraints forming a finite-volume polyhedron inside +the nuclear norm cylinder, making its constraint redundant. The optimal point representing the signal to +be recovered is shown by yellow. +VI. LOW-RANK MATRIX RECOVERY VIA ORKA +As mentioned earlier, low-rank matrix recovery is an excellent example for problems that assume the +form in (1), and that can be tackled using our methodology. In this section, at first, we will briefly +introduce the nuclear norm minimization form of the problem. Accordingly, we will apply ORKA to this +problem without considering the associated costly constraints. At the end, the recovery performance of +ORKA will be numerically evaluated considering different matrix ranks and sample sizes to investigate +the existence of a sample abundance scenario. +January 10, 2023 +DRAFT + +0.5 +0 +.50.5 +N +-0.5 +0 +0.5 +1 +-0 +X1.51 +0.5 +0 +.5 +YN +0.5- +0 +-0.5 +-0.5 +0 +0.5 +1 +-0 +X1.51 +0.5 +X0.5 +N +0 +-0.5 +1 +0.5 +0 +Y +-0.5 +-0.5SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +22 +A. Problem Formulation +The problem of the low-rank matrix recovery can be formulated as: +find +X +s.t. +A (X) = y, +rank (X) ≤ M, +X ∈ Ωc, +(57) +where X ∈ Cn1×n2 is the matrix of unknowns, y ∈ Rn is the measurement vector, and A is a linear +transformation mapping n1 × n2 into Rn. In general, Ωc can be chosen such as the set of semi-definite +matrices, symmetric matrices, upper or lower triangle matrices, Hessenberg matrices and a specific +constraint on the matrix elements ∥X∥∞ ≤ α or on its eigenvalues, i.e., λi ≤ ǫ where {λi} are eigenvalues +of X [1], [4], [53]. +The problem (57) can be rewritten as an optimization problem: +min +X +rank (X) +s.t. +A (X) = y, +X ∈ Ωc. +(58) +This problem is known to be NP-hard, whose solution is difficult to approximate [15], [56]. Recall that +the rank of X is equal to the number of nonzero singular values. In the case when the singular values +are all equal to one, the sum of the singular values is equal to the rank. When the singular values are +less than or equal to one, the sum of the singular values is a convex function that is strictly less than +the rank. Therefore, it is been popular for this problem to replace the rank function with the sum of the +singular values of X; i.e., its nuclear norm. The nuclear norm minimization alternative of the problem +is given by [2], [15], [57]: +min +X +∥X∥⋆ +s.t. +A (X) = y, +X ∈ Ωc. +(59) +In this problem, the feasible set FX is obtained as +FX = {P⋆ ∩ Ωc} , +(60) +where P⋆ is defined as follows +P⋆ = {X | ∥X∥⋆ ≤ τ} , +τ ∈ R+. +(61) +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +23 +10 +20 +30 +40 +50 +60 +-4 +-3.5 +-3 +-2.5 +-2 +-1.5 +PrSKM +Block SKM +(a) +10 +20 +30 +40 +50 +60 +-4 +-3.5 +-3 +-2.5 +-2 +-1.5 +-1 +PrSKM +Block SKM +(b) +Figure 3: Average NMSE for the Frobenius norm of error for the recovery of the matrix X associated +with different time-varying sampling threshold sequences sizes when the PrSKM and the block SKM are +utilized in ORKA: (a) rank (X) = 1, (b) rank (X) = 4. +Next, we will apply ORKA to (58) to make its costly constraints redundant by using abundant number +of time-varying sampling thresholds m. +A numerical investigation of (37) when it is achieved for the nuclear norm minimization, reveals that +by increasing the number of time-varying sampling threshold sequences m, the space formed by the +intersection of half-spaces (inequality constraints) can fully shrink to the desired signal X⋆ inside the +feasible region of (61) which is shown by the cylindrical space [15]—see Fig. 2 for an illustrative example +of this phenomenon. As can be seen in this figure, the blue lines displaying the linear feasibility form +a finite-volume space around the optimal point displayed by the yellow circle inside the cylinder (the +elliptical region) by growing the number of threshold sequences or one-bit samples. In (a)/(d), constraints +are not enough to create a finite-volume space, whereas in (b)/(e) such constraints can create the desired +finite-volume polyhedron space which, however, is not fully inside the cylinder. Lastly, in (c)/(f), the +created finite-volume space shrinks to be fully inside the cylinder. +B. Numerical Illustrations +In this section, we numerically scrutinize the capability of the ORKA in the nuclear norm minimization +problem (59) instead of (59) by the squared Frobenius norm of the error normalized by the squared +Frobenius norm of the desired matrix X⋆, defined as +NMSE ≜ +��X⋆ − ¯X +��2 +F +∥X⋆∥2 +F +. +(62) +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +24 +0 +50 +100 +150 +200 +-25 +-20 +-15 +-10 +-5 +0 +ORKA with a random threshold +ORKA with an adaptive threshold +(a) +0 +50 +100 +150 +200 +-25 +-20 +-15 +-10 +-5 +0 +ORKA with a random threshold +ORKA with an adaptive threshold +(b) +Figure 4: Comparing the average NMSE for the Frobenius norm of error for the recovery of the matrix +X using ORKA when (i) a random threshold and (ii) the adaptive sampling threshold are adopted when +the PrSKM and the block SKM are utilized in ORKA: (a) rank (X) = 1, (b) rank (X) = 4. +We solve the overdetermined one-bit polyhedron in (37) via the PrSKM and the Block SKM. To make this +happen, we obtain the one-bit polyhedron from a linear feasibility problem Ax = y, where A ∈ R200×25, +x ∈ R25 (x = vec (X) where X ∈ R5×5), and y ∈ R200. We consider the number of time-varying sampling +threshold sequences to be m ∈ {10, 20, 30, 40, 50, 60}. Each row of A is generated as aj ∼ N (0, I25). +For the desired matrix X, we generate X = KK⊤ where (i) K ∈ R5×4 is the Gaussian matrix, and (ii) +K ∈ R5×1 is the Gaussian vector. Also, each time-varying sampling threshold τ(ℓ) is considered to have +the distribution τ(ℓ) ∼ N (0, I200). Fig. 3 appears to confirm the possibility of recovering the desired +matrix X⋆ in the one-bit polyhedron (37) by ORKA. As expected, the performance of the recovery will +be significantly enhanced as the number of time-varying sampling threshold sequences grows large. Also, +similar to before, it can be seen that the Block SKM outperforms the PrSKM in the low-rank matrix +recovery problem. +To improve the recovery performance, we proposed the adaptive time-varying sampling threshold in +Section V. Fig. 4 illustrates the performance of the Block SKM in the low rank matrix recovery in the +one-bit polyhedron (37) when we have the high-dimensional input signal x ∈ R128 and A ∈ R20000×128, +with (i) a random threshold, and (ii) an adaptive time-varying threshold. As can be seen, the recovery +performance is significantly enhanced when the Block SKM is equipped with the adaptive time-varying +threshold. +VII. ONE-BIT COMPRESSED SENSING: +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +25 +10 +20 +30 +40 +50 +60 +-7 +-6 +-5 +-4 +-3 +-2 +-1 +PrSKM +Block SKM +(a) +10 +20 +30 +40 +50 +60 +-6 +-5 +-4 +-3 +-2 +-1 +PrSKM +Block SKM +(b) +Figure 5: Average NMSE for the error between the desired signal x⋆ and its recovered version ¯x for +different time-varying sampling threshold sequences sizes when the PrSKM and the block SKM are +utilized in ORKA with (a) k = 2, (b) k = 4. +FROM OPTIMIZATION TO LINEAR FEASIBILITY +Compressed sensing (CS) is an interesting and rapidly growing area of research that has attracted +considerable attention in electrical engineering, applied mathematics, statistics, and computer science [4], +[29]. In CS, a sparse high-dimensional signal is to be recovered by incomplete measurements such a +recovery may be formulated as [29]: +min +x +∥x∥1 +s.t. +Ax = y, +(63) +where A ∈ Rm×n, and m ≪ n. One of the important applications of CS emerges in the signal recovery +from a sequence of acquisitions {yi} obtained from a sparse linear transformation (wavelet transformations +are known for such a property, for instance) in the magnetic resonance imaging (MRI). The reconstruction +problem of the desired signal x⋆ is given by +min +x +∥x∥1 +s.t. +Ai (x) = yi, +i ∈ {1, · · · , n} . +(64) +In this section, we first formulate the optimization problem of the one-bit compressed sensing. Then, +by taking advantage of one-bit sampling, we increase sample size in (64) and create an associated one-bit +polyhedron. +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +26 +A. Problem Formulation +Let τ denotes the time-varying threshold vector. The one-bit samples are generated as +ri = + + + + + ++1 +a⊤ +i x ≥ τi, +−1 +a⊤ +i x < τi, +(65) +where Ai (x) = a⊤ +i x. The occurrence probability vector p for the one-bit measurement r is given as +[16], +pi = + + + + + +Φ +� +a⊤ +i x +� +for +{ri = +1}, +1 − Φ +� +a⊤ +i x +� +for +{ri = −1}, +(66) +where Φ(.) is the CDF of τ. The log-likelihood function of the sign data r is given by +Lr(µ, x) = +m +� +i=1 +� +I(ri=+1) log +� +Φ(a⊤ +i x) +� ++I(ri=−1) log +� +1 − Φ(a⊤ +i x) +�� +. +(67) +Therefore, the maximum likelihood estimation (MLE) for the one-bit compressed sensing can be written +as +min +x Lr(µ, x) + λ∥x∥1. +(68) +The alternative formulations for one-bit compressed sensing can be found in [58]. +Nevertheless, as discussed earlier, by deploying one-bit sampling, the opportunity exists to increase +the number of samples in (64). The one-bit compressed sensing is thus solely accomplished by creating +a highly-constrained one-bit polyhedron. In other words, instead of solving an optimization problem with +costly constraints, the problem may be tackled by the proposed accelerated Kaczmarz algorithms; namely, +PrSKM and the block SKM. +B. Numerical results +To examine the performance of ORKA in CS and to validate the theoretical results described in this +paper, we consider signal recovery with different number of time-varying sampling threshold sequences +m ∈ {10, 20, 30, 40, 50, 60}. Input signals x⋆ ∈ R10 are generated with sparsity orders k = 2 and +k = 4, respectively. The sparsity order k is defined as the number of non-zero elements in a vector. +Time-varying sampling thresholds and the constraint matrix A are generated as in Subsection VI-B. To +compare two proposed algorithms, the NMSE defined in (39) is utilized and the results are averaged over +15 experiments. +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +27 +0 +50 +100 +150 +200 +-25 +-20 +-15 +-10 +-5 +0 +ORKA with a random threshold +ORKA with an adaptive threshold +(a) +0 +50 +100 +150 +200 +-25 +-20 +-15 +-10 +-5 +0 +ORKA with a random threshold +ORKA with an adaptive threshold +(b) +Figure 6: Comparing the average NMSE for the desired signal x⋆ and its recovered signal using ORKA +when (i) a random threshold and (ii) the adaptive sampling threshold are adopted with (a) k = 20, (b) +k = 40 +Table I: Comparing CPU times and NMSE of ORKA and ℓ1-minimization. +Algorithm +m⋆ +CPU time (s) +NMSE +ORKA +500 +3.1240e − 04 +3.2052e − 12 +ℓ1 +100 +0.0071 +2.4572e − 11 +As can be seen in Fig. 5, by increasing the number of time-varying sampling threshold sequences, the +performance of our method is improved. Beside the possibility of increasing the number of measurements +n, the higher number of samples are available in ORKA by comparing the measurements with multiple +threshold sequences ℓ ∈ {1, · · · , m}. In other words, we have the opportunity to increase n and m +simultaneously, when the number of samples is m′ = mn. +Same as Subsection VI-B, the adaptive thresholding algorithm is applied to ORKA for the high- +dimensional input signal x ∈ R128 in order to enhance its recovery performance, whose outcome is +presented in Fig. 6. The NMSE results are reported with sparsity orders k = 20 and k = 40. +To further investigate the efficacy of ORKA in CS, we compare our proposed approach with the well- +known ℓ1-minimization approach formulated in (64) in terms of NMSE and CPU time. As presented +in Table I, ORKA outperforms ℓ1-minimization in terms of both NMSE and CPU time. The results are +obtained for x ∈ R128 when the optimal number of samples are utilized, and where m⋆ = 4k log(n/k) +and m⋆ = 500 are considered for the high-resolution method and ORKA, respectively. Herein, optimality +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +28 +of sample sizes means that the number of samples utilized by algorithms leads to their best performance, +i.e. satisfying the criterion ∥xi − x⋆∥2 +2 ≤ 5× 10−11 ∥x⋆∥2 +2. By this comparison, we remove the burden of +the large number of samples from the ℓ1-minimization to fairly compare their optimal shape deploying +incomplete measurements with that of ORKA. +It is worth pointing out that for a 64-bit ADC, m = 100 corresponds to 6400 bits of information while +ORKA solely employs 500 bits. Therefore, it appears from Table I that ORKA achieves a better accuracy +in terms of NMSE with not only fewer information bits but also a smaller computational cost. +VIII. CONCLUSION +We proposed a novel algorithm, ORKA, that takes advantage of the abundant number of samples avail- +able in one-bit sampling with time-varying thresholds to efficiently and globally solve some well-studied +problems in the form of (1); including low-rank matrix recovery and compressed sensing. Moreover, +two state-of-the-art randomized Kaczmarz algorithms are proposed to use in ORKA to find the desired +signal inside the emerging confined feasible regions, named the one-bit polyhedron, with an enhanced +convergence rate. The numerical results showcased the effectiveness of the proposed approaches for the +low-rank matrix recovery and compressed sensing problems. +REFERENCES +[1] E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM review, vol. 57, +no. 2, pp. 225–251, 2015. +[2] J. Cai, E.J. Cand`es, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM Journal on +optimization, vol. 20, no. 4, pp. 1956–1982, 2010. +[3] M. A. Davenport, Y. Plan, E. Van Den Berg, and M. Wootters, “1-bit matrix completion,” Information and Inference: A +Journal of the IMA, vol. 3, no. 3, pp. 189–223, 2014. +[4] M.A. Davenport and J. Romberg, “An overview of low-rank matrix recovery from incomplete observations,” IEEE Journal +of Selected Topics in Signal Processing, vol. 10, no. 4, pp. 608–622, 2016. +[5] E.J. Candes and Y. Plan, “Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random +measurements,” IEEE Transactions on Information Theory, vol. 57, no. 4, pp. 2342–2359, 2011. +[6] Y. Chen, A. Jalali, S. Sanghavi, and C. Caramanis, “Low-rank matrix recovery from errors and erasures,” IEEE Transactions +on Information Theory, vol. 59, no. 7, pp. 4324–4337, 2013. +[7] X. Li, L. Huang, H. So, and B. Zhao, “A survey on matrix completion: Perspective of signal processing,” arXiv preprint +arXiv:1901.10885, 2019. +[8] B. Haeffele, E. Young, and R. Vidal, “Structured low-rank matrix factorization: Optimality, algorithm, and applications to +image processing,” in International conference on machine learning. PMLR, 2014, pp. 2007–2015. +[9] F. Nie, H. Huang, and C. Ding, “Low-rank matrix recovery via efficient schatten p-norm minimization,” in Twenty-sixth +AAAI conference on artificial intelligence, 2012. +[10] David M Blei, “Probabilistic topic models,” Communications of the ACM, vol. 55, no. 4, pp. 77–84, 2012. +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +29 +[11] G. Obozinski, B. Taskar, and M. Jordan, “Joint covariate selection and joint subspace selection for multiple classification +problems,” Statistics and Computing, vol. 20, no. 2, pp. 231–252, 2010. +[12] M. Pontile, T. Evgeniou, and A. Argyriou, “Convex multi-task feature learning,” Journal of Machine Learning, vol. 10, +pp. 243–272, 2007. +[13] A. Argyriou, T. Evgeniou, and M. Pontil, “Multi-task feature learning,” Advances in neural information processing systems, +vol. 19, 2006. +[14] C. Tomasi and T. Kanade, “Shape and motion from image streams under orthography: a factorization method,” International +journal of computer vision, vol. 9, no. 2, pp. 137–154, 1992. +[15] B. Recht, W. Xu, and B. Hassibi, “Null space conditions and thresholds for rank minimization,” Mathematical programming, +vol. 127, no. 1, pp. 175–202, 2011. +[16] A. Eamaz, F. Yeganegi, and M. Soltanalian, “One-bit phase retrieval: More samples means less complexity?,” +IEEE +Transactions on Signal Processing, vol. 70, pp. 4618–4632, 2022. +[17] R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A, vol. 7, no. 3, pp. 394–411, Mar 1990. +[18] W. Kim and M. H. Hayes, “The phase retrieval problem in x-ray crystallography,” in Proceedings of the Acoustics, Speech, +and Signal Processing, 1991. ICASSP-91., 1991 International Conference, USA, 1991, ICASSP ’91, p. 1765–1768, IEEE +Computer Society. +[19] J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval +algorithms,” Appl. Opt., vol. 32, no. 10, pp. 1747–1767, Apr 1993. +[20] J. E. Krist and C. J. Burrows, “Phase-retrieval analysis of pre- and post-repair hubble space telescope images,” Appl. Opt., +vol. 34, no. 22, pp. 4951–4964, Aug 1995. +[21] A. M. Sarnik, “Phase retrieval: A practical application for the space telescope,” in Inverse Optics II, R.H.T. Bates and +A. J. Devaney, Eds. International Society for Optics and Photonics, 1985, vol. 0558, pp. 85 – 94, SPIE. +[22] R. W. Gerchberg and W. O. Saxton, +“Phase determination for image and diffraction plane pictures in the electron +microscope,” Optik, vol. 34, pp. 275–284, 1971. +[23] R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction +plane pictures,” Optik, vol. 35, pp. 227–246, 1972. +[24] J. R. Fienup, “Comments on ”the reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier +transform”,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 31, no. 3, pp. 738–739, Jun 1983. +[25] J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt., vol. 21, no. 15, pp. 2758–2769, Aug 1982. +[26] J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett., vol. 3, no. 1, pp. 27–29, +Jul 1978. +[27] E. J. Candes, T. Strohmer, and V. Voroninski, “PhaseLift: Exact and stable signal recovery from magnitude measurements +via convex programming,” Communications on Pure and Applied Mathematics, vol. 66, no. 8, pp. 1241–1274, 2013. +[28] E.J. Cand`es and X. Li, “Solving quadratic equations via PhaseLift when there are about as many equations as unknowns,” +Foundations of Computational Mathematics, vol. 14, no. 5, pp. 1017–1026, 2014. +[29] Y.C. Eldar and G. Kutyniok, Compressed sensing: theory and applications, Cambridge university press, 2012. +[30] M. Lustig, D. Donoho, J.M. Santos, and J.M. Pauly, “Compressed sensing MRI,” IEEE signal processing magazine, vol. +25, no. 2, pp. 72–82, 2008. +[31] J.A. Fessler, “Model-based image reconstruction for MRI,” IEEE signal processing magazine, vol. 27, no. 4, pp. 81–89, +2010. +[32] A. Eamaz, F. Yeganegi, and M. Soltanalian, “Modified arcsine law for one-bit sampled stationary signals with time-varying +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +30 +thresholds,” in ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). +IEEE, 2021, pp. 5459–5463. +[33] Texas Instruments, “Analog embedded processing,” Active Filter Design Techniques, Literature Number SLOA088 Texas +Instruments, excerpted from OP-Amps for Everyone, Literature Number SLOD006A. +[34] A. Mezghani and A. L. Swindlehurst, “Blind estimation of sparse broadband massive MIMO channels with ideal and +one-bit ADCs,” IEEE Transactions on Signal Processing, vol. 66, no. 11, pp. 2972–2983, 2018. +[35] S. Sedighi, B. Shankar, M. Soltanalian, and B. Ottersten, “One-bit DoA estimation via sparse linear arrays,” in IEEE +International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2020, pp. 9135–9139. +[36] C. Qian and J. Li, “ADMM for harmonic retrieval from one-bit sampling with time-varying thresholds,” in 2017 IEEE +International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2017, pp. 3699–3703. +[37] C. Gianelli, Luzhou Xu, Jian Li, and Petre Stoica, “One-bit compressive sampling with time-varying thresholds for sparse +parameter estimation,” in 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM). IEEE, 2016, +pp. 1–5. +[38] Arian Eamaz, Farhang Yeganegi, and Mojtaba Soltanalian, “Covariance recovery for one-bit sampled non-stationary signals +with time-varying sampling thresholds,” IEEE Transactions on Signal Processing, vol. 70, pp. 5222–5236, 2022. +[39] P. Wang, J. Li, M. Pajovic, P. T. Boufounos, and Philip V Orlik, “On angular-domain channel estimation for one-bit +massive MIMO systems with fixed and time-varying thresholds,” in 51st Asilomar Conference on Signals, Systems, and +Computers. IEEE, 2017, pp. 1056–1060. +[40] F. Xi, Y. Xiang, S. Chen, and A. Nehorai, “Gridless parameter estimation for one-bit MIMO radar with time-varying +thresholds,” IEEE Transactions on Signal Processing, vol. 68, pp. 1048–1063, 2020. +[41] S. Kaczmarz, “Angen¨aherte aufl¨osung von systemen linearer gleichungen (english translation by Jason Stockmann): Bulletin +international de l’acad´emie polonaise des sciences et des lettres,” 1937. +[42] H.G. Feichtinger and T. Strohmer, “A Kaczmarz-based approach to nonperiodic sampling on unions of rectangular lattices,” +in SampTA’95: 1995 Workshop on Sampling Theory and Applications, 1995, pp. 32–37. +[43] M. Sezan and H. Stark, “Applications of convex projection theory to image recovery in tomography and related areas,” +Image Recovery: Theory and Application, pp. 155–270, 1987. +[44] T. Strohmer and R. Vershynin, “A randomized Kaczmarz algorithm with exponential convergence,” Journal of Fourier +Analysis and Applications, vol. 15, no. 2, pp. 262–278, 2009. +[45] D. Leventhal and A. S. Lewis, +“Randomized methods for linear constraints: convergence rates and conditioning,” +Mathematics of Operations Research, vol. 35, no. 3, pp. 641–654, 2010. +[46] D. Needell and J.A. Tropp, “Paved with good intentions: Analysis of a randomized block Kaczmarz method,” Linear +Algebra and its Applications, vol. 441, pp. 199–221, 2014. +[47] J. Briskman and D. Needell, “Block Kaczmarz method with inequalities,” Journal of Mathematical Imaging and Vision, +vol. 52, no. 3, pp. 385–396, 2015. +[48] J. De Loera, J. Haddock, and D. Needell, “A sampling Kaczmarz–Motzkin algorithm for linear feasibility,” SIAM Journal +on Scientific Computing, vol. 39, no. 5, pp. S66–S87, 2017. +[49] I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity +constraint,” Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical +Sciences, vol. 57, no. 11, pp. 1413–1457, 2004. +[50] P.L. Combettes and V. Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale modeling & simulation, +vol. 4, no. 4, pp. 1168–1200, 2005. +January 10, 2023 +DRAFT + +SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 +31 +[51] L. Dai, M. Soltanalian, and K. Pelckmans, “On the randomized Kaczmarz algorithm,” IEEE Signal Processing Letters, +vol. 21, no. 3, pp. 330–333, 2013. +[52] M. Sarowar Morshed and M. Saiful Islam, +“Sampling Kaczmarz Motzkin method for linear feasibility problems: +Generalization & acceleration,” arXiv e-prints, pp. arXiv–2002, 2020. +[53] C. F. Van Loan and G. Golub, “Matrix computations (Johns Hopkins studies in mathematical sciences),” 1996. +[54] A. Bhandari, F. Krahmer, and R. Raskar, “On unlimited sampling and reconstruction,” IEEE Transactions on Signal +Processing, vol. 69, pp. 3827–3839, 2020. +[55] T. Elfving, “Block-iterative methods for consistent and inconsistent linear equations,” Numerische Mathematik, vol. 35, +no. 1, pp. 1–12, 1980. +[56] R. Meka, P. Jain, C. Caramanis, and I.S. Dhillon, “Rank minimization via online learning,” in Proceedings of the 25th +International Conference on Machine learning, 2008, pp. 656–663. +[57] B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm +minimization,” SIAM review, vol. 52, no. 3, pp. 471–501, 2010. +[58] S. Khobahi and M. Soltanalian, “Model-based deep learning for one-bit compressive sensing,” IEEE Transactions on +Signal Processing, vol. 68, pp. 5292–5307, 2020. +January 10, 2023 +DRAFT + diff --git a/ndE1T4oBgHgl3EQf1QXp/content/tmp_files/load_file.txt b/ndE1T4oBgHgl3EQf1QXp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..36fd92bc285673948d3ce4e873f580d949f42388 --- /dev/null +++ b/ndE1T4oBgHgl3EQf1QXp/content/tmp_files/load_file.txt @@ -0,0 +1,955 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf,len=954 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='03467v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='SP] 8 Dec 2022 SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2022 1 ORKA: Accelerated Kaczmarz Algorithms for Signal Recovery from One-Bit Samples Arian Eamaz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Student Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Farhang Yeganegi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Deanna Needell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' and Mojtaba Soltanalian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Senior Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' IEEE Abstract One-bit quantization with time-varying sampling thresholds has recently found significant utilization potential in statistical signal processing applications due to its relatively low power consumption and low implementation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In addition to such advantages, an attractive feature of one-bit analog-to- digital converters (ADCs) is their superior sampling rates as compared to their conventional multi- bit counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This characteristic endows one-bit signal processing frameworks with what we refer to as sample abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' On the other hand, many signal recovery and optimization problems are formulated as (possibly non-convex) quadratic programs with linear feasibility constraints in the one- bit sampling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' We demonstrate, with a particular focus on the nuclear norm minimization, that the sample abundance paradigm allows for the transformation of such quadratic problems to merely a linear feasibility problem by forming a large-scale overdetermined linear system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' thus removing the need for costly optimization constraints and objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To make this achievable, we propose enhanced randomized Kaczmarz algorithms to tackle these highly overdetermined feasibility problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Several numerical results are presented to illustrate the effectiveness of the proposed methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Index Terms Convex-relaxed problems, nuclear norm minimization, one-bit quantization, one-bit ADCs, random- ized Kaczmarz algorithm, statistical signal processing, time-varying sampling thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This work was supported in part by National Science Foundation Grant CCF-1704401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The first two authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Eamaz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Yeganegi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Soltanalian are with the Department of Electrical and Computer Engineering, University of Illinois Chicago, Chicago, IL 60607, USA (Corresponding author: Arian Eamaz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Needell is with the Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095 USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' INTRODUCTION We consider an optimization problem of the form min X f(X) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' A (X) = y, X ∈ Ωc, (1) where f(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=') is a cost function, X ∈ Cn1×n2 is the matrix of unknowns, y ∈ Rn is the measurement vector, and A is a linear transformation mapping Cn1×n2 into Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This problem has been used as a relaxed version of some well-known NP-hard problems, and emerging in wide variety of statistical signal processing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Although many problems can be expressed in the form in (1), the applications we will focus on in this paper include some specific problems of interest in statistical signal processing, which can take advantage of low-resolution (and particularly one bit) sampling and processing: Low-rank matrix recovery: The task of recovering a low-rank matrix from its linear measurements plays a central role in computational science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The problem occurs in many areas of applied math- ematics such as signal processing [1]–[7], machine learning [8]–[13], and computer vision [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In this scenario, the cost function of (1), f(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' ), is typically to be the nuclear norm or the Frobenius norm, and the constraint set Ωc would be a amplitude restriction on the elements of matrix X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' see [2], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Phase retrieval: Phase retrieval has received a great deal of interest as it aims to recover an unknown signal solely from phaseless measurements that depend on the signal through a linear observation, commanding numerous applications in applied physics and statistical signal processing communities over the past decades [16]–[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To have a convex formulation, the phase retrieval has been relaxed into semi-definite programs where the problem boils down to a trace minimization while considering the positive semi-definite constraint [1], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Compressed sensing: Compressed sensing (CS) offers a framework for simultaneous sensing and compression of finite dimensional vectors, that relies on linear dimensionality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Through a CS formulation, sparse signals may be recovered from highly incomplete measurements [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The problem (1) can be adopted in the CS content when f (X) = ∥vec (X)∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Magnetic resonance imaging: Reconstructing magnetic resonance images commonly involves col- lecting a series of frames of data in which a radio frequency excitation produces new transverse magnetization, which is then sampled along a particular trajectory in k-sparse representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Due to meet various physical and physiological constraints, most MRI methods utilize a sequence of January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 3 acquisitions, each of which partially samples the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Let the acquired sequence of measurements be represented by yi, where i is the sequence index, and {Ai (X)} denote a linear transformation, chosen in a manner that promotes sparsity in the range space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In this example, the cost function can be considered to be the ℓ1-norm, and the sequence of acquisitions are used as linear constraints in (1) [30], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Sampling the signals of interest at high data rates with high-resolution ADCs would dramatically increase the overall manufacturing cost and power consumption of such ADCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In multi-bit sampling scenarios, a very large number of quantization levels is necessary in order to represent the original continuous signal in with high accuracy, which in turn leads to a considerable reduction in sampling rate [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This attribute of multi-bit sampling is the key reason for the general emergence of underdetermined systems n1n2 ≥ n in (1) [1], [16], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' An alternative solution to such challenges is to deploy one-bit quantization which is an extreme sampling scenario, where the signals are merely compared with given threshold levels at the ADCs, producing sign data (±1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This enables signal processing equipments to sample at a very high rate, with a considerably lower cost and energy consumption, compared to their counterparts which employ multi-bit ADCs [32]–[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In traditional one-bit sampling schemes, the signal recovery is accomplished by comparing the signal with a fixed threshold, usually zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This creates some difficulties in estimating signal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In contrast, recent works have employed time-varying sampling thresholds, which exhibit enhanced recovery performance for the signal parameters [16], [32], [36]–[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In this paper, we consider the deployment of one-bit sampling with time-varying thresholds,leading to an increased sample size and a highly overdetermined system as a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The proposed One-bit aided Randomized Kaczmarz Algorithm, which we refer to as ORKA, can find the desired signal X⋆ in (1) by (i) generating abundant one-bit measurements, in order to define a large scale overdetermined system where a finite volume feasible set is created for (1), and (ii) solving this obtained linear feasibility problem by leveraging one of the efficient solver families of overdetermined systems, Kaczmarz algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The Kaczmarz method [41] is an iterative projection algorithm for solving linear systems of equations and inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' It is usually applied to highly overdetermined systems because of its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Each iteration projects onto the solution space corresponding to one row in the linear system, in a sequential regimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The method has been applied to various applications in image reconstruction, digital signal processing, and computer tomography [16], [42], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Many variants of this iterative method and their convergence rates have been proposed and studied in recent decades for both consistent and inconsistent systems including the randomized Kaczmarz algorithm, the randomized block Kaczmarz algorithm and most recently, the sampling Kaczmarz-Motzkin method [44]–[48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Contributions of the Paper In [16], we showed that the sheer number of measurements acquired in one bit sampling facilitates recovering the signal of interest in a less costly manner by making costly constraints such as semidef- initeness and rank redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Then, a simple randomized Kaczmarz algorithm (RKA) was utilized to solve the obtained linear feasibility problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This idea is generalized in this paper to (1) where we generate the abundant samples and eventually introduce a one-bit linear feasibility region named the one-bit polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In other words, by using this technique, we make (1) a large-scale overdetermined system which is the desired application setting for Kaczmarz algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To solve our highly overdetermined system, we propose two novel variants of RKA which will be compared with the existing RKA variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Furthermore, an algorithm is proposed based on our model to adaptively evaluate the time-varying sampling thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The convergence rate of the proposed algorithm is investigated based on the moments generating function of recovery errors and the scaled condition number of the constraint matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Finally, the performance of the proposed method is examined in nuclear norm minimization-based problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Organization of the Paper Section II is dedicated to a review of proximal methods which have been utilized to tackle (1) by projecting the final solution on the desired feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In Section III, we will introduce our algorithm to solve (1), ORKA, which tackles the problem as a large-scale overdetermined system and finds the optimal point in the one-bit polyhedron by an accelerated Kaczmarz approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Moreover, two new variants of the Kaczmarz algorithms are proposed that enhance the convergence rate and the computational complexity of these solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To investigate the convergence rate of ORKA, at first, we will introduce a penalty function in Section IV based on the Chernoff bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Section V discusses an iterative algorithm to achieve optimized time-varying sampling threshold sequences which benefit the signal recovery process with enhanced accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' As a representative application, in Section VI, ORKA and other proposed algorithms will be applied in the context of low-rank matrix recovery in the form of a nuclear norm minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Finally, Section IX concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Notation: We use bold lowercase letters for vectors and bold uppercase letters for matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' C and R represent the set of complex and real numbers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (·)⊤ and (·)H denote the vector/matrix transpose, and the Hermitian transpose, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' IN ∈ RN×N is the identity matrix of size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Tr(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=') denotes the trace of the matrix argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' ⟨B1, B2⟩ = Tr(BH 1 B2) is the standard inner product between two matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The nuclear norm of a matrix B ∈ CN1×N2 is denoted ∥B∥⋆ = �M i=1 σi where M and {σi} are the rank and singular values of B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The Frobenius norm of a matrix B is defined January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 5 as ∥B∥F= ��N1 r=1 �N2 s=1 |brs|2 where {brs} are elements of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The ℓk-norm of a vector b is defined as ∥b∥k k= � i|b|k i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The Hadamard (element-wise) product of two matrices B1 and B2 is denoted as B1⊙B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Additionally, the Kronecker product is denoted as B1 ⊗B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The vectorized form of a matrix B is written as vec(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1s is the s-dimensional all-one vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Given a scalar x, we define (x)+ as max {x, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' f ≍ g means f and g are asymptotically equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Diag {b} denotes a diagonal matrix with {bi} as its diagonal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' PROJECTIONS ON CONVEX SETS: DEALING WITH COSTLY CONSTRAINTS To tackle (1), many non-convex and local optimization algorithms have been developed over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Nevertheless, in recent decades, convex programming formulations via relaxation have come to the fore to approximate global solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In the convex framework, various iterative methods have been proposed to tackle the problem with a Lagrangian formulation such as Uzawa’s algorithm and the proximal forward- backward splitting method (PFBS) [2], [49], [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Moreover, to keep the problem solution inside the constraint set Ωc, the orthogonal projection PΩc is applied to solutions in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This process is briefly explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The Lagrangian for (1) is written as [2], L (X, λ) = f (X) + ⟨λ, y − A (X)⟩ , (2) where λ ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Uzawa’s algorithm aims to find a saddle point (X⋆, λ⋆), where supλ infX L (X, λ) = infX supλ L (X, λ), with the iterative procedure: \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 L � Xk, λk−1� = minX L � X, λk−1� , λk = PΩc � λk−1 + αk � y − A � Xk��� , (3) where αk is the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This iterative steps can be rewritten as \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Xk = Proxf � A⋆ � λk−1�� , λk = PΩc � λk−1 + αk � y − A � Xk��� , (4) where Proxf is the proximal operator minimizing the Lagrangian function, and A⋆ is the adjoint of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Since every linear equation can be reformulated in standard form, we recast A (X) = y as Ax = y, where A ∈ Cn×n1n2 is a matrix version of the operator A, and x = vec (X) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The optimization problem (1) is equivalently given by [1], [2] min X g (X) = 1 2 ∥y − A vec (X)∥2 2 + λf(X) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' X ∈ Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (5) January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 6 To solve this problem, instead of using proximal methods, a projected gradient method such as Nesterov iterative approach may be utilized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=', Xk = PΩc � Xk−1 − αk∇g � Xk−1��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Famous examples for Proxf and PΩc, are the singular value thresholding operator (SVT) and the semi- definite orthogonal projector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' SVT is useful when f (X) = ∥X∥⋆, mathematically defined as [2]: Dδ = U Diag � (σk − δ)+� V⊤, (6) where U and V are unitary matrices from singular value decomposition (SVD), and {σk} are the singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Furthermore, the semi-definite projector emerges in semi-definite programming where the convex constraint set is a positive semi-definite (PSD) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' It compares eigenvalues of the solution in each iteration with zero or a fixed threshold [1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=', PΩc = U Diag � (λk − δ)+� U⊤, (7) where U is the unitary matrix coming from the Schur decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In the case of both operators, the approximate solution should be projected onto a feasible convex set at each iteration via recovering all singular values and eigenvalues and comparing their smaller elements with a threshold, which is quite expensive [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' An interesting alternative to enforcing the feasible set FX = {Proxf ∩ Ωc} in (1) emerges when one increases the number of samples n, and solves the overdetermined linear system of equations with n ≥ n1n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In this sample abundance regimen, the linear constraint A (X) = y may actually yield the optimum inside FX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' As a result of increasing the number of samples, it is possible that the intersection of these hyperplanes will achieve the optimal point without the need to consider costly constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' However, this idea may face practical limitations in the case of multi-bit quantization systems since ADCs capable of ultra-high rate sampling are difficult and expensive to produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Moreover, one cannot necessarily expect these constraints to intersect with FX in such a way to form a finite-volume space before the optimum is obtained [16], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In the next section, by deploying the idea of one-bit sampling with time-varying thresholds, linear equality constraints are superseded by a massive array of linear inequalities in forming the feasible polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Therefore, by increasing the number of samples, a finite-volume space may be created inside FX with shrinking size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' making projections on Ωc redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' From a practical point of view, one-bit sampling is done efficiently at a very high rate with a significantly lower cost compared to its high- resolution counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' It has been examined in [16] that even though only partial information is made available to one-bit signal processing algorithms, they can achieve acceptable recovery performance with less complexity compared to the high-resolution scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Thus, it is both practical and necessary to study January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 7 the ground-breaking opportunities that emerge in the context of the wide array of problems formulated as (1) due to the availability of a large number of one-bit samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' PROPOSED ALGORITHM In this section, at first we begin by presenting a summarized review of randomized Kaczmarz algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Then, we propose a novel Kaczmarz method variant formulated based on the sampling Kaczmarz- Motzkin algorithm (SKM) and a preconditioning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' One-bit sampling via time-varying thresholds will be combined with the proposed randomized Kaczmarz method to create highly overdetermined linear inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This paves the way for the recovery of the desired signal X⋆ in (1) without solving the original optimization problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' merely by tacking accounts of its linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' We name our algorithm One-bit aided Randomized Kaczmarz Algorithm (ORKA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Due to the block structure of the linear feasibility in ORKA, we will propose a block-based Kaczmarz algorithm accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Randomized Kaczmarz Algorithm (RKA) The randomized Kaczmarz algorithm (RKA) is a sub-conjugate gradient method to solve a linear feasibility problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='e, Cx ⪯ b where C is a m × n matrix with m > n [44], [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Conjugate-gradient methods immediately turn the mentioned inequality to an equality in the following form: (Cx − b)+ = 0, (8) and then, approach the solution by the same process as used for systems of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Without any loss of generality, consider (8) to be a polyhedron: \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 cjx ≤ bj (j ∈ I≤) , cjx = bj (j ∈ I=) , (9) where the disjoint index sets I≤ and I= partition our sample index set J , and {cj} denote the rows of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Based on this problem, the projection coefficient βi of the RKA is defined as [45], [47], [51]: βi = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (cjxi − bj)+ (j ∈ I≤) , cjxi − bj (j ∈ I=) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (10) Also, the unknown column vector x is iteratively updated as xi+1 = xi − βi ∥cj∥2 2 cH j , (11) where, at each iteration i, the index j is chosen independently at random from the set J , following the distribution P{j = k} = ∥ck∥2 2 ∥C∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (12) January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2022 8 If the system (9) is consistent and its feasible region is nonempty,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' RKA converges linearly in expectation [44],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [45]: E {¯h (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' x⋆)} ≤ qi ¯h (x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' x⋆) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (13) where ¯h (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' x⋆) = ∥xi − x⋆∥2 2 is the distance function between two points in the space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' x⋆ is a desired point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' i is the number of required iterations for RKA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' and q ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1) is given as q = 1 − 1 κ2 (C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (14) with κ (C) = ∥C∥F∥C†∥2 denoting the scaled condition number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Sampling Kaczmarz-Motzkin Algorithm (SKM) The SKM combines the ideas of both the RKA and the Motzkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Its generalized convergence theorem, and a validation of feasibility, which has been formulated based on the convergence analysis of RKA and sampling Motzkin method for solving linear feasibility problem have been fully explored in [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The central contribution of SKM lies in its innovative way of projection plane selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The hyperplane selection is done as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' At iteration i the SKM algorithm selects a collection of γ (denoted by the set τi), uniformly at random out of m rows of the constraint matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Then, out of these γ rows, the row with maximum positive residual is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Finally, the solution is updated as [48], [52]: xi+1 = xi − λi βi ∥cj⋆∥2 2 cH j⋆, (15) where j⋆ = argmax � (cjxi − bj)+� , j ∈ τi, and λi is a relaxation parameter which for consistent systems must satisfy [44], 0 ≤ lim i→∞ inf λi ≤ lim i→∞ sup λi < 2, (16) to ensure convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The convergence bound for SKM is given by E {¯h (xi, x⋆)} ≤ � 1 − 2λi − λ2 i κ2 (C) �i ¯h (x0, x⋆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (17) In the case where the constraint matrix is normalized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' ∥cj∥2 2= 1, si is the number of satisfied constraints after iteration i, and Li = max {m − si, m − γ}, for the (i + 1)th iteration we have [48], E {¯h (xi, x⋆)} ≤ � 1 − σ2 min � 2λi − λ2 i � Vi �i ¯h (x0, x⋆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (18) This recovery error bound is tighter than (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 9 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Our Contribution: Preconditioned SKM (PrSKM) According to the convergence rate formula of RKA, if we can reduce the value of the scaled condition number, the convergence is accelerated, and the upper bound of the recovery error E � ∥xi − x⋆∥2 2 � decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Moreover, by having a lower value of q, a lower number of iterations is required to achieve a specific recovery error bound, usually considered to be the algorithm’s termination criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Consequently, let I is the number of iterations, the computational cost of RKA which behaves as O (In), is diminished as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To make this happen, one can start from reducing q = 1 − 1 κ2(C) which occurs when the scaled condition number κ (C) is diminished;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' a condition that can be satisfied by considering the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The infimum scaled condition number of a matrix C ∈ Rm×n is given by inf C κ (C) = √n, (19) which is achieved if and only if C is of the form C = αU, where U is an orthonormal matrix and α ∈ R is a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The condition number of the matrix C is defined as ̺ (C) = σmax σmin , where σmax and σmin are its minimum and maximum singular values, respectively [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The scaled condition number can be written as κ (C) = ∥C∥F σmin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Therefore, the scaled condition number has the following relation with ̺ (C): κ (C) = ∥C∥F σmax ̺ (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (20) Furthermore, the condition number can be considered to be an upper bound for the scaled condition number as well based on the readily-known inequality relation between norm-2 and the Frobenius norm [53]: ∥C∥F ≤ √n∥C∥2, ∥C∥F σmin ≤ √n∥C∥2 σmin , (21) or equivalently, κ (C) ≤ √n̺ (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (22) Thus, lowering ̺ (C) generally leads to a decreasing scaled condition number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Additionally, the lowest possible value for ̺ is 1 which is achieved for scaled unitary matrices U as if we let S = αU, and O = S⊤S = α2In, then, σmin = σmax = α, and ̺ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Vice versa, if ̺ = 1, it means σmin = σmax which leads to a diagonal matrix O = α2In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' It is straightforward to verify that the decomposition of O results in an S that is a scaled-version of an orthonormal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' As a result, the lowest achievable January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 10 upper bound for the scaled condition number is obtained as κ (C) ≤ √n, and according to (20), κ (C) = α∥U∥F α = √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Accordingly, it would be enough to make our matrix C unitary by a process which is referred to as the preconditioning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In preconditioning, the linear feasibility is rewritten as CMz ⪯ b, (23) where M is the preconditioner and x = Mz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The straightforward way to approach this task is to use QR decomposition where the constraint matrix is decomposed as C = QcRc, with unitary Qc ∈ Rm×n, and Rc ∈ Rn×n is an upper triangular matrix, leading to Qc = CR−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (24) Thus, the good choice for the preconditioner is M = R−1 c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To find the desired point z⋆, the SKM is selected in order to apply to the linear feasibility (23), then the desired signal x⋆ is obtained from x⋆ = R−1 c z⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' We refer to this method Preconditioned SKM (PrSKM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Proposition 1 (PrSKM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The proposed algorithm, PrSKM, can be summarized as follows: 1) Calculate the QR decomposition of the constraint matrix C to obtain the preconditioner M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2) Using the change of variables, x = Mz, obtain Hz ⪯ b, where H = CM and M = R−1 c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3) Choose a sample set of γ constraints (denoted as τi) uniformly at random from the rows of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4) From these γ constraints, choose j⋆ = argmax � (hjzi − bj)+� , j ∈ τi where hj is the jth row of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 5) Update the solution via the iterations zi+1 = zi − λi (hj⋆zi−bj⋆)+ ∥hj⋆∥2 2 hH j⋆ until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 6) Recover the desired signal from the final solution of SKM as x⋆ = Mz⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The scaled condition number of PrSKM is obtained as κ (H) = √n, which implies q = n−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' One-Bit Polyhedron Consider a bandlimited signal y ∈ L2 to be represented by its samples via the standard sampling formula [54], 0 < T ⩽ π Ω, y(t) = k=+∞ � k=−∞ y(kT) sinc � t T − k � , (25) where 1/T is the sampling rate and sinc(t) = sin(πt) (πt) is an ideal low-pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Suppose yk = y(kT) denotes the uniform samples of y(t) with the sampling rate T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Let rk denote the quantized version of y[k] with the formulation rk = Q(yk), (26) January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 11 Algorithm 1 Block SKM Algorithm Input: Matrix B ∈ Rmn×d where B = � B⊤ 1 · · B⊤ m �⊤ , right-hand side b with dimension mn, initial value of x0 with dimension n, convergence tolerance ǫ > 0, and relaxation parameter λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Output: An estimate x for the solution to the linear feasibility problem Bx ⪯ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1: Initiate the following loop by setting i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2: while ��(Bxi − b)+�� 2 ≤ ǫ do 3: Choose a block Bj uniformly at random with the probability P{j = k} = ∥Bk∥2 F ∥B∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4: Let e′ denote the sorted version of e from emax (the maximum element of e) to emin (the minimum element of e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 5: Select the first k′ < d element of e′ and construct the problem B′ jx ⪯ b′ j, where B′ j ∈ Rk′×d and b′ j ∈ Rk′×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 6: Compute the Moore-Penrose of B′ j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=', B′† j ← B′⊤ j � B′ jB′⊤ j �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 7: Update the solution xi+1 as: xi+1 ← xi − λiB′† j � B′ jx − b′ j �+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 8: Increase i by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 9: end while where Q denotes the quantization effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Consider a non-zero time-varying Gaussian threshold τ = [τk] with the distribution τ ∼ N (d = 1d, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In the case of one-bit quantization with such time-varying sampling thresholds, (26) is simply written as rk = sgn (yk − τk) , (27) where sgn(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=') is the sign function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The information gathered through the one-bit sampling with time- varying thresholds presented here may be formulated in terms of an overdetermined linear system of inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (27), rk = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 +1 yk > τk, −1 yk < τk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (28) Therefore, one can formulate the geometric location of the signal as rk (yk − τk) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (29) January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 12 Let y = [yk] and r = [rk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Then, the vectorized representation of (29) is r ⊙ (y − τ) ≥ 0, (30) or equivalently Ωy ⪰ r ⊙ τ, (31) where Ω ≜ diag {r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Suppose y, τ ∈ Rn, and that τ(ℓ) denotes the time-varying sampling threshold sequence in ℓ-th experiment where ℓ ∈ L = {1, · · · , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' According to (31), we have \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 Ω(1)y ⪰ r(1) ⊙ τ(1) r(1) = sgn � y − τ(1)� , Ω(2)y ⪰ r(2) ⊙ τ(2) r(2) = sgn � y − τ(2)� , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Ω(m)y ⪰ r(m) ⊙ τ(m) r(m) = sgn � y − τ(m)� , (32) where Ω(ℓ) = diag � r(ℓ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (32), we have m linear system of inequalities which can also be merged in one inequality as ˜Ωy ⪰ vec (R) ⊙ vec (Γ) , (33) where R and Γ are matrices with � r(ℓ)� and � τ(ℓ)� representing their columns, respectively, and ˜Ω is ˜Ω = � Ω(1) · · Ω(m) �⊤ , ˜Ω ∈ Rmn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (34) Assuming a large number of samples which is a common situation in one-bit sampling scenarios, hereafter, we consider (33) as an overdetermined linear system of inequalities associated with the sampling scheme presented in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The inequality (33) can be recast by a polyhedron as Py = � y | ˜Ωy ⪰ vec (R) ⊙ vec (Γ) � , (35) which we refer to as the one-bit polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' ORKA: Towards Circumventing Costly Constraints If one applies one-bit sampling with time-varying sampling thresholds to the measurement y ∈ Rn from (1) following the process defined in Subsection III-D, the arising inequality system is simply given by ˜ΩAx ⪰ vec (R) ⊙ vec (Γ) , (36) where Ax = y, and x = vec (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Consequently, the one-bit polyhedron for this problem is obtained as Px = {x | Px ⪰ vec (R) ⊙ vec (Γ)} , (37) January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 13 Algorithm 2 Architecture of ORKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Input: The measurement vector y obtained as A (X) = y from (1), m sequences of time-varying sampling thresholds generated as � τ(ℓ) ∼ N (0, I) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' ℓ ∈ L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Output: Recovered optimal signal x⋆ = vec (X⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1: Apply one-bit sampling on y and generate sequences of one-bit measurements from: r(ℓ) ← sgn � x − τ(ℓ)� , ℓ ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2: Construct a linear feasibility region from the one-bit sampled data as: Ω(ℓ)y ⪰ r(ℓ) ⊙ τ(ℓ), ℓ ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3: Define a highly-overdetermined system, the one-bit polyhedron, based on obtained inequalities: Px = {x | Px ⪰ vec (R) ⊙ vec (Γ)} , where R and Γ are matrices with � r(ℓ)� and � τ(ℓ)� representing their columns, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4: Employ RKA variants (PrSKM or Block SKM) to recover X⋆ within the one-bit polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' where P = ˜ΩA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' By taking advantage of one-bit sampling, in the asymptotic scenario of with sample abundance, the space restricted by the one-bit polyhedron Px, shrinks to become contained inside the feasible set FX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Note that this shrinking space always contains the global minima, with a volume that is diminished with an increased sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' As a result, instead of using proximal operators and orthogonal projectors, it is enough to find the desired signal x⋆ in (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To do so, one can use the PrSKM algorithm proposed in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' It is worth noting that the PrSKM is a row-based algorithm, where at each iteration, the row index is chosen independently at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' However, the matrix P in (37) has a block structure with the following formulation P = � A⊤Ω(1) · · A⊤Ω(m) �⊤ , P ∈ Rmn×d, (38) where d = n1n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Therefore, it is useful to investigate the block-based RKA methods to find the desired signal in Px for further efficiency enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Our proposed algorithm, Block SKM, is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Proposition 2 (Block SKM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' We have a linear feasibility problem Bx ⪯ b where B = � B⊤ 1 · · B⊤ m �⊤ , and b = � b⊤ 1 · · b⊤ m �⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The proposed algorithm for feasible signal recovery, Block SKM, can be summarized as follows: January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 14 0 100 200 300 Iteration 30 25 20 15 10 5 0 RKA SKM PrSKM Block SKM (a) 0 50 100 150 200 Iteration 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 0 RKA SKM PrSKM Block SKM (b) Figure 1: Comparing the NMSE recovery performance of the two proposed Kaczmarz algorithms, namely the PrSKM and the block SKM, with that of SKM and RKA for: (a) a linear equation system, (b) a linear inequality system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1) Choose a block Bj uniformly at random with the probability P{j = k} = ∥Bk∥2 F ∥B∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2) Compute e = Bjx − bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3) Let e′ denote the sorted version of e from emax (the maximum element of e) to emin (the minimum element of e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This step is inspired by the idea of the Motzkin sampling, presented in Section III-B, to have an accelerated convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4) Select the first k′ < d element of e′ and construct the sub-problem B′ jx ⪯ b′ j, where B′ j ∈ Rk′×d and b′ j ∈ Rk′×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The reason behind choosing k′ < d is due to the computation of � B′ jB′⊤ j �−1 in the next step (Step 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' For k′ > d, the matrix B′ jB′⊤ j is rank-deficient and its inverse is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 5) Compute the Moore-Penrose of B′ j, B′† j = B′⊤ j � B′ jB′⊤ j �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 6) Update the solution xi+1 = xi − λiB′† j � B′ jx − b′ j �+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This update process is inspired from the randomized block Kaczmarz method [46], [55] which takes advantage of the efficient matrix-vector multiplication, thus giving the method a significant reduction in computational cost [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The steps of the block SKM and ORKA are summarized in Algorithm 1 and Algorithm 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To examine the performance of the block SKM, we compare it to the PrSKM, SKM and RKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Comparing RKA, SKM, PrSKM and Block SKM In this section, we numerically compare the RKA, SKM, PrSKM, and Block SKM in linear systems of equalities as well as those formed by inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 15 Linear feasibility of equalities: Herein, we consider a block linear system of equalities Ax = b, where A = � A⊤ 1 · · A⊤ 100 �⊤ , Ai ∈ R10×10, x ∈ R10, and b ∈ R1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Each row of Ai is generated as ai j ∼ N (0, I10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Also, the unknown signal x is generated as x ∼ N (0, I10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The normalized mean square error (NMSE) is defined as NMSE ≜ ∥x⋆ − ¯x∥2 2 ∥x⋆∥2 2 , (39) where x⋆ and ¯x denote the true discretized signal and its recovered version, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1 illustrates the performance of RKA, SKM, PrSKM, and Block SKM in the recovery of x from the system Ax = b with NMSE results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' As can be observed, the Block SKM outperforms the other three approaches in the recovery task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Also, it can be seen that the PrSKM has a better recovery performance compared to that of the RKA and the SKM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Linear feasibility of inequalities: We utilize ORKA to make a linear equation Bx = y linear inequal- ities system, where the number of time-varying sampling threshold sequences is m = 40, B ∈ R100×10, x ∈ R10, and y ∈ R100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Each row of B is generated as bj ∼ N (0, I10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Also, the desired signal x is generated as x ∈∼ N (0, I10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Each time-varying sampling threshold sequence τ(ℓ) is considered to have the distribution τ(ℓ) ∼ N (0, I10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The performance of the RKA, SKM, PrSKM, and Block SKM is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Similar to the linear feasibility of equalities, it can be seen that the Block SKM has a better accuracy in the recovery of the desired signal x in the one-bit polyhedron (37) compared to the other three approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The NMSE results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1 are averaged over 15 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' PROBABILISTIC EFFECT OF SAMPLE ABUNDANCE IN ORKA An integral part of our proposed recovery algorithm is RKA, whose recovery error was readily given in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' As shown in [44], the convergence rate of RKA does not depend on the number of equations in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' We will show that the convergence rate of ORKA for linear feasibility is the same as RKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Nevertheless, when we face a non-linear constraint in our problem, as is generally the case in (1), it is desirable are made redundant by using the opportunity of having a large number of samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' as typically provided via one-bit sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In such a case, The offered convergence rate appears to be insufficient since we must have enough number of samples to fulfill costly constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' So, an extra term as a penalty must be considered to present the importance of sample size in our algorithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' By adding more inequality constraints in (37) as a result of extra one-bit samples, the shrinkage of the said polyhedron will put a downward pressure on the distance between the desired signal x⋆ and its surrounding hyperplanes, each presenting an informative measurement that will shrink the feasibility January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 16 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' We will show that by judicious sampling, the average of these distances will be bounded, which may be considered to be a finite-volume space created around x⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Moreover, as a result of using an overdetermined linear system of inequalities, the convergence of the RKA is guaranteed [16], [45], [46], [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Recovery Error Upper Bound for ORKA As the scaled condition number is the central parameter governing the recovery error of Kaczmarz algorithms, we will evaluate it for ORKA-created matrix P in the following, starting with σmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The singular values of P may be determined based on the following theorem, which thus unveils the value of σmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In ORKA, the rank of P is equal to that of the constraint matrix A, and its singular values are given by {σi} = √m {σiA} , (40) where {σiA} are singular values of A, and m is the number of time-varying sampling threshold sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Moreover, the scaled condition number of the ORKA-created matrix P is equal to that of the constraint matrix A: κ (P) = κ (A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (41) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To obtain the singular values of P, the matrix W = P⊤P is computed as W = P⊤P, = � A⊤Ω(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='A⊤Ω(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='A⊤Ω(m) � \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 Ω(1)A · · Ω(2)A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' · · Ω(m)A \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , = A⊤Ω(1)Ω(1)A + · · · + A⊤Ω(m)Ω(m)A, = mA⊤IA = mA⊤A, (42) which means the singular values of P are {σi} = √m {σiA}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Also, the Frobenius norm of P is obtained as ∥P∥2 F = Tr � P⊤P � , = Tr � mA⊤A � = m∥A∥2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (43) January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 17 Consequently, the scaled condition number is independent of the number of time-varying thresholds sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' It follows that κ (P) = κ (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' For A = I, corresponding to the one-bit sampled signal recovery problem, the scaled condition number of Ω = � Ω(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='Ω(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='Ω(m) � is κ (Ω) = √n, which is the infimum of the scaled condition number as shown in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Note that the convergence bound (17) for ORKA is independent of the number of time-varying sampling threshold sequences m, and it cannot take into account the effect of an increasing number of time-varying threshold sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Inspired by [16], we augment (17) with a sample size-dependent penalty function to make it useful in a sample abundance scenario: Proposition 3 (Convergence rate of ORKA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In the proposed recovery approach, it is deemed necessary to have a sufficient number of samples (inequalities) in order to guarantee a finite-volume feasible region and a bounded recovery error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Once our search area is located inside FX, we may effectively deploy (17) for the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The convergence rate of the Kaczmarz variants is useful when we have a linear feasibility problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' On the other hand, in (1), the main constraints are non-linear and they may be considered to be redundant by deploying enough samples [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' A sample size-aware convergence rate for ORKA may be formulated as: E {¯h (xi, x⋆)} ≤ � 1 − 2λi − λ2 i κ2 (A) �i ¯h (x0, x⋆) + Υ (m) , (44) where Υ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=') is an asymptotically decreasing function, such that if the number of time-varying threshold sequences is enough for the one-bit polyhedron to fit inside FX, the sample size-dependent penalty function Υ (m) approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To propose an appropriate sample size-dependent penalty function, we will utilize the first theorem in [16], which studies the possibility of creating a finite-volume space around the optimal signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Sample Size-Dependent Penalty Function via Moment Generating Functions We investigate the convergence of ORKA through a probabilistic lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To do so, define the distance between the optimal point x⋆ and the j-th hyperplane presented in (37) as dj � x⋆, τ(ℓ)� = ���rj ⊙ � ajx⋆ − τ (ℓ) j ���� 2 , j ∈ � 1, · · · , m′� , (45) where rj ⊙ aj is the j-th row of P, aj is the j-th row of A, and aj = aj+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' It is easy to observe that by generally reducing the distances between x⋆ and the constraint-associated hyperplanes, the possibility of January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 18 capturing the optimal point is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' For a specific sample size m′ = mn, when the volume of the finite space around the optimal point is reduced, the average of � dj �˜x⋆, τ(ℓ)�� is diminished as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This average of distance can be written as [45]: Tave = 1 m′ m′ � j=1 dj � ˜x⋆, τ(ℓ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (46) The possibility of creating a finite-volume, and the importance of the number of samples in the recovery performance of RKA, can be captured by the Chernoff bound as illustrated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Theorem 3 (See [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Assume the distances � dj � ˜x⋆, τ(ℓ)�� between the desired point ˜x⋆ and the hyperplanes of the polyhedron defined in (12) are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Then: The Chernoff bound of Tave is given by Pr \uf8eb \uf8ed 1 m′ m′ � j=1 dj � ˜x⋆, τ(ℓ)� ≤ a \uf8f6 \uf8f8 ≥ 1 − inf t≥0 ΨT eta , (47) where a is an average distance point in space at which the finite-volume space around the desired signal is created, and ΨT is the moment generating function (MGF) of the error recovery, given as ΨT = \uf8eb \uf8ed1 + t µ(1) dj m′ + · · · + tκ µ(κ) dj κ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' m′κ + R � m′� \uf8f6 \uf8f8 m′ , (48) with µ(κ) dj = E � dκ j � , and R denoting a bounded remainder associated with truncating the Taylor series expansion of ΨT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' ΨT is decreasing with an increasing sample size in the sample abundance scenario, leading to an increasing lower bound in (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The MGF ΨT contains two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The first part has an increasing trend until a specific sample size m⋆, which indicates the existence of an abundant number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' After that, the function has a decreasing behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Therefore, ΨT with m ≥ m⋆ can be a good choice for a sample size-dependent penalty function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Particularly, the penalty function can be chosen as ΨT −Ψ∞, where Ψ∞ = limm→∞ ΨT , to ensure Υ(m) → 0 as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Since, we do not have access to the probability density function of {dj}, thus, the MGF must be evaluated by the truncated Taylor series expansion, which may be accurately approximated by a rational January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 19 function such as a Pad´e approximation (PA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The decreasing part of ΨT in m > m⋆ with PA is modeled as follows [32], [38]: Υ(m) ≍ \uf8eb \uf8ed1 + · · · + tκ µ(κ) dj κ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' m′κ \uf8f6 \uf8f8 m′ − ΨT, = a0 + a1 m b0 + b1 m − a0 b0 , (49) where {a0, a1, b0, b1} are the PA coefficients as given by a0 = eu � 12u2 − 24v � , a1 = eu � −3u4 + 8u3 + 12u2v − 24uv − 12v2� , b0 = 12u2 − 24v, b1 = 3u4 + 8u3 − 12u2v − 24uv + 12v2, (50) where u = µ(1) dj t and v = µ(2) dj t2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' JUDICIOUS SAMPLING WITH ADAPTIVE THRESHOLDING FOR ORKA By the spirit of using the iterative RKA, a suitable time-varying threshold can be selected in order to enhance the recovery performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In ORKA, we face a highly overdetermined linear feasibility problem creating a finite-volume space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To capture the desired signal x⋆ more efficiently, the right-hand side of the inequalities in (37), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' vec (R) ⊙ vec (Γ), must be determined in a way that each associated hyperplane passes through the desired feasible region within FX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Therefore, an algorithm is proposed to ensure that this occurs in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' We propose an iterative algorithm generating an adaptive sampling threshold to accurately obtain the desired solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To have the smaller area of the finite-volume space around the desired signal x⋆, one can somehow choose thresholds to reduce distances between them and the desired point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To do so, we update the time-varying threshold for ℓ ∈ {1, · · · , m} as Axk − r(ℓ) k ⊙ ǫ(ℓ) k = τ(ℓ) k+1, (51) where ǫ(ℓ) k are positive vectors in the k-th iteration of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' This updating process is based on rj = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 +1 ajx > τj, −1 ajx < τj, (52) where {aj} are the rows of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The one-bit measurements � r(ℓ) k � are updated in the way to satisfy (37) in iteration k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' the inequalities Ω(ℓ) k Ax⋆ ≥ r(ℓ) k ⊙ τ(ℓ) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The reason behind this updating is to ensure January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 20 that each halfspace associated with a threshold in iteration k is getting closer to the optimal point in the correct direction which means the main side of the halfspace is forced to cover the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Proposition 4 (ORKA with Adaptive Thresholding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Consider applying ORKA to a linear feasibility problem Ax = y as part of the linear constraints of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Suppose the initial time-varying threshold sequences are � τ(ℓ) 0 � ∼ N (0, 1) (with the same length as r(ℓ)), and � δ(ℓ)� are positive numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Also, xk, τ(ℓ) k , r(ℓ) k and ǫk denote their associated values at iteration k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The proposed sampling algorithm is summarized as follows: 1) Find a point inside the following polyhedron with proposed accelerated Kaczmarz algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' the PrSKM or the block SKM for τ = � τ(ℓ) k � : Pk = � xk | ˜ΩkAxk ⪰ bk � , (53) where bk = vec (Rk) ⊙ vec (Γk), Rk and Γk are matrices with � r(ℓ) k � and � τ(ℓ) k � representing their columns, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2) Update Γk+1 as: vec (Rk) ⊙ vec (Γk+1) = ˜ΩkAxk − ǫk 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (54) 3) Compute ǫk, a block vector containing � ǫ(ℓ) k � , as: ǫk = ˜ΩkAxk − vec (Rk) ⊙ vec (Γk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (55) 4) Update Rk+1 based on (52): r(ℓ) k+1 = sgn � y − τ(ℓ) k+1 � , ℓ ∈ {1, · · · , m} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (56) 5) Increase k by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 6) Stop when ���τ(ℓ) k+1 − τ(ℓ) k ��� 2 ≤ δ(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' One can observe that by deploying this adaptive thresholding algorithm, smaller values of {dj} will emerge which leads to their moments � µ(κ) dj � to further diminish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Therefore, ΨT is smaller in this scenario and a smaller number of time-varying sampling threshold sequences can be utilized in ORKA with similar recovery performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Additionally, non-informative sampling thresholds, which appear as extra inequality constraints in the random time-varying sampling thresholds scenario, may be efficiently removed by choosing the adaptive thresholds with closer hyperplanes to the desired point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 21 (a) m = 2 (b) m = 6 (c) m = 60 (d) m = 2 (e) m = 6 (f) m = 60 Figure 2: Shrinkage of the one-bit polyhedron (37) in blue, ultimately placed within the unit ball of the nuclear norm ∥X∥⋆ ≤ 1 shown with black cylindrical region and its red contours, when the number constraints (samples) grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The arrows point to the half-space associated with each inequality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The evolution of the feasible regime is depicted with increasing samples in three cases: (a) and (d) small sample-size regime, constraints not forming a finite-value polyhedron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (b) and (e) medium sample-size regime, constraints forming a finite-volume polyhedron, parts of which are outside the cylinder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (c) and (f) large sample-size regime, constraints forming a finite-volume polyhedron inside the nuclear norm cylinder, making its constraint redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The optimal point representing the signal to be recovered is shown by yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' LOW-RANK MATRIX RECOVERY VIA ORKA As mentioned earlier, low-rank matrix recovery is an excellent example for problems that assume the form in (1), and that can be tackled using our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In this section, at first, we will briefly introduce the nuclear norm minimization form of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Accordingly, we will apply ORKA to this problem without considering the associated costly constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' At the end, the recovery performance of ORKA will be numerically evaluated considering different matrix ranks and sample sizes to investigate the existence of a sample abundance scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' January 10, 2023 DRAFT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 1 0 X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 YN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5- 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 1 0 X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 N 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 0 Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 22 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Problem Formulation The problem of the low-rank matrix recovery can be formulated as: find X s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' A (X) = y, rank (X) ≤ M, X ∈ Ωc, (57) where X ∈ Cn1×n2 is the matrix of unknowns, y ∈ Rn is the measurement vector, and A is a linear transformation mapping n1 × n2 into Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In general, Ωc can be chosen such as the set of semi-definite matrices, symmetric matrices, upper or lower triangle matrices, Hessenberg matrices and a specific constraint on the matrix elements ∥X∥∞ ≤ α or on its eigenvalues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=', λi ≤ ǫ where {λi} are eigenvalues of X [1], [4], [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The problem (57) can be rewritten as an optimization problem: min X rank (X) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' A (X) = y, X ∈ Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (58) This problem is known to be NP-hard, whose solution is difficult to approximate [15], [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Recall that the rank of X is equal to the number of nonzero singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In the case when the singular values are all equal to one, the sum of the singular values is equal to the rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' When the singular values are less than or equal to one, the sum of the singular values is a convex function that is strictly less than the rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Therefore, it is been popular for this problem to replace the rank function with the sum of the singular values of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=', its nuclear norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The nuclear norm minimization alternative of the problem is given by [2], [15], [57]: min X ∥X∥⋆ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' A (X) = y, X ∈ Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (59) In this problem, the feasible set FX is obtained as FX = {P⋆ ∩ Ωc} , (60) where P⋆ is defined as follows P⋆ = {X | ∥X∥⋆ ≤ τ} , τ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (61) January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 23 10 20 30 40 50 60 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 PrSKM Block SKM (a) 10 20 30 40 50 60 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 1 PrSKM Block SKM (b) Figure 3: Average NMSE for the Frobenius norm of error for the recovery of the matrix X associated with different time-varying sampling threshold sequences sizes when the PrSKM and the block SKM are utilized in ORKA: (a) rank (X) = 1, (b) rank (X) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Next, we will apply ORKA to (58) to make its costly constraints redundant by using abundant number of time-varying sampling thresholds m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' A numerical investigation of (37) when it is achieved for the nuclear norm minimization, reveals that by increasing the number of time-varying sampling threshold sequences m, the space formed by the intersection of half-spaces (inequality constraints) can fully shrink to the desired signal X⋆ inside the feasible region of (61) which is shown by the cylindrical space [15]—see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2 for an illustrative example of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' As can be seen in this figure, the blue lines displaying the linear feasibility form a finite-volume space around the optimal point displayed by the yellow circle inside the cylinder (the elliptical region) by growing the number of threshold sequences or one-bit samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In (a)/(d), constraints are not enough to create a finite-volume space, whereas in (b)/(e) such constraints can create the desired finite-volume polyhedron space which, however, is not fully inside the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Lastly, in (c)/(f), the created finite-volume space shrinks to be fully inside the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Numerical Illustrations In this section, we numerically scrutinize the capability of the ORKA in the nuclear norm minimization problem (59) instead of (59) by the squared Frobenius norm of the error normalized by the squared Frobenius norm of the desired matrix X⋆, defined as NMSE ≜ ��X⋆ − ¯X ��2 F ∥X⋆∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (62) January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='ORKA with a random threshold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='ORKA with an adaptive threshold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='ORKA with a random threshold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='ORKA with an adaptive threshold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='Figure 4: Comparing the average NMSE for the Frobenius norm of error for the recovery of the matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='X using ORKA when (i) a random threshold and (ii) the adaptive sampling threshold are adopted when ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='the PrSKM and the block SKM are utilized in ORKA: (a) rank (X) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (b) rank (X) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' We solve the overdetermined one-bit polyhedron in (37) via the PrSKM and the Block SKM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To make this happen, we obtain the one-bit polyhedron from a linear feasibility problem Ax = y, where A ∈ R200×25, x ∈ R25 (x = vec (X) where X ∈ R5×5), and y ∈ R200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' We consider the number of time-varying sampling threshold sequences to be m ∈ {10, 20, 30, 40, 50, 60}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Each row of A is generated as aj ∼ N (0, I25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' For the desired matrix X, we generate X = KK⊤ where (i) K ∈ R5×4 is the Gaussian matrix, and (ii) K ∈ R5×1 is the Gaussian vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Also, each time-varying sampling threshold τ(ℓ) is considered to have the distribution τ(ℓ) ∼ N (0, I200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3 appears to confirm the possibility of recovering the desired matrix X⋆ in the one-bit polyhedron (37) by ORKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' As expected, the performance of the recovery will be significantly enhanced as the number of time-varying sampling threshold sequences grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Also, similar to before, it can be seen that the Block SKM outperforms the PrSKM in the low-rank matrix recovery problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To improve the recovery performance, we proposed the adaptive time-varying sampling threshold in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4 illustrates the performance of the Block SKM in the low rank matrix recovery in the one-bit polyhedron (37) when we have the high-dimensional input signal x ∈ R128 and A ∈ R20000×128, with (i) a random threshold, and (ii) an adaptive time-varying threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' As can be seen, the recovery performance is significantly enhanced when the Block SKM is equipped with the adaptive time-varying threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' ONE-BIT COMPRESSED SENSING: January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 25 10 20 30 40 50 60 7 6 5 4 3 2 1 PrSKM Block SKM (a) 10 20 30 40 50 60 6 5 4 3 2 1 PrSKM Block SKM (b) Figure 5: Average NMSE for the error between the desired signal x⋆ and its recovered version ¯x for different time-varying sampling threshold sequences sizes when the PrSKM and the block SKM are utilized in ORKA with (a) k = 2, (b) k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' FROM OPTIMIZATION TO LINEAR FEASIBILITY Compressed sensing (CS) is an interesting and rapidly growing area of research that has attracted considerable attention in electrical engineering, applied mathematics, statistics, and computer science [4], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In CS, a sparse high-dimensional signal is to be recovered by incomplete measurements such a recovery may be formulated as [29]: min x ∥x∥1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Ax = y, (63) where A ∈ Rm×n, and m ≪ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' One of the important applications of CS emerges in the signal recovery from a sequence of acquisitions {yi} obtained from a sparse linear transformation (wavelet transformations are known for such a property, for instance) in the magnetic resonance imaging (MRI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The reconstruction problem of the desired signal x⋆ is given by min x ∥x∥1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Ai (x) = yi, i ∈ {1, · · · , n} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (64) In this section, we first formulate the optimization problem of the one-bit compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Then, by taking advantage of one-bit sampling, we increase sample size in (64) and create an associated one-bit polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Problem Formulation Let τ denotes the time-varying threshold vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The one-bit samples are generated as ri = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 +1 a⊤ i x ≥ τi, −1 a⊤ i x < τi, (65) where Ai (x) = a⊤ i x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The occurrence probability vector p for the one-bit measurement r is given as [16], pi = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Φ � a⊤ i x � for {ri = +1}, 1 − Φ � a⊤ i x � for {ri = −1}, (66) where Φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=') is the CDF of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The log-likelihood function of the sign data r is given by Lr(µ, x) = m � i=1 � I(ri=+1) log � Φ(a⊤ i x) � +I(ri=−1) log � 1 − Φ(a⊤ i x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (67) Therefore, the maximum likelihood estimation (MLE) for the one-bit compressed sensing can be written as min x Lr(µ, x) + λ∥x∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (68) The alternative formulations for one-bit compressed sensing can be found in [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Nevertheless, as discussed earlier, by deploying one-bit sampling, the opportunity exists to increase the number of samples in (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The one-bit compressed sensing is thus solely accomplished by creating a highly-constrained one-bit polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In other words, instead of solving an optimization problem with costly constraints, the problem may be tackled by the proposed accelerated Kaczmarz algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' namely, PrSKM and the block SKM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Numerical results To examine the performance of ORKA in CS and to validate the theoretical results described in this paper, we consider signal recovery with different number of time-varying sampling threshold sequences m ∈ {10, 20, 30, 40, 50, 60}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Input signals x⋆ ∈ R10 are generated with sparsity orders k = 2 and k = 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The sparsity order k is defined as the number of non-zero elements in a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Time-varying sampling thresholds and the constraint matrix A are generated as in Subsection VI-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To compare two proposed algorithms, the NMSE defined in (39) is utilized and the results are averaged over 15 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2022 27 0 50 100 150 200 25 20 15 10 5 0 ORKA with a random threshold ORKA with an adaptive threshold (a) 0 50 100 150 200 25 20 15 10 5 0 ORKA with a random threshold ORKA with an adaptive threshold (b) Figure 6: Comparing the average NMSE for the desired signal x⋆ and its recovered signal using ORKA when (i) a random threshold and (ii) the adaptive sampling threshold are adopted with (a) k = 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' (b) k = 40 Table I: Comparing CPU times and NMSE of ORKA and ℓ1-minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Algorithm m⋆ CPU time (s) NMSE ORKA 500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='1240e − 04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='2052e − 12 ℓ1 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='0071 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='4572e − 11 As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 5, by increasing the number of time-varying sampling threshold sequences, the performance of our method is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Beside the possibility of increasing the number of measurements n, the higher number of samples are available in ORKA by comparing the measurements with multiple threshold sequences ℓ ∈ {1, · · · , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' In other words, we have the opportunity to increase n and m simultaneously, when the number of samples is m′ = mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Same as Subsection VI-B, the adaptive thresholding algorithm is applied to ORKA for the high- dimensional input signal x ∈ R128 in order to enhance its recovery performance, whose outcome is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The NMSE results are reported with sparsity orders k = 20 and k = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' To further investigate the efficacy of ORKA in CS, we compare our proposed approach with the well- known ℓ1-minimization approach formulated in (64) in terms of NMSE and CPU time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' As presented in Table I, ORKA outperforms ℓ1-minimization in terms of both NMSE and CPU time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The results are obtained for x ∈ R128 when the optimal number of samples are utilized, and where m⋆ = 4k log(n/k) and m⋆ = 500 are considered for the high-resolution method and ORKA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Herein, optimality January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 28 of sample sizes means that the number of samples utilized by algorithms leads to their best performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' satisfying the criterion ∥xi − x⋆∥2 2 ≤ 5× 10−11 ∥x⋆∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' By this comparison, we remove the burden of the large number of samples from the ℓ1-minimization to fairly compare their optimal shape deploying incomplete measurements with that of ORKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' It is worth pointing out that for a 64-bit ADC, m = 100 corresponds to 6400 bits of information while ORKA solely employs 500 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Therefore, it appears from Table I that ORKA achieves a better accuracy in terms of NMSE with not only fewer information bits but also a smaller computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' CONCLUSION We proposed a novel algorithm, ORKA, that takes advantage of the abundant number of samples avail- able in one-bit sampling with time-varying thresholds to efficiently and globally solve some well-studied problems in the form of (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' including low-rank matrix recovery and compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Moreover, two state-of-the-art randomized Kaczmarz algorithms are proposed to use in ORKA to find the desired signal inside the emerging confined feasible regions, named the one-bit polyhedron, with an enhanced convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' The numerical results showcased the effectiveness of the proposed approaches for the low-rank matrix recovery and compressed sensing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' REFERENCES [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Candes, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Eldar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Strohmer, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Voroninski, “Phase retrieval via matrix completion,” SIAM review, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 57, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 225–251, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Cai, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Cand`es, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Shen, “A singular value thresholding algorithm for matrix completion,” SIAM Journal on optimization, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1956–1982, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Davenport, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Plan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Van Den Berg, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Wootters, “1-bit matrix completion,” Information and Inference: A Journal of the IMA, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 189–223, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Davenport and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Romberg, “An overview of low-rank matrix recovery from incomplete observations,” IEEE Journal of Selected Topics in Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 608–622, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Candes and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Plan, “Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 57, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2342–2359, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Jalali, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Sanghavi, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Caramanis, “Low-rank matrix recovery from errors and erasures,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4324–4337, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [7] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' So, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Zhao, “A survey on matrix completion: Perspective of signal processing,” arXiv preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='10885, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [8] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Haeffele, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Young, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Vidal, “Structured low-rank matrix factorization: Optimality, algorithm, and applications to image processing,” in International conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' PMLR, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2007–2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Nie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Huang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Ding, “Low-rank matrix recovery via efficient schatten p-norm minimization,” in Twenty-sixth AAAI conference on artificial intelligence, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [10] David M Blei, “Probabilistic topic models,” Communications of the ACM, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 77–84, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 29 [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Obozinski, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Taskar, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Jordan, “Joint covariate selection and joint subspace selection for multiple classification problems,” Statistics and Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 231–252, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Pontile, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Evgeniou, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Argyriou, “Convex multi-task feature learning,” Journal of Machine Learning, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 243–272, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Argyriou, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Evgeniou, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Pontil, “Multi-task feature learning,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 19, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Tomasi and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Kanade, “Shape and motion from image streams under orthography: a factorization method,” International journal of computer vision, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 137–154, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [15] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Recht, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Xu, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Hassibi, “Null space conditions and thresholds for rank minimization,” Mathematical programming, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 127, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 175–202, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Eamaz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Yeganegi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Soltanalian, “One-bit phase retrieval: More samples means less complexity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=',” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 70, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4618–4632, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [17] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Millane, “Phase retrieval in crystallography and optics,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 394–411, Mar 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [18] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Kim and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Hayes, “The phase retrieval problem in x-ray crystallography,” in Proceedings of the Acoustics, Speech, and Signal Processing, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' ICASSP-91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=', 1991 International Conference, USA, 1991, ICASSP ’91, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1765–1768, IEEE Computer Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Fienup, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Marron, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Schulz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1747–1767, Apr 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Krist and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Burrows, “Phase-retrieval analysis of pre- and post-repair hubble space telescope images,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4951–4964, Aug 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Sarnik, “Phase retrieval: A practical application for the space telescope,” in Inverse Optics II, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Bates and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Devaney, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' International Society for Optics and Photonics, 1985, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 0558, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 85 – 94, SPIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Gerchberg and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 34, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 275–284, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Gerchberg and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 35, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 227–246, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Fienup, “Comments on ”the reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform”,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 738–739, Jun 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Fienup, “Phase retrieval algorithms: a comparison,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 15, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2758–2769, Aug 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 27–29, Jul 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Candes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Strohmer, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Voroninski, “PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming,” Communications on Pure and Applied Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1241–1274, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [28] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Cand`es and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Li, “Solving quadratic equations via PhaseLift when there are about as many equations as unknowns,” Foundations of Computational Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1017–1026, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [29] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Eldar and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Kutyniok, Compressed sensing: theory and applications, Cambridge university press, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Lustig, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Donoho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Santos, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Pauly, “Compressed sensing MRI,” IEEE signal processing magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 72–82, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Fessler, “Model-based image reconstruction for MRI,” IEEE signal processing magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 81–89, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Eamaz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Yeganegi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Soltanalian, “Modified arcsine law for one-bit sampled stationary signals with time-varying January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 30 thresholds,” in ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 5459–5463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [33] Texas Instruments, “Analog embedded processing,” Active Filter Design Techniques, Literature Number SLOA088 Texas Instruments, excerpted from OP-Amps for Everyone, Literature Number SLOD006A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Mezghani and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Swindlehurst, “Blind estimation of sparse broadband massive MIMO channels with ideal and one-bit ADCs,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2972–2983, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [35] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Sedighi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Shankar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Soltanalian, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Ottersten, “One-bit DoA estimation via sparse linear arrays,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 9135–9139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [36] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Qian and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Li, “ADMM for harmonic retrieval from one-bit sampling with time-varying thresholds,” in 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' IEEE, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3699–3703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [37] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Gianelli, Luzhou Xu, Jian Li, and Petre Stoica, “One-bit compressive sampling with time-varying thresholds for sparse parameter estimation,” in 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' IEEE, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [38] Arian Eamaz, Farhang Yeganegi, and Mojtaba Soltanalian, “Covariance recovery for one-bit sampled non-stationary signals with time-varying sampling thresholds,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 70, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 5222–5236, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [39] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Pajovic, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Boufounos, and Philip V Orlik, “On angular-domain channel estimation for one-bit massive MIMO systems with fixed and time-varying thresholds,” in 51st Asilomar Conference on Signals, Systems, and Computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' IEEE, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1056–1060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [40] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Xi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Xiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Chen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Nehorai, “Gridless parameter estimation for one-bit MIMO radar with time-varying thresholds,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 68, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1048–1063, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Kaczmarz, “Angen¨aherte aufl¨osung von systemen linearer gleichungen (english translation by Jason Stockmann): Bulletin international de l’acad´emie polonaise des sciences et des lettres,” 1937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [42] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Feichtinger and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Strohmer, “A Kaczmarz-based approach to nonperiodic sampling on unions of rectangular lattices,” in SampTA’95: 1995 Workshop on Sampling Theory and Applications, 1995, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 32–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Sezan and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Stark, “Applications of convex projection theory to image recovery in tomography and related areas,” Image Recovery: Theory and Application, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 155–270, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [44] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Strohmer and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Vershynin, “A randomized Kaczmarz algorithm with exponential convergence,” Journal of Fourier Analysis and Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 262–278, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [45] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Leventhal and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Lewis, “Randomized methods for linear constraints: convergence rates and conditioning,” Mathematics of Operations Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 641–654, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [46] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Needell and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Tropp, “Paved with good intentions: Analysis of a randomized block Kaczmarz method,” Linear Algebra and its Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 441, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 199–221, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [47] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Briskman and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Needell, “Block Kaczmarz method with inequalities,” Journal of Mathematical Imaging and Vision, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 385–396, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [48] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' De Loera, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Haddock, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Needell, “A sampling Kaczmarz–Motzkin algorithm for linear feasibility,” SIAM Journal on Scientific Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 39, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' S66–S87, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [49] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Daubechies, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Defrise, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 57, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1413–1457, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [50] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Combettes and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale modeling & simulation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1168–1200, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' January 10, 2023 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022 31 [51] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Dai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Soltanalian, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Pelckmans, “On the randomized Kaczmarz algorithm,” IEEE Signal Processing Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 330–333, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [52] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Sarowar Morshed and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Saiful Islam, “Sampling Kaczmarz Motzkin method for linear feasibility problems: Generalization & acceleration,” arXiv e-prints, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' arXiv–2002, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [53] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Van Loan and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Golub, “Matrix computations (Johns Hopkins studies in mathematical sciences),” 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [54] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Bhandari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Krahmer, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Raskar, “On unlimited sampling and reconstruction,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 69, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3827–3839, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [55] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Elfving, “Block-iterative methods for consistent and inconsistent linear equations,” Numerische Mathematik, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 1–12, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [56] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Meka, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Jain, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Caramanis, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Dhillon, “Rank minimization via online learning,” in Proceedings of the 25th International Conference on Machine learning, 2008, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 656–663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [57] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Recht, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Fazel, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM review, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 471–501, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' [58] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Khobahi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' Soltanalian, “Model-based deep learning for one-bit compressive sensing,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 68, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' 5292–5307, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQf1QXp/content/2301.03467v1.pdf'} +page_content=' January 10, 2023 DRAFT' metadata={'source': 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Fluid Mech., DOI: 10.1017/jfm.2022.976 +1 +Precession-driven flows in stress-free +ellipsoids +J´er´emie Vidal† & David C´ebron +Universit´e Grenoble Alpes, CNRS, ISTerre, 38000 Grenoble, France +(Received 22 March 2022; revised 23 August 2022; accepted 11 November 2022) +Motivated by modelling rotating turbulence in planetary fluid layers, we investigate +precession-driven flows in ellipsoids subject to stress-free boundary conditions (SF- +BC). The SF-BC could indeed unlock numerical constraints associated with the no-slip +boundary conditions (NS-BC), but are also relevant for some astrophysical applications. +Although SF-BC have been employed in the pioneering work of Lorenzani & Tilgner (J. +Fluid Mech., 2003, 492, pp. 363–379), they have scarcely been used due to the discovery +of some specific mathematical issues associated with angular momentum conservation. +We revisit the problem using asymptotic analysis in the low-viscosity regime, which +is validated with numerical simulations. First, we extend the reduced model of uniform- +vorticity flows in ellipsoids to account for SF-BC. We show that the long-term evolution of +angular momentum is affected by viscosity in triaxial geometries, but also in axisymmetric +ellipsoids when the mean rotation axis of the fluid is not the symmetry axis. In a regime +relevant to planets, we analytically obtain the primary forced flow in triaxial geometries, +which exhibits a second inviscid resonance. Then, we investigate the bulk instabilities +existing in precessing ellipsoids. We show that using SF-BC would be useful to explore +the non-viscous instabilities (e.g. Kerswell, Geophys. Astrophys. Fluid Dyn., 1993, 72, +pp. 107-144), which are presumably relevant for planetary applications but are often +hampered in experiments or simulations with NS-BC. +Key words: rotating flows, waves in rotating fluids, geophysical and geological flows +1. Introduction +Motivated by numerous natural applications (e.g. Le Bars et al. 2015), we aim to +explore the long-term dynamics of rapidly rotating fluids enclosed in ellipsoids subject to +(harmonic) mechanical forcings. Global rotation is indeed ubiquitous in many planetary +fluid layers or stars, which are usually ellipsoidal at the leading order (e.g. due to +the combined action of centrifugal effects and gravitational interactions with nearby +orbital partners, see Chandrasekhar 1969). In particular, mechanically driven flows in +ellipsoids (e.g. flows driven by precession or tides) have received much attention in the +fluid community. Mechanical forcings can indeed sustain bulk instabilities (e.g. Kerswell +1993, 2002), turbulence (e.g. Grannan et al. 2017; Le Reun et al. 2019) and possibly +dynamo magnetic fields (e.g. Reddy et al. 2018; Vidal et al. 2018). These works have also +renewed interest in a key fundamental question in the theory of rotating fluids, which is +the generation of two-dimensional geostrophic motions (Greenspan 1969). However, this +problem has only received scant attention in global geometries exhibiting the so-called +† Email address for correspondence: jeremie.vidal@univ-grenoble-alpes.fr +arXiv:2301.03254v1 [physics.flu-dyn] 9 Jan 2023 + +2 +J. Vidal & D. C´ebron +topographic beta effect (which strongly modifies the geostrophic flows, e.g. Greenspan +1968). Exploring rotating turbulence thus deserves further work using global models. +The incompressible Navier-Stokes equation is commonly adopted to explore the tur- +bulence driven by mechanical forcings, together with the no-slip boundary conditions +(NS-BC). The latter are appropriate to model the flow dynamics in the presence of +a rigid boundary (e.g. the solid interface between a liquid core and a solid overlying +mantle in planetary interiors). However, the range of parameters that is accessible to +global simulations with NS-BC is severely limited, in particular for the Ekman number E +(which crucially controls the dynamics of rapidly rotating flows). Typical values in natural +systems are E ⩽ O(10−12), whereas direct numerical simulations (DNS) and laboratory +experiments of mechanically driven rotating turbulence can only reach much larger values +E ≳ 10−6 (e.g. Grannan et al. 2017; Le Reun et al. 2019). As a consequence, the Ekman +boundary layer is often a prominent feature in the models (whereas the smallness of +E in planetary systems suggests that viscosity should rather play a minor dynamical +role), and its resolution requires considerable computational resources when E is lowered. +Moreover, the overestimated viscous torque at the boundary can also largely inhibit the +fluid response to mechanical forcings (which is primarily driven by the shape deformation +of the fluid boundary, combined with non-stationary effects due to the possibly oscillatory +angular velocity of the container). Therefore, different modelling approaches are worth +considering to simulate such flows at more realistic parameters for planetary applications. +One natural way to avoid the physical and computational disadvantages of NS-BC is +to employ stress-free boundary conditions (SF-BC). A thin outer Ekman boundary layer +is still present for stress-free boundaries (e.g. Livermore et al. 2016), but its dynamical +role is expected to be less important because the boundary-layer flow is much weaker in +amplitude than the bulk flow (e.g. Rieutord 1992). Moreover, SF-BC are also commonly +employed in astrophysical modelling since they are often believed to yield similar results +to those obtained with a realistic free surface (Barker 2016a). However, SF-BC have +scarcely been used in spheres and ellipsoids because of mathematical difficulties. The +most serious one is related to angular momentum conservation. Angular momentum can +indeed be arbitrary in axisymmetric geometries, leading to spurious solutions on long +time scales (e.g. Jones et al. 2011; Guermond et al. 2013). The usefulness of SF-BC for +simulating rotating flows in ellipsoids has thus been questioned, but we believe that this +mathematical set-up deserves further analysis. +In this paper, we thus revisit the influence of SF-BC for rotating ellipsoids using +asymptotic analysis when E ≪ 1 and targeted numerical simulations. The paper is +organised as follows. The model is presented in §2 and applied to precessing ellipsoids in +§3. The results are discussed in §4, and we end the paper in §5. +2. Mathematical modelling +2.1. Fluid dynamic equations +We consider a fluid-filled ellipsoid of uniform density and volume V , which is assumed +to co-rotate with the surrounding mantle at the angular velocity Ωc(t) = Ω0 [Ω + δ(t)] +with respect to the inertial frame (δ(t) being the time-dependent departure from the +steady global rotation Ω along the unit vector 1Ω = Ω/|Ω|). To have a tractable +mathematical problem, we seek mechanically driven flows in the mantle reference frame +in which the ellipsoidal boundary S is steady and δ(t) ̸= 0. This set-up allows us to +model flows driven by precession or librations, which have already received consideration +using NS-BC (e.g. Noir & C´ebron 2013; Zhang et al. 2012, 2014; Vantieghem et al. 2015). + +Precession-driven flows in stress-free ellipsoids +3 +We non-dimensionalise the problem using Ω−1 +0 +as the time scale, and a typical length R +as the length scale (which is here arbitrary). Considering a Newtonian fluid of uniform +kinematic viscosity ν, the dimensionless equations for the velocity v are +∂tv + (v · ∇) v + 2Ωc × v = −∇p + 2E ∇ · ϵ(v) + r × dtδ, +(2.1a) +∇ · v = 0, +(2.1b) +where r is the position vector, ϵ(v) = (1/2)[∇v + (∇v)⊤] is the strain-rate tensor, and +E = ν/(Ω0R2) is the Ekman number. The ellipsoidal geometry, which is assumed to be +steady in the mantle frame, is given by the dimensionless equation +(x/a)2 + (y/b)2 + (z/c)2 = 1 +(2.2) +where [a, b, c] are the (dimensionless) ellipsoidal semi-axes and [x, y, z] are the Cartesian +coordinates. In the following, axisymmetric geometries refer to ellipsoids with a revolution +symmetry axis (i.e. when either a = b, b = c or a = c). Finally, spheroids will refer to +the particular axisymmetric geometries for which the revolution symmetry axis is aligned +with the rotation axis (with a = b and Ω ∝ 1z in this study). We aim to consider the +SF-BC given in the mantle frame by +v · 1n|S = 0, +[ϵ(v) · 1n] × 1n|S = 0, +(2.3a,b) +where 1n is the outward normal unit vector at the boundary, instead of the NS-BC +v · 1n|S = 0, +v × 1n|S = 0. +(2.4a,b) +It is obvious from SF-BC (2.3) and NS-BC (2.4) that the tangential velocity at the +boundary will differ between the two cases (since the flow is allowed to freely slip on +the boundary with the SF-BC). One may thus wonder in which circumstances the above +conditions will lead to similar flows in the bulk (i.e. far from the boundary region). +A necessary condition is that the mechanical forcings can sustain flows against viscous +dissipation for the two BC in the mantle frame. This is evidenced by the conservation +equation for the volume-averaged kinetic energy Ek. In a frame where the fluid boundary +is steady, it is given by (e.g. equation 5 in Wu & Roberts 2009) +dtEk = +� +V +v · [r × dtδ] dV + 2E +�� +S +v · T dS − Dν +� +(2.5) +where T = ϵ(v) · 1n is the surface traction and Dv ⩾ 0 is a volume-averaged viscous +dissipation (for both the NS-BC and SF-BC). For a velocity satisfying the no-penetration +condition such that v = (v ·1n) 1n −1n ×(1n ×v) = −1n ×(1n ×v), the surface integral +can actually be written as +� +S +v · T dS = − +� +S +T · [1n × (1n × v)] dS = − +� +S +[T × 1n] · [v × 1n] dS +(2.6) +where we have used a property of the scalar triple product to obtain the last expression. +Thus, the above surface integral exactly vanishes for both SF-BC (2.3) and NS-BC (2.4) +in the mantle frame. Then, equation (2.5) shows that we can have dtEk ⩾ 0 for both +SF-BC and NS-BC if the mechanical forcings are oscillatory in the mantle frame (i.e. +when dtδ ̸= 0). Harmonic mechanical forcings, such as precession or librations, can thus +sustain flows against viscous dissipation in the mantle frame (even with the SF-BC). +Note that a very different conclusion is obtained for steady forcings, such as precession +viewed in the frame of precession for spheroidal geometries (Lorenzani & Tilgner 2003; +Wu & Roberts 2009). We indeed have dtEk < 0 at every time for the SF-BC in the + +4 +J. Vidal & D. C´ebron +precession frame, whereas precession could sustain non-vanishing flows against viscous +dissipation for the NS-BC (since v × 1n|S ̸= 0 for a no-slip boundary in the precession +frame, e.g. C´ebron et al. 2019). In the following, we will only investigate the dynamics +driven by oscillatory forcings in the mantle frame with SF-BC. +2.2. Angular momentum +The angular momentum L = +� +V r × v dV of the flow plays a central dynamical role +for mechanically driven flows in ellipsoids. Actually, the Cartesian components of the +angular momentum L = (Lx, Ly, Lz)⊤ are exactly given for incompressible flows by +Lx = +� +V +(yvz − zvy) dV = +� +V +v · (1x × r + ∇Ψx) dV, +(2.7a) +Ly = +� +V +(zvx − xvz) dV = +� +V +v · (1y × r + ∇Ψy) dV, +(2.7b) +Lx = +� +V +(xvy − yvx) dV = +� +V +v · (1z × r + ∇Ψz) dV, +(2.7c) +where [Ψx, Ψy, Ψz] are arbitrary scalar potentials if ∇ · v = 0 and if the flow obeys +the no-penetration BC in rigid ellipsoids. The scalar potentials are thus often discarded +to simply express the angular momentum as projections onto the solid-body rotations +1i × r (e.g. Guermond et al. 2013). Yet, the solid-body rotations are not admissible flow +solutions in non-spherical geometries (even without viscosity), since they do not satisfy +the no-penetration condition. +A more appropriate definition of the angular momentum for incompressible flows is +thus given in ellipsoids by +L · 1i = +� +V +ei · v dV, +(2.8) +where {ei}i∈{x,y,z} is the set of uniform-vorticity (flow) elements defined by +ei = 1i × r + ∇Ψi, +∇ · ei = 0, +ei · 1n|S = 0. +(2.9a–c) +The scalar functions Ψi allow the elements ei to satisfy the no-penetration condition. In +ellipsoidal geometries, they are explicitly given by (e.g. Noir & C´ebron 2013) +Ψx = c2 − b2 +b2 + c2 yz, +Ψy = a2 − c2 +a2 + c2 xz, +Ψz = b2 − a2 +a2 + b2 xy. +(2.10a–c) +It is worth noting that definition (2.8) is purely kinematic. It thus remains valid in the +presence of additional effects, for instance without global rotation or with magnetic effects +(e.g Gerick et al. 2020). Moreover, this definition can also be generalised for compressible +flows under the anelastic approximation (see Appendix A). Consequently, we can always +rigorously expand incompressible velocity fields in ellipsoids as +v(r, t) = U(r, t) + vf(r, t), +� +V +r × vf dV = 0, +(2.11a,b) +where the uniform-vorticity flow U carrying the angular momentum is given by +U(r, t) = ωx(t) ex(r) + ωy(t) ey(r) + ωz(t) ez(r), +U · 1n|S = 0, +(2.12a,b) +and with the effective rotation vector of the fluid ω(t) = (ωx(t), ωy(t), ωz(t))⊤. The ve- +locity vf, which does not carry angular momentum by definition since +� +V U ·vf dV = 0, +contains bulks flows of higher spatial complexity (e.g. flow instabilities or turbulence) and + +Precession-driven flows in stress-free ellipsoids +5 +0 +2 +4 +6 +8 +E t +0 +0.2 +0.4 +0.6 +100 Lz +E = 5 × 10−3 +E = 5 × 10−4 +0 +2 +4 +6 +8 +10 +12 +−0.02 +0.00 +100 Lz +0 +2 +4 +6 +8 +10 +12 +E t +16.00 +16.25 +100 Lz +(a) Non-rotating Ω ≃ 0 +(b) Rotating Ω ∝ 1z +Figure 1. Non-convergence of the angular momentum Lz in DNS after several viscous time +units. Precession forcing given by definition (3.1) with Px = 10−2 in stress-free spheroids +(a = b = 1, c = 0.95). At t = 0, [ωx, ωy] are chosen to match asymptotic solution (3.7). +(a) DNS at Po = −1 for the two values of the Ekman number E = 5 × 10−3 (e.g. as considered +in Wu & Roberts 2009) and E = 5 × 10−4. At t = 0, ωz ≈ 0 for the two simulations. (b) DNS +at Po = −1.8 and E = 5 × 10−4 for ωz ≈ 0 (top panel) and ωz = 0.1 (bottom panel) at t = 0. +also viscous structures (e.g. the Ekman boundary layer, Rieutord 1992). The Cartesian +components of L are then exactly given by +L = L−1 ω, +L−1 = 16π +15 abc diag +� b2c2 +b2 + c2 , +a2c2 +a2 + c2 , +a2b2 +a2 + b2 +� +. +(2.13a,b) +Finally, the time evolution of the angular momentum (or equivalently that of ω) is +affected by viscosity through the action of the viscous torque Γ ν on long time scales. We +have for example Γ ν = 0 in spheres, such that angular momentum has to be conserved +for uniformly rotating fluids in the inertial frame (e.g. Jones et al. 2011). To clarify the +dynamical role of SF-BC in ellipsoids, it is worth computing the viscous torque. +2.3. Viscous torque in stress-free ellipsoids +Because of definition (2.8), the Cartesian components of the viscous torque Γ ν = +(Γ ν · 1x, Γ ν · 1y, Γ ν · 1z)⊤ are exactly given for SF-BC (2.3) by +Γ ν · 1i = 2E +� +V +ei · ∇ · ϵ(v) dV = −2E +� +V +ϵ(ei) : ϵ(v) dV, +(2.14) +where we have used integration by parts and the decomposition ei = (1n · ei) 1n − 1n × +(1n×ei) = −1n×(1n×ei) to cancel out the surface integral for SF-BC (e.g. see the proof +of proposition 2.1 in Guermond et al. 2013). We recover from the formula that Γ ν = 0 in +spheres since ϵ(ei) exactly vanishes when ei is a solid-body rotation, but we also obtain +that Γ ν ̸= 0 in triaxial geometries (because ϵ(ei) ̸= 0 when a ̸= b ̸= c). Moreover, +it shows that Γ ν · 1i = 0 when the Cartesian vector 1i is an axis of revolution of the +geometry (irrespective of the fluid global rotation, as ei is then a solid-body rotation). +We can now inspect the long-term evolution of angular momentum since pathological +behaviours have been reported in some axisymmetric configurations (Guermond et al. +2013). To illustrate this behaviour, we expand the angular momentum as L = L0 + L1, +where L0 is the angular momentum of a dynamical solution of the problem and L1 +is a modification of L0 associated with an additional uniform-vorticity flow. The time + +6 +J. Vidal & D. C´ebron +evolution of L1 is then given in the rotating frame by (e.g. Roberts & Aurnou 2012) +dtL1 + Ωc × L1 = Γ p,1 + Γ ν,1, +(2.15) +where Γ p,1 = +� +S p1 1n ×r dS is the pressure torque and Γ ν,1 is the viscous torque. Since +the viscous and pressure torques are non-zero when a ̸= b ̸= c, equation (2.15) shows +that the angular momentum is affected by viscosity in triaxial ellipsoids. The situation +is possibly different in axisymmetric geometries. If the fluid is not globally rotating (i.e. +when Ω = 0), then the component L1 · 1i carried by the uniform-vorticity element ei +is arbitrary when 1i is a revolution symmetry axis (since Γ p,1 · 1i = Γ ν,1 · 1i = 0). +Similarly, if the fluid is globally rotating along the revolution symmetry axis 1i, then +the perturbation angular momentum L1 ∝ 1i is arbitrary (it will depend on the initial +conditions, e.g. as shown in Guermond et al. 2013). +The two situations are illustrated numerically in figure 1 for a spheroid a = b subject +to the precession forcing (see its definition below in §3). We have performed DNS using +the standard finite-element method as implemented in the commercial software comsol. +The latter has already been employed to simulate precession-driven flows in ellipsoids +with NS-BC (e.g. Noir & C´ebron 2013) and can also account for SF-BC (e.g. for tidal +flows in C´ebron et al. 2013). The geometry is modelled by an unstructured mesh with +tetrahedral elements in the bulk, surrounded by a boundary-layer mesh (made of prism +elements) to ensure the convergence of the thin Ekman layer. We have employed Lagrange +elements P2-P3 (i.e. quadratic for the pressure field and cubic for the velocity field). The +total number of degrees of freedom ranges between 3 × 105 and 5 × 105, such that every +targeted simulation took a few days to run in parallel on a cluster (to investigate the +long-term evolution of L). We observe that the axial angular momentum Lz does not +converge in time for the considered stress-free spheroid (it is still growing or decaying +even after several viscous time scales) if either the fluid is non-rotating in average as +in panel (a) or Ω ∝ 1z as in panel (b). However, a definitive conclusion should not be +drawn for every axisymmetric geometry. The situation is indeed different if the global +rotation is not aligned with the revolution axis, since the three components of the angular +momentum should be strongly coupled in equation (2.15) for such configurations (even +if Γ ν · 1i = 0, see §3). +3. Application to precession-driven flows +We consider precession-driven flows in ellipsoids, which have only received scant +attention with SF-BC (Lorenzani & Tilgner 2003; Wu & Roberts 2009; Guermond et al. +2013). We work in the mantle frame rotating with respect to the inertial frame at the +dimensionless angular velocity (e.g. Noir & C´ebron 2013) +Ωc(t) = (1 + Pz)1z +� +�� +� +Ω ++ δ(t), +δ(t) = Px [cos(t)1x − sin(t)1y] , +(3.1a,b) +with Px = Po sin(α) and Pz = Po cos(α), where Po = Ωp/Ω0 is the Poincar´e number +(Ωp being the angular velocity of precession and Ω0 that of the mantle) and α is the angle +of precession measured from 1z. Because the Poincar´e force r ×dtδ is linear in Cartesian +coordinates, the primary response of the fluid is a laminar uniform-vorticity flow (e.g. +Noir & C´ebron 2013; Kida 2020), on top of which secondary flows and turbulence can +develop. For analytical progress, we expand the velocity field as v += v0 + v1, where +v0 is the primary forced flow (which is mainly of uniform vorticity) and v1 represents +small-amplitude additional flows such that |v1| ≪ |v0|. We first seek analytical solutions + +Precession-driven flows in stress-free ellipsoids +7 +of the primary flow in §3.1, which are compared with DNS in §3.2. Then, we explore the +flow instabilities v1 growing upon the forced flow in §3.3. +3.1. Laminar forced flows +The forced laminar flows, which have been explored for a long time after the seminal +work of Poincar´e (1910), can be obtained using boundary-layer theory (BLT) in the +low-viscosity regime E ≪ 1 for SF-BC. To do so, we seek v0 as +v0(r, t) ≃ ωx(t)ex + ωy(t)ey + ωz(t)ez +� +�� +� +U(r,t) ++E1/2 �U(r, t) +(3.2) +where U(r, t) is a forced uniform-vorticity flow carrying angular momentum, and �U(r, t) +is the viscous flow within the boundary layer at the leading order in E1/2 (e.g. Rieutord +1992). A direct consequence of asymptotic expansion (3.2) is that the bulk flow for SF- +BC can be determined without explicitly solving for the boundary-layer flow (since the +latter has an amplitude that is E1/2 smaller than the bulk flow amplitude). The exact +viscous torque given by formula (2.14) can then be approximated as +Γ ν ≃ −16π +3 abc E diag +�(b2 − c2)2 +(b2 + c2)2 , (a2 − c2)2 +(a2 + c2)2 , (a2 − b2)2 +(a2 + b2)2 +� +ω. +(3.3) +The viscous flow E1/2 �U in expansion (3.2) has a contribution of amplitude O(E3/2) to +the viscous torque (since |ϵ( �U)| = O(E−1/2) and the volume scales as O(E1/2) within the +Ekman layer), which can be neglected compared with expression (3.3) in the asymptotic +regime E ≪ 1. We recover from formula (3.3) that Γ ν ·1i = 0 when the Cartesian vector +1i is a revolution symmetry axis, but also that the three components of the viscous torque +are non-zero when a ̸= b ̸= c. Then, the momentum equation reduces to +dtω − [(ω + Ωc) · ∇] U = −dtΩc + LΓ ν, +(3.4) +where Γ ν is the viscous torque given by formula (3.3) and L is the matrix given by the +inverse of expression (2.13b). The approximated viscous term is thus +LΓ ν = −5Ediag +�(b/c − c/b)2 +b2 + c2 +, (a/c − c/a)2 +a2 + c2 +, (a/b − b/a)2 +a2 + b2 +� +ω. +(3.5) +Equations (3.4) and (3.5) extend the asymptotic viscous model of Noir & C´ebron (2013) +to stress-free ellipsoids, but we remind the reader that this stress-free model is not valid +in spheres (since the angular momentum would be arbitrary in spheres because of Γ ν = +0). The close similarity between the no-slip and stress-free cases, for which only the +expression of the viscous term in equation (3.4) differs, suggests that the same interior +solution should be approached when E → 0 in no-slip and stress-free ellipsoids. +Precession is often characterised by |Px| ≪ 1 in planetary liquid cores (e.g. Noir & +C´ebron 2013). Hence, we seek asymptotic solutions of equation (3.4) in powers of Px as +ω(t) = ω(0)(t) + Px ω(1)(t) + P 2 +x ω(2)(t) + . . . +(3.6) +Since the mean rotation axis is Ω ∝ 1z when |Px| ≪ 1, we assume that a ̸= b (to avoid +the pathological situations outlined in §2 for the angular momentum conservation). The +zeroth-order solution ω(0)(t) corresponds to a decaying transient when t → ∞ (because +of viscosity). We thus discard ω(0)(t) in the following and solve the first-order problem +in Px. In the regime of vanishing viscosity E → 0, we obtain the first-order solution +ω(1) +x (t) ≃ − 1 + [1 + Pz]A1 +1 − [1 + Pz]2λ2so +cos(t), +ω(1) +y (t) ≃ 1 + [1 + Pz]B2 +1 − [1 + Pz]2λ2so +sin(t) +(3.7a,b) + +8 +J. Vidal & D. C´ebron +10−2 +10−1 +100 +101 +E t +−0.03 +−0.02 +−0.01 +0.00 +0.01 +0.02 +0.03 +ω · 1x +ω = (1/2) +� +V ∇ × v dV +ω = LL +0 +2 +4 +6 +E t +0 +0.01 +0.02 +0.03 +|ω| +ω = (1/2) +� +V ∇ × v dV +ω = LL +(a) +(b) +Figure 2. DNS of precessing ellipsoids with SF-BC at Po = −1.8, E = 5 × 10−4 and +Px = Po sin(α) = 10−2. Axisymmetric geometry a = 1.5 and b = c = 1. (a) Time evolution of +the Cartesian component ω · 1x and (b) absolute value |ω| of the angular velocity, computed +in the DNS either from the volume-averaged vorticity as ω = (1/2) +� +V ∇ × v dV or using the +angular momentum as ω = LL using expression (2.13). +and ω(1) +z (t) → 0, with A1 = 2a2/(a2 + c2), B2 = 2b2/(b2 + c2), and λso = √A1B2. We +have finally to compute the second-order solution ω(2), accounting for weakly nonlinear +interactions in the viscous interior, to estimate the axial angular velocity (since it is +undefined at the first order). An analytical solution can be obtained when E ̸= 0, showing +that ω(2) = ω(2) +z 1z, but the general expression of ω(2) +z +is too lengthy to be given here. In +the regime of vanishing viscosity E → 0, it simplifies into +ω(2) +z (t) = +c2 +4 D2 +2 +� +ω(2) +z ++ δω(2) +z +cos(2t) +� +(3.8a) +with the denominator D2 = a2b2 �Pz (Pz + 1/2) − c2 � +a2 + b2 + c2� +/4 and �Pz = Pz + 3/2, +where the amplitude of the mean geostrophic flow is given by +ω(2) +z += − +�c2 +2 + a2 �Pz +� �c2 +2 + b2 �Pz +� +a2 + b2 +(a/b − b/a)2 +��a +c − c +a +�2 ++ +�b +c − c +b +�2� +(3.8b) +and that of the oscillatory component by δω(2) +z += (Pz + 1)(a2 − b2)(a2b2 �P 2 +z − c4/4). It is +worth noting that the mean geostrophic flow ω(2) +z +has an amplitude that is independent of +E in the vanishing regime E → 0, which is somehow similar to the mean geostrophic flows +driven by nonlinear boundary-layer interactions for NS-BC (e.g. C´ebron et al. 2021). +A striking property of the asymptotic solution is that it exhibits two inviscid direct +resonances, which occur when the common denominator in expressions (3.7a,b) vanishes +at the two resonant values Po± given by λso [1 + Po± cos(α)] = ±1. The resonance +associated with Po+ actually corresponds to the inviscid resonance initially predicted by +Poincar´e (1910), which has been observed for no-slip boundaries (e.g. Vormann & Hansen +2018; Nobili et al. 2021; Burmann & Noir 2022). However, the second resonance at Po− +is new, although precession-driven flows have been explored for more than a century in +triaxial ellipsoids (e.g. Poincar´e 1910; Noir & C´ebron 2013). + +Precession-driven flows in stress-free ellipsoids +9 +3.2. Numerical simulations +We have checked that the analytic expressions are in excellent agreement with the +numerical integration of the exact uniform-vorticity model (3.4) when E → 0 (not shown). +Yet, it remains to confirm the validity of the asymptotic solutions against DNS with SF- +BC. We first show in figure 2 the time evolution of the rotation vector ω(t) in the DNS +(performed with comsol, as explained in §2). We illustrate the DNS at Px = 10−2 +with Po = −1.8 and E = 5 × 10−4, in the particular axisymmetric geometry a = 1.5 +and b = c = 1 (other parameters yield similar results, not shown). The fluid angular +velocity ω has been computed in the DNS using either the volume-averaged vorticity or +formula (2.13a) after having computed the angular momentum. Both methods are found +to be in excellent quantitative agreement for the SF-BC (as observed in the figure). +For such an axisymmetric geometry, we may naively think (before any computation) +that the long-term evolution of ωx (or equivalently that of Lx) is unconstrained due to +the vanishing component of the viscous torque Γ ν · 1x = 0 according to formula (3.3). +We observe that ωx initially displays a complicated transient (panel a), which dies out +because of viscosity as expected from the asymptotic theory. Then, it converges towards +a well-defined oscillatory state after a few viscous time scales (i.e. when E t ≫ 1 in +dimensionless units). The total angular velocity ω, which exhibits no long-term spurious +dynamics (panel b), has a small amplitude compared with the mean rotation axis of the +fluid Ω = 1z with respect to the inertial frame. We have checked that the final state +is robust, as it is recovered by varying the numerical resolution and adopting different +initial conditions for a few values of Po and E (although multiple solutions may exist +close to the inviscid resonances, as shown for sufficiently small Ekman numbers with +NS-BC in C´ebron 2015). +The comparison between the asymptotic results and the DNS is further illustrated in +figure 3, still considering the illustrative axisymmetic geometry a = 1.5 and b = c = 1 +(other geometries with a ̸= b give again similar results, not shown). The DNS are in +excellent quantitative agreement with the asymptotic solution, although the latter has +been obtained assuming E → 0, for both the time-averaged and the instantaneous angular +velocity (see panel b after seven viscous time scales). We also have checked that δω(2) +z +is accurately recovered in the DNS (not shown). The observed excellent quantitative +agreement with theoretical precession-driven flows has not been obtained using NS-BC +in ellipsoids, both in DNS (e.g. Noir & C´ebron 2013) and laboratory experiments (e.g. +Nobili et al. 2021; Burmann & Noir 2022). Finally, the DNS also confirm the physical +existence of the two inviscid resonances of solutions (3.7). +3.3. Asymptotic theory of flow instabilities +The forced laminar flow U(r, t), given by equation (3.4) when E ≪ 1, can be desta- +bilised by various hydrodynamic instabilities in ellipsoids. Precession-driven instabilities +are classified either as viscously driven if they only exist when E ̸= 0, or as inertial if they +survive when E = 0. Viscous instabilities exist in no-slip spheres, such as boundary-layer +instabilities (e.g. Lorenzani & Tilgner 2001; Buffett 2021) or the conical-shear instability +(e.g. Lin et al. 2015; C´ebron et al. 2019). On the contrary, the inertial instabilities +only exist in non-spherical geometries (e.g. Kerswell 1993; Wu & Roberts 2011; Vidal & +C´ebron 2017). In the following, we extend the prior inviscid linear analyses of the inertial +instabilities, which all considered precession at α = π/2 and in the precession frame (i.e. +only for spheroids), to account for the SF-BC and the time-dependent background flow +(3.7) in the mantle frame. To do so, we expand the governing equations with respect to +U (discarding the small-amplitude viscous flow E1/2 �U in the bulk, which is negligible + +10 +J. Vidal & D. C´ebron +−2 +−1 +0 +1 +Po +10−2 +10−1 +100 +ϵ +Asymptotic E → 0 +Non rotating +DNS +7.50 +7.51 +7.52 +7.53 +E t +−0.01 +−0.005 +0 +0.005 +0.01 +ω · 1x +Asymptotic E → 0 +DNS +(a) +(b) +Figure 3. Precession-driven flows (SF-BC) at Px = Po sin(α) = 10−2 for a = 1.5 and b = c = 1. +Comparison between asymptotic solution (3.7) and DNS at E = 5 × 10−4. (a) Time-averaged +angular velocity ϵ = |ω| as a function of Po. The fluid is not globally rotating when Po ≃ −1 +if |Px| ≪ 1 (grey area). Vertical dashed lines show the two resonances of asymptotic solutions +(3.7) at Po± = − +� +P 2 +x + (1/λso ∓ 1)2. Teal vertical line shows the region |Po| < 10−2 where no +α can satisfy Px = 10−2. (b) Value of |ω| as a function of the re-scaled time E t at Po = −1.8. +when E ≪ 1 as found in the DNS). The perturbation velocity v1, which is assumed to +be of small amplitude compared with U, is governed in the mantle frame by +∂tv1 + 2Ωc × v1 = L(v1) + 2E∇ · ϵ(v1) − ∇p, +(3.9a) +∇ · v1 = 0, +(3.9b) +with the linearised advection operator L(a) = −(a · ∇) U − (U · ∇) a. The perturbation +velocity v1 then satisfies the SF-BC (Mason & Kerswell 2002; Wu & Roberts 2009) +v1 · 1n|S = 0, +[ϵ(v1) · 1n] × 1n|S = 0. +(3.10a,b) +To explore the low-viscosity regime E ≪ 1, which is difficult to probe using DNS, we +develop an asymptotic model. We seek v1 using BLT as (e.g. Rieutord 1992) +v1(r, t) ≃ u(r, t) + E1/2 �u(r, t), +∇ · u = 0, +u · 1n|S = 0, +(3.11a–c) +where u(r, t) represents the inviscid bulk flow and �u(r, t) is the leading-order viscous flow +within the Ekman layer to satisfy SF-BC (3.10). Because the boundary-layer flow has +an amplitude that is E1/2 smaller than the bulk flow amplitude, SF-BC strongly weaken +the viscous instabilities in ellipsoids. In particular, the critical shear layers spawned by +the Ekman layer at the critical latitudes are almost suppressed in stress-free ellipsoids +without an inner core (Tilgner 1999). Consequently, the inertial instabilities triggered +in the (nearly) inviscid bulk are expected to be largely favoured in stress-free ellipsoids +(compared with viscous instabilities). +To solve problem (3.11), we introduce the finite-dimensional polynomial vector space +Vn spawned by the global real-valued incompressible elements {uk}, made of Cartesian +monomials xiyjzk of maximum degree i + j + k ⩽ n and satisfying the no-penetration +BC (e.g. Vidal et al. 2020; Vidal & C´ebron 2021a). Such vector elements are indeed +known to form a complete basis for smooth velocity fields in ellipsoids when n → ∞ (e.g. +Lebovitz 1989; Backus & Rieutord 2017). Then, we seek the bulk flow using the Galerkin + +Precession-driven flows in stress-free ellipsoids +11 +expansion (written using Einstein’s convention) +u(r, t) = αk(t)uk(r), +∇ · uk = 0, +uk · 1n|S = 0, +(3.12a–c) +where α = (α1, α2, . . . , αN)⊤ is the state vector of the modal coefficients. The number of +elements N for a given maximum degree n in expansion (3.12) is N = n(n+1)(2n+7)/6. +In practice, we truncate the polynomial expansion at the maximum degree n, substitute +the truncated expansion into equation (3.9) and, finally, project the resulting equations +onto every basis element ui to minimise the residual with respect to the real-valued inner +product defined by ⟨a, b⟩V = +� +V a · b dV . The governing equations then reduce to +M dtα = (L − C − D) α, +(3.13) +where M ij = ⟨ui, uj⟩V is the mass matrix, Cij = ⟨ui, 2Ωc×uj⟩V represents the Coriolis +force, Lij = ⟨ui, L(uj)⟩V is the matrix representing the linearised advection terms and +the viscous matrix D is given by (after integration by parts) +Dij = 2E +� +V +ϵ(ui) : ϵ(uj) dV +(3.14) +in which we have enforced SF-BC (3.10) in the projection to simplify the integration (e.g. +Guermond et al. 2013). As already noticed for the forced flow, a useful consequence of +expansion (3.11) is that the bulk flow u can be determined in equation (3.13) without +an explicit solution of �u for SF-BC. This has also been reported for asymptotic models +of thermal convection or waves in rotating stress-free spheres (Liao et al. 2001; Zhang & +Liao 2004). This is a noticeable difference from asymptotic models using NS-BC, which +require a matching between the boundary-layer flow and the interior solution (which are +of the same order of magnitude, e.g. Zhang et al. 2007, 2014). +Since asymptotic solution (3.7) is periodic of period T = 2π, we investigate the linear +stability using Floquet theory. We first compute the eigenvalues χ of the monodromy +matrix Φ(2π) given by +M dtΦ = (L − C − D) Φ, +Φ(0) = I, +(3.15a,b) +where I is the identity matrix. Then, we compute the complex-valued Lyapunov expo- +nents as µ = (1/T) log χ whose real part Re(µ) = σ is the growth rate of the instability. +As initially noticed by Kerswell (1993) and Wu & Roberts (2011), the finite-dimensional +polynomial is left invariant by the linear operator in the momentum equation, that is +L(Vn) ∈ Vn. Therefore, we can construct exact polynomial solutions of equation (3.9) +giving sufficient conditions for linear instability in the inviscid regime E = 0. +We show in figure 4(a) the results of the linear inviscid stability analysis at Px = 10−2. +We have numerically solved equation (3.15) using a fourth-order Runge-Kutta solver and +standard linear algebra routines. As in Kerswell (1993) and Wu & Roberts (2011), there +are no instabilities associated with the linear elements n = 1. The first instabilities, which +are here associated with the quadratic modes with n = 2, only occur near the resonance +at Po+. When n is increased, additional tongues of inertial (topographic) instabilities +appear with a growth rate scaling in the inviscid regime as +σtopo = O(ϵη), +(3.16) +where ϵ = |ω| is the mean value of the differential rotation between the fluid and the +mantle and η = a2/c2 − 1 is the polar flattening. The numerical prefactor is found to +be σtopo/(ϵη) ≈ 0.1 when n ⩽ 20 (as shown in the figure). Moreover, when n → ∞, +the growth is expected to approach the upper bound given in the unbounded short- + +12 +J. Vidal & D. C´ebron +−2 +−1.5 +−1 +−0.5 +0 +0.5 +Po +0 +0.02 +0.04 +0.06 +0.08 +0.1 +σ +n = 2 +n = 15 +n = 20 +σtopo = 0.1 ϵη +Kerswell (1993) +Non-rotating +−2 +−1.5 +−1 +−0.5 +0 +0.5 +Po +10−4 +10−3 +10−2 +10−1 +100 +σ +σtopo = 0.1 ϵη +SF-BC (E = 3 × 10−6) +SF-BC (E = 5 × 10−4) +NS-BC (E = 3 × 10−6, K = 10) +NS-BC (E = 3 × 10−6, K = 4) +Non-rotating +Figure 4. Growth rate σ of the inertial (topographic) instabilities growing upon flow (3.7) at +Px = Po sin(α) = 10−2, as a function of Po (using sampled values). Teal vertical line shows the +interval |Po| < 10−2 in which no α can satisfy Px = 10−2. The fluid is not globally rotating +near Po ≃ −1 when |Px| ≪ 1 (grey area). (a) Inviscid growth rate for various degrees n of the +global modes. Dashed black curve is obtained in the unbounded short-wavelength limit (Kerswell +1993). (b) Viscous effects. Dotted blue line shows the upper bound of the inviscid growth rate. +Olive coloured area shows the unstable region for SF-BC at E = 3 × 10−6, and thick red line +shows the unstable zone for SF-BC at E = 5×10−4 (both computed at n = 20). Purple coloured +curves show viscous growth rate (3.17) for NS-BC, with K ∈ [4, 10] to account for the Ekman +damping of the large-scale modes (see figure 5). +wavelength approximation (Kerswell 1993). This shows that the forced laminar flow is +generically unstable to short-wavelength perturbations without viscosity. +However, the short-wavelength modes are more damped by viscosity than the large- +scale ones. Consequently, viscous effects will select the allowable unstable modes for a +given value of the Ekman number. To show this, we have explored the linear stability +including viscous damping in figure 4(b). At E = 5 × 10−4, the forced flow is only +unstable in extremely thin tongues near the two resonances at Po± for the SF-BC. This +is consistent with the absence of instabilities in the DNS performed at E = 5×10−4 (see + +Precession-driven flows in stress-free ellipsoids +13 +figure 3). More challenging DNS with SF-BC at smaller values E = O(10−6), which are +beyond the scope of the present paper, could allow us to obtain instabilities for values +of Po in a larger interval. Finally, it is also useful to compare the stability of the forced +flow with SF-BC and NS-BC. A proper asymptotic theory for the no-slip case, rooted +in the BLT of the inertial modes (e.g. Greenspan 1968), will be considered elsewhere. +Nonetheless, an upper bound for the viscous growth rate of the inertial instabilities can +be estimated as +σtopo ≈ 0.1 ϵη − K +� +E[1 + Pz], +(3.17) +assuming that the fluid is rotating on average at 1 + Pz in the mantle frame. Here, the +numerical prefactor K = 4 − 10 heuristically accounts for the Ekman damping of the +large-scale flow structures with NS-BC (see figure 5). For the small value E = 3 × 10−6, +we observe that the forced flow at Px = 10−2 would be mainly stable with NS-BC (except +near the resonance Po+), whereas it would be unstable for other values of Po with SF- +BC. Therefore, the figure clearly illustrates that adopting SF-BC (instead of NS-BC) can +be useful to explore the turbulence driven by inertial instabilities in the bulk of the fluid. +4. Discussion +4.1. Physical insight from the Coriolis eigenmodes +We have illustrated with the case of precession-driven flows that the long-term evo- +lution of angular momentum is damped by viscosity in triaxial ellipsoids. Similarly, +viscosity affects the angular momentum in axisymmetric rotating ellipsoids if the mean +rotation axis Ω is not aligned with the revolution symmetry axis (even if Γ i · 1i = 0 in +such geometries, where 1i is the revolution axis along one of the principal semi-axes). +Asymptotic analysis offers a physical understanding of why the cases Ω ∝ 1i and Ω ̸∝ 1i +strongly differ in axisymmetric ellipsoids. +When E ≪ 1, the solutions of equations (2.1a,b) in stress-free or no-slip ellipsoids can +be rigorously expanded onto a combination of the inviscid eigenmodes of the (steady) +Coriolis operator given by (e.g. Backus & Rieutord 2017) +iλk∇ × Qk = −2∇ × (Ω × Qk), +∇ · Qk = 0, +Qk · 1n|S = 0, +(4.1a–c) +where [λk, Qk(r)] is the kth eigenvalue-eigenfunction pair. Only three of these eigenmodes +carry a non-zero angular momentum in ellipsoids (by virtue of the orthogonality of the +eigenmodes, see Ivers 2017), namely the spin over mode Qso, its complex conjugate +Q† +so and the zero-frequency geostrophic mode Qsup associated with axial (differential) +rotation along Ω. Because these three modes are uniform-vorticity flows such as Qk = +ωk,x ex + ωk,y ey + ωk,z ez, they are given by the matrix eigenvalue problem +� +� +0 +2a2Ωz/(a2 + c2) +−2a2Ωy/(a2 + b2) +−2b2Ωz/(b2 + c2) +0 +2b2Ωx/(a2 + b2) +2c2Ωy/(b2 + c2) +−2c2Ωx/(a2 + c2) +0 +� +� ωk = iλkωk +(4.2) +with Ω = (Ωx, Ωy, Ωz)⊤, where the rotation vector ωk = (ωk,x, ωk,y, ωk,z)⊤ of the +eigenmode Qk is given by the kth eigenvector of matrix (4.2). Consequently, the uniform- +vorticity components ωi(t) ei of the flow in expansion (2.11) are not mutually independent +in rotating ellipsoids but, instead, are tied to the dynamics of these modes. More precisely, +the equatorial components of the angular momentum L × 1Ω are coupled through the +dynamics of the two spin-over modes. Similarly, the axial angular momentum L · 1Ω +(related to the fluid spin-up) is piloted by the dynamics of the geostrophic mode Qsup. +From a physical viewpoint, whether viscosity affects the long-term evolution of angular + +14 +J. Vidal & D. C´ebron +−2 +−1 +0 +1 +2 +λk +2 +4 +6 +8 +10 +12 +−Re (τk)//E1/2 +−1 +−0.5 +0 +0.5 +1 +Im(τk)/E1/2 +−2 +−1 +0 +1 +2 +λk +100 +101 +102 +103 +104 +−τk/E +(a) NS-BC +(b) SF-BC +Figure 5. Viscous decay rates τk of the inertial modes of maximum polynomial degree n = 20, as +a function of the inviscid eigenfrequency λk ̸= 0. Axisymmetric ellipsoid with semi-axes a = 1.5 +and b = c = 1 rotating at the angular frequency Ω = 1z. (a) Complex-valued τk for NS-BC. +Colour bar shows the normalised imaginary part Im(τk). (b) Real-valued τk (i.e. Im(τk) = 0) +given by formula (4.6) for SF-BC. +momentum or not is thus deeply rooted in the viscous dynamics of these three eigen- +modes. We can quantify how viscosity impacts the inviscid eigenmodes by estimating the +global viscous decay rates τk of the Coriolis modes as +∂tQk|t=0 ≃ τk Qk. +(4.3) +For NS-BC, τk is a complex-valued quantity with a real part Re(τk) ⩽ 0 representing +the volume-averaged viscous decay rate, and an imaginary part Im(τk) characterising the +frequency shift due to viscous effects (e.g. Greenspan 1968). Typical values are illustrated +in figure 5 for a particular ellipsoidal geometry. It has also been recognised for a long time +that, for NS-BC (2.4), the viscous torque in the mantle frame is related to the viscous +damping of these three eigenmodes (e.g. Rochester 1976). In no-slip spherical geometries, +it is given by (see formula 35 in Rochester 1976) +Γ ν ∝ E1/2 [Re(τso) ω⊥ − Im(τso) 1z × ω⊥ + τsup ωz1z] +(4.4) +at the leading order in E (assuming Ω = 1z), where ω = ω⊥ + ωz1z = (ωx, ωy, ωz)⊤ +is the uniform vorticity of the forced flow. Note that similar expressions have been later +rediscovered for the particular case of precession as viewed in the precession frame (e.g. +Noir et al. 2003; Noir & C´ebron 2013). Formula (4.4) clearly shows that the equatorial +components L × 1z are damped by viscosity when Re(τso) ̸= 0 and, similarly, τsup ̸= 0 +(which is a real number for this mode) ensures that the axial angular momentum L·1z is +affected by viscosity. Since Re(τso) ̸= 0 and τsup ̸= 0 in no-slip spheres and ellipsoids, we +have Γ ν ̸= 0 from formula (4.4) such that the angular momentum is affected by viscosity +on long time scales for NS-BC. +Similar reasoning can be applied to the stress-free rotating case. It can be shown that +leading-order viscous torque (3.3) depends on the viscous decay rates [τso, τsup] for the +SF-BC (not given here, since it vainly makes the expression more complex because a full +description of the viscous cross-interactions between Qso and Qsup is required contrary +to the no-slip case). We can thus get physical insight into formula (3.3) by computing +the viscous decay rates for SF-BC. To do so, we expand the velocity as Qk + E1/2 �Qk +(Rieutord 1992), where �Qk is the boundary-layer flow such that Qk + E1/2 �Qk satisfies + +Precession-driven flows in stress-free ellipsoids +15 +SF-BC (2.3). The viscous decay rate for SF-BC is then given at the leading order in E +by (e.g. Liao et al. 2001) +τk +� +V +|Qk|2 dV = E +� +V +Q† +k · ∇2(Qk + E1/2 �Qk) dV. +(4.5) +Contrary to the no-slip case (for which the boundary-layer flow is of the same order +of magnitude as the inviscid flow, e.g. Greenspan 1968), an explicit solution of �Qk for +SF-BC is not required to estimate τk in equation (4.5). Indeed, the representative volume- +averaged viscous decay rate of all the eigenmodes is given at leading order in E for our +SF-BC by (e.g. Rieutord & Zahn 1997) +τk +� +V +|Qk|2 dV = −2E +� +V +ϵ(Qk) : ϵ(Q† +k) dV. +(4.6) +Expression (4.6) generalises formula (3.14) in Liao et al. (2001), which is only valid for +spheres (see Appendix B), to triaxial ellipsoids. Since the right-hand side of equation +(4.5) is real, we have τk ⩽ 0 for SF-BC. Consequently, there is no viscous correction of +the inviscid eigenfrequency λk at the leading order in E for SF-BC (as initially reported +in Liao et al. 2001). Formula (4.6) is illustrated in figure 5 for a particular configuration. +We recover from the formula that τso = τsup = 0 in spherical geometries (since Qso and +Qsup are exact solid-body rotations in spheres), which agrees with the fact that Γ ν = 0 +in spheres (e.g. Jones et al. 2011). +Explicit expressions of τso and τsup can be obtained for the uniform-vorticity modes +in ellipsoids, because the eigenvectors [ωso, ωsup] of matrix (4.2) can be analytically +obtained. The analytical formula of τso, which is too lengthy to be given here, shows that +τso ̸= 0 in every non-spherical geometry. The mathematical reason is that the spin-over +mode Qso is no longer a solid-body rotation in ellipsoids (i.e. ϵ(Qk) is non-zero for the +spin-over mode in ellipsoids). Thus, from a physical viewpoint, a non-zero boundary-layer +flow �Qso is required to match the SF-BC within a thin Ekman boundary layer. Since the +spin-over mode is damped by viscosity in ellipsoids, the equatorial angular momentum +L×1Ω is affected by viscosity on long time scales (even in axisymmetric ellipsoids). After +little algebra, the decay rate τsup is explicitly given by +τsup +E +� +V +|Qsup|2 dV = −16π +3 abc +� +Ω2 +x(b2 − c2)2 + Ω2 +y(a2 − c2)2 + Ω2 +z(a2 − b2)2� +, +(4.7) +where the axial geostrophic mode is Qsup = ωsup,x ex +ωsup,y ey +ωsup,z ez with ωsup,x = +Ωx(b2+c2), ωsup,y = Ωy(a2+c2) and ωsup,z = Ωz(a2+b2). Formula (4.7) shows that τsup ̸= +0 when a ̸= b ̸= c, illustrating that the axial geostrophic mode is damped by viscosity +in triaxial geometries. Therefore, the physical reason why Γ ν ̸= 0 in triaxial stress- +free ellipsoids is that the spin-over and geostrophic modes are damped by viscosity (as +evidenced by the non-zero decay rates τso ̸= 0 and τsup ̸= 0 in such geometries). Moreover, +formula (4.7) shows that τsup = 0 when Ω is an axis of revolution of the geometry (i.e. +when Ω ∝ 1x if b = c, Ω ∝ 1y if a = c, or Ω ∝ 1z if a = b). The axial geostrophic mode +is thus unaffected by viscous dissipation, which explains why the long-term evolution +of L · 1Ω is physically unconstrained in such pathological configurations. This was the +situation previously considered for precession-driven flows in spheroids (Lorenzani & +Tilgner 2003; Wu & Roberts 2009; Guermond et al. 2013). Yet, the conclusion is not +valid for every axisymmetric geometry with global rotation. Indeed, we have τsup ̸= 0 in +axisymmetric geometries if Ω is not the revolution symmetry axis (such that L · 1Ω will +be damped by viscosity). +The BLT of Coriolis eigenmodes has thus explained why the long-term angular mo- + +16 +J. Vidal & D. C´ebron +−2.2 +−2 +−1.8 +Po +0 +0.2 +0.4 +0.6 +0.8 +1 +ϵ +b = 0.7 +b = 0.9 +b = 1.1 +b = 1.3 +−0.2 +0 +0.2 +Po +0 +0.2 +0.4 +0.6 +0.8 +1 +ϵ +b = 0.7 +b = 0.9 +b = 1.1 +b = 1.3 +(a) Po− +(b) Po+ +Figure 6. Double resonance at Po± of the forced precession-driven flow in ellipsoids for SF-BC +with a = 1 and c = 0.9 (values of b given in the legend). Time-averaged differential rotation +ϵ = |ω| of numerical solutions of equation (3.4) at E = 10−3 and small precession angle α = 3◦. +Vertical dashed lines show Po± predicted by equation (4.8) at b = 1. +mentum evolution is damped by viscosity in triaxial geometries, but also in axisymmetric +ellipsoids if the mean rotation axis Ω is not the revolution symmetry axis. +4.2. Resonance conditions for mechanical forcings +A key property of the primary uniform-vorticity flow is its ability to enter in direct +resonance with the precession forcing (as evidenced by the divergent amplitude of +the asymptotic solution). A direct resonance requires a close spatial and temporal +matching between the Poincar´e force and the flow response (e.g. Greenspan 1968). The +spatial matching is ensured by the fact that both the Poincar´e force and the forced +uniform-vorticity flow are linear in the Cartesian coordinates. Heuristically, the temporal +resonance condition requires that the frequency ωp of the forcing (for monochromatic +forcings) must be equal (or close) to the angular frequency f of the forced flow in +the mantle frame, which gives f = ± ωp. The latter condition generally predicts the +existence of two resonances for mechanically driven flows in ellipsoids (if the spatial +resonance conditions are satisfied). A quick inspection of equation (3.4) shows that the +uniform-vorticity dynamics roughly corresponds to that of a harmonic oscillator driven +by the Poincar´e force in the inviscid regime E = 0. Consequently, direct resonances occur +when the forced flow corresponds to a free oscillatory eigenmode of the unforced system, +namely the spin-over mode Qso such that f ∝ λso (up to a normalisation prefactor). +For this reason, longitudinal librations (which only directly excite the zero-frequency +geostrophic mode) do not exhibit any inviscid resonance in spheres (e.g. Zhang et al. +2013) or ellipsoids. On the contrary, latitudinal librations can trigger the spin-over mode +and the corresponding forced laminar flow exhibits two inviscid resonances occurring +at λso = ± ωp in non-spherical geometries (Zhang et al. 2012; Vantieghem et al. 2015), +where ωp is the libration angular frequency. Similarly, a second resonance has already +been found for the interaction between tides and precession in triaxial ellipsoids (C´ebron +et al. 2010). A second resonance for pure precession is thus also expected in ellipsoids +from simple theoretical arguments. Assuming that the forced uniform-vorticity flow is +oscillating in the mantle frame at the effective angular frequency f ≃ [1 + Pz] λso when +|Px| ≪ 1, the temporal resonance condition predicts two direct resonances for precession + +Precession-driven flows in stress-free ellipsoids +17 +−2.0 +−1.9 +−1.8 +−1.7 +−1.6 +Po +10−2 +10−1 +100 +ϵ +a = 2, c ≃ 1.10 +a = 1.5, c = 1 +a = 1.1, c ≃ 0.876 +Poincar´e +DNS +10−3 +10−2 +10−1 +100 +101 +a − b +10−2 +10−1 +ϵ +SF-BC (a ̸= b) +Poincar´e +(a) +(b) +Figure 7. Behaviour of the forced flow near the second resonance Po− in stress-free ellipsoids +with b = 1 and Px = 10−2. The resonant value is fixed at the value Po− obtained with +a = 1.5 and c = 1 (as in figure 3). To maintain a fixed resonance when a is varied, the +polar axis is given by c = 0.5[−2a2 − 2 + 2 +� +−32a2 ∆1/2 + 1 + a4 + 2 (8∆ + 7)a2]1/2 with +∆ = (Po− − Px)(Po− + Px). In the two panels, the dashed teal line shows the expected inviscid +value from Poincar´e solution (4.9) for a = b = 1. (a) Comparison between DNS at E = 5 × 10−4 +and asymptotic solution (3.7). (b) Numerical solutions of equation (3.4) for SF-BC at Po = Po− +and E = 10−3. +at the resonant Poincar´e numbers Po± given by +1 + Po± cos(α) = ±1/λso, +(4.8) +where λso = 2ab/ +� +(a2 + c2)(b2 + c2) is here the eigenfrequency of the spin-over mode in +equation (4.2) with Ω = 1z (see also formula 3.21 in Vantieghem 2014). The above +condition is exactly the resonance condition of asymptotic solution (3.7). The two +resonances at [Po−, Po+] are thus robust features of precession-driven flows, but it +remains to elucidate why the second resonance at Po− has not been reported before. +We have numerically solved equation (3.4) in time to explore the behaviour of the +solutions near the double resonances in figure 6. The two resonances at Po± are con- +tinuously shifted when b is varied and, at b = 1, the resonant value Po− differs from +Po+ as observed in panel (b). This directly results from condition (4.8), which predicts +that the two resonances are linked by [Po+ + Po−] cos(α) = −2. This clearly shows that +the two direct resonances do not merge together in ellipsoids. We further explore the +behaviour near Po− in figure 7. We have fixed the resonant value Po− at its value given +in figure 3 for a = 1.5 and b = c = 1 and, then, adjusted the polar axis c to maintain the +resonance at Po− for different values of a. We observe that the width of the resonance +peak decreases when a → b (panel a). This is a purely inviscid feature of the asymptotic +solution, which is recovered in the DNS. The particular case a = b is not formally defined +for SF-BC, but it can be approached by decreasing a − b (panel b). The amplitude of +the stress-free solution at Po = Po− is limited by the viscosity and approaches, when +a → b, the inviscid Poincar´e solution for a = b. The differential rotation ϵ of the inviscid +Poincar´e solution is given by (assuming ωz = 1, see Appendix B in Wu & Roberts 2011) +ϵ = +���� +Px(2 + η) +η + 2(1 + η)Pz +���� , +(4.9) +which is non-divergent when Po = Po−. This agrees with a lengthy mathematical analysis + +18 +J. Vidal & D. C´ebron +0 +0.2 +0.4 +0.6 +0.8 +1 +z +0 +0.2 +0.4 +0.6 +(vf · 1z)/(UE1/2) +Ekman layer ≃ 5 E1/2 +(a) +(b) +Figure 8. DNS of precession-driven flow with SF-BC at Po = −1.8, Px = Po sin(α) = 10−2 and +E = 5 × 10−4. Axisymmetric geometry a = 1.5 and b = c = 1. Normalised velocity vf/(UE1/2), +as defined in expansion (2.11), at time t = 39530 where U = 0.0129 is the maximum of |vf|. (a) +Three-dimensional rendering of the velocity magnitude using a linear scale. (b) Axial velocity +component as a function of z along the c-axis. +of the behaviour near the inviscid resonances (not given here), which shows that the +second inviscid resonance at Po− disappears in spheroids with a = b contrary to the +other resonance at Po+ (e.g. Busse 1968; Noir & C´ebron 2013). +We have thus understood why precession-driven flows are subject to two inviscid +resonances in triaxial ellipsoids, which occur at the resonant Poincar´e numbers Po± +given by equation (4.8) when |Px| ≪ 1. Since the two resonances are inviscid features of +the forced flow in ellipsoids, they exist for both SF-BC and NS-BC. The second resonance +actually disappears in spheroidal geometries a = b (i.e. its amplitude is vanishing), which +explains why previous works in spheroids have not observed it (e.g. C´ebron 2015; Nobili +et al. 2021). Previous studies in triaxial geometries (e.g. Noir & C´ebron 2013; Burmann +& Noir 2022) have also overlooked it, because it usually occurs at |Po−| ≫ |Po+|. +4.3. Implications for DNS +We have shown that the long-term evolution of angular momentum is affected by +viscosity, due to the existence of an Ekman boundary layer in rapidly rotating ellipsoids. +The uniform-vorticity elements carrying angular momentum in expansion (2.11) do not +indeed satisfy the SF-BC in triaxial geometries. Thus, they are associated with an Ekman +boundary layer to match the boundary conditions. This is a noticeable difference with +the more usual spherical geometry, in which Γ ν = 0 (e.g. Jones et al. 2011). The Ekman +boundary layer in ellipsoids is clearly observed in figure 8. Its typical thickness is still +O(E1/2) but, contrary to the case of NS-BC, the amplitude of the boundary-layer flow +is O(E1/2) smaller than the bulk flow amplitude (in agreement with Rieutord 1992). +This could have implications for numerical studies using stress-free boundaries. A +numerical strategy has to be employed to ensure the conservation of angular momentum +in spherical codes (e.g. Jones et al. 2011). This is no longer necessary in triaxial ellipsoids +since Γ ν ̸= 0 (albeit such a strategy may be considered to ensure the conservation of +the axial angular momentum if the mean rotation axis is an axis of revolution symmetry, +as proposed in Guermond et al. 2013). However, for the moderate values of the Ekman +number achievable in DNS, the flow within the Ekman layer will modify the value of +the viscous torque (which pilots the long-term evolution of angular momentum). Indeed, + +1.2 +1 +0.8 +0.6 +0.4 +0.2Precession-driven flows in stress-free ellipsoids +19 +instead of using expression (2.14), the viscous torque is usually computed with the surface +integral Γ ν = 2 E +� +S r × (∇ · ϵ) dS as +Γ ν = 2 E +� +S +r × T dS = 2 E +� +S +r × [(T · 1n) 1n] dS, +(4.10) +in which we have used formula (9) in Rochester (1962) for a symmetric tensor to obtain +the first equality and, then, have written the surface traction as T = (T · 1n) 1n − 1n × +(1n × T ) = (T · 1n) 1n on the boundary for SF-BC (2.3). Formula (4.10) shows that +the normal component of the surface traction, which is non-zero in the presence of an +Ekman layer in stress-free ellipsoids, contributes to the viscous torque. Hence, numerical +and local approximations of SF-BC (2.3) have no reasons to yield a vanishing torque +component in formula (4.10) for axisymmetric ellipsoids if the boundary layer is not +sufficiently resolved (as observed in some DNS, not shown). Using a refined boundary- +layer mesh may thus be required to properly describe the Ekman layer in ellipsoids and +ensure sufficient torque accuracy (which can be used to check the numerical convergence). +4.4. Scaling laws +Despite the existence of a thin Ekman layer, we believe that adopting SF-BC in +global simulations is useful to probe bulk mechanisms that can be hampered by viscous +effects when NS-BC are employed. The case of precession is illuminating in this respect. +Indeed, the laminar precession-driven flow can be destabilised by several hydrodynamic +instabilities in no-slip ellipsoids, such as the inertial (topographic) instabilities outlined +in §3.3 and the conical-shear instability (CSI). The former are due to the ellipticity +of the boundary and survive in the inviscid regime E = 0. On the contrary, the CSI +is a parametric instability existing because of the viscous conical shear layers spawned +from the Ekman layer at the critical latitudes (Lin et al. 2015). In addition, precession +also often triggers boundary-layer instabilities within the Ekman layer for NS-BC (e.g. +Lorenzani & Tilgner 2001; C´ebron et al. 2019; Buffett 2021). A comprehensive study of +these instabilities deserves further work, but we can estimate their relevance as follows. +As outlined in §3.3, the typical inviscid growth rate of the precession-driven inertial +instabilities is given by formula (3.16) for the large-scale modes. For the CSI, the growth +rate in full spheres and ellipsoids is given by (Lin et al. 2015; Horimoto et al. 2020) +σCSI = O(ϵE1/5). +(4.11) +Quantitatively, a necessary condition for the existence of the two instabilities is that +growth rates (3.16) and (4.11) are larger than the viscous damping. For the NS-BC, this +damping is mainly due to the Ekman layer and its amplitude is of the order O(E1/2) +(Greenspan 1968). Actually, it appears that large-scale inertial instabilities are difficult +to obtain for the moderately small values of the Ekman number usually considered in +experiments or DNS (as outlined in figure 4). +A linear analysis is, however, not sufficient to determine the physical relevance of +these instabilities. In particular, scaling laws are worth finding to estimate the strength +of the precession-driven flows driven by such instabilities. Indeed, the inertial instabilities +have presumably a saturation amplitude almost independent of the Ekman number (as +found for the turbulence driven by tidal instabilities, e.g. Grannan et al. 2017), whereas +the CSI amplitude could decrease when E → 0 as the instability results from viscous +effects. A rigorous description of the nonlinear regimes requires dedicated simulations, +but the saturation amplitudes can be crudely estimated using simple order-of-magnitude +arguments (which have already proven useful for tidal flows, e.g. in Barker & Lithwick +2013; Barker 2016b). We assume that the flow amplitude U resulting from the primary + +20 +J. Vidal & D. C´ebron +10−2 +10−1 +100 +ϵ − ϵc +100 +101 +U E−2/5 +Ef = 10−4 +Ef = 3 × 10−5 +Ef = 10−5 +−3.0 +−2.5 +−2.0 +−1.5 +−1.0 +log10 |Po| +10−17 +10−14 +10−11 +10−8 +10−5 +E +10−6 +10−4 +10−2 +1 +η = a2/c2 − 1 +Early Earth +Moon +Early Moon +Horimoto et al. (2020) +Nobili et al. (2021) +η = E1/5 +Utopo ≫ UCSI +Utopo ∼ UCSI +(a) +(b) +Figure 9. (a) Comparison between scaling law (4.13) and DNS for |Po| < 0.1 in no-slip full +spheres from the database of figure 7 in C´ebron et al. (2019), with Ef = E/|1 + Po| ≃ E +when +|Po| +≪ +1. +Colour +bar +indicates +log10 |Po|. +Grey +area +shows +the +scaling +law +U E−2/5 = (6.5 ± 1.5) (ϵ − ϵc) and dashed line U E−2/5 = 6.5 (ϵ − ϵc), where ϵc ≈ 7E3/10 +f +is an +estimate of the onset value (see equation 17 in C´ebron et al. 2019). (b) Competition between the +inertial instabilities and the CSI in precessing ellipsoids. Empty blue squares □ show conditions +for which inertial (topographic) instabilities are expected, and red crosses × indicate where the +CSI is expected or observed. Grey area shows η ∝ E2/5 with a (unknown) prefactor chosen in +the range [1, 100], in which we expect Utopo ∼ UCSI when both instabilities exist. Hatched area +is the region where σtopo ≳ σCSI (if the two instabilities coexist). White area is the region where +Utopo ≪ UCSI. Estimates for the early Moon and Earth taken from Appendix C with η ≈ 2f. +instability grows until secondary instabilities, characterised by the growth rate σsec, +become strong enough to prevent further growth of the primary instability. A saturated +turbulent regime would then be obtained when U ∼ σsecℓ, where ℓ is a characteristic +length scale of the primary unstable flow. The nonlinear saturation of the inertial +(topographic) instabilities would thus be given by (in dimensionless units) +Utopo = O(ϵη) +(4.12) +with ℓ ∼ 1 for a large-scale instability. A good agreement with the above scaling law +has been found using DNS in shearing periodic boxes (Barker 2016b), but the scaling +law might be different for short-wavelength instabilities with ℓ ≪ 1. Using the same +reasoning for the CSI, the relevant length scale is likely the width of the critical shear +layer ℓ ∼ E1/5 (Lin et al. 2015). Assuming that the CSI is limited by secondary CSI +within the critical shear layers, we obtain the (dimensionless) scaling law +UCSI = O(ϵE2/5). +(4.13) +We compare in figure 9(a) the above scaling law with previously published DNS in no-slip +full spheres (Lin et al. 2015; C´ebron et al. 2019). Considering the full sphere geometry +allows us to discard the possible CSI resulting from the inner boundary (which would give +a different scaling). However, even in the full sphere, identifying the instability mechanism +is difficult due to the competition between the CSI and the boundary-layer instabilities. +Moreover, due to the non-trivial dependence of the two viscously driven instabilities +with the forcing parameters, it is unlikely that a single scaling law could fully describe +the entire simulation dataset. The onset distance is indeed difficult to estimate (e.g. see +figure 6 in C´ebron et al. 2019). Besides, the simulations may not be in the asymptotic + +Precession-driven flows in stress-free ellipsoids +21 +regime E ≪ 1. Despite such uncertainties, a fairly good agreement is found between the +DNS and scaling law (4.13) sufficiently far from the onset. This suggests that the CSI +was present in the nonlinear regime and that its saturation amplitude obeys scaling law +(4.13) for sufficiently small values of the Ekman number. +Finally, the comparison between scaling laws (4.12) and (4.13) shows that the inertial +instability would have a larger amplitude than the CSI when η ≫ E2/5 (if the two +instabilities were simultaneously triggered). The resulting regime diagram is illustrated +in figure 9(b), using planetary estimates given in Appendix C. Precession-driven inertial +instabilities may only have been excited in the primitive liquid cores of the Earth and +Moon, whereas the CSI is expected to be present (respectively absent) in the core of +the Moon (respectively the Earth) during its whole history (Lin et al. 2015; Landeau +et al. 2022). In the early Moon, the inertial instabilities may have dominated the CSI +in flow amplitude (although the CSI may have had a larger growth rate than the +inertial instabilities according to previous formulas, not shown). Therefore, the inertial +instabilities may actually be more relevant than the CSI for some planetary conditions +(although they have not been convincingly observed yet in experiments, e.g. Nobili et al. +2021; Burmann & Noir 2022). This could be key for the generation of planetary magnetic +fields, as initially postulated for the geodynamo (Malkus 1968). Preliminary estimates of +the dynamo capability of the precession-driven instabilities, obtained using (speculative) +order-of-magnitude arguments, are given in Appendix C. +5. Conclusion +5.1. Summary +We have investigated precession-driven flows in stress-free ellipsoids, using asymptotic +analysis and targeted DNS. We have developed a reduced model for SF-BC to determine +the forced uniform-vorticity flows, which carry angular momentum. We have shown that +angular momentum is affected on long time scales by viscosity in triaxial ellipsoids, +but also in axisymmetric geometries if the mean rotation axis is not a revolution +symmetry axis. This is a noticeable difference from spherical geometries, in which angular +momentum is unaffected by viscosity. The fundamental reason is that the flows carrying +a non-zero angular momentum in ellipsoids are associated with an Ekman boundary +layer in rotating ellipsoids. From a numerical viewpoint, a boundary-layer mesh may be +necessary to get numerical convergence of the angular momentum in rotating ellipsoids. +We also have obtained the analytical solution of the time-dependent laminar flow forced +by precession in the mantle frame, which is valid for planetary parameters and triaxial +geometries. The comparison with the DNS has shown that, even for moderately small +values of the Ekman number, the forced laminar flow in the DNS converges to the +asymptotic solution in the vanishing viscosity regime. Moreover, we have uncovered a +second (inviscid) resonance of the forced laminar flow in triaxial ellipsoids. +Then, we have explored the inertial instabilities growing upon the forced laminar flow +in the bulk, which survive in the inviscid regime E = 0. We have shown that these +instabilities could be more easily observed in stress-free ellipsoids than in no-slip ones (at +least for the moderate values E ≳ 10−6 considered in DNS). We have finally proposed +scaling laws for the velocity amplitude of the inertial instabilities and of the CSI, which +are in good agreement with previous DNS. The comparison between the two scaling +laws confirms that replacing NS-BC with SF-BC in the mantle frame could be useful +to directly probe scenarios of bulk turbulence in the low-viscosity regime (which are of +interest for planetary modelling). + +22 +J. Vidal & D. C´ebron +5.2. Perspectives +Despite the presence of a thin Ekman boundary layer, we believe that SF-BC are +relevant for global models of mechanically driven flows. The stress-free model could be +used to investigate the saturated flows driven by the inertial (topographic) instabilities +in precession ellipsoids and, then, their dynamo capability for planetary applications (as +outlined in Appendix C). Stress-free models could indeed shed new light on alternative +mechanisms giving birth to dynamo fields in planetary interiors. For instance, the past +dynamo of the Moon may have been driven by precession (e.g. Dwyer et al. 2011). Yet, +previous numerical investigations of precession-driven dynamos failed to reproduce large- +scale magnetic fields in spherical geometries (C´ebron et al. 2019). This could result from +the fact that the turbulence was driven in those simulations by viscous flows (e.g. the +CSI or boundary-layer instabilities), which may be negligible in amplitude compared +with the turbulence driven by the inertial (topographic) instabilities in the early Moon +(as discussed in §4.4). This hypothesis could be tested in simulations using stress-free +ellipsoids. Similarly, energetic arguments suggest that the dynamo of the early Earth +may have been sustained by tidal flows (Landeau et al. 2022). However, the associated +fluid dynamics remains to be quantitatively studied to go beyond prior proof-of-concept +simulations (Reddy et al. 2018; Vidal et al. 2018). Precessing stress-free ellipsoids are +also relevant for short-period hot Jupiters (Barker 2016b), or gaseous planets with a big +moon outside the equatorial plane (e.g. the Neptune/Triton pair, Wicht & Tilgner 2010). +Finally, SF-BC could also be used to revisit the long-standing problem associated +with the generation of geostrophic flows in rotating fluids (Greenspan 1969). Nonlinear +interactions within the Ekman boundary layers for NS-BC (e.g. Busse 1968; C´ebron +et al. 2021) or in the bulk through the action of the Reynolds stresses (e.g. Zhang +& Liao 2004; Livermore et al. 2016), are usually invoked, but geostrophic flows can +also result from bulk turbulence. However, it remains unclear whether two- or three- +dimensional rotating bulk turbulence is established in natural systems (e.g. Le Reun +et al. 2019). This fundamental problem has been attacked in cylindrical or plane-layer +geometries (e.g. Kerswell 1999; Brunet et al. 2020; Le Reun et al. 2020). Yet, the latter +geometries are not directly relevant for planetary modelling, due to the absence of the +so-called topographic beta effect that strongly modifies the geostrophic flows in spheres +and ellipsoids (e.g. Greenspan 1968). We believe that using SF-BC opens the way for +new fundamental studies dealing with the interplay between waves and geostrophic flows +in global geometries. +Acknowledgements. We acknowledge the three anonymous referees for their constructive +criticisms, which considerably improved the quality of the manuscript. We also acknowledge +the editor, N. Balmforth, for his careful editorial work. +Funding. This work received funding from the European Research Council (ERC) under the +European Union’s Horizon 2020 research and innovation programme via the theia project (grant +agreement no. 847433). ISTerre is part of Labex OSUG@2020 (ANR10 LABX56). +Declaration of interest. The authors report no conflict of interest. +Author ORCIDs. +ID J´er´emie Vidal https://orcid.org/0000-0002-3654-6633; +ID David C´ebron https://orcid.org/0000-0002-3579-8281. +Author contributions. The paper is an idea of J.V., who designed the study, conducted the +asymptotic theory and developed the bespoke numerical code. D.C. conducted the finite-element +computations using comsol, and analytically obtained the second-order geostrophic flow. Both +authors discussed and approved the results presented in the article. J.V. drafted the paper, and +both authors gave final approval for submission. + +Precession-driven flows in stress-free ellipsoids +23 +Appendix A. Angular momentum for compressible fluids +We investigate whether alternative definition (2.8), which has proven useful for incom- +pressible flows, can be extended to compressible flows with a spatially varying density +ρ(r). For mathematical tractability, we assume that the density does not vanish on +the ellipsoidal boundary. Then, we expand the velocity of compressible flows using the +weighted Helmholtz decomposition in rigid ellipsoids as (e.g. Vidal & C´ebron 2020) +v = (1/ρ) ∇ × A + ∇Φ, +v · 1n|S = 0, +(A 1a,b) +where A is a vector potential and Φ is a scalar potential. The first subspace represents +anelastic flows satisfying ∇ · (ρv) = 0 (e.g. Jones et al. 2011), whereas the irrotational +subspace represents compressible flows with ∇ · (ρv) ̸= 0 (such as the acoustic modes +without rotation, e.g. Vidal & C´ebron 2021a). This spectral decomposition has the great +advantage of being compatible with the natural inner product of the fully compressible +(and anelastic) problem (e.g. Sobouti 1981; Clausen & Tilgner 2014) +⟨a, b⟩V = +� +V +ρa† · b dV, +(A 2) +where a† is the complex conjugate of the vector a, contrary to the usual Helmholtz +decomposition v = ∇ × A + ∇Φ. Consequently, the two subspaces in decomposition +(A 1) are mutually orthogonal with respect to inner product (A 2). Guided by planetary +applications, we only consider in the following density profiles of the form +ρ(r) = ρ0(x2/a2 + y2/b2 + z2/c2), +(A 3) +for which the density is constant on every homothetic ellipsoidal shell in the interior. +Such density profiles are indeed often assumed in compressible planetary models, where +they represent background density profiles (e.g. in ellipsoids Clausen & Tilgner 2014; +Vidal & C´ebron 2020). +A.1. Direct calculation +The angular momentum is defined for compressible fluids as L = +� +V r × (ρ0v) dV . As +in the incompressible case, the anelastic subspace has elements with non-zero angular +momentum (e.g. in spheres Jones et al. 2011). Hence, it only remains to calculate the +angular momentum associated with the compressible subspace in decomposition (A 1). +A direct calculation gives (using formula B26 in Mathews et al. 1991) +� +V +r × (ρ0∇Φ) dV = − +� +V +Φ (r × ∇ρ0) dV − +� +V +∇ × (ρ0Φ r) dV, +(A 4a) += − +� +V +Φ (r × ∇ρ0) dV + +� +S +ρ0Φ (r × 1n) dS. +(A 4b) +It shows that, if the density is of the form (A 3), the compressible subspace has no angular +momentum in spheres (since ∇ρ0 ∝ r). On the contrary, the compressible subspace in +spectral decomposition (A 1) has always a non-zero angular momentum in ellipsoids. +A.2. Projection approach +We have outlined that the two subspaces in decomposition (A 1) have a non-zero +angular momentum in compressible ellipsoids. The remaining question is whether, as for +incompressible flows, this angular momentum is solely carried by the uniform-vorticity +elements ei(r) given by formula (2.9) in rigid ellipsoids. We project the velocity onto the + +24 +J. Vidal & D. C´ebron +three uniform-vorticity elements with respect to inner product (A 2), obtaining +� +V +ei · (ρ0v) dV = +� +V +(1i × r) · ρ0v dV +� +�� +� +L·1i ++ +� +V +∇Ψi · (ρ0v) dV +(A 5) +where L · 1i are the Cartesian components of the angular momentum. We recover from +the above expression that the compressible angular momentum is the projection onto +the solid-body rotations 1i × r in spherical geometries (for which Ψi = 0). An admissible +decomposition for compressible spherical flows is thus (e.g. Mathews et al. 1991) +v(r, t) = ω(t) × r + vf(r, t), +� +V +r × (ρ0vf) dV = 0, +(A 6a,b) +where the compressible flow vf has no angular momentum by definition since ⟨ω × +r, vf⟩ = 0. In ellipsoids, the last volume integral in equation (A 5) can be simplified by +using the divergence theorem and decomposition (A 1). It gives +� +V +∇Ψi · (ρ0v) dV = +� +S +Ψx (ρ0v) · 1n dS +� +�� +� +0 +− +� +V +Ψi ∇ · (ρ0v) dV, +(A 7a) += +� +0 +if +∇ · (ρ0v) = 0, +− +� +V Ψi ∇ · (ρ0∇Φ) dV +if +∇ · (ρ0v) ̸= 0. +(A 7b) +Equation (A 7) shows that the angular momentum of anelastic flows with ∇ · (ρ0v) = 0 +is rigorously given by L · 1i = ⟨ei, v⟩, as in the incompressible case. We can thus extend +formula (2.11) to anelastic flows as +v(r, t) = U(r, t) + vf(r, t), +∇ · (ρ0vf) = 0, +� +V +r × (ρ0vf) dV = 0, +(A 8a–c) +where U(r, t) is the uniform-vorticity flow given by expression (2.12), and vf is an +anelastic flow with ⟨U, vf⟩ = 0 by definition. However, in the fully compressible case, the +angular momentum cannot be obtained as the projections of the compressible flow onto +the uniform-vorticity elements in ellipsoids (because (A 7) does not vanish). Moreover, +we have by virtue of the divergence theorem +� +V +ei · (ρ0∇Φ) dV = − +� +V +Φ ∇ · (ρ0ei) dV = − +� +V +φ (ei · ∇ρ0) dV = 0 +(A 9) +if the density is of the form (A 3) because ei · ∇ρ0 ∝ ei · 1n = 0 on every homothetic +ellipsoidal shell in the volume (i.e. not only on the outer ellipsoidal boundary). Thus, +the compressible subspace can have a non-zero angular momentum that is not carried by +the uniform-vorticity elements in ellipsoids (since we have simultaneously ⟨ei, ∇Φ⟩ = 0 +and +� +V r × (ρ0∇Φ) dV ̸= 0). In such configurations, a possible generalisation of anelastic +expansion (A 8) to the compressible case could be +v(r, t) = U(r, t) + vf(r, t) + ∇Φ, +∇ · (ρ0vf) = 0, +� +V +r × (ρ0vf) dV = 0, (A 10a–c) +where U(r, t) is a uniform-vorticity flow given by expression (2.12) in rigid ellipsoids, +vf is an anelastic flow having no angular momentum (i.e. ρ0vf = ∇ × A but with +⟨U, vf⟩ = 0), and ∇Φ is a potential flow carrying a non-zero angular momentum even if +⟨U, ∇Φ⟩ = 0 according to equation (A 9). +The anelastic and fully compressible cases may thus give different results for the + +Precession-driven flows in stress-free ellipsoids +25 +0 +0.5 +1 +1.5 +2 +2.5 +3 +c +10−3 +10−2 +10−1 +100 +101 +102 +103 +104 +|τso/E| +Formula (B1b) +Formula (B2) +Formula (4.6) +0 +1 +2 +3 +b +10−3 +10−2 +10−1 +100 +101 +102 +103 +104 +|τsup/E| +Ω = 1z +Ω = 0.1 1x + 0.5 1y + 1z +(a) +(b) +Figure 10. (a) Decay rate |τso/E| for Ω = 1z as a function of the semi-axis c, in spheroids with +a = b = 1. Comparison between correct formula (B 1b) and erroneous one (B 2) for the surface +integral in expression (B 1a). Note that |τso| → 0 when c → 1. (b) Decay rate |τsup/E| computed +from formula (4.6) as a function of the semi-axis b, for two rotation vectors Ω in ellipsoids with +a = 1.5 and c = 1. Note that |τsup| → 0 when b → a (i.e. in the spheroid). +evolution of angular momentum in rotating compressible ellipsoids. Differences between +the two formulations can be expected when the compressible subspace significantly +interacts with the anelastic one in spectral decomposition (A 1). This for instance happens +in the presence of global rotation when MΩ = O(10−1), where MΩ = RΩ0/C0 is +the rotational Mach number (Vidal & C´ebron 2020, 2021a) and C0 is the speed of +sound. Planetary estimates give MΩ = O(10−3) for planetary moons, but larger values +MΩ = O(10−1) are obtained in Jupiter-like gaseous planets (which are also non- +spherical because of centrifugal gravity, e.g. Zhang et al. 2017). Investigating the long- +term evolution of angular momentum in such strongly compressible rotating bodies +certainly deserves further work. +Appendix B. Viscous decay rates +We present an alternative formula for the viscous decay rate of the Coriolis eigenmodes +in stress-free ellipsoids, which is equivalent to formula (4.6). To enforce SF-BC (2.3) in +equation (4.5), we employ the curvilinear orthogonal coordinates [q1, q2, q3] (such that the +boundary is given by a constant value of q1). Then, the volume integral can be rewritten +using the divergence theorem as +τk +E +� +V +|Qk|2 dV = IS − +� +V +|∇ × Qk|2 dV +(B 1a) +with the surface integral (dS = h2h3 dq2dq3 being the surface element) +IS = 2 +� +S +� +1 +h1h2 +∂h2 +∂q1 +|Qk · 1q2|2 + +1 +h1h3 +∂h3 +∂q1 +|Qk · 1q3|2 +� +dS, +(B 1b) +where [h1, h2, h3] are the curvilinear scale factors and [1q1, 1q2, 1q3] are the orthogonal +basis vectors. In the sphere, expression (B 1b) reduces to +IS = 2 +� +S +|Qk × 1n|2 dS, +(B 2) + +26 +J. Vidal & D. C´ebron +Body +E +f = η/2 +α [◦] +Po +Px +ϵ +Earth +10−15 +2.5 × 10−3 +23.5 −1.1 × 10−7 −4.4 × 10−8 1.7 × 10−5 +Early Earth 5 × 10−16 1.0 × 10−2 +17.5 +−3.3 × 10−8 2.5 × 10−6 +Moon +10−12 +2.5 × 10−5 +1.54 −4.0 × 10−3 −2.2 × 10−4 2.7 × 10−2 +3.0 × 10−4 +3.0 × 10−2 +Early Moon 5 × 10−13 1.0 × 10−4 +33.2 −3.7 × 10−4 −2.0 × 10−4 5.5 × 10−1 +1.2 × 10−3 +6.4 × 10−1 +Table 1. Precession forcing in the liquid core of the Earth and Moon. Ekman number E based +on the typical viscosity value ν = 10−6 m2.s−1, polar flattening f = (a − c)/a, precession angle +α. Currently, f is well enough known for the Earth (Mathews et al. 2002), but the lunar values of +f vary from f = 2.5×10−5 for a purely hydrostatic Moon (Le Bars et al. 2011) to f = 3.0×10−4 +when considering the present-day non-hydrostatic lithosphere and a liquid core of radius 350 km +(Viswanathan et al. 2019). Parameters for the Early Moon and Earth, estimated ∼ 4 Gy ago, +are deduced from the current values by considering a spin rate Ω0 two times larger, leading to +values of E twice smaller and of f fourth time larger than the present estimates (due to the +centrifugal acceleration in Ω2 +0). Typical estimates for the Moon’s history from C´ebron et al. +(2019) and the orbital evolution model of Touma & Wisdom (1994), and for the Early Earth +from the low-obliquity scenario in Landeau et al. (2022). +recovering formula (3.14) of Liao et al. (2001) in the sphere. Note that vector expression +(B 2) has been erroneously employed in the spheroid (see formula (6.21) in Maffei et al. +2017, which is incorrect because of the missing curvature terms). Expression (B 1a) is +very difficult to implement in practice (because of the curvilinear coordinates), contrary +to formula (4.6) in which the volume integral can be performed fully analytically in +ellipsoids (e.g. see formula 50 in Lebovitz 1989). +For a numerical (cross-validation) benchmark of formulas (4.6) and (B 1a), we can +compute the decay rate Qso of the spin-over mode in spheroidal geometries (i.e. with +a = b = 1). To do so, we take the formula (3.25) in Vantieghem (2014), giving Qso for +Ω = 1z in triaxial ellipsoids, and express it using the curvilinear spheroidal coordinates +(e.g. equation 3.1 in C´ebron et al. 2021) +x = ηT sin(q2) cos(q3), +y = ηT sin(q2) sin(q3), +z = η (dq1T ) cos(q2), +(B 3a–c) +with η = |1 − (c/a)2|1/2 and T += cosh(q1) for oblate spheroids (i.e. a ⩾ c) or +T = sinh(q1) for prolate spheroids (i.e. a ⩽ c). The scale factors are then h1 = h2 = +η[sinh2(q1) + cos2(q2)]1/2 when a ⩾ c or h1 = h2 = η[cosh2(q1) − cos2(q2)]1/2 when +a ⩽ c, and h3 = ηT sin(q2). The differences between formulas (B 1b) and (B 2) are +illustrated in figure 10(a). For the particular geometry a = b = 1 and c = 0.9, we +have +� +V |Qk|2 dV +≃ +3.36965, +� +V |∇ × Qk|2 dV ≃ 37.64855 and IS ≃ 37.23369 from +formula (B 1b). Formulas (4.6) and (B 1a) then both predict that τso/E ≃ −0.12312 in +this spheroidal geometry (as observed in the figure). On the contrary, we would get +IS ≃ 35.30153 with formula (B 2), yielding the erroneous value τso/E ≃ −0.69652. +Finally, we show in figure 10(b) the decay rate τsup for different orientations of the +mean rotation axis in triaxial ellipsoids. + +Precession-driven flows in stress-free ellipsoids +27 +Appendix C. Planetary extrapolation for dynamo action +We can crudely estimate the dynamo capability of precession-driven flows using ener- +getic arguments. To do so, we compute a magnetic Reynolds number Rm as +Rm = Utopo/Em, +Em = νm/(Ω0R2), +(C 1a,b) +where Em is the magnetic Ekman number and νm ∼ 0.5 − 4 m2.s−1 is the magnetic +diffusivity of the fluid at typical core conditions (estimated from measurements and +computations of the electrical conductivity, e.g. see figure 1 in Ohta & Hirose 2021). +A necessary condition for large-scale dynamo action is that Rm ⩾ O(102) in spheres or +ellipsoids (e.g. Chen et al. 2018; Holdenried-Chernoff et al. 2019; Vidal & C´ebron 2021b). +Estimating the magnetic Reynolds number thus crucially depends on the scaling law for +the flow strength Utopo, whose order of magnitude is expected to be given by formula +(4.12). To be more quantitative, we rewrite formula (4.12) using asymptotic flow (3.7) in +the planetary regime |Px| ≪ 1, which gives at the leading order in η ≪ 1 +Utopo ≃ Kϵη ∼ K +� +2|Po| +when +α = π/2, +| tan(α)| η +when +α ̸= π/2, +(C 2) +where α is the precession angle measured from 1z, and K is an unknown numerical +prefactor that must be determined for planetary extrapolation. We recover from our +asymptotic solution that the quantity ϵη is actually independent of η at the leading +order when α = π/2 (e.g. see formula 9.b in Horimoto et al. 2020) and that, when +α ̸= π/2, the differential rotation ϵ becomes independent of Po in the regime |Px| ≪ 1 +(e.g. Williams et al. 2001; C´ebron et al. 2019). Moreover, local DNS in periodic shearing +boxes, performed at α = π/2, are actually consistent with the scaling law Utopo ∝ 0.1|Po| +(see figure 7 in Barker 2016b), which is of the form (C 2) with the numerical constant +K ≃ 0.05. Assume that K is a constant (without further numerical results), we can +crudely estimate the dynamo capability of the flows driven by the (topographic) inertial +instabilities for realistic planetary conditions by combining equations (C 1) and (C 2). +Using acceptable scenarios for the lunisolar precession over time (see table 1), we obtain +Rm ⩽ O(10) in the Earth’s core over geological ages, showing that precession was not +strong enough to drive dynamo action (even billion years ago, which agrees with the +conclusions of Landeau et al. 2022). Similarly, the estimate Rm ⩽ O(1) in the current +Moon’s core shows precession is not presently dynamo capable (in agreement with the +end of the lunar dynamo observed in paleomagnetic studies, e.g. Mighani et al. 2020). +However, we can obtain larger values Rm ⩽ 140 for the liquid core of the early Moon +(depending on the uncertainties on the polar flattening η and the magnetic diffusivity). +Our estimate thus suggests that precession might have been dynamo capable in the early +Moon (as initially suggested by Dwyer et al. 2011). Further work is obviously needed to +rigorously assess the relevance of scaling law (C 2) in precessing ellipsoids, which is key +for planetary extrapolation. Adopting SF-BC would be particularly useful to strongly +weaken the viscous turbulent flows (which are a priori not well suited to sustain large- +scale dynamo fields, see C´ebron et al. 2019) and extract a robust scaling law for the +saturation amplitude of the inertial (topographic) instabilities. + +28 +J. Vidal & D. C´ebron +REFERENCES +Backus, G. & Rieutord, M. 2017 Completeness of inertial modes of an incompressible inviscid +fluid in a corotating ellipsoid. Phys. Rev. E 95 (5), 053116. +Barker, A. J. 2016a Non-linear tides in a homogeneous rotating planet or star: global +simulations of the elliptical instability. Mon. Not. R. Astron. Soc. 459 (1), 939–956. +Barker, A. J. 2016b On turbulence driven by axial precession and tidal evolution of the spin- +orbit angle of close-in giant planets. Mon. Not. R. Astron. Soc. 460 (3), 2339–2350. +Barker, A. J. & Lithwick, Y. M. 2013 Non-linear evolution of the tidal elliptical instability +in gaseous planets and stars. Mon. Not. R. Astron. Soc. 435 (4), 3614–3626. +Brunet, M., Gallet, B. & Cortet, P.-P. 2020 Shortcut to geostrophy in wave-driven +rotating turbulence: the quartetic instability. Phys. Rev. Lett. 124 (12), 124501. +Buffett, B. A. 2021 Conditions for turbulent Ekman layers in precessionally driven flow. +Geophys. J. Int. 226 (1), 56–65. +Burmann, F. & Noir, J. 2022 Experimental study of the flows in a non-axisymmetric ellipsoid +under precession. J. Fluid Mech. 932, A24. +Busse, F. H. 1968 Steady fluid flow in a precessing spheroidal shell. J. Fluid Mech. 33 (4), +739–751. +C´ebron, D. 2015 Bistable flows in precessing spheroids. Fluid Dyn. Res. 47 (2), 025504. +C´ebron, D., Laguerre, R., Noir, J. & Schaeffer, N. 2019 Precessing spherical shells: flows, +dissipation, dynamo and the lunar core. Geophys. J. Int. 219 (Supplement 1), S34–S57. +C´ebron, D., Le Bars, M., Le Gal, P., Moutou, C., Leconte, J. & Sauret, A. 2013 +Elliptical instability in hot Jupiter systems. Icarus 226 (2), 1642–1653. +C´ebron, D., Le Bars, M. & Meunier, P. 2010 Tilt-over mode in a precessing triaxial ellipsoid. +Phys. Fluids 22 (11), 116601. +C´ebron, D., Vidal, J., Schaeffer, N., Borderies, A. & Sauret, A. 2021 Mean zonal flows +induced by weak mechanical forcings in rotating spheroids. J. Fluid Mech. 916, A39. +Chandrasekhar, S. 1969 Ellipsoidal Figures of Equilibrium. Dover Publications. +Chen, L., Herreman, W., Li, K., Livermore, P. W., Luo, J. W. & Jackson, A. 2018 The +optimal kinematic dynamo driven by steady flows in a sphere. J. Fluid Mech. 839, 1–32. +Clausen, N. & Tilgner, A. 2014 Elliptical instability of compressible flow in ellipsoids. Astron. +Astrophys. 562, A25. +Dwyer, C. A., Stevenson, D. J. & Nimmo, F. 2011 A long-lived lunar dynamo driven by +continuous mechanical stirring. Nature 479 (7372), 212–214. +Gerick, F., Jault, D., Noir, J. & Vidal, J. 2020 Pressure torque of torsional Alfv´en modes +acting on an ellipsoidal mantle. Geophys. J. Int. 222 (1), 338–351. +Grannan, A. M., Favier, B., Le Bars, M. & Aurnou, J. M. 2017 Tidally forced turbulence +in planetary interiors. Geophys. J. Int. 208 (3), 1690–1703. +Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press. +Greenspan, H. P. 1969 On the non-linear interaction of inertial modes. J. Fluid Mech. 36 (2), +257–264. +Guermond, J.-L., L´eorat, J., Luddens, F. & Nore, C. 2013 Remarks on the stability of the +Navier–Stokes equations supplemented with stress boundary conditions. Eur. J. Mech. B +Fluids 39, 1–10. +Holdenried-Chernoff, D., Chen, L. & Jackson, A. 2019 A trio of simple optimized +axisymmetric kinematic dynamos in a sphere. Proc. R. Soc. A 475 (2229), 20190308. +Horimoto, Y., Katayama, A. & Goto, S. 2020 Conical shear-driven parametric instability +of steady flow in precessing spheroids. Phys. Rev. Fluids 5 (6), 063901. +Ivers, D. 2017 Enumeration, orthogonality and completeness of the incompressible Coriolis +modes in a tri-axial ellipsoid. Geophys. Astrophys. Fluid Dyn. 111 (5), 333–354. +Jones, C. A., Boronski, P., Brun, A. S., Glatzmaier, G. A., Gastine, T., Miesch, M. S. +& Wicht, J. 2011 Anelastic convection-driven dynamo benchmarks. Icarus 216 (1), 120– +135. +Kerswell, R. R. 1993 The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72 (1- +4), 107–144. +Kerswell, R. R. 1999 Secondary instabilities in rapidly rotating fluids: inertial wave +breakdown. J. Fluid Mech. 382, 283–306. + +Precession-driven flows in stress-free ellipsoids +29 +Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mch. 34 (1), 83–113. +Kida, S. 2020 Steady flow in a rapidly rotating spheroid with weak precession: I. Fluid Dyn. +Res. 52 (1), 015513. +Landeau, M., Fournier, A., Nataf, H.-C., C´ebron, D. & Schaeffer, N. 2022 Sustaining +Earth’s magnetic dynamo. Nat. Rev. Earth Environ. 3 (4), 255–269. +Le Bars, M., C´ebron, D. & Le Gal, P. 2015 Flows driven by libration, precession, and tides. +Annu. Rev. Fluid Mech. 47, 163–193. +Le Bars, M., Wieczorek, M. A., Karatekin, ¨O., C´ebron, D. & Laneuville, M. 2011 An +impact-driven dynamo for the early Moon. Nature 479 (7372), 215–218. +Le Reun, T., Favier, B. & Le Bars, M. 2019 Experimental study of the nonlinear saturation +of the elliptical instability: inertial wave turbulence versus geostrophic turbulence. J. Fluid +Mech. 879, 296–326. +Le Reun, T., Gallet, B., Favier, B. & Le Bars, M. 2020 Near-resonant instability of +geostrophic modes: beyond Greenspan’s theorem. J. Fluid Mech. 900, R2. +Lebovitz, N. R. 1989 The stability equations for rotating, inviscid fluids: Galerkin methods +and orthogonal bases. Geophys. Astrophys. Fluid Dyn. 46 (4), 221–243. +Liao, X., Zhang, K. & Earnshaw, P. 2001 On the viscous damping of inertial oscillation in +planetary fluid interiors. Phys. Earth Planet. Int. 128 (1-4), 125–136. +Lin, Y., Marti, P. & Noir, J. 2015 Shear-driven parametric instability in a precessing sphere. +Phys. Fluids 27 (4), 046601. +Livermore, P. W., Bailey, L. M. & Hollerbach, R. 2016 A comparison of no-slip, stress- +free and inviscid models of rapidly rotating fluid in a spherical shell. Sci. Rep. 6 (1), +1–11. +Lorenzani, S. & Tilgner, A. 2001 Fluid instabilities in precessing spheroidal cavities. J. Fluid +Mech. 447, 111–128. +Lorenzani, S. & Tilgner, A. 2003 Inertial instabilities of fluid flow in precessing spheroidal +shells. J. Fluid Mech. 492, 363–379. +Maffei, S., Jackson, A. & Livermore, P. W. 2017 Characterization of columnar inertial +modes in rapidly rotating spheres and spheroids. Proc. R. Soc. A 473 (2204), 20170181. +Malkus, W. V. R. 1968 Precession of the earth as the cause of geomagnetism. Science +160 (3825), 259–264. +Mason, R. M. & Kerswell, R. R. 2002 Chaotic dynamics in a strained rotating flow: a +precessing plane fluid layer. J. Fluid Mech. 471, 71–106. +Mathews, P. M., Buffett, B. A., Herring, T. A. & Shapiro, I. I. 1991 Forced nutations +of the Earth: Influence of inner core dynamics: 1. Theory. J. Geophys. Res. Solid Earth +96 (B5), 8219–8242. +Mathews, P. M., Herring, T. A. & Buffett, B. A. 2002 Modeling of nutation and +precession: New nutation series for nonrigid Earth and insights into the Earth’s interior. +J. Geophys. Res. Solid Earth 107 (B4), ETG–3. +Mighani, S., Wang, H., Shuster, D. L., Borlina, C. S., Nichols, C. I. O. & Weiss, B. P. +2020 The end of the lunar dynamo. Sci. Adv. 6 (1), eaax0883. +Nobili, C., Meunier, P., Favier, B. & Le Bars, M. 2021 Hysteresis and instabilities in a +spheroid in precession near the resonance with the tilt-over mode. J. Fluid Mech. 909, +A17. +Noir, J., Cardin, P., Jault, D. & Masson, J.-P. 2003 Experimental evidence of non-linear +resonance effects between retrograde precession and the tilt-over mode within a spheroid. +Geophys. J. Int. 154 (2), 407–416. +Noir, J. & C´ebron, D. 2013 Precession-driven flows in non-axisymmetric ellipsoids. J. Fluid +Mech. 737, 412–439. +Ohta, K. & Hirose, K. 2021 The thermal conductivity of the Earth’s core and implications +for its thermal and compositional evolution. Nat. Sci. Rev. 8 (4), 1–3. +Poincar´e, H. 1910 Sur la pr´ecession des corps d´eformables. Bull. Astro. 27, 321–356. +Reddy, K. S., Favier, B. & Le Bars, M. 2018 Turbulent kinematic dynamos in ellipsoids +driven by mechanical forcing. Geophys. Res. Lett. 45 (4), 1741–1750. +Rieutord, M. 1992 Ekman circulation and the synchronization of binary stars. Astron. +Astrophys. 259, 581–584. + +30 +J. Vidal & D. C´ebron +Rieutord, M. & Zahn, J.-P. 1997 Ekman pumping and tidal dissipation in close binaries: A +refutation of Tassoul’s mechanism. Astrophys. J. 474 (2), 760. +Roberts, P.H. & Aurnou, J.M. 2012 On the theory of core-mantle coupling. Geophys. +Astrophys. Fluid Dyn. 106 (2), 157–230. +Rochester, M. G. 1962 Geomagnetic core-mantle coupling. J. Geophys. Res. 67 (12), 4833– +4836. +Rochester, M. G. 1976 The secular decrease of obliquity due to dissipative core—mantle +coupling. Geophys. J. Int. 46 (1), 109–126. +Sobouti, Y. 1981 The potentials for the g-, p- and the toroidal modes of self-gravitating fluids. +Astron. Astrophys. 100, 319–322. +Tilgner, A. 1999 Non-axisymmetric shear layers in precessing fluid ellipsoidal shells. Geophys. +J. Int. 136 (3), 629–636. +Touma, J. & Wisdom, J. 1994 Evolution of the Earth–Moon system. The Astron. J. 108 (5), +1943–1961. +Vantieghem, S. 2014 Inertial modes in a rotating triaxial ellipsoid. Proc. R. Soc. A 470 (2168), +20140093. +Vantieghem, S., C´ebron, D. & Noir, J. 2015 Latitudinal libration driven flows in triaxial +ellipsoids. J. Fluid Mech. 771, 193–228. +Vidal, J. & C´ebron, D. 2017 Inviscid instabilities in rotating ellipsoids on eccentric Kepler +orbits. J. Fluid Mech. 833, 469–511. +Vidal, J. & C´ebron, D. 2020 Acoustic and inertial modes in planetary-like rotating ellipsoids. +Proc. R. Soc. A 476 (2239), 20200131. +Vidal, J. & C´ebron, D. 2021a Acoustic modes of rapidly rotating ellipsoids subject to +centrifugal gravity. J. Acoust. Soc. Am. 150 (2), 1467–1478. +Vidal, J. & C´ebron, D. 2021b Kinematic dynamos in triaxial ellipsoids. Proc. R. Soc. A +477 (2252), 20210252. +Vidal, J., C´ebron, D., Schaeffer, N. & Hollerbach, R. 2018 Magnetic fields driven by +tidal mixing in radiative stars. Mon. Not. R. Astron. Soc. 475 (4), 4579–4594. +Vidal, J., Su, S. & C´ebron, D. 2020 Compressible fluid modes in rigid ellipsoids: towards +modal acoustic velocimetry. J. Fluid Mech. 885, A39. +Viswanathan, V., Rambaux, N., Fienga, A., Laskar, J. & Gastineau, M. 2019 +Observational constraint on the radius and oblateness of the lunar core-mantle boundary. +Geophys. Res. Lett. 46 (13), 7295–7303. +Vormann, J. & Hansen, U. 2018 Numerical simulations of bistable flows in precessing +spheroidal shells. Geophys. J. Int. 213 (2), 786–797. +Wicht, J. & Tilgner, A. 2010 Theory and modeling of planetary dynamos. Space Sci. Rev. +152 (1), 501–542. +Williams, J. G., Boggs, D.H., Yoder, C. F., Ratcliff, J. T. & Dickey, J. O. 2001 Lunar +rotational dissipation in solid body and molten core. J. Geophys. Res. Planets 106 (E11), +27933–27968. +Wu, C.-C. & Roberts, P. H. 2009 On a dynamo driven by topographic precession. Geophys. +Astrophys. Fluid Dyn. 103 (6), 467–501. +Wu, C.-C. & Roberts, P. H. 2011 High order instabilities of the Poincar´e solution for +precessionally driven flow. Geophys. Astrophys. Fluid Dyn. 105 (2-3), 287–303. +Zhang, K., Chan, K. H. & Liao, X. 2012 Asymptotic theory of resonant flow in a spheroidal +cavity driven by latitudinal libration. J. Fluid Mech. 692, 420–445. +Zhang, K., Chan, K. H. & Liao, X. 2014 On precessing flow in an oblate spheroid of arbitrary +eccentricity. J. Fluid Mech. 743, 358–384. +Zhang, K., Chan, K. H., Liao, X. & Aurnou, J. M. 2013 The non-resonant response of +fluid in a rapidly rotating sphere undergoing longitudinal libration. J. Fluid Mech. 720, +212–235. +Zhang, K., Kong, D. & Schubert, G. 2017 Shape, internal structure, zonal winds, and +gravitational field of rapidly rotating Jupiter-like planets. Annu. Rev. Earth Planet. Sci. +45 (1), 416–446. +Zhang, K. & Liao, X. 2004 A new asymptotic method for the analysis of convection in a +rapidly rotating sphere. J. Fluid Mech. 518, 319–346. + +Precession-driven flows in stress-free ellipsoids +31 +Zhang, K., Liao, X. & Busse, F. H. 2007 Asymptotic solutions of convection in rapidly +rotating non-slip spheres. J. Fluid Mech. 578, 371–380. + diff --git a/qtE1T4oBgHgl3EQfiwQZ/content/tmp_files/load_file.txt b/qtE1T4oBgHgl3EQfiwQZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6fb35f1833aa7c3e0e472a12d1889191990dae8 --- /dev/null +++ b/qtE1T4oBgHgl3EQfiwQZ/content/tmp_files/load_file.txt @@ -0,0 +1,1890 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf,len=1889 +page_content='Accepted for publication in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1017/jfm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='976 1 Precession-driven flows in stress-free ellipsoids J´er´emie Vidal† & David C´ebron Universit´e Grenoble Alpes, CNRS, ISTerre, 38000 Grenoble, France (Received 22 March 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' revised 23 August 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' accepted 11 November 2022) Motivated by modelling rotating turbulence in planetary fluid layers, we investigate precession-driven flows in ellipsoids subject to stress-free boundary conditions (SF- BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The SF-BC could indeed unlock numerical constraints associated with the no-slip boundary conditions (NS-BC), but are also relevant for some astrophysical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Although SF-BC have been employed in the pioneering work of Lorenzani & Tilgner (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', 2003, 492, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 363–379), they have scarcely been used due to the discovery of some specific mathematical issues associated with angular momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We revisit the problem using asymptotic analysis in the low-viscosity regime, which is validated with numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' First, we extend the reduced model of uniform- vorticity flows in ellipsoids to account for SF-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We show that the long-term evolution of angular momentum is affected by viscosity in triaxial geometries, but also in axisymmetric ellipsoids when the mean rotation axis of the fluid is not the symmetry axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In a regime relevant to planets, we analytically obtain the primary forced flow in triaxial geometries, which exhibits a second inviscid resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Then, we investigate the bulk instabilities existing in precessing ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We show that using SF-BC would be useful to explore the non-viscous instabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Kerswell, Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', 1993, 72, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 107-144), which are presumably relevant for planetary applications but are often hampered in experiments or simulations with NS-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Key words: rotating flows, waves in rotating fluids, geophysical and geological flows 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Introduction Motivated by numerous natural applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Le Bars et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015), we aim to explore the long-term dynamics of rapidly rotating fluids enclosed in ellipsoids subject to (harmonic) mechanical forcings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Global rotation is indeed ubiquitous in many planetary fluid layers or stars, which are usually ellipsoidal at the leading order (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' due to the combined action of centrifugal effects and gravitational interactions with nearby orbital partners, see Chandrasekhar 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In particular, mechanically driven flows in ellipsoids (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' flows driven by precession or tides) have received much attention in the fluid community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mechanical forcings can indeed sustain bulk instabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Kerswell 1993, 2002), turbulence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Grannan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Le Reun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019) and possibly dynamo magnetic fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' These works have also renewed interest in a key fundamental question in the theory of rotating fluids, which is the generation of two-dimensional geostrophic motions (Greenspan 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' However, this problem has only received scant attention in global geometries exhibiting the so-called † Email address for correspondence: jeremie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='vidal@univ-grenoble-alpes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='fr arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='03254v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='flu-dyn] 9 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron topographic beta effect (which strongly modifies the geostrophic flows, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Greenspan 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Exploring rotating turbulence thus deserves further work using global models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The incompressible Navier-Stokes equation is commonly adopted to explore the tur- bulence driven by mechanical forcings, together with the no-slip boundary conditions (NS-BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The latter are appropriate to model the flow dynamics in the presence of a rigid boundary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' the solid interface between a liquid core and a solid overlying mantle in planetary interiors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' However, the range of parameters that is accessible to global simulations with NS-BC is severely limited, in particular for the Ekman number E (which crucially controls the dynamics of rapidly rotating flows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Typical values in natural systems are E ⩽ O(10−12), whereas direct numerical simulations (DNS) and laboratory experiments of mechanically driven rotating turbulence can only reach much larger values E ≳ 10−6 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Grannan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Le Reun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' As a consequence, the Ekman boundary layer is often a prominent feature in the models (whereas the smallness of E in planetary systems suggests that viscosity should rather play a minor dynamical role), and its resolution requires considerable computational resources when E is lowered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Moreover, the overestimated viscous torque at the boundary can also largely inhibit the fluid response to mechanical forcings (which is primarily driven by the shape deformation of the fluid boundary, combined with non-stationary effects due to the possibly oscillatory angular velocity of the container).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Therefore, different modelling approaches are worth considering to simulate such flows at more realistic parameters for planetary applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' One natural way to avoid the physical and computational disadvantages of NS-BC is to employ stress-free boundary conditions (SF-BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A thin outer Ekman boundary layer is still present for stress-free boundaries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Livermore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2016), but its dynamical role is expected to be less important because the boundary-layer flow is much weaker in amplitude than the bulk flow (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rieutord 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Moreover, SF-BC are also commonly employed in astrophysical modelling since they are often believed to yield similar results to those obtained with a realistic free surface (Barker 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' However, SF-BC have scarcely been used in spheres and ellipsoids because of mathematical difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The most serious one is related to angular momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Angular momentum can indeed be arbitrary in axisymmetric geometries, leading to spurious solutions on long time scales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Guermond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The usefulness of SF-BC for simulating rotating flows in ellipsoids has thus been questioned, but we believe that this mathematical set-up deserves further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In this paper, we thus revisit the influence of SF-BC for rotating ellipsoids using asymptotic analysis when E ≪ 1 and targeted numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The model is presented in §2 and applied to precessing ellipsoids in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The results are discussed in §4, and we end the paper in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mathematical modelling 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid dynamic equations We consider a fluid-filled ellipsoid of uniform density and volume V , which is assumed to co-rotate with the surrounding mantle at the angular velocity Ωc(t) = Ω0 [Ω + δ(t)] with respect to the inertial frame (δ(t) being the time-dependent departure from the steady global rotation Ω along the unit vector 1Ω = Ω/|Ω|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To have a tractable mathematical problem, we seek mechanically driven flows in the mantle reference frame in which the ellipsoidal boundary S is steady and δ(t) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This set-up allows us to model flows driven by precession or librations, which have already received consideration using NS-BC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir & C´ebron 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2012, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vantieghem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precession-driven flows in stress-free ellipsoids 3 We non-dimensionalise the problem using Ω−1 0 as the time scale, and a typical length R as the length scale (which is here arbitrary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Considering a Newtonian fluid of uniform kinematic viscosity ν, the dimensionless equations for the velocity v are ∂tv + (v · ∇) v + 2Ωc × v = −∇p + 2E ∇ · ϵ(v) + r × dtδ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1a) ∇ · v = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1b) where r is the position vector, ϵ(v) = (1/2)[∇v + (∇v)⊤] is the strain-rate tensor, and E = ν/(Ω0R2) is the Ekman number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The ellipsoidal geometry, which is assumed to be steady in the mantle frame, is given by the dimensionless equation (x/a)2 + (y/b)2 + (z/c)2 = 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2) where [a, b, c] are the (dimensionless) ellipsoidal semi-axes and [x, y, z] are the Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In the following, axisymmetric geometries refer to ellipsoids with a revolution symmetry axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' when either a = b, b = c or a = c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Finally, spheroids will refer to the particular axisymmetric geometries for which the revolution symmetry axis is aligned with the rotation axis (with a = b and Ω ∝ 1z in this study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We aim to consider the SF-BC given in the mantle frame by v · 1n|S = 0, [ϵ(v) · 1n] × 1n|S = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3a,b) where 1n is the outward normal unit vector at the boundary, instead of the NS-BC v · 1n|S = 0, v × 1n|S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4a,b) It is obvious from SF-BC (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3) and NS-BC (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) that the tangential velocity at the boundary will differ between the two cases (since the flow is allowed to freely slip on the boundary with the SF-BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' One may thus wonder in which circumstances the above conditions will lead to similar flows in the bulk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' far from the boundary region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A necessary condition is that the mechanical forcings can sustain flows against viscous dissipation for the two BC in the mantle frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This is evidenced by the conservation equation for the volume-averaged kinetic energy Ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In a frame where the fluid boundary is steady, it is given by (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' equation 5 in Wu & Roberts 2009) dtEk = � V v · [r × dtδ] dV + 2E �� S v · T dS − Dν � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5) where T = ϵ(v) · 1n is the surface traction and Dv ⩾ 0 is a volume-averaged viscous dissipation (for both the NS-BC and SF-BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' For a velocity satisfying the no-penetration condition such that v = (v ·1n) 1n −1n ×(1n ×v) = −1n ×(1n ×v), the surface integral can actually be written as � S v · T dS = − � S T · [1n × (1n × v)] dS = − � S [T × 1n] · [v × 1n] dS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6) where we have used a property of the scalar triple product to obtain the last expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Thus, the above surface integral exactly vanishes for both SF-BC (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3) and NS-BC (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) in the mantle frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Then, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5) shows that we can have dtEk ⩾ 0 for both SF-BC and NS-BC if the mechanical forcings are oscillatory in the mantle frame (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' when dtδ ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Harmonic mechanical forcings, such as precession or librations, can thus sustain flows against viscous dissipation in the mantle frame (even with the SF-BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Note that a very different conclusion is obtained for steady forcings, such as precession viewed in the frame of precession for spheroidal geometries (Lorenzani & Tilgner 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Wu & Roberts 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We indeed have dtEk < 0 at every time for the SF-BC in the 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron precession frame, whereas precession could sustain non-vanishing flows against viscous dissipation for the NS-BC (since v × 1n|S ̸= 0 for a no-slip boundary in the precession frame, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In the following, we will only investigate the dynamics driven by oscillatory forcings in the mantle frame with SF-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Angular momentum The angular momentum L = � V r × v dV of the flow plays a central dynamical role for mechanically driven flows in ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Actually, the Cartesian components of the angular momentum L = (Lx, Ly, Lz)⊤ are exactly given for incompressible flows by Lx = � V (yvz − zvy) dV = � V v · (1x × r + ∇Ψx) dV, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7a) Ly = � V (zvx − xvz) dV = � V v · (1y × r + ∇Ψy) dV, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7b) Lx = � V (xvy − yvx) dV = � V v · (1z × r + ∇Ψz) dV, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7c) where [Ψx, Ψy, Ψz] are arbitrary scalar potentials if ∇ · v = 0 and if the flow obeys the no-penetration BC in rigid ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The scalar potentials are thus often discarded to simply express the angular momentum as projections onto the solid-body rotations 1i × r (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Guermond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Yet, the solid-body rotations are not admissible flow solutions in non-spherical geometries (even without viscosity), since they do not satisfy the no-penetration condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A more appropriate definition of the angular momentum for incompressible flows is thus given in ellipsoids by L · 1i = � V ei · v dV, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8) where {ei}i∈{x,y,z} is the set of uniform-vorticity (flow) elements defined by ei = 1i × r + ∇Ψi, ∇ · ei = 0, ei · 1n|S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9a–c) The scalar functions Ψi allow the elements ei to satisfy the no-penetration condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In ellipsoidal geometries, they are explicitly given by (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir & C´ebron 2013) Ψx = c2 − b2 b2 + c2 yz, Ψy = a2 − c2 a2 + c2 xz, Ψz = b2 − a2 a2 + b2 xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='10a–c) It is worth noting that definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8) is purely kinematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' It thus remains valid in the presence of additional effects, for instance without global rotation or with magnetic effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g Gerick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Moreover, this definition can also be generalised for compressible flows under the anelastic approximation (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Consequently, we can always rigorously expand incompressible velocity fields in ellipsoids as v(r, t) = U(r, t) + vf(r, t), � V r × vf dV = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='11a,b) where the uniform-vorticity flow U carrying the angular momentum is given by U(r, t) = ωx(t) ex(r) + ωy(t) ey(r) + ωz(t) ez(r), U · 1n|S = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='12a,b) and with the effective rotation vector of the fluid ω(t) = (ωx(t), ωy(t), ωz(t))⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The ve- locity vf, which does not carry angular momentum by definition since � V U ·vf dV = 0, contains bulks flows of higher spatial complexity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' flow instabilities or turbulence) and Precession-driven flows in stress-free ellipsoids 5 0 2 4 6 8 E t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6 100 Lz E = 5 × 10−3 E = 5 × 10−4 0 2 4 6 8 10 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='00 100 Lz 0 2 4 6 8 10 12 E t 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='00 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='25 100 Lz (a) Non-rotating Ω ≃ 0 (b) Rotating Ω ∝ 1z Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Non-convergence of the angular momentum Lz in DNS after several viscous time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precession forcing given by definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1) with Px = 10−2 in stress-free spheroids (a = b = 1, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' At t = 0, [ωx, ωy] are chosen to match asymptotic solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (a) DNS at Po = −1 for the two values of the Ekman number E = 5 × 10−3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' as considered in Wu & Roberts 2009) and E = 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' At t = 0, ωz ≈ 0 for the two simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (b) DNS at Po = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8 and E = 5 × 10−4 for ωz ≈ 0 (top panel) and ωz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 (bottom panel) at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' also viscous structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' the Ekman boundary layer, Rieutord 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The Cartesian components of L are then exactly given by L = L−1 ω, L−1 = 16π 15 abc diag � b2c2 b2 + c2 , a2c2 a2 + c2 , a2b2 a2 + b2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='13a,b) Finally, the time evolution of the angular momentum (or equivalently that of ω) is affected by viscosity through the action of the viscous torque Γ ν on long time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have for example Γ ν = 0 in spheres, such that angular momentum has to be conserved for uniformly rotating fluids in the inertial frame (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To clarify the dynamical role of SF-BC in ellipsoids, it is worth computing the viscous torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Viscous torque in stress-free ellipsoids Because of definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8), the Cartesian components of the viscous torque Γ ν = (Γ ν · 1x, Γ ν · 1y, Γ ν · 1z)⊤ are exactly given for SF-BC (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3) by Γ ν · 1i = 2E � V ei · ∇ · ϵ(v) dV = −2E � V ϵ(ei) : ϵ(v) dV, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='14) where we have used integration by parts and the decomposition ei = (1n · ei) 1n − 1n × (1n×ei) = −1n×(1n×ei) to cancel out the surface integral for SF-BC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' see the proof of proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 in Guermond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We recover from the formula that Γ ν = 0 in spheres since ϵ(ei) exactly vanishes when ei is a solid-body rotation, but we also obtain that Γ ν ̸= 0 in triaxial geometries (because ϵ(ei) ̸= 0 when a ̸= b ̸= c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Moreover, it shows that Γ ν · 1i = 0 when the Cartesian vector 1i is an axis of revolution of the geometry (irrespective of the fluid global rotation, as ei is then a solid-body rotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We can now inspect the long-term evolution of angular momentum since pathological behaviours have been reported in some axisymmetric configurations (Guermond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To illustrate this behaviour, we expand the angular momentum as L = L0 + L1, where L0 is the angular momentum of a dynamical solution of the problem and L1 is a modification of L0 associated with an additional uniform-vorticity flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The time 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron evolution of L1 is then given in the rotating frame by (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Roberts & Aurnou 2012) dtL1 + Ωc × L1 = Γ p,1 + Γ ν,1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='15) where Γ p,1 = � S p1 1n ×r dS is the pressure torque and Γ ν,1 is the viscous torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Since the viscous and pressure torques are non-zero when a ̸= b ̸= c, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='15) shows that the angular momentum is affected by viscosity in triaxial ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The situation is possibly different in axisymmetric geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' If the fluid is not globally rotating (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' when Ω = 0), then the component L1 · 1i carried by the uniform-vorticity element ei is arbitrary when 1i is a revolution symmetry axis (since Γ p,1 · 1i = Γ ν,1 · 1i = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Similarly, if the fluid is globally rotating along the revolution symmetry axis 1i, then the perturbation angular momentum L1 ∝ 1i is arbitrary (it will depend on the initial conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' as shown in Guermond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The two situations are illustrated numerically in figure 1 for a spheroid a = b subject to the precession forcing (see its definition below in §3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have performed DNS using the standard finite-element method as implemented in the commercial software comsol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The latter has already been employed to simulate precession-driven flows in ellipsoids with NS-BC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir & C´ebron 2013) and can also account for SF-BC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' for tidal flows in C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The geometry is modelled by an unstructured mesh with tetrahedral elements in the bulk, surrounded by a boundary-layer mesh (made of prism elements) to ensure the convergence of the thin Ekman layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have employed Lagrange elements P2-P3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' quadratic for the pressure field and cubic for the velocity field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The total number of degrees of freedom ranges between 3 × 105 and 5 × 105, such that every targeted simulation took a few days to run in parallel on a cluster (to investigate the long-term evolution of L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We observe that the axial angular momentum Lz does not converge in time for the considered stress-free spheroid (it is still growing or decaying even after several viscous time scales) if either the fluid is non-rotating in average as in panel (a) or Ω ∝ 1z as in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' However, a definitive conclusion should not be drawn for every axisymmetric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The situation is indeed different if the global rotation is not aligned with the revolution axis, since the three components of the angular momentum should be strongly coupled in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='15) for such configurations (even if Γ ν · 1i = 0, see §3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Application to precession-driven flows We consider precession-driven flows in ellipsoids, which have only received scant attention with SF-BC (Lorenzani & Tilgner 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Wu & Roberts 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Guermond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We work in the mantle frame rotating with respect to the inertial frame at the dimensionless angular velocity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir & C´ebron 2013) Ωc(t) = (1 + Pz)1z � �� � Ω + δ(t), δ(t) = Px [cos(t)1x − sin(t)1y] , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1a,b) with Px = Po sin(α) and Pz = Po cos(α), where Po = Ωp/Ω0 is the Poincar´e number (Ωp being the angular velocity of precession and Ω0 that of the mantle) and α is the angle of precession measured from 1z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Because the Poincar´e force r ×dtδ is linear in Cartesian coordinates, the primary response of the fluid is a laminar uniform-vorticity flow (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir & C´ebron 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Kida 2020), on top of which secondary flows and turbulence can develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' For analytical progress, we expand the velocity field as v = v0 + v1, where v0 is the primary forced flow (which is mainly of uniform vorticity) and v1 represents small-amplitude additional flows such that |v1| ≪ |v0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We first seek analytical solutions Precession-driven flows in stress-free ellipsoids 7 of the primary flow in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1, which are compared with DNS in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Then, we explore the flow instabilities v1 growing upon the forced flow in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Laminar forced flows The forced laminar flows, which have been explored for a long time after the seminal work of Poincar´e (1910), can be obtained using boundary-layer theory (BLT) in the low-viscosity regime E ≪ 1 for SF-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To do so, we seek v0 as v0(r, t) ≃ ωx(t)ex + ωy(t)ey + ωz(t)ez � �� � U(r,t) +E1/2 �U(r, t) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2) where U(r, t) is a forced uniform-vorticity flow carrying angular momentum, and �U(r, t) is the viscous flow within the boundary layer at the leading order in E1/2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rieutord 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A direct consequence of asymptotic expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2) is that the bulk flow for SF- BC can be determined without explicitly solving for the boundary-layer flow (since the latter has an amplitude that is E1/2 smaller than the bulk flow amplitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The exact viscous torque given by formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='14) can then be approximated as Γ ν ≃ −16π 3 abc E diag �(b2 − c2)2 (b2 + c2)2 , (a2 − c2)2 (a2 + c2)2 , (a2 − b2)2 (a2 + b2)2 � ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3) The viscous flow E1/2 �U in expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2) has a contribution of amplitude O(E3/2) to the viscous torque (since |ϵ( �U)| = O(E−1/2) and the volume scales as O(E1/2) within the Ekman layer), which can be neglected compared with expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3) in the asymptotic regime E ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We recover from formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3) that Γ ν ·1i = 0 when the Cartesian vector 1i is a revolution symmetry axis, but also that the three components of the viscous torque are non-zero when a ̸= b ̸= c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Then, the momentum equation reduces to dtω − [(ω + Ωc) · ∇] U = −dtΩc + LΓ ν, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) where Γ ν is the viscous torque given by formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3) and L is the matrix given by the inverse of expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='13b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The approximated viscous term is thus LΓ ν = −5Ediag �(b/c − c/b)2 b2 + c2 , (a/c − c/a)2 a2 + c2 , (a/b − b/a)2 a2 + b2 � ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5) Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5) extend the asymptotic viscous model of Noir & C´ebron (2013) to stress-free ellipsoids, but we remind the reader that this stress-free model is not valid in spheres (since the angular momentum would be arbitrary in spheres because of Γ ν = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The close similarity between the no-slip and stress-free cases, for which only the expression of the viscous term in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) differs, suggests that the same interior solution should be approached when E → 0 in no-slip and stress-free ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precession is often characterised by |Px| ≪ 1 in planetary liquid cores (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir & C´ebron 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Hence, we seek asymptotic solutions of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) in powers of Px as ω(t) = ω(0)(t) + Px ω(1)(t) + P 2 x ω(2)(t) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6) Since the mean rotation axis is Ω ∝ 1z when |Px| ≪ 1, we assume that a ̸= b (to avoid the pathological situations outlined in §2 for the angular momentum conservation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The zeroth-order solution ω(0)(t) corresponds to a decaying transient when t → ∞ (because of viscosity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We thus discard ω(0)(t) in the following and solve the first-order problem in Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In the regime of vanishing viscosity E → 0, we obtain the first-order solution ω(1) x (t) ≃ − 1 + [1 + Pz]A1 1 − [1 + Pz]2λ2so cos(t), ω(1) y (t) ≃ 1 + [1 + Pz]B2 1 − [1 + Pz]2λ2so sin(t) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7a,b) 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron 10−2 10−1 100 101 E t −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='03 ω · 1x ω = (1/2) � V ∇ × v dV ω = LL 0 2 4 6 E t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='03 |ω| ω = (1/2) � V ∇ × v dV ω = LL (a) (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' DNS of precessing ellipsoids with SF-BC at Po = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8, E = 5 × 10−4 and Px = Po sin(α) = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Axisymmetric geometry a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 and b = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (a) Time evolution of the Cartesian component ω · 1x and (b) absolute value |ω| of the angular velocity, computed in the DNS either from the volume-averaged vorticity as ω = (1/2) � V ∇ × v dV or using the angular momentum as ω = LL using expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' and ω(1) z (t) → 0, with A1 = 2a2/(a2 + c2), B2 = 2b2/(b2 + c2), and λso = √A1B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have finally to compute the second-order solution ω(2), accounting for weakly nonlinear interactions in the viscous interior, to estimate the axial angular velocity (since it is undefined at the first order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' An analytical solution can be obtained when E ̸= 0, showing that ω(2) = ω(2) z 1z, but the general expression of ω(2) z is too lengthy to be given here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In the regime of vanishing viscosity E → 0, it simplifies into ω(2) z (t) = c2 4 D2 2 � ω(2) z + δω(2) z cos(2t) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8a) with the denominator D2 = a2b2 �Pz (Pz + 1/2) − c2 � a2 + b2 + c2� /4 and �Pz = Pz + 3/2, where the amplitude of the mean geostrophic flow is given by ω(2) z = − �c2 2 + a2 �Pz � �c2 2 + b2 �Pz � a2 + b2 (a/b − b/a)2 ��a c − c a �2 + �b c − c b �2� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8b) and that of the oscillatory component by δω(2) z = (Pz + 1)(a2 − b2)(a2b2 �P 2 z − c4/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' It is worth noting that the mean geostrophic flow ω(2) z has an amplitude that is independent of E in the vanishing regime E → 0, which is somehow similar to the mean geostrophic flows driven by nonlinear boundary-layer interactions for NS-BC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A striking property of the asymptotic solution is that it exhibits two inviscid direct resonances, which occur when the common denominator in expressions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7a,b) vanishes at the two resonant values Po± given by λso [1 + Po± cos(α)] = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The resonance associated with Po+ actually corresponds to the inviscid resonance initially predicted by Poincar´e (1910), which has been observed for no-slip boundaries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vormann & Hansen 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Nobili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Burmann & Noir 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' However, the second resonance at Po− is new, although precession-driven flows have been explored for more than a century in triaxial ellipsoids (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Poincar´e 1910;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir & C´ebron 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precession-driven flows in stress-free ellipsoids 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Numerical simulations We have checked that the analytic expressions are in excellent agreement with the numerical integration of the exact uniform-vorticity model (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) when E → 0 (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Yet, it remains to confirm the validity of the asymptotic solutions against DNS with SF- BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We first show in figure 2 the time evolution of the rotation vector ω(t) in the DNS (performed with comsol, as explained in §2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We illustrate the DNS at Px = 10−2 with Po = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8 and E = 5 × 10−4, in the particular axisymmetric geometry a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 and b = c = 1 (other parameters yield similar results, not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The fluid angular velocity ω has been computed in the DNS using either the volume-averaged vorticity or formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='13a) after having computed the angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Both methods are found to be in excellent quantitative agreement for the SF-BC (as observed in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' For such an axisymmetric geometry, we may naively think (before any computation) that the long-term evolution of ωx (or equivalently that of Lx) is unconstrained due to the vanishing component of the viscous torque Γ ν · 1x = 0 according to formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We observe that ωx initially displays a complicated transient (panel a), which dies out because of viscosity as expected from the asymptotic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Then, it converges towards a well-defined oscillatory state after a few viscous time scales (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' when E t ≫ 1 in dimensionless units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The total angular velocity ω, which exhibits no long-term spurious dynamics (panel b), has a small amplitude compared with the mean rotation axis of the fluid Ω = 1z with respect to the inertial frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have checked that the final state is robust, as it is recovered by varying the numerical resolution and adopting different initial conditions for a few values of Po and E (although multiple solutions may exist close to the inviscid resonances, as shown for sufficiently small Ekman numbers with NS-BC in C´ebron 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The comparison between the asymptotic results and the DNS is further illustrated in figure 3, still considering the illustrative axisymmetic geometry a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 and b = c = 1 (other geometries with a ̸= b give again similar results, not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The DNS are in excellent quantitative agreement with the asymptotic solution, although the latter has been obtained assuming E → 0, for both the time-averaged and the instantaneous angular velocity (see panel b after seven viscous time scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We also have checked that δω(2) z is accurately recovered in the DNS (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The observed excellent quantitative agreement with theoretical precession-driven flows has not been obtained using NS-BC in ellipsoids, both in DNS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir & C´ebron 2013) and laboratory experiments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Nobili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Burmann & Noir 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Finally, the DNS also confirm the physical existence of the two inviscid resonances of solutions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Asymptotic theory of flow instabilities The forced laminar flow U(r, t), given by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) when E ≪ 1, can be desta- bilised by various hydrodynamic instabilities in ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precession-driven instabilities are classified either as viscously driven if they only exist when E ̸= 0, or as inertial if they survive when E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Viscous instabilities exist in no-slip spheres, such as boundary-layer instabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Lorenzani & Tilgner 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Buffett 2021) or the conical-shear instability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' On the contrary, the inertial instabilities only exist in non-spherical geometries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Kerswell 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Wu & Roberts 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & C´ebron 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In the following, we extend the prior inviscid linear analyses of the inertial instabilities, which all considered precession at α = π/2 and in the precession frame (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' only for spheroids), to account for the SF-BC and the time-dependent background flow (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7) in the mantle frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To do so, we expand the governing equations with respect to U (discarding the small-amplitude viscous flow E1/2 �U in the bulk, which is negligible 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron −2 −1 0 1 Po 10−2 10−1 100 ϵ Asymptotic E → 0 Non rotating DNS 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='51 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='52 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='53 E t −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='01 ω · 1x Asymptotic E → 0 DNS (a) (b) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precession-driven flows (SF-BC) at Px = Po sin(α) = 10−2 for a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 and b = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Comparison between asymptotic solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7) and DNS at E = 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (a) Time-averaged angular velocity ϵ = |ω| as a function of Po.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The fluid is not globally rotating when Po ≃ −1 if |Px| ≪ 1 (grey area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vertical dashed lines show the two resonances of asymptotic solutions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7) at Po± = − � P 2 x + (1/λso ∓ 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Teal vertical line shows the region |Po| < 10−2 where no α can satisfy Px = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (b) Value of |ω| as a function of the re-scaled time E t at Po = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' when E ≪ 1 as found in the DNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The perturbation velocity v1, which is assumed to be of small amplitude compared with U, is governed in the mantle frame by ∂tv1 + 2Ωc × v1 = L(v1) + 2E∇ · ϵ(v1) − ∇p, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9a) ∇ · v1 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9b) with the linearised advection operator L(a) = −(a · ∇) U − (U · ∇) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The perturbation velocity v1 then satisfies the SF-BC (Mason & Kerswell 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Wu & Roberts 2009) v1 · 1n|S = 0, [ϵ(v1) · 1n] × 1n|S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='10a,b) To explore the low-viscosity regime E ≪ 1, which is difficult to probe using DNS, we develop an asymptotic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We seek v1 using BLT as (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rieutord 1992) v1(r, t) ≃ u(r, t) + E1/2 �u(r, t), ∇ · u = 0, u · 1n|S = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='11a–c) where u(r, t) represents the inviscid bulk flow and �u(r, t) is the leading-order viscous flow within the Ekman layer to satisfy SF-BC (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Because the boundary-layer flow has an amplitude that is E1/2 smaller than the bulk flow amplitude, SF-BC strongly weaken the viscous instabilities in ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In particular, the critical shear layers spawned by the Ekman layer at the critical latitudes are almost suppressed in stress-free ellipsoids without an inner core (Tilgner 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Consequently, the inertial instabilities triggered in the (nearly) inviscid bulk are expected to be largely favoured in stress-free ellipsoids (compared with viscous instabilities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To solve problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='11), we introduce the finite-dimensional polynomial vector space Vn spawned by the global real-valued incompressible elements {uk}, made of Cartesian monomials xiyjzk of maximum degree i + j + k ⩽ n and satisfying the no-penetration BC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & C´ebron 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Such vector elements are indeed known to form a complete basis for smooth velocity fields in ellipsoids when n → ∞ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Lebovitz 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Backus & Rieutord 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Then, we seek the bulk flow using the Galerkin Precession-driven flows in stress-free ellipsoids 11 expansion (written using Einstein’s convention) u(r, t) = αk(t)uk(r), ∇ · uk = 0, uk · 1n|S = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='12a–c) where α = (α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' , αN)⊤ is the state vector of the modal coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The number of elements N for a given maximum degree n in expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='12) is N = n(n+1)(2n+7)/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In practice, we truncate the polynomial expansion at the maximum degree n, substitute the truncated expansion into equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9) and, finally, project the resulting equations onto every basis element ui to minimise the residual with respect to the real-valued inner product defined by ⟨a, b⟩V = � V a · b dV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The governing equations then reduce to M dtα = (L − C − D) α, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='13) where M ij = ⟨ui, uj⟩V is the mass matrix, Cij = ⟨ui, 2Ωc×uj⟩V represents the Coriolis force, Lij = ⟨ui, L(uj)⟩V is the matrix representing the linearised advection terms and the viscous matrix D is given by (after integration by parts) Dij = 2E � V ϵ(ui) : ϵ(uj) dV (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='14) in which we have enforced SF-BC (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='10) in the projection to simplify the integration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Guermond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' As already noticed for the forced flow, a useful consequence of expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='11) is that the bulk flow u can be determined in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='13) without an explicit solution of �u for SF-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This has also been reported for asymptotic models of thermal convection or waves in rotating stress-free spheres (Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Zhang & Liao 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This is a noticeable difference from asymptotic models using NS-BC, which require a matching between the boundary-layer flow and the interior solution (which are of the same order of magnitude, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2007, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Since asymptotic solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7) is periodic of period T = 2π, we investigate the linear stability using Floquet theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We first compute the eigenvalues χ of the monodromy matrix Φ(2π) given by M dtΦ = (L − C − D) Φ, Φ(0) = I, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='15a,b) where I is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Then, we compute the complex-valued Lyapunov expo- nents as µ = (1/T) log χ whose real part Re(µ) = σ is the growth rate of the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' As initially noticed by Kerswell (1993) and Wu & Roberts (2011), the finite-dimensional polynomial is left invariant by the linear operator in the momentum equation, that is L(Vn) ∈ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Therefore, we can construct exact polynomial solutions of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9) giving sufficient conditions for linear instability in the inviscid regime E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We show in figure 4(a) the results of the linear inviscid stability analysis at Px = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have numerically solved equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='15) using a fourth-order Runge-Kutta solver and standard linear algebra routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' As in Kerswell (1993) and Wu & Roberts (2011), there are no instabilities associated with the linear elements n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The first instabilities, which are here associated with the quadratic modes with n = 2, only occur near the resonance at Po+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' When n is increased, additional tongues of inertial (topographic) instabilities appear with a growth rate scaling in the inviscid regime as σtopo = O(ϵη), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='16) where ϵ = |ω| is the mean value of the differential rotation between the fluid and the mantle and η = a2/c2 − 1 is the polar flattening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The numerical prefactor is found to be σtopo/(ϵη) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 when n ⩽ 20 (as shown in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Moreover, when n → ∞, the growth is expected to approach the upper bound given in the unbounded short- 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron −2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 Po 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 σ n = 2 n = 15 n = 20 σtopo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 ϵη Kerswell (1993) Non-rotating −2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 Po 10−4 10−3 10−2 10−1 100 σ σtopo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 ϵη SF-BC (E = 3 × 10−6) SF-BC (E = 5 × 10−4) NS-BC (E = 3 × 10−6, K = 10) NS-BC (E = 3 × 10−6, K = 4) Non-rotating Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Growth rate σ of the inertial (topographic) instabilities growing upon flow (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7) at Px = Po sin(α) = 10−2, as a function of Po (using sampled values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Teal vertical line shows the interval |Po| < 10−2 in which no α can satisfy Px = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The fluid is not globally rotating near Po ≃ −1 when |Px| ≪ 1 (grey area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (a) Inviscid growth rate for various degrees n of the global modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Dashed black curve is obtained in the unbounded short-wavelength limit (Kerswell 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (b) Viscous effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Dotted blue line shows the upper bound of the inviscid growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Olive coloured area shows the unstable region for SF-BC at E = 3 × 10−6, and thick red line shows the unstable zone for SF-BC at E = 5×10−4 (both computed at n = 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Purple coloured curves show viscous growth rate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='17) for NS-BC, with K ∈ [4, 10] to account for the Ekman damping of the large-scale modes (see figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' wavelength approximation (Kerswell 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This shows that the forced laminar flow is generically unstable to short-wavelength perturbations without viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' However, the short-wavelength modes are more damped by viscosity than the large- scale ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Consequently, viscous effects will select the allowable unstable modes for a given value of the Ekman number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To show this, we have explored the linear stability including viscous damping in figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' At E = 5 × 10−4, the forced flow is only unstable in extremely thin tongues near the two resonances at Po± for the SF-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This is consistent with the absence of instabilities in the DNS performed at E = 5×10−4 (see Precession-driven flows in stress-free ellipsoids 13 figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' More challenging DNS with SF-BC at smaller values E = O(10−6), which are beyond the scope of the present paper, could allow us to obtain instabilities for values of Po in a larger interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Finally, it is also useful to compare the stability of the forced flow with SF-BC and NS-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A proper asymptotic theory for the no-slip case, rooted in the BLT of the inertial modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Greenspan 1968), will be considered elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Nonetheless, an upper bound for the viscous growth rate of the inertial instabilities can be estimated as σtopo ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 ϵη − K � E[1 + Pz], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='17) assuming that the fluid is rotating on average at 1 + Pz in the mantle frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Here, the numerical prefactor K = 4 − 10 heuristically accounts for the Ekman damping of the large-scale flow structures with NS-BC (see figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' For the small value E = 3 × 10−6, we observe that the forced flow at Px = 10−2 would be mainly stable with NS-BC (except near the resonance Po+), whereas it would be unstable for other values of Po with SF- BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Therefore, the figure clearly illustrates that adopting SF-BC (instead of NS-BC) can be useful to explore the turbulence driven by inertial instabilities in the bulk of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Physical insight from the Coriolis eigenmodes We have illustrated with the case of precession-driven flows that the long-term evo- lution of angular momentum is damped by viscosity in triaxial ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Similarly, viscosity affects the angular momentum in axisymmetric rotating ellipsoids if the mean rotation axis Ω is not aligned with the revolution symmetry axis (even if Γ i · 1i = 0 in such geometries, where 1i is the revolution axis along one of the principal semi-axes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Asymptotic analysis offers a physical understanding of why the cases Ω ∝ 1i and Ω ̸∝ 1i strongly differ in axisymmetric ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' When E ≪ 1, the solutions of equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1a,b) in stress-free or no-slip ellipsoids can be rigorously expanded onto a combination of the inviscid eigenmodes of the (steady) Coriolis operator given by (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Backus & Rieutord 2017) iλk∇ × Qk = −2∇ × (Ω × Qk), ∇ · Qk = 0, Qk · 1n|S = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1a–c) where [λk, Qk(r)] is the kth eigenvalue-eigenfunction pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Only three of these eigenmodes carry a non-zero angular momentum in ellipsoids (by virtue of the orthogonality of the eigenmodes, see Ivers 2017), namely the spin over mode Qso, its complex conjugate Q† so and the zero-frequency geostrophic mode Qsup associated with axial (differential) rotation along Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Because these three modes are uniform-vorticity flows such as Qk = ωk,x ex + ωk,y ey + ωk,z ez, they are given by the matrix eigenvalue problem � � 0 2a2Ωz/(a2 + c2) −2a2Ωy/(a2 + b2) −2b2Ωz/(b2 + c2) 0 2b2Ωx/(a2 + b2) 2c2Ωy/(b2 + c2) −2c2Ωx/(a2 + c2) 0 � � ωk = iλkωk (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2) with Ω = (Ωx, Ωy, Ωz)⊤, where the rotation vector ωk = (ωk,x, ωk,y, ωk,z)⊤ of the eigenmode Qk is given by the kth eigenvector of matrix (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Consequently, the uniform- vorticity components ωi(t) ei of the flow in expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='11) are not mutually independent in rotating ellipsoids but, instead, are tied to the dynamics of these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' More precisely, the equatorial components of the angular momentum L × 1Ω are coupled through the dynamics of the two spin-over modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Similarly, the axial angular momentum L · 1Ω (related to the fluid spin-up) is piloted by the dynamics of the geostrophic mode Qsup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' From a physical viewpoint, whether viscosity affects the long-term evolution of angular 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron −2 −1 0 1 2 λk 2 4 6 8 10 12 −Re (τk)//E1/2 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 1 Im(τk)/E1/2 −2 −1 0 1 2 λk 100 101 102 103 104 −τk/E (a) NS-BC (b) SF-BC Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Viscous decay rates τk of the inertial modes of maximum polynomial degree n = 20, as a function of the inviscid eigenfrequency λk ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Axisymmetric ellipsoid with semi-axes a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 and b = c = 1 rotating at the angular frequency Ω = 1z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (a) Complex-valued τk for NS-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Colour bar shows the normalised imaginary part Im(τk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (b) Real-valued τk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Im(τk) = 0) given by formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6) for SF-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' momentum or not is thus deeply rooted in the viscous dynamics of these three eigen- modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We can quantify how viscosity impacts the inviscid eigenmodes by estimating the global viscous decay rates τk of the Coriolis modes as ∂tQk|t=0 ≃ τk Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3) For NS-BC, τk is a complex-valued quantity with a real part Re(τk) ⩽ 0 representing the volume-averaged viscous decay rate, and an imaginary part Im(τk) characterising the frequency shift due to viscous effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Greenspan 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Typical values are illustrated in figure 5 for a particular ellipsoidal geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' It has also been recognised for a long time that, for NS-BC (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4), the viscous torque in the mantle frame is related to the viscous damping of these three eigenmodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rochester 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In no-slip spherical geometries, it is given by (see formula 35 in Rochester 1976) Γ ν ∝ E1/2 [Re(τso) ω⊥ − Im(τso) 1z × ω⊥ + τsup ωz1z] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) at the leading order in E (assuming Ω = 1z), where ω = ω⊥ + ωz1z = (ωx, ωy, ωz)⊤ is the uniform vorticity of the forced flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Note that similar expressions have been later rediscovered for the particular case of precession as viewed in the precession frame (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir & C´ebron 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) clearly shows that the equatorial components L × 1z are damped by viscosity when Re(τso) ̸= 0 and, similarly, τsup ̸= 0 (which is a real number for this mode) ensures that the axial angular momentum L·1z is affected by viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Since Re(τso) ̸= 0 and τsup ̸= 0 in no-slip spheres and ellipsoids, we have Γ ν ̸= 0 from formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) such that the angular momentum is affected by viscosity on long time scales for NS-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Similar reasoning can be applied to the stress-free rotating case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' It can be shown that leading-order viscous torque (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3) depends on the viscous decay rates [τso, τsup] for the SF-BC (not given here, since it vainly makes the expression more complex because a full description of the viscous cross-interactions between Qso and Qsup is required contrary to the no-slip case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We can thus get physical insight into formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3) by computing the viscous decay rates for SF-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To do so, we expand the velocity as Qk + E1/2 �Qk (Rieutord 1992), where �Qk is the boundary-layer flow such that Qk + E1/2 �Qk satisfies Precession-driven flows in stress-free ellipsoids 15 SF-BC (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The viscous decay rate for SF-BC is then given at the leading order in E by (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2001) τk � V |Qk|2 dV = E � V Q† k · ∇2(Qk + E1/2 �Qk) dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5) Contrary to the no-slip case (for which the boundary-layer flow is of the same order of magnitude as the inviscid flow, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Greenspan 1968), an explicit solution of �Qk for SF-BC is not required to estimate τk in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Indeed, the representative volume- averaged viscous decay rate of all the eigenmodes is given at leading order in E for our SF-BC by (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rieutord & Zahn 1997) τk � V |Qk|2 dV = −2E � V ϵ(Qk) : ϵ(Q† k) dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6) Expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6) generalises formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='14) in Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (2001), which is only valid for spheres (see Appendix B), to triaxial ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Since the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5) is real, we have τk ⩽ 0 for SF-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Consequently, there is no viscous correction of the inviscid eigenfrequency λk at the leading order in E for SF-BC (as initially reported in Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6) is illustrated in figure 5 for a particular configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We recover from the formula that τso = τsup = 0 in spherical geometries (since Qso and Qsup are exact solid-body rotations in spheres), which agrees with the fact that Γ ν = 0 in spheres (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Explicit expressions of τso and τsup can be obtained for the uniform-vorticity modes in ellipsoids, because the eigenvectors [ωso, ωsup] of matrix (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2) can be analytically obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The analytical formula of τso, which is too lengthy to be given here, shows that τso ̸= 0 in every non-spherical geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The mathematical reason is that the spin-over mode Qso is no longer a solid-body rotation in ellipsoids (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' ϵ(Qk) is non-zero for the spin-over mode in ellipsoids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Thus, from a physical viewpoint, a non-zero boundary-layer flow �Qso is required to match the SF-BC within a thin Ekman boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Since the spin-over mode is damped by viscosity in ellipsoids, the equatorial angular momentum L×1Ω is affected by viscosity on long time scales (even in axisymmetric ellipsoids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' After little algebra, the decay rate τsup is explicitly given by τsup E � V |Qsup|2 dV = −16π 3 abc � Ω2 x(b2 − c2)2 + Ω2 y(a2 − c2)2 + Ω2 z(a2 − b2)2� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7) where the axial geostrophic mode is Qsup = ωsup,x ex +ωsup,y ey +ωsup,z ez with ωsup,x = Ωx(b2+c2), ωsup,y = Ωy(a2+c2) and ωsup,z = Ωz(a2+b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7) shows that τsup ̸= 0 when a ̸= b ̸= c, illustrating that the axial geostrophic mode is damped by viscosity in triaxial geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Therefore, the physical reason why Γ ν ̸= 0 in triaxial stress- free ellipsoids is that the spin-over and geostrophic modes are damped by viscosity (as evidenced by the non-zero decay rates τso ̸= 0 and τsup ̸= 0 in such geometries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Moreover, formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7) shows that τsup = 0 when Ω is an axis of revolution of the geometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' when Ω ∝ 1x if b = c, Ω ∝ 1y if a = c, or Ω ∝ 1z if a = b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The axial geostrophic mode is thus unaffected by viscous dissipation, which explains why the long-term evolution of L · 1Ω is physically unconstrained in such pathological configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This was the situation previously considered for precession-driven flows in spheroids (Lorenzani & Tilgner 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Wu & Roberts 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Guermond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Yet, the conclusion is not valid for every axisymmetric geometry with global rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Indeed, we have τsup ̸= 0 in axisymmetric geometries if Ω is not the revolution symmetry axis (such that L · 1Ω will be damped by viscosity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The BLT of Coriolis eigenmodes has thus explained why the long-term angular mo- 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2 −2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8 Po 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8 1 ϵ b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9 b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2 Po 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8 1 ϵ b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9 b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3 (a) Po− (b) Po+ Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Double resonance at Po± of the forced precession-driven flow in ellipsoids for SF-BC with a = 1 and c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9 (values of b given in the legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Time-averaged differential rotation ϵ = |ω| of numerical solutions of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) at E = 10−3 and small precession angle α = 3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vertical dashed lines show Po± predicted by equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8) at b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' mentum evolution is damped by viscosity in triaxial geometries, but also in axisymmetric ellipsoids if the mean rotation axis Ω is not the revolution symmetry axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Resonance conditions for mechanical forcings A key property of the primary uniform-vorticity flow is its ability to enter in direct resonance with the precession forcing (as evidenced by the divergent amplitude of the asymptotic solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A direct resonance requires a close spatial and temporal matching between the Poincar´e force and the flow response (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Greenspan 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The spatial matching is ensured by the fact that both the Poincar´e force and the forced uniform-vorticity flow are linear in the Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Heuristically, the temporal resonance condition requires that the frequency ωp of the forcing (for monochromatic forcings) must be equal (or close) to the angular frequency f of the forced flow in the mantle frame, which gives f = ± ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The latter condition generally predicts the existence of two resonances for mechanically driven flows in ellipsoids (if the spatial resonance conditions are satisfied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A quick inspection of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) shows that the uniform-vorticity dynamics roughly corresponds to that of a harmonic oscillator driven by the Poincar´e force in the inviscid regime E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Consequently, direct resonances occur when the forced flow corresponds to a free oscillatory eigenmode of the unforced system, namely the spin-over mode Qso such that f ∝ λso (up to a normalisation prefactor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' For this reason, longitudinal librations (which only directly excite the zero-frequency geostrophic mode) do not exhibit any inviscid resonance in spheres (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013) or ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' On the contrary, latitudinal librations can trigger the spin-over mode and the corresponding forced laminar flow exhibits two inviscid resonances occurring at λso = ± ωp in non-spherical geometries (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vantieghem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015), where ωp is the libration angular frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Similarly, a second resonance has already been found for the interaction between tides and precession in triaxial ellipsoids (C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A second resonance for pure precession is thus also expected in ellipsoids from simple theoretical arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Assuming that the forced uniform-vorticity flow is oscillating in the mantle frame at the effective angular frequency f ≃ [1 + Pz] λso when |Px| ≪ 1, the temporal resonance condition predicts two direct resonances for precession Precession-driven flows in stress-free ellipsoids 17 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6 Po 10−2 10−1 100 ϵ a = 2, c ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='10 a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5, c = 1 a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1, c ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='876 Poincar´e DNS 10−3 10−2 10−1 100 101 a − b 10−2 10−1 ϵ SF-BC (a ̸= b) Poincar´e (a) (b) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Behaviour of the forced flow near the second resonance Po− in stress-free ellipsoids with b = 1 and Px = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The resonant value is fixed at the value Po− obtained with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 and c = 1 (as in figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To maintain a fixed resonance when a is varied, the polar axis is given by c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5[−2a2 − 2 + 2 � −32a2 ∆1/2 + 1 + a4 + 2 (8∆ + 7)a2]1/2 with ∆ = (Po− − Px)(Po− + Px).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In the two panels, the dashed teal line shows the expected inviscid value from Poincar´e solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9) for a = b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (a) Comparison between DNS at E = 5 × 10−4 and asymptotic solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (b) Numerical solutions of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) for SF-BC at Po = Po− and E = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' at the resonant Poincar´e numbers Po± given by 1 + Po± cos(α) = ±1/λso, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8) where λso = 2ab/ � (a2 + c2)(b2 + c2) is here the eigenfrequency of the spin-over mode in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2) with Ω = 1z (see also formula 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='21 in Vantieghem 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The above condition is exactly the resonance condition of asymptotic solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The two resonances at [Po−, Po+] are thus robust features of precession-driven flows, but it remains to elucidate why the second resonance at Po− has not been reported before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have numerically solved equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4) in time to explore the behaviour of the solutions near the double resonances in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The two resonances at Po± are con- tinuously shifted when b is varied and, at b = 1, the resonant value Po− differs from Po+ as observed in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This directly results from condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8), which predicts that the two resonances are linked by [Po+ + Po−] cos(α) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This clearly shows that the two direct resonances do not merge together in ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We further explore the behaviour near Po− in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have fixed the resonant value Po− at its value given in figure 3 for a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 and b = c = 1 and, then, adjusted the polar axis c to maintain the resonance at Po− for different values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We observe that the width of the resonance peak decreases when a → b (panel a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This is a purely inviscid feature of the asymptotic solution, which is recovered in the DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The particular case a = b is not formally defined for SF-BC, but it can be approached by decreasing a − b (panel b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The amplitude of the stress-free solution at Po = Po− is limited by the viscosity and approaches, when a → b, the inviscid Poincar´e solution for a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The differential rotation ϵ of the inviscid Poincar´e solution is given by (assuming ωz = 1, see Appendix B in Wu & Roberts 2011) ϵ = ���� Px(2 + η) η + 2(1 + η)Pz ���� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9) which is non-divergent when Po = Po−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This agrees with a lengthy mathematical analysis 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8 1 z 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6 (vf · 1z)/(UE1/2) Ekman layer ≃ 5 E1/2 (a) (b) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' DNS of precession-driven flow with SF-BC at Po = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8, Px = Po sin(α) = 10−2 and E = 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Axisymmetric geometry a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 and b = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Normalised velocity vf/(UE1/2), as defined in expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='11), at time t = 39530 where U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='0129 is the maximum of |vf|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (a) Three-dimensional rendering of the velocity magnitude using a linear scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (b) Axial velocity component as a function of z along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' of the behaviour near the inviscid resonances (not given here), which shows that the second inviscid resonance at Po− disappears in spheroids with a = b contrary to the other resonance at Po+ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Busse 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir & C´ebron 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have thus understood why precession-driven flows are subject to two inviscid resonances in triaxial ellipsoids, which occur at the resonant Poincar´e numbers Po± given by equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8) when |Px| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Since the two resonances are inviscid features of the forced flow in ellipsoids, they exist for both SF-BC and NS-BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The second resonance actually disappears in spheroidal geometries a = b (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' its amplitude is vanishing), which explains why previous works in spheroids have not observed it (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Nobili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Previous studies in triaxial geometries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir & C´ebron 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Burmann & Noir 2022) have also overlooked it, because it usually occurs at |Po−| ≫ |Po+|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Implications for DNS We have shown that the long-term evolution of angular momentum is affected by viscosity, due to the existence of an Ekman boundary layer in rapidly rotating ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The uniform-vorticity elements carrying angular momentum in expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='11) do not indeed satisfy the SF-BC in triaxial geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Thus, they are associated with an Ekman boundary layer to match the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This is a noticeable difference with the more usual spherical geometry, in which Γ ν = 0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The Ekman boundary layer in ellipsoids is clearly observed in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Its typical thickness is still O(E1/2) but, contrary to the case of NS-BC, the amplitude of the boundary-layer flow is O(E1/2) smaller than the bulk flow amplitude (in agreement with Rieutord 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This could have implications for numerical studies using stress-free boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A numerical strategy has to be employed to ensure the conservation of angular momentum in spherical codes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This is no longer necessary in triaxial ellipsoids since Γ ν ̸= 0 (albeit such a strategy may be considered to ensure the conservation of the axial angular momentum if the mean rotation axis is an axis of revolution symmetry, as proposed in Guermond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' However, for the moderate values of the Ekman number achievable in DNS, the flow within the Ekman layer will modify the value of the viscous torque (which pilots the long-term evolution of angular momentum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Indeed, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2Precession-driven flows in stress-free ellipsoids 19 instead of using expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='14), the viscous torque is usually computed with the surface integral Γ ν = 2 E � S r × (∇ · ϵ) dS as Γ ν = 2 E � S r × T dS = 2 E � S r × [(T · 1n) 1n] dS, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='10) in which we have used formula (9) in Rochester (1962) for a symmetric tensor to obtain the first equality and, then, have written the surface traction as T = (T · 1n) 1n − 1n × (1n × T ) = (T · 1n) 1n on the boundary for SF-BC (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='10) shows that the normal component of the surface traction, which is non-zero in the presence of an Ekman layer in stress-free ellipsoids, contributes to the viscous torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Hence, numerical and local approximations of SF-BC (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3) have no reasons to yield a vanishing torque component in formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='10) for axisymmetric ellipsoids if the boundary layer is not sufficiently resolved (as observed in some DNS, not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Using a refined boundary- layer mesh may thus be required to properly describe the Ekman layer in ellipsoids and ensure sufficient torque accuracy (which can be used to check the numerical convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Scaling laws Despite the existence of a thin Ekman layer, we believe that adopting SF-BC in global simulations is useful to probe bulk mechanisms that can be hampered by viscous effects when NS-BC are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The case of precession is illuminating in this respect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Indeed, the laminar precession-driven flow can be destabilised by several hydrodynamic instabilities in no-slip ellipsoids, such as the inertial (topographic) instabilities outlined in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3 and the conical-shear instability (CSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The former are due to the ellipticity of the boundary and survive in the inviscid regime E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' On the contrary, the CSI is a parametric instability existing because of the viscous conical shear layers spawned from the Ekman layer at the critical latitudes (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In addition, precession also often triggers boundary-layer instabilities within the Ekman layer for NS-BC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Lorenzani & Tilgner 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Buffett 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A comprehensive study of these instabilities deserves further work, but we can estimate their relevance as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' As outlined in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3, the typical inviscid growth rate of the precession-driven inertial instabilities is given by formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='16) for the large-scale modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' For the CSI, the growth rate in full spheres and ellipsoids is given by (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Horimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020) σCSI = O(ϵE1/5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='11) Quantitatively, a necessary condition for the existence of the two instabilities is that growth rates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='16) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='11) are larger than the viscous damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' For the NS-BC, this damping is mainly due to the Ekman layer and its amplitude is of the order O(E1/2) (Greenspan 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Actually, it appears that large-scale inertial instabilities are difficult to obtain for the moderately small values of the Ekman number usually considered in experiments or DNS (as outlined in figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A linear analysis is, however, not sufficient to determine the physical relevance of these instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In particular, scaling laws are worth finding to estimate the strength of the precession-driven flows driven by such instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Indeed, the inertial instabilities have presumably a saturation amplitude almost independent of the Ekman number (as found for the turbulence driven by tidal instabilities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Grannan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2017), whereas the CSI amplitude could decrease when E → 0 as the instability results from viscous effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A rigorous description of the nonlinear regimes requires dedicated simulations, but the saturation amplitudes can be crudely estimated using simple order-of-magnitude arguments (which have already proven useful for tidal flows, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' in Barker & Lithwick 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Barker 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We assume that the flow amplitude U resulting from the primary 20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron 10−2 10−1 100 ϵ − ϵc 100 101 U E−2/5 Ef = 10−4 Ef = 3 × 10−5 Ef = 10−5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='0 log10 |Po| 10−17 10−14 10−11 10−8 10−5 E 10−6 10−4 10−2 1 η = a2/c2 − 1 Early Earth Moon Early Moon Horimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (2020) Nobili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (2021) η = E1/5 Utopo ≫ UCSI Utopo ∼ UCSI (a) (b) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (a) Comparison between scaling law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='13) and DNS for |Po| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 in no-slip full spheres from the database of figure 7 in C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (2019), with Ef = E/|1 + Po| ≃ E when |Po| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Colour bar indicates log10 |Po|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Grey area shows the scaling law U E−2/5 = (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5) (ϵ − ϵc) and dashed line U E−2/5 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 (ϵ − ϵc), where ϵc ≈ 7E3/10 f is an estimate of the onset value (see equation 17 in C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (b) Competition between the inertial instabilities and the CSI in precessing ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Empty blue squares □ show conditions for which inertial (topographic) instabilities are expected, and red crosses × indicate where the CSI is expected or observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Grey area shows η ∝ E2/5 with a (unknown) prefactor chosen in the range [1, 100], in which we expect Utopo ∼ UCSI when both instabilities exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Hatched area is the region where σtopo ≳ σCSI (if the two instabilities coexist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' White area is the region where Utopo ≪ UCSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Estimates for the early Moon and Earth taken from Appendix C with η ≈ 2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' instability grows until secondary instabilities, characterised by the growth rate σsec, become strong enough to prevent further growth of the primary instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A saturated turbulent regime would then be obtained when U ∼ σsecℓ, where ℓ is a characteristic length scale of the primary unstable flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The nonlinear saturation of the inertial (topographic) instabilities would thus be given by (in dimensionless units) Utopo = O(ϵη) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='12) with ℓ ∼ 1 for a large-scale instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A good agreement with the above scaling law has been found using DNS in shearing periodic boxes (Barker 2016b), but the scaling law might be different for short-wavelength instabilities with ℓ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Using the same reasoning for the CSI, the relevant length scale is likely the width of the critical shear layer ℓ ∼ E1/5 (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Assuming that the CSI is limited by secondary CSI within the critical shear layers, we obtain the (dimensionless) scaling law UCSI = O(ϵE2/5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='13) We compare in figure 9(a) the above scaling law with previously published DNS in no-slip full spheres (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Considering the full sphere geometry allows us to discard the possible CSI resulting from the inner boundary (which would give a different scaling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' However, even in the full sphere, identifying the instability mechanism is difficult due to the competition between the CSI and the boundary-layer instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Moreover, due to the non-trivial dependence of the two viscously driven instabilities with the forcing parameters, it is unlikely that a single scaling law could fully describe the entire simulation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The onset distance is indeed difficult to estimate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' see figure 6 in C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Besides, the simulations may not be in the asymptotic Precession-driven flows in stress-free ellipsoids 21 regime E ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Despite such uncertainties, a fairly good agreement is found between the DNS and scaling law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='13) sufficiently far from the onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This suggests that the CSI was present in the nonlinear regime and that its saturation amplitude obeys scaling law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='13) for sufficiently small values of the Ekman number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Finally, the comparison between scaling laws (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='13) shows that the inertial instability would have a larger amplitude than the CSI when η ≫ E2/5 (if the two instabilities were simultaneously triggered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The resulting regime diagram is illustrated in figure 9(b), using planetary estimates given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precession-driven inertial instabilities may only have been excited in the primitive liquid cores of the Earth and Moon, whereas the CSI is expected to be present (respectively absent) in the core of the Moon (respectively the Earth) during its whole history (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Landeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In the early Moon, the inertial instabilities may have dominated the CSI in flow amplitude (although the CSI may have had a larger growth rate than the inertial instabilities according to previous formulas, not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Therefore, the inertial instabilities may actually be more relevant than the CSI for some planetary conditions (although they have not been convincingly observed yet in experiments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Nobili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Burmann & Noir 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This could be key for the generation of planetary magnetic fields, as initially postulated for the geodynamo (Malkus 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Preliminary estimates of the dynamo capability of the precession-driven instabilities, obtained using (speculative) order-of-magnitude arguments, are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Conclusion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Summary We have investigated precession-driven flows in stress-free ellipsoids, using asymptotic analysis and targeted DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have developed a reduced model for SF-BC to determine the forced uniform-vorticity flows, which carry angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have shown that angular momentum is affected on long time scales by viscosity in triaxial ellipsoids, but also in axisymmetric geometries if the mean rotation axis is not a revolution symmetry axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This is a noticeable difference from spherical geometries, in which angular momentum is unaffected by viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The fundamental reason is that the flows carrying a non-zero angular momentum in ellipsoids are associated with an Ekman boundary layer in rotating ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' From a numerical viewpoint, a boundary-layer mesh may be necessary to get numerical convergence of the angular momentum in rotating ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We also have obtained the analytical solution of the time-dependent laminar flow forced by precession in the mantle frame, which is valid for planetary parameters and triaxial geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The comparison with the DNS has shown that, even for moderately small values of the Ekman number, the forced laminar flow in the DNS converges to the asymptotic solution in the vanishing viscosity regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Moreover, we have uncovered a second (inviscid) resonance of the forced laminar flow in triaxial ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Then, we have explored the inertial instabilities growing upon the forced laminar flow in the bulk, which survive in the inviscid regime E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have shown that these instabilities could be more easily observed in stress-free ellipsoids than in no-slip ones (at least for the moderate values E ≳ 10−6 considered in DNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We have finally proposed scaling laws for the velocity amplitude of the inertial instabilities and of the CSI, which are in good agreement with previous DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The comparison between the two scaling laws confirms that replacing NS-BC with SF-BC in the mantle frame could be useful to directly probe scenarios of bulk turbulence in the low-viscosity regime (which are of interest for planetary modelling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Perspectives Despite the presence of a thin Ekman boundary layer, we believe that SF-BC are relevant for global models of mechanically driven flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The stress-free model could be used to investigate the saturated flows driven by the inertial (topographic) instabilities in precession ellipsoids and, then, their dynamo capability for planetary applications (as outlined in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Stress-free models could indeed shed new light on alternative mechanisms giving birth to dynamo fields in planetary interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' For instance, the past dynamo of the Moon may have been driven by precession (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Dwyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Yet, previous numerical investigations of precession-driven dynamos failed to reproduce large- scale magnetic fields in spherical geometries (C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This could result from the fact that the turbulence was driven in those simulations by viscous flows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' the CSI or boundary-layer instabilities), which may be negligible in amplitude compared with the turbulence driven by the inertial (topographic) instabilities in the early Moon (as discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This hypothesis could be tested in simulations using stress-free ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Similarly, energetic arguments suggest that the dynamo of the early Earth may have been sustained by tidal flows (Landeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' However, the associated fluid dynamics remains to be quantitatively studied to go beyond prior proof-of-concept simulations (Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precessing stress-free ellipsoids are also relevant for short-period hot Jupiters (Barker 2016b), or gaseous planets with a big moon outside the equatorial plane (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' the Neptune/Triton pair, Wicht & Tilgner 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Finally, SF-BC could also be used to revisit the long-standing problem associated with the generation of geostrophic flows in rotating fluids (Greenspan 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Nonlinear interactions within the Ekman boundary layers for NS-BC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Busse 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021) or in the bulk through the action of the Reynolds stresses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Zhang & Liao 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Livermore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2016), are usually invoked, but geostrophic flows can also result from bulk turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' However, it remains unclear whether two- or three- dimensional rotating bulk turbulence is established in natural systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Le Reun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This fundamental problem has been attacked in cylindrical or plane-layer geometries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Kerswell 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Brunet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Le Reun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Yet, the latter geometries are not directly relevant for planetary modelling, due to the absence of the so-called topographic beta effect that strongly modifies the geostrophic flows in spheres and ellipsoids (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Greenspan 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We believe that using SF-BC opens the way for new fundamental studies dealing with the interplay between waves and geostrophic flows in global geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We acknowledge the three anonymous referees for their constructive criticisms, which considerably improved the quality of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We also acknowledge the editor, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Balmforth, for his careful editorial work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme via the theia project (grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 847433).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' ISTerre is part of Labex OSUG@2020 (ANR10 LABX56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Declaration of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The authors report no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Author ORCIDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' ID J´er´emie Vidal https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='org/0000-0002-3654-6633;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' ID David C´ebron https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='org/0000-0002-3579-8281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Author contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The paper is an idea of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', who designed the study, conducted the asymptotic theory and developed the bespoke numerical code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' conducted the finite-element computations using comsol, and analytically obtained the second-order geostrophic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Both authors discussed and approved the results presented in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' drafted the paper, and both authors gave final approval for submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precession-driven flows in stress-free ellipsoids 23 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Angular momentum for compressible fluids We investigate whether alternative definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='8), which has proven useful for incom- pressible flows, can be extended to compressible flows with a spatially varying density ρ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' For mathematical tractability, we assume that the density does not vanish on the ellipsoidal boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Then, we expand the velocity of compressible flows using the weighted Helmholtz decomposition in rigid ellipsoids as (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & C´ebron 2020) v = (1/ρ) ∇ × A + ∇Φ, v · 1n|S = 0, (A 1a,b) where A is a vector potential and Φ is a scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The first subspace represents anelastic flows satisfying ∇ · (ρv) = 0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011), whereas the irrotational subspace represents compressible flows with ∇ · (ρv) ̸= 0 (such as the acoustic modes without rotation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & C´ebron 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This spectral decomposition has the great advantage of being compatible with the natural inner product of the fully compressible (and anelastic) problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Sobouti 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Clausen & Tilgner 2014) ⟨a, b⟩V = � V ρa† · b dV, (A 2) where a† is the complex conjugate of the vector a, contrary to the usual Helmholtz decomposition v = ∇ × A + ∇Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Consequently, the two subspaces in decomposition (A 1) are mutually orthogonal with respect to inner product (A 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Guided by planetary applications, we only consider in the following density profiles of the form ρ(r) = ρ0(x2/a2 + y2/b2 + z2/c2), (A 3) for which the density is constant on every homothetic ellipsoidal shell in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Such density profiles are indeed often assumed in compressible planetary models, where they represent background density profiles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' in ellipsoids Clausen & Tilgner 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & C´ebron 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Direct calculation The angular momentum is defined for compressible fluids as L = � V r × (ρ0v) dV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' As in the incompressible case, the anelastic subspace has elements with non-zero angular momentum (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' in spheres Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Hence, it only remains to calculate the angular momentum associated with the compressible subspace in decomposition (A 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A direct calculation gives (using formula B26 in Mathews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1991) � V r × (ρ0∇Φ) dV = − � V Φ (r × ∇ρ0) dV − � V ∇ × (ρ0Φ r) dV, (A 4a) = − � V Φ (r × ∇ρ0) dV + � S ρ0Φ (r × 1n) dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (A 4b) It shows that, if the density is of the form (A 3), the compressible subspace has no angular momentum in spheres (since ∇ρ0 ∝ r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' On the contrary, the compressible subspace in spectral decomposition (A 1) has always a non-zero angular momentum in ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Projection approach We have outlined that the two subspaces in decomposition (A 1) have a non-zero angular momentum in compressible ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The remaining question is whether, as for incompressible flows, this angular momentum is solely carried by the uniform-vorticity elements ei(r) given by formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9) in rigid ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We project the velocity onto the 24 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron three uniform-vorticity elements with respect to inner product (A 2), obtaining � V ei · (ρ0v) dV = � V (1i × r) · ρ0v dV � �� � L·1i + � V ∇Ψi · (ρ0v) dV (A 5) where L · 1i are the Cartesian components of the angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We recover from the above expression that the compressible angular momentum is the projection onto the solid-body rotations 1i × r in spherical geometries (for which Ψi = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' An admissible decomposition for compressible spherical flows is thus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mathews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1991) v(r, t) = ω(t) × r + vf(r, t), � V r × (ρ0vf) dV = 0, (A 6a,b) where the compressible flow vf has no angular momentum by definition since ⟨ω × r, vf⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In ellipsoids, the last volume integral in equation (A 5) can be simplified by using the divergence theorem and decomposition (A 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' It gives � V ∇Ψi · (ρ0v) dV = � S Ψx (ρ0v) · 1n dS � �� � 0 − � V Ψi ∇ · (ρ0v) dV, (A 7a) = � 0 if ∇ · (ρ0v) = 0, − � V Ψi ∇ · (ρ0∇Φ) dV if ∇ · (ρ0v) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (A 7b) Equation (A 7) shows that the angular momentum of anelastic flows with ∇ · (ρ0v) = 0 is rigorously given by L · 1i = ⟨ei, v⟩, as in the incompressible case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We can thus extend formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='11) to anelastic flows as v(r, t) = U(r, t) + vf(r, t), ∇ · (ρ0vf) = 0, � V r × (ρ0vf) dV = 0, (A 8a–c) where U(r, t) is the uniform-vorticity flow given by expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='12), and vf is an anelastic flow with ⟨U, vf⟩ = 0 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' However, in the fully compressible case, the angular momentum cannot be obtained as the projections of the compressible flow onto the uniform-vorticity elements in ellipsoids (because (A 7) does not vanish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Moreover, we have by virtue of the divergence theorem � V ei · (ρ0∇Φ) dV = − � V Φ ∇ · (ρ0ei) dV = − � V φ (ei · ∇ρ0) dV = 0 (A 9) if the density is of the form (A 3) because ei · ∇ρ0 ∝ ei · 1n = 0 on every homothetic ellipsoidal shell in the volume (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' not only on the outer ellipsoidal boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Thus, the compressible subspace can have a non-zero angular momentum that is not carried by the uniform-vorticity elements in ellipsoids (since we have simultaneously ⟨ei, ∇Φ⟩ = 0 and � V r × (ρ0∇Φ) dV ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In such configurations, a possible generalisation of anelastic expansion (A 8) to the compressible case could be v(r, t) = U(r, t) + vf(r, t) + ∇Φ, ∇ · (ρ0vf) = 0, � V r × (ρ0vf) dV = 0, (A 10a–c) where U(r, t) is a uniform-vorticity flow given by expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='12) in rigid ellipsoids, vf is an anelastic flow having no angular momentum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' ρ0vf = ∇ × A but with ⟨U, vf⟩ = 0), and ∇Φ is a potential flow carrying a non-zero angular momentum even if ⟨U, ∇Φ⟩ = 0 according to equation (A 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The anelastic and fully compressible cases may thus give different results for the Precession-driven flows in stress-free ellipsoids 25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 3 c 10−3 10−2 10−1 100 101 102 103 104 |τso/E| Formula (B1b) Formula (B2) Formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6) 0 1 2 3 b 10−3 10−2 10−1 100 101 102 103 104 |τsup/E| Ω = 1z Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 1x + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 1y + 1z (a) (b) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (a) Decay rate |τso/E| for Ω = 1z as a function of the semi-axis c, in spheroids with a = b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Comparison between correct formula (B 1b) and erroneous one (B 2) for the surface integral in expression (B 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Note that |τso| → 0 when c → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (b) Decay rate |τsup/E| computed from formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6) as a function of the semi-axis b, for two rotation vectors Ω in ellipsoids with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 and c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Note that |τsup| → 0 when b → a (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' in the spheroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' evolution of angular momentum in rotating compressible ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Differences between the two formulations can be expected when the compressible subspace significantly interacts with the anelastic one in spectral decomposition (A 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' This for instance happens in the presence of global rotation when MΩ = O(10−1), where MΩ = RΩ0/C0 is the rotational Mach number (Vidal & C´ebron 2020, 2021a) and C0 is the speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Planetary estimates give MΩ = O(10−3) for planetary moons, but larger values MΩ = O(10−1) are obtained in Jupiter-like gaseous planets (which are also non- spherical because of centrifugal gravity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Investigating the long- term evolution of angular momentum in such strongly compressible rotating bodies certainly deserves further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Viscous decay rates We present an alternative formula for the viscous decay rate of the Coriolis eigenmodes in stress-free ellipsoids, which is equivalent to formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To enforce SF-BC (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3) in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5), we employ the curvilinear orthogonal coordinates [q1, q2, q3] (such that the boundary is given by a constant value of q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Then, the volume integral can be rewritten using the divergence theorem as τk E � V |Qk|2 dV = IS − � V |∇ × Qk|2 dV (B 1a) with the surface integral (dS = h2h3 dq2dq3 being the surface element) IS = 2 � S � 1 h1h2 ∂h2 ∂q1 |Qk · 1q2|2 + 1 h1h3 ∂h3 ∂q1 |Qk · 1q3|2 � dS, (B 1b) where [h1, h2, h3] are the curvilinear scale factors and [1q1, 1q2, 1q3] are the orthogonal basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' In the sphere, expression (B 1b) reduces to IS = 2 � S |Qk × 1n|2 dS, (B 2) 26 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron Body E f = η/2 α [◦] Po Px ϵ Earth 10−15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 × 10−3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 × 10−7 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4 × 10−8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7 × 10−5 Early Earth 5 × 10−16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='0 × 10−2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='3 × 10−8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 × 10−6 Moon 10−12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='54 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='0 × 10−3 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='0 × 10−4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='0 × 10−2 Early Moon 5 × 10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='0 × 10−4 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7 × 10−4 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='0 × 10−4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='2 × 10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='4 × 10−1 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precession forcing in the liquid core of the Earth and Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Ekman number E based on the typical viscosity value ν = 10−6 m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='s−1, polar flattening f = (a − c)/a, precession angle α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Currently, f is well enough known for the Earth (Mathews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2002), but the lunar values of f vary from f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5×10−5 for a purely hydrostatic Moon (Le Bars et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011) to f = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='0×10−4 when considering the present-day non-hydrostatic lithosphere and a liquid core of radius 350 km (Viswanathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Parameters for the Early Moon and Earth, estimated ∼ 4 Gy ago, are deduced from the current values by considering a spin rate Ω0 two times larger, leading to values of E twice smaller and of f fourth time larger than the present estimates (due to the centrifugal acceleration in Ω2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Typical estimates for the Moon’s history from C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (2019) and the orbital evolution model of Touma & Wisdom (1994), and for the Early Earth from the low-obliquity scenario in Landeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' recovering formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='14) of Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' (2001) in the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Note that vector expression (B 2) has been erroneously employed in the spheroid (see formula (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='21) in Maffei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2017, which is incorrect because of the missing curvature terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Expression (B 1a) is very difficult to implement in practice (because of the curvilinear coordinates), contrary to formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6) in which the volume integral can be performed fully analytically in ellipsoids (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' see formula 50 in Lebovitz 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' For a numerical (cross-validation) benchmark of formulas (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6) and (B 1a), we can compute the decay rate Qso of the spin-over mode in spheroidal geometries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' with a = b = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To do so, we take the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='25) in Vantieghem (2014), giving Qso for Ω = 1z in triaxial ellipsoids, and express it using the curvilinear spheroidal coordinates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1 in C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021) x = ηT sin(q2) cos(q3), y = ηT sin(q2) sin(q3), z = η (dq1T ) cos(q2), (B 3a–c) with η = |1 − (c/a)2|1/2 and T = cosh(q1) for oblate spheroids (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' a ⩾ c) or T = sinh(q1) for prolate spheroids (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' a ⩽ c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The scale factors are then h1 = h2 = η[sinh2(q1) + cos2(q2)]1/2 when a ⩾ c or h1 = h2 = η[cosh2(q1) − cos2(q2)]1/2 when a ⩽ c, and h3 = ηT sin(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The differences between formulas (B 1b) and (B 2) are illustrated in figure 10(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' For the particular geometry a = b = 1 and c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='9, we have � V |Qk|2 dV ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='36965, � V |∇ × Qk|2 dV ≃ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='64855 and IS ≃ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='23369 from formula (B 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Formulas (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='6) and (B 1a) then both predict that τso/E ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='12312 in this spheroidal geometry (as observed in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' On the contrary, we would get IS ≃ 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='30153 with formula (B 2), yielding the erroneous value τso/E ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='69652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Finally, we show in figure 10(b) the decay rate τsup for different orientations of the mean rotation axis in triaxial ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precession-driven flows in stress-free ellipsoids 27 Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Planetary extrapolation for dynamo action We can crudely estimate the dynamo capability of precession-driven flows using ener- getic arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To do so, we compute a magnetic Reynolds number Rm as Rm = Utopo/Em, Em = νm/(Ω0R2), (C 1a,b) where Em is the magnetic Ekman number and νm ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='5 − 4 m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='s−1 is the magnetic diffusivity of the fluid at typical core conditions (estimated from measurements and computations of the electrical conductivity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' see figure 1 in Ohta & Hirose 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A necessary condition for large-scale dynamo action is that Rm ⩾ O(102) in spheres or ellipsoids (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Holdenried-Chernoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & C´ebron 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Estimating the magnetic Reynolds number thus crucially depends on the scaling law for the flow strength Utopo, whose order of magnitude is expected to be given by formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' To be more quantitative, we rewrite formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='12) using asymptotic flow (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='7) in the planetary regime |Px| ≪ 1, which gives at the leading order in η ≪ 1 Utopo ≃ Kϵη ∼ K � 2|Po| when α = π/2, | tan(α)| η when α ̸= π/2, (C 2) where α is the precession angle measured from 1z, and K is an unknown numerical prefactor that must be determined for planetary extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' We recover from our asymptotic solution that the quantity ϵη is actually independent of η at the leading order when α = π/2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' see formula 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='b in Horimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020) and that, when α ̸= π/2, the differential rotation ϵ becomes independent of Po in the regime |Px| ≪ 1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Moreover, local DNS in periodic shearing boxes, performed at α = π/2, are actually consistent with the scaling law Utopo ∝ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='1|Po| (see figure 7 in Barker 2016b), which is of the form (C 2) with the numerical constant K ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Assume that K is a constant (without further numerical results), we can crudely estimate the dynamo capability of the flows driven by the (topographic) inertial instabilities for realistic planetary conditions by combining equations (C 1) and (C 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Using acceptable scenarios for the lunisolar precession over time (see table 1), we obtain Rm ⩽ O(10) in the Earth’s core over geological ages, showing that precession was not strong enough to drive dynamo action (even billion years ago, which agrees with the conclusions of Landeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Similarly, the estimate Rm ⩽ O(1) in the current Moon’s core shows precession is not presently dynamo capable (in agreement with the end of the lunar dynamo observed in paleomagnetic studies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mighani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' However, we can obtain larger values Rm ⩽ 140 for the liquid core of the early Moon (depending on the uncertainties on the polar flattening η and the magnetic diffusivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Our estimate thus suggests that precession might have been dynamo capable in the early Moon (as initially suggested by Dwyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Further work is obviously needed to rigorously assess the relevance of scaling law (C 2) in precessing ellipsoids, which is key for planetary extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Adopting SF-BC would be particularly useful to strongly weaken the viscous turbulent flows (which are a priori not well suited to sustain large- scale dynamo fields, see C´ebron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019) and extract a robust scaling law for the saturation amplitude of the inertial (topographic) instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 28 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron REFERENCES Backus, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Rieutord, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2017 Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' E 95 (5), 053116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Barker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2016a Non-linear tides in a homogeneous rotating planet or star: global simulations of the elliptical instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 459 (1), 939–956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Barker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2016b On turbulence driven by axial precession and tidal evolution of the spin- orbit angle of close-in giant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 460 (3), 2339–2350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Barker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Lithwick, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013 Non-linear evolution of the tidal elliptical instability in gaseous planets and stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 435 (4), 3614–3626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Brunet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Gallet, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Cortet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020 Shortcut to geostrophy in wave-driven rotating turbulence: the quartetic instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 124 (12), 124501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Buffett, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021 Conditions for turbulent Ekman layers in precessionally driven flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 226 (1), 56–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Burmann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Noir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2022 Experimental study of the flows in a non-axisymmetric ellipsoid under precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 932, A24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Busse, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1968 Steady fluid flow in a precessing spheroidal shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 33 (4), 739–751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015 Bistable flows in precessing spheroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 47 (2), 025504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Laguerre, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Noir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Schaeffer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019 Precessing spherical shells: flows, dissipation, dynamo and the lunar core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 219 (Supplement 1), S34–S57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Le Bars, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Le Gal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Moutou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Leconte, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Sauret, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013 Elliptical instability in hot Jupiter systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Icarus 226 (2), 1642–1653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Le Bars, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Meunier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2010 Tilt-over mode in a precessing triaxial ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluids 22 (11), 116601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Vidal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Schaeffer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Borderies, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Sauret, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021 Mean zonal flows induced by weak mechanical forcings in rotating spheroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 916, A39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Chandrasekhar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1969 Ellipsoidal Figures of Equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Dover Publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Herreman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Livermore, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Jackson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2018 The optimal kinematic dynamo driven by steady flows in a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 839, 1–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Clausen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Tilgner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2014 Elliptical instability of compressible flow in ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 562, A25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Dwyer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Stevenson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Nimmo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011 A long-lived lunar dynamo driven by continuous mechanical stirring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Nature 479 (7372), 212–214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Gerick, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Jault, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Noir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Vidal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020 Pressure torque of torsional Alfv´en modes acting on an ellipsoidal mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 222 (1), 338–351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Grannan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Favier, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Le Bars, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Aurnou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2017 Tidally forced turbulence in planetary interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 208 (3), 1690–1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Greenspan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1968 The Theory of Rotating Fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Greenspan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1969 On the non-linear interaction of inertial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 36 (2), 257–264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Guermond, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', L´eorat, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Luddens, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Nore, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013 Remarks on the stability of the Navier–Stokes equations supplemented with stress boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' B Fluids 39, 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Holdenried-Chernoff, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Jackson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019 A trio of simple optimized axisymmetric kinematic dynamos in a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A 475 (2229), 20190308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Horimoto, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Katayama, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Goto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020 Conical shear-driven parametric instability of steady flow in precessing spheroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluids 5 (6), 063901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Ivers, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2017 Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a tri-axial ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 111 (5), 333–354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Jones, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Boronski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Brun, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Glatzmaier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Gastine, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Miesch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Wicht, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011 Anelastic convection-driven dynamo benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Icarus 216 (1), 120– 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Kerswell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1993 The instability of precessing flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 72 (1- 4), 107–144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Kerswell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1999 Secondary instabilities in rapidly rotating fluids: inertial wave breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 382, 283–306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precession-driven flows in stress-free ellipsoids 29 Kerswell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2002 Elliptical instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 34 (1), 83–113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Kida, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020 Steady flow in a rapidly rotating spheroid with weak precession: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 52 (1), 015513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Landeau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Fournier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Nataf, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Schaeffer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2022 Sustaining Earth’s magnetic dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Earth Environ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 3 (4), 255–269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Le Bars, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Le Gal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015 Flows driven by libration, precession, and tides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 47, 163–193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Le Bars, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Wieczorek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Karatekin, ¨O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Laneuville, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011 An impact-driven dynamo for the early Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Nature 479 (7372), 215–218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Le Reun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Favier, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Le Bars, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019 Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 879, 296–326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Le Reun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Gallet, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Favier, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Le Bars, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020 Near-resonant instability of geostrophic modes: beyond Greenspan’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 900, R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Lebovitz, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1989 The stability equations for rotating, inviscid fluids: Galerkin methods and orthogonal bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 46 (4), 221–243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Liao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Earnshaw, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2001 On the viscous damping of inertial oscillation in planetary fluid interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Earth Planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 128 (1-4), 125–136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Marti, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Noir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015 Shear-driven parametric instability in a precessing sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluids 27 (4), 046601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Livermore, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Bailey, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Hollerbach, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2016 A comparison of no-slip, stress- free and inviscid models of rapidly rotating fluid in a spherical shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 6 (1), 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Lorenzani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Tilgner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2001 Fluid instabilities in precessing spheroidal cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 447, 111–128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Lorenzani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Tilgner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2003 Inertial instabilities of fluid flow in precessing spheroidal shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 492, 363–379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Maffei, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Jackson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Livermore, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2017 Characterization of columnar inertial modes in rapidly rotating spheres and spheroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A 473 (2204), 20170181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Malkus, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1968 Precession of the earth as the cause of geomagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Science 160 (3825), 259–264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mason, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Kerswell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2002 Chaotic dynamics in a strained rotating flow: a precessing plane fluid layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 471, 71–106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mathews, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Buffett, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Herring, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Shapiro, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1991 Forced nutations of the Earth: Influence of inner core dynamics: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Solid Earth 96 (B5), 8219–8242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mathews, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Herring, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Buffett, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2002 Modeling of nutation and precession: New nutation series for nonrigid Earth and insights into the Earth’s interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Solid Earth 107 (B4), ETG–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mighani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Shuster, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Borlina, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Nichols, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Weiss, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020 The end of the lunar dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 6 (1), eaax0883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Nobili, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Meunier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Favier, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Le Bars, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021 Hysteresis and instabilities in a spheroid in precession near the resonance with the tilt-over mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 909, A17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Cardin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Jault, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Masson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2003 Experimental evidence of non-linear resonance effects between retrograde precession and the tilt-over mode within a spheroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 154 (2), 407–416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Noir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013 Precession-driven flows in non-axisymmetric ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 737, 412–439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Ohta, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Hirose, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021 The thermal conductivity of the Earth’s core and implications for its thermal and compositional evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 8 (4), 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Poincar´e, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1910 Sur la pr´ecession des corps d´eformables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 27, 321–356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Reddy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Favier, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Le Bars, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2018 Turbulent kinematic dynamos in ellipsoids driven by mechanical forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 45 (4), 1741–1750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rieutord, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1992 Ekman circulation and the synchronization of binary stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 259, 581–584.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 30 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' C´ebron Rieutord, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Zahn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1997 Ekman pumping and tidal dissipation in close binaries: A refutation of Tassoul’s mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 474 (2), 760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Roberts, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Aurnou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2012 On the theory of core-mantle coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 106 (2), 157–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rochester, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1962 Geomagnetic core-mantle coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 67 (12), 4833– 4836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rochester, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1976 The secular decrease of obliquity due to dissipative core—mantle coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 46 (1), 109–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Sobouti, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1981 The potentials for the g-, p- and the toroidal modes of self-gravitating fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 100, 319–322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Tilgner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1999 Non-axisymmetric shear layers in precessing fluid ellipsoidal shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 136 (3), 629–636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Touma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Wisdom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 1994 Evolution of the Earth–Moon system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' The Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 108 (5), 1943–1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vantieghem, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2014 Inertial modes in a rotating triaxial ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A 470 (2168), 20140093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vantieghem, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Noir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2015 Latitudinal libration driven flows in triaxial ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 771, 193–228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2017 Inviscid instabilities in rotating ellipsoids on eccentric Kepler orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 833, 469–511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020 Acoustic and inertial modes in planetary-like rotating ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A 476 (2239), 20200131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021a Acoustic modes of rapidly rotating ellipsoids subject to centrifugal gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Acoust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 150 (2), 1467–1478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2021b Kinematic dynamos in triaxial ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' A 477 (2252), 20210252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Schaeffer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Hollerbach, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2018 Magnetic fields driven by tidal mixing in radiative stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 475 (4), 4579–4594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vidal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Su, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & C´ebron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2020 Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 885, A39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Viswanathan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Rambaux, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Fienga, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Laskar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Gastineau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2019 Observational constraint on the radius and oblateness of the lunar core-mantle boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 46 (13), 7295–7303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Vormann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Hansen, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2018 Numerical simulations of bistable flows in precessing spheroidal shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 213 (2), 786–797.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Wicht, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Tilgner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2010 Theory and modeling of planetary dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 152 (1), 501–542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Boggs, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Yoder, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Ratcliff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Dickey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2001 Lunar rotational dissipation in solid body and molten core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Planets 106 (E11), 27933–27968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Roberts, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2009 On a dynamo driven by topographic precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 103 (6), 467–501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Roberts, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2011 High order instabilities of the Poincar´e solution for precessionally driven flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 105 (2-3), 287–303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Chan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Liao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2012 Asymptotic theory of resonant flow in a spheroidal cavity driven by latitudinal libration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 692, 420–445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Chan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Liao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2014 On precessing flow in an oblate spheroid of arbitrary eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 743, 358–384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Chan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Liao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Aurnou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2013 The non-resonant response of fluid in a rapidly rotating sphere undergoing longitudinal libration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 720, 212–235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Kong, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Schubert, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2017 Shape, internal structure, zonal winds, and gravitational field of rapidly rotating Jupiter-like planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Earth Planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 45 (1), 416–446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Liao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2004 A new asymptotic method for the analysis of convection in a rapidly rotating sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 518, 319–346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Precession-driven flows in stress-free ellipsoids 31 Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=', Liao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' & Busse, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 2007 Asymptotic solutions of convection in rapidly rotating non-slip spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} +page_content=' 578, 371–380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQfiwQZ/content/2301.03254v1.pdf'} diff --git a/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf b/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..619263a8c1537732d6b976383a6762a11e5bc116 --- /dev/null +++ b/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:56d2472cf56652511b4470c48404fef259ec7f91ed7ded64ed88c354fabd5bde +size 4432231 diff --git a/rtFST4oBgHgl3EQfQDhb/vector_store/index.faiss b/rtFST4oBgHgl3EQfQDhb/vector_store/index.faiss new file mode 100644 index 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the role of GxE interactions in human disease + +Amanda J. Lea*, Andrew G. Clark, Andrew W. Dahl, Orrin Devinsky, Angela R. Garcia, +Christopher D. Golden, Joseph Kamau, Thomas S. Kraft, Yvonne A. L. Lim, Dino Martins, +Donald Mogoi, Paivi Pajukanta, George Perry, Herman Pontzer, Benjamin C. Trumble, Samuel +S. Urlacher, Vivek V. Venkataraman, Ian J. Wallace, Michael Gurven†, Daniel Lieberman†, +Julien F. Ayroles* + +†These authors contributed equally +*Corresponding authors’ e-mail: amanda.j.lea@vanderbilt.edu and jayroles@princeton.edu + +AL - Department of Biological Sciences, Vanderbilt University, Nashville, TN, USA +AL - Child and Brain Development, Canadian Institute for Advanced Research, Toronto, Canada +AC - Department of Computational Biology, Cornell University, Ithaca, NY, USA +AC - Department of Molecular Biology and Genetics, Cornell University, Ithaca, NY, USA +AD - Section of Genetic Medicine, University of Chicago, Chicago, IL, USA +OD - Department of Neurology, NYU Langone Medical Center, New York, NY, USA +OD - Comprehensive Epilepsy Center, NYU Langone Medical Center, New York, NY, USA +AG - Center for Evolution and Medicine, Arizona State University, Tempe, United States +CG - Department of Nutrition, Harvard T.H. Chan School of Public Health, Boston, MA, USA +JK - Department of Biochemistry, School of Medicine, University of Nairobi, Nairobi, Kenya +JK - Institute of Primate Research, National Museums of Kenya, Nairobi, Kenya +TK - Department of Anthropology, University of Utah, Salt Lake City, USA +YL - Department of Parasitology, Faculty of Medicine, Universiti Malaya, Kuala Lumpur, +Malaysia +DM1 - Turkana Basin Research Institute, Turkana, Kenya +DM1 - Department of Ecology and Evolution, Princeton University, Princeton, NJ, USA +DM2 - Director at County Government of Laikipia, Nanyuki, Kenya +PP - Department of Human Genetics, David Geffen School of Medicine at UCLA, Los Angeles, +CA, USA +PP - Institute for Precision Health, David Geffen School of Medicine at UCLA, Los Angeles, CA, +USA +GP - Department of Anthropology, Pennsylvania State University, University Park, PA, USA +GP - Department of Biology, Pennsylvania State University, University Park, PA, USA +GP - Huck Institutes of the Life Sciences, Pennsylvania State University, University Park, PA, +USA +HP - Evolutionary Anthropology, Duke University, Durham, NC, USA +HP - Duke Global Health Institute, Duke University, Durham, NC, USA +BT - School of Human Evolution and Social Change, Arizona State University, Tempe, US +BT - Center for Evolution and Medicine, Arizona State University, Tempe, United States +SU - Department of Anthropology, Baylor University, Waco, TX, USA +SU - Child and Brain Development, Canadian Institute for Advanced Research, Toronto, +Canada +VV - Department of Anthropology and Archaeology, University of Calgary, Calgary, Canada +IW - Department of Anthropology, University of New Mexico, Albuquerque, USA +MG - Department of Anthropology, University of California: Santa Barbara, Santa Barbara, CA, +USA +DL - Department of Human Evolutionary Biology, Harvard University, Cambridge, MA, USA +JA - Department of Ecology and Evolution, Princeton University, Princeton, NJ, USA +JA - Lewis Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ, USA + + + +2 +Abstract +Globally, we are witnessing the rise of complex, non-communicable diseases (NCDs) +related to changes in our daily environments. Obesity, asthma, cardiovascular disease, and type +2 diabetes are part of a long list of “lifestyle” diseases that were rare throughout human history +but are now common. A key idea from anthropology and evolutionary biology—the evolutionary +mismatch hypothesis—seeks to explain this phenomenon. It posits that humans evolved in +environments that radically differ from the ones experienced by most people today, and thus traits +that were advantageous in past environments may now be “mismatched” and disease-causing. +This hypothesis is, at its core, a genetic one: it predicts that loci with a history of selection will +exhibit “genotype by environment” (GxE) interactions and have differential health effects in +ancestral versus modern environments. Here, we discuss how this concept could be leveraged +to uncover the genetic architecture of NCDs in a principled way. Specifically, we advocate for +partnering with small-scale, subsistence-level groups that are currently transitioning from +environments that are arguably more “matched” with their recent evolutionary history to those that +are more “mismatched”. These populations provide diverse genetic backgrounds as well as the +needed levels and types of environmental variation necessary for mapping GxE interactions in an +explicit mismatch framework. Such work would make important contributions to our understanding +of environmental and genetic risk factors for NCDs across diverse ancestries and sociocultural +contexts. + + + +3 +Introduction +Non-communicable diseases (NCDs) such as cardiovascular disease (CVD), type II +diabetes, and Alzheimer’s are among the leading causes of death worldwide (Figure 1). NCDs +are often difficult to prevent and treat, because they result from complex and poorly understood +interactions between a person’s genetic makeup and their environment. For example, +cardiovascular disease (CVD) has a heritability of 40-50%, with dozens of loci now mapped +through genome-wide association studies (1–3). However, when tallied together in an additive +framework, these loci explain only a small fraction of the heritable genetic effect. This has led +many to conclude that environmental risk factors—such as a diet high in processed foods and +low levels of physical activity—interact with genetic variation to shape NCD risk (4, 5). In other +words, genetic variation may predispose individuals toward physiological sensitivity or resilience +in the face of environmental perturbations, a phenomenon known as “genotype x environment” +(GxE) interactions (Box 1). +Despite major interest in GxE interactions in the context of NCDs, scientists have +struggled in practice to identify them. There are many complex reasons for this, including that the +relevant environmental factors are often unknown, difficult to measure, or minimally variable within +the study population (e.g., most individuals in high income countries (HICs) consume processed +foods). Further, large sample sizes are needed to test for interaction effects, and even more so +to overcome the multiple testing burden incurred by testing for interactions between many genetic +variants and many environments (6, 7). To overcome power issues, current state-of-the-art +approaches have leveraged very large studies such as the UK Biobank to scan for interactions +between genome-wide genetic variation and key lifestyle factors (e.g., smoking, diet, or physical +activity) (8–11). However, these studies have not delivered as expected, and have only uncovered +a handful of GxE interactions for NCDs like obesity, type II diabetes, and depression. +Here, we argue for a complementary approach informed by anthropological traditions, +genomic tools, and evolutionary theory. In particular, we believe there is much to learn by 1) +viewing GxE interactions through the lens of the “evolutionary mismatch” hypothesis and 2) +partnering with genetically and environmentally diverse small-scale, subsistence-level +populations to map them. The evolutionary mismatch hypothesis posits that traits that evolved +under past selection regimes are often imperfectly or inadequately suited to modern +environments, leading to “mismatches” in the form of NCDs (12–15). At the genetic level, we +would thus expect that previously neutral or beneficial alleles are now disease-causing. +While we cannot go back in time to evaluate genotype-phenotype relationships in past +environments, we can collaborate with populations that practice non-industrial, subsistence-level + +4 +lifestyles that are arguably more “matched” to their recent evolutionary history (though we caution +that, of course, no modern population is perfectly representative of ancestral conditions). +Importantly, many subsistence-level populations are currently exposed to globalizing forces +causing rapid environmental shifts; this situation creates a quasi-natural experiment for studying +the transition from traditional to modern lifeways within a single group (16) (Figure 2A). +Additionally, many subsistence-level groups have already been well-characterized ecologically +and phenotypically through long-term work with anthropologists (Figure 2B; Box 2), setting the +stage for integration of genomic studies. +In this Consensus, we argue that uniting an evolutionary mismatch framework, long-term +anthropological work with subsistence-level groups, and cutting-edge genomic tools can increase +our power to identify and understand GxE interactions. Specifically, because the mismatch +framework provides clear expectations for the types of loci and environments we expect to affect +NCDs, we can narrow the search space considerably. Further, by focusing on populations where +Western diets and lifestyles are the exception rather than the norm, we can design studies that +explicitly sample environmental extremes, thereby boosting power. Finally, by studying many +genetically distinct populations under a uniting intellectual framework, we can identify new loci +that have so far been invisible to studies focused on individuals of European descent. With these +goals in mind, we first review the evolutionary mismatch hypothesis and discuss its current +support at the phenotypic and genetic levels. Second, we propose consensus recommendations +for integrating mismatch principles with molecular and genomic techniques, focusing on +collaborations with subsistence-level groups. Third, we discuss the payoffs for scientists and +study communities that would come from implementing these partnerships. + + +Figure 1. Non-communicable diseases are the leading cause of death worldwide. A) Proportion of worldwide +deaths attributable to non-communicable diseases, communicable (infectious) diseases, and injuries through time. B) +Proportion of deaths within the US in 2019, broken down by the top 10 causes of death. NCDs are highlighted in +green. For both panels, data were sourced from ourworldindata.org and represent all ages. + + +A +Cause +Infectiousdiseases +Injuries +NCDs +B +Cause +Other +NCDs +Proportion of total deaths +1.00 +CVD - +Cancers +0.75 +Chronicrespiratorydisease +Alzheimer's and dementia +0.50 +Digestive diseases +Chronic kidney disease +Respiratory infections +0.25 +Diabetesmellitus +Chronic liver disease +0.00 +Drug use disorders +1990 +2000 +2010 +2020 +0.0 +0.1 +0.2 +0.3 +Year +Proportionoftotal deaths(2019)5 + + +Figure 2. Subsistence-level groups experiencing lifestyle change are a potential model for uncovering GxE +interactions. A) Subsistence-level groups faced with urbanization, market-integration, and modernization experience +extreme variation in diet and physical activity levels, pathogen and toxin exposures, and social conditions. This list of +environmental components for which there is extreme variation is not exhaustive, and in many cases will also be +population specific. We highlight a few broad categories that tend to change consistently during lifestyle transitions. +B) Studies such as The Turkana Health and Genomics Project (17, 18), The Orang Asli Health and Lifeways Project +(19), The Pacific Planetary Health Initiative, Madagascar Health and Environmental Research (20–22), The Tsimane +Health and Life History Project (23), and The Shuar Health and Life History Project (24, 25) all combine +anthropological and biomedical data collection in transitioning societies, and are thus poised to uncover GxE +interactions in the context of evolutionary mismatch. We note that this list is meant to be illustrative and only includes +projects directed by authors of this Consensus; it does not by any means cover all ongoing projects of small-scale, +subsistence-level groups. + +Overview of the evolutionary mismatch hypothesis +An evolutionary mismatch is a condition that is more common or severe in an organism +because it is imperfectly or inadequately adapted to a novel environment (26). While mismatches +are not unique to humans, their frequency may be unusually high in our species. This is because +human culture can generate rapid and profound environmental change: in just a few generations, +industrialization has transformed human diets, physical activity patterns, and toxin exposure +landscapes, especially in HICs, and these changes presumably contribute to the long list of NCDs +that used to be rare or nonexistent (27–29). +For at least a century, a wide range of conditions have been assumed to be “diseases of +civilization” or “lifestyle diseases” (30, 31), but mismatches need to be explicitly and rigorously + +A +Subsistence-level +Urban ++processed foods +Iprocessed foods +Tphysical activity +tphysical activity +tpathogen exposure +tpathogen exposure +↓toxin exposure +↓toxin exposure +ftl socioeconomic inequality +↑l socioeconomic inequality +t social support + social support +B +TheTurkana Health +and Genomics Project +The Orang Asli Health +and Lifeways Project +TheShuar Health and +Life History Project +ThePacificPlanetary +The Tsimane Health +Health Initiative +and Life History Project, +Madagascar Health and +Environmental Research6 +tested according to three criteria (32). First, a mismatch condition should be more common or +severe in the “novel” (e.g., post-industrial, HICs) relative to the “ancestral” environment (Figure +3A). Small-scale, subsistence-level societies typically stand in as the best available, though often +imperfect, proxy for the “ancestral” condition in humans; this is because they experience a closer +“match” between their recent evolutionary history and their current environments relative to +individuals in HICs, though we caution they are not themselves “ancestral” populations. +In addition to the hypothesized mismatch condition being more prevalent in post-industrial +versus subsistence-level groups, the second criteria is that it should also be tied to some +environmental variable that differs between these groups (Figure 3B). One complication for +achieving this is that NCDs arise from complex multifactorial causes, and thus, while between- +population comparisons are necessary, they can be confounded by many covariates that must +also be taken into account (e.g., sanitation, access to medical care, age structure). +The third criteria is that it is necessary to establish a molecular or physiological mechanism +by which the environmental shift generates the proposed mismatch condition. At the genetic level, +this should manifest as a locus for which 1) a variant exhibits a past history of positive selection +and is associated with health benefits in the ancestral environment but health detriments in the +novel environment or 2) past stabilizing selection has created a situation where two intermediate +alleles have similar fitness in the ancestral environment, but one allele becomes associated with +health detriments in the novel environment (Figure 3C; see also Box 1). + +Figure 3. Mismatch diseases must be tested according to three criteria. A) Health phenotypes related to the +hypothesized mismatch disease must be more common or severe in the novel versus ancestral environment. B) + +A +B +Health phenotype +Health phenotype +Novel +Ancestral +Environmentalvariable +c +Ancestral +Health phenotype +Before +After +AA +AT +TT +selection +selection +Genotype7 +These health phenotypes must be attributable to an environmental variable, which will most often differ in mean and +range between groups (e.g., physical activity influences cardiovascular health and is consistently higher in +subsistence-level groups relative to HICs). C) It is necessary to establish a mechanism by which an environmental +shift generates health issues. At the genetic level, this could manifest as a locus for which a variant exhibits a past +history of positive selection and is associated with health benefits in the ancestral environment but health detriments +in the novel environment. In panel C, horizontal lines represent haplotypes and the dark orange circle represents the +selected variant. In all panels, dark blue represents the novel environment and light blue represents the ancestral +environment. + +Current evidence for evolutionary mismatch at the phenotypic level +Scientists have been relatively successful at testing the first two criteria for mismatch, +especially in the context of CVD, the single largest cause of mortality worldwide (33). In support +of the first criteria, subsistence-level groups experience remarkably low rates of CVD (29, 34, 35) +relative to HICs, as well as minimal age-associated increases in CVD or its biomarkers (e.g., +hypertension, cholesterol) (36–38) (Figure 4A). Studies of small-scale societies in the midst of +socioeconomic transition have demonstrated within-population effects of industrialization (17, 39, +40), strengthening the findings from between-population comparisons. +In support of the second criteria, recent work has also isolated salient environmental +changes by which industrialization promotes CVD. People in subsistence-level communities are +generally very physically active, accruing 5-10 times more daily physical activity than adults in +Europe, the U.S., and other HICs (41, 42). Moderate to vigorous physical activity increases +cardiac output promoting nitric oxide production and arterial elasticity (43, 44), it also decreases +baseline levels of inflammation, which plays a critical role in all aspects of CVD (45). Within +industrialized populations, individuals accruing daily physical activity similar to those of +subsistence-level individuals experience similarly low rates of CVD as well as NCD-related +mortality (46) (Figure 4B). However, while physical activity plays a critical role in averting CVD, it +is not a panacea and several other factors are surely important. For example, relative to HICs, +subsistence-level groups subsist on diets dominated by unprocessed or minimally processed +foods and experience different types and degrees of social integration and inequality—all of which +impact CVD risk (47–49). +Finally, we note that while we have focused this section on CVD as an illustrative example +of the type of comprehensive evidence required for diagnosing a mismatch disease, several other +conditions also have relatively clear evidence for the first two criteria for mismatch. For example, +inflammatory and autoimmune disorders have increased during the twentieth century, which has +been linked to a reduced exposure to parasites and microorganisms (a phenomenon attributed to +the “hygiene hypothesis” or “old friends hypothesis”) (50–52). + + +8 + +Figure 4. Evidence for evolutionary mismatch at the phenotypic level. A) Mean levels of total cholesterol are +much lower in select subsistence-level populations relative to US adults (>18 years old) profiled as part of the +National Health and Nutrition Examination Survey (NHANES) (56) (subsistence-level data from (17)). B) Evidence +that, within industrialized populations, individuals accruing daily physical activity similar to those of men and women in +subsistence-level societies experience similarly low rates of CVD as well as all-cause mortality from NCDs. Dose +response relationship between minutes/week of moderate to vigorous leisure time physical activity and age-adjusted +relative risk of death from a sample of 661,137 adult Americans and Europeans (57). The arrow for physical activity +estimates in subsistence-level groups is based on studies of the Hadza (estimated at x=944 minutes (35)) and the +Tsimane (x=924 minutes (58)). + +Current evidence for evolutionary mismatch at the genetic level + As mentioned above, to fulfill the third criteria for mismatch, we would need to identify a +locus for which 1) there is evidence of past selection and 2) performance of at least one allele +varies across environments and confers inflated risk of an NCD in the novel environment (see +also Figure 1B and Box 1). One would think this would be easy to find, but in fact there are only +a handful of clear cases, despite good evidence for the existence of GxE interactions in general +(59–62). One clear example of mismatch involves variants in the APOL1 gene, which provides +resistance to trypanosome infections. Given the prevalence of trypanosomes across Africa, +beneficial alleles are found at high frequency in African populations as well as African Americans. +However, these same variants confer elevated kidney disease risk in African Americans living in +the US (63, 64). + +Another example is related to the “thrifty genotype” hypothesis (14), which suggests that +individuals living in environments where food is unpredictably and periodically scarce should +experience selection to store body fat in times of plenty. Recently, an intriguing variant was found +in Samoans, who are also susceptible to extreme obesity when eating a Western diet: a single +amino acid variant (p.Arg475Gln) in the CREBRF gene exhibits signatures of past selection and +is currently associated with a 1.3-fold increased risk of obesity (though puzzlingly, also a 1.6-fold +decreased risk of type 2 diabetes). Subsequent functional work in cell culture models + +Sex +Both +Female +Male +250 +B +1.0 +Age-adjusted relative +risk of all-cause death +200 +I cholesterol +levels (mg/dL) +0.8 +WHOminimum +recommendation +150 +0.6 +Total +100 +Subsistence- +0.4 +levelgroups +50 +0.2 +0 +NHANES +Bantu +Evenki - +Hadza +Maasai- +Shuar +Tsimane +Turkana +100 +300 +500 +700 +900 +1100 +1300 +1500 +1700 +1900 +Minutes perweek of moderate to vigorous physical activity +Subsistence-levelgroups9 +demonstrated that p.Arg475Gln has direct effects on metabolism, reducing energy use while +increasing lipid storage (65). + +In addition to these well-characterized examples (see also Figure 2 of (66)), recent +genomic work has shown that, in aggregate, variants that serve as modern-day risk alleles for +particular NCDs (namely CVD and autoimmune diseases) are more likely to show signatures of +past selection relative to non-risk alleles (67–69). More broadly, there is now ample evidence that +human populations can adapt to their unique ecologies quite quickly (70), setting the stage for +mismatches when local conditions shift. For example, within the last 10,000 years, the high P. +vivax malaria risk experienced by West Africans has selected for changes to a key chemokine +receptor encoded by the DARC gene (71, 72), while the spread of dairying in Europe has selected +for lactase persistence through changes in the regulation of the LCT gene (73, 74). As pathogen +environments and diets inevitably change, local adaptation sets the stage for mismatches to +occur. + +Consensus recommendations for a new path forward: integrating genomic tools and +partnerships with transitioning populations + +In principle, GxE interactions are most simply identifiable using a mismatch framework by +testing for environmentally-dependent genetic effects in transitioning populations. However, in +practice, this would be difficult because most NCDs arise from many small genetic effects +distributed across the genome. Further, the standard approach to resolve this needle-in-a- +haystack problem—using a massive sample size—is difficult in small-scale groups who typically +have modest population sizes. Instead, we discuss how advanced genomic methods can be +combined with the mismatch framework in a principled way to quantify the role of GxE interactions +in NCDs in subsistence-level settings. + +First, we can improve GxE test power in transitioning populations by focusing on genetic +loci with already demonstrated evidence for phenotypic relevance, for example, 1) those with +evidence for recent selection in the study group or 2) those that have already been discovered in +urban/industrialized environments. For example, recent work on the APOE locus found that the +E4 variant—a well-known risk factor for CVD and Alzheimer’s disease in HICs—is associated with +lower innate inflammation and may have beneficial effects on lipid moderation and cognition in a +high pathogen/low obesity environment (75–77). We might expect similar successes in elucidating +GxE mismatches at other well-known risk loci that replicate across HICs (e.g., FTO, ADCY3, +BRCA1/2). A related approach is to test for GxE enrichment at the level of known genes or +pathways, generalizing single SNP tests. These set-based approaches (i.e., that target predefined + +10 +sets of loci) may also perform well in transitioning populations, even if the specific causal variants +are not shared. + +Second, polygenic approaches that integrate GxE signals across the genome can improve +power when studying complex traits like NCDs. For example, recent methodological +developments have extended the popular polygenic risk score (PRS) framework to allow for PRS- +environment interaction tests, thus providing a polygenic GxE test (78–80). This approach has so +far been used to show how diet and other lifestyle factors modulate the genetic risk of obesity +(81–83). While polygenic approaches such as PRS sacrifice variant-level resolution, they yield +much greater power to detect GxE interactions, an invaluable exchange for quantifying +evolutionary mismatch in transitioning populations. Three downsides however are that: 1) +compared to single, large-effect allele results, one can be left with no suggestion of underlying +mechanism; 2) for PRS-environment interaction tests, power unavoidably depends on the +predictive power of the PRS as well as its portability across contexts and ancestries, which is a +clear problem given that most PRS work has focused on European ancestry individuals in HICs; +and 3) again for PRS-environment interaction tests, an underlying assumption is that risk effects +are systematically stronger in one environment than another (84). +Finally, we can add power and interpretability for GxE interactions using intermediate +phenotypes like gene expression, DNA methylation, and chromatin accessibility. One approach +is to impute these functional genomic features from genotype data and then test them for +environmental interaction (e.g., akin to a GxE version of transcriptome-wide association studies +(TWAS) (85, 86). The imputation step can use large, publicly available functional genomic +datasets from HICs, but will improve when similar datasets are available for the study populations. +A second approach is to test GxE in the map from genotype to functional genomic feature by +identifying environmentally-sensitive variants that impact nearby gene expression, DNA +methylation, chromatin accessibility, etc; this “molecular QTL” framework has so far proven very +powerful and could be extended to transitioning populations (59, 87, 88). Moreover, GxE +molecular QTLs can be validated experimentally by exposing cell lines or model organisms to +stimuli that mimic aspects of the environmental gradients experienced by transitioning +populations; indeed, this can pinpoint key components of the incredibly complex environmental +shifts that drive GxE. Finally, a third option is to use functional genomic experiments to narrow +the search space, by first identifying regulatory elements that respond to mismatch-relevant +environments. For example, Garske and colleagues recently identified chromatin elements that +respond to dietary fatty acids in adipocytes and then focused GxE follow up work on variants in +these responsive elements. By doing so, they were able to gain power to search for interaction + +11 +effects between genotype and dietary saturated fat intake on BMI (89). Similar in vitro functional +genomic experiments (using field-collected samples) could be leveraged to target regions of the +genome that may be most important for responding to key aspects of lifestyle transitions. + +Payoffs for NCD prevention and treatment +Testing the degree to which GxE interactions arise from evolutionary mismatch would +answer mechanistic questions about how GxE interactions manifest. For example, are loci that +were involved in adaptation to a population’s past environment more likely to exhibit GxE effects +when the environment shifts? To what degree does the nature of GxE interactions vary across +ancestries with distinct evolutionary histories? What is the envelope of “optimal” human +environmental conditions that do not provoke mismatch? Molecular insights into evolutionary +mismatch would allow us to prioritize the study of genetic variants that may adversely affect health +outcomes in novel environments. It would also enable prediction of potential future adverse +environments that could accelerate the onset of disease, and it could help us refine explanations +for already observed ancestry-related differences in disease susceptibility. +The studies we recommend would also advance our understanding of health issues in +minority, indigenous, and other underrepresented groups. Most subsistence-level populations in +low- and middle-income countries (LMICs) are facing rapid rises in NCD risk, and the limited +reports from these counties suggest that population responses to urbanization and market- +integration are highly variable. Studies of European ancestry individuals in HICs are not well- +suited to explain why. Partnering with transitioning groups to conduct evolutionarily and culturally +informed studies is needed to better serve their health concerns. + +Conclusions and future directions +The basic argument of this review is that we can further our understanding of evolution as +well as the genetic architecture of human disease by combining genomic tools with studies of +transitioning populations (as has been discussed previously (16), though not in the context of +genomics). This recommended path improves on current approaches, which typically rely on +“brute forcing” GxE scans across many SNPs and many environments. Instead, we advocate for +using evolutionary theory to parse a priori which G and E we expect to interact. Doing so would +boost power, better position us to understand and predict GxE interactions in the etiology of +NCDs, and provide much needed insight into urgent health issues affecting vulnerable +populations around the world. + +12 +Because the interdisciplinary perspective we take here necessarily touches on several +fields, we did not attempt an exhaustive review of research on either evolutionary mismatch or +GxE interactions (instead, we refer readers to excellent existing work (6, 12, 13, 15, 93, 94)). +However, there are several interesting new directions in these fields that are worth highlighting. +For example, a growing body of work has begun to conceptualize the human microbiome as an +evolved trait that is currently “mismatched” to its environment, often with serious health +implications (95). Given that 1) the microbiome is under host genetic control and can therefore be +a target of natural selection (96), and 2) industrialization can induce large scale changes in gut +microbial communities (97–99), this is an exciting area in which to investigate GxE interactions +that generate mismatch diseases. Another emerging research topic is sex differences in the +response to lifestyle change: several recent studies have found that women experience greater +NCD risk following economic and nutritional transitions than men (17, 24, 100, 101), yet how sex- +specific genetic, physiological, or environmental variation interact to produce this phenomenon is +still unknown. Finally, it is well-established that early life experiences are important for predicting +NCD risk later in life (102–104), and the timing of lifestyle change, as well as the degree to which +individuals experience environmental mismatches within their lifetimes, may therefore be +important to consider and to intersect with GxE frameworks (Box 3). In many cases, long-term +partnerships with focal communities have already led to the creation of longitudinal datasets well +positioned to take a lifecourse approach. Moving forward, we expect that longitudinal perspectives +on environmental change, NCD risk, and GxE interactions will be especially fruitful. + + +Acknowledgments + +This work was supported by a postdoctoral research fellowship to AJL from the Helen Hay +Whitney Foundation as well as grants from the Searle Scholars Program, Canadian Institute for +Advanced Research, and the National Institutes of Health (R35-GM147267). This work was also +supported by grants from the National Institutes of Health to JFA (R01-ES029929 and R35- +GM124881). We thank all participants from the “Evolutionary Mismatch Hypothesis in the +Genomics Era” symposium, which generated many of the ideas discussed here. + + +Author contributions + +AJL and JFA conceived the idea for the review. All authors drafted and edited the review. + + +13 + +Competing interests + +The authors declare no competing interests. + + + +14 +Boxes +Box 1. GxE interactions in population genetics: definitions and related concepts +In population genetics, the simplest conceptualization of a GxE interaction involves three +genotypes for a single bi-allelic locus, with each of the three genotypes found in two different +environments and with fitnesses varying across these six conditions (Figure 3C). At equilibrium, +this population will harbor, among other types of genetic variation, 1) alleles that have been +selected to high frequency as a consequence of positive selection (i.e., selection on a trait value +in a particular direction) and 2) alleles that are at intermediate frequency as a consequence of +stabilizing selection (i.e., selection to keep trait values near an optimum). Now let’s suppose the +environment changes quickly: previously selected alleles may now be associated with a trait that +is no longer beneficial, and even disease-causing, but they will remain at high frequency for some +time before selection is able to purge them. Note that loci with no genetic variation (e.g., fixed +beneficial mutations) could still be involved in mismatches in the new environment, but in the +absence of genetic variation we will be unable to identify them. +In addition to GxE interactions, another population genetic concept relating to evolutionary +mismatch and the modern increase in NCDs is decanalization (105). Canalization refers to the +process of stabilizing selection that minimizes genetic variation associated with fitness-related +traits in a given environment. Decanalization, then, is a perturbation from this state that reveals +genetic variation for health- or disease-associated phenotypes (106). Though similar, evolutionary +mismatch is more specific than decanalization. Evolutionary mismatch can occur without having +a previously canalized trait, and is a more general term not necessarily linked to stabilizing +selection. Decanalization is always a form of evolutionary mismatch, but not the other way around. +A final term that is distinct from all of these is robustness. Robustness refers to a property of +individual genotypes, wherein they are able to retain an advantageous phenotype despite genetic +or environmental hazards. In contrast, evolutionary mismatch and decanalization are population- +level phenomena. + + +Box 2. Ethical considerations of conducting genomic work in diverse populations +Community engagement and ethical research is fundamental to achieving the broader +vision of this Consensus. There is widespread consensus that broader population representation +in biomedical research is critical for reducing health disparities (107), but moving forward on this +agenda requires that we simultaneously acknowledge and learn from past mistakes and abuses. + +15 +At the heart of ethical considerations in genetics research is a situation in which diverse +populations are dually under-represented and under-consulted (108). Recent work has outlined +best practices for overcoming these issues (108–115). For example, Claw et al. (109) suggest six +principles of research ethics: 1) understand community sovereignty and research regulations; 2) +engage and collaborate; 3) build cultural competencies; 4) improve transparency; 5) build local +research capacity; and 6) disseminate research in accessible formats. The common thread +behind these principles is the importance of building trustful and long-term relationships based on +principles of dynamic consent, reciprocity, beneficence, and sovereignty. In our own experience, +building these sorts of relationships takes time (typically years) but is essential to do before +engaging in research. +Basic research with populations in LMICs can lead to important insights, yet the value- +added benefits from basic research (e.g., shaping health policy based on epidemiological trends, +and/or the development of novel treatment strategies) often can take decades to materialize. +Mechanisms for participant community involvement in these longer-term benefits should be +explicitly embedded in initial plans (107). It is also important to recognize that community benefits +can extend beyond the research itself. The needs and desires of local communities will vary +widely, but populations in LMICs may face problems that are deeply inter-connected and often +stem from systemic discrimination: poor nutrition and sanitation (often due to environmental +degradation), minimal access to education, few economic opportunities, and loss of land rights. +The priorities of communities will seldom match perfectly with the aims of scientists, especially +when participant communities lack basic infrastructure and face discrimination. Prioritizing +solutions to these problems is an opportunity to have great impact that will require cooperation +between researchers, study participants, universities, NGOs, governments, and funding bodies. + + +Box 3. Life course perspectives on NCD risk +Development is a period of heightened environmental sensitivity, and challenging +experiences early in life increase lifelong risk of most NCDs (102, 104, 116). Subsistence-level +societies are an under-utilized yet potentially powerful model for studying early life influences on +NCD risk. Many of these groups are currently experiencing rapid lifestyle changes leading to 1) +extreme variation in early life conditions within a single population and 2) frequent mismatch +between early life and adult environments—a situation that is thought to put individuals at risk for +later life health issues (117-119). Point #1 provides a clear opportunity to leverage the +distributional extremes to study early life effects on health (25, 122). Further, point #2 affords us + +16 +the opportunity to compare outcomes when individuals experience within-lifetime environmental +“matches” versus “mismatches”. To date, studies of industrial transitions have come to mixed +conclusions about the importance of within-lifetime mismatches (17, 39, 123, 124). More work in +this area is urgently needed to understand when, why, and how early life experiences shape adult +health in these groups. +Genomic tools applied to populations undergoing lifestyle change could also provide +valuable insight into how early life experiences become “embedded” into lifelong physiology. At +the molecular level, this process is thought to be mediated by stable changes in gene regulation +(e.g., DNA methylation, chromatin accessibility, and gene expression). However, many gene +regulatory elements are also dynamic and responsive to environmental perturbations throughout +life. This fact leads to challenges in disentangling the effects of early versus later life +environments, especially when the two are highly correlated (as is common in HICs). In contrast, +subsistence-level groups in transition often experience decoupled early life and adult experiences, +which could be leveraged to disentangle early versus later life influences. Genotype data collected +for the same individuals could also be used to identify rarely studied GxE interactions where the +“E” encompasses early life experiences (125–127). Overall, integrative studies of transitioning +populations are primed to reveal which individuals will be most susceptible to NCDs during +lifestyle transitions as well as when in the life course these exposures matter most. + + + + +17 +References +1. S. Kathiresan, D. Srivastava, Genetics of human cardiovascular disease. Cell 148, 1242– +1257 (2012). +2. A. V. Khera, S. Kathiresan, Genetics of coronary artery disease: discovery, biology and +clinical translation. Nat. Rev. Genet. 18, 331–344 (2017). +3. R. McPherson, A. Tybjaerg-Hansen, Genetics of Coronary Artery Disease. Circ. Res. 118, +564–578 (2016). +4. T. A. Manolio, et al., Finding the missing heritability of complex diseases. Nature 461, 747– +753 (2009). +5. P. M. Visscher, M. A. Brown, M. I. McCarthy, J. Yang, Five years of GWAS discovery. Am. +J. Hum. Genet. 90, 7–24 (2012). +6. K. McAllister, et al., Current Challenges and New Opportunities for Gene-Environment +Interaction Studies of Complex Diseases. Am. J. Epidemiol. 186, 753–761 (2017). +7. D. Thomas, Gene–environment-wide association studies: emerging approaches. Nat. Rev. +Genet. 11, 259–272 (2010). +8. H. Wang, et al., Genotype-by-environment interactions inferred from genetic effects on +phenotypic variability in the UK Biobank. Science Advances 5, eaaw3538 (2019). +9. J. Tyrrell, et al., Gene–obesogenic environment interactions in the UK Biobank study. +International Journal of Epidemiology, dyw337 (2017). +10. A. Nag, A. Mahajan, M. Mccarthy, 306-OR: Gene-Environment Interactions for Type 2 +Diabetes in UK Biobank. Diabetes 68, 306–OR (2019). +11. A. Arnau-Soler, et al., Genome-wide by environment interaction studies of depressive +symptoms and psychosocial stress in UK Biobank and Generation Scotland. Transl. +Psychiatry 9, 14 (2019). +12. D. Lieberman, The Story of the Human Body: Evolution, Health, and Disease (Vintage, +2014). +13. R. M. Nesse, G. C. Williams, Why We Get Sick: The New Science of Darwinian Medicine +(Vintage, 2012). +14. J. V. Neel, Diabetes Mellitus: A “Thrifty” Genotype Rendered Detrimental by “Progress”? +Am. J. Hum. Genet., 353–362 (1962). +15. S. Corbett, A. Courtiol, V. Lummaa, J. Moorad, S. Stearns, The transition to modernity and +chronic disease: mismatch and natural selection. Nat. Rev. Genet. 19, 419–430 (2018). +16. R. M. Garruto, M. A. Little, G. D. James, D. E. Brown, Natural experimental models: the +global search for biomedical paradigms among traditional, modernizing, and modern +populations. Proc. Natl. Acad. Sci. U. S. A. 96, 10536–10543 (1999). +17. A. J. Lea, D. Martins, J. Kamau, M. Gurven, J. F. Ayroles, Urbanization and market + +18 +integration have strong, nonlinear effects on cardiometabolic health in the Turkana. Sci Adv +6 (2020). +18. A. J. Lea, et al., Socioeconomic status effects on health vary between rural and urban +Turkana. Evol Med Public Health 9, 406–419 (2021). +19. I. J. Wallace, et al., Orang Asli Health and Lifeways Project (OA HeLP): Study Protocol. +bioRxiv (2021) https:/doi.org/10.1101/2021.10.24.21265442. +20. C. D. Golden, et al., Cohort Profile: The Madagascar Health and Environmental Research +(MAHERY) study in north-eastern Madagascar. Int. J. Epidemiol. 46, 1747–1748d (2017). +21. C. D. Golden, et al., Cohort Description of the Madagascar Health and Environmental +Research–Antongil (MAHERY–Antongil) Study in Madagascar. Frontiers in Nutrition 6 +(2019). +22. C. D. Golden, et al., Study Protocol: A Cross-Sectional Examination of Socio-Demographic +and Ecological Determinants of Nutrition and Disease Across Madagascar. Front Public +Health 8, 500 (2020). +23. M. Gurven, et al., The Tsimane Health and Life History Project: Integrating anthropology +and biomedicine. Evol. Anthropol. 26, 54–73 (2017). +24. M. A. Liebert, et al., Implications of market integration for cardiovascular and metabolic +health among an indigenous Amazonian Ecuadorian population. Ann. Hum. Biol. 40, 228– +242 (2013). +25. S. S. Urlacher, et al., Heterogeneous effects of market integration on sub-adult body size +and nutritional status among the Shuar of Amazonian Ecuador. Ann. Hum. Biol. 43, 316– +329 (2016). +26. D. E. Lieberman, The Story of the Human Body: Evolution. Health, and Disease (Pantheon, +New York) (2013). +27. N. B. Holowka, I. J. Wallace, D. E. Lieberman, Foot strength and stiffness are related to +footwear use in a comparison of minimally- vs. conventionally-shod populations. Sci. Rep. +8, 3679 (2018). +28. B. C. Trumble, C. E. Finch, THE EXPOSOME IN HUMAN EVOLUTION: FROM DUST TO +DIESEL. Q. Rev. Biol. 94, 333–394 (2019). +29. H. Pontzer, B. M. Wood, D. A. Raichlen, Hunter-gatherers as models in public health. +Obes. Rev. 19 Suppl 1, 24–35 (2018). +30. W. A. Price, T. Nguyen, Nutrition and Physical Degeneration: A Comparison of Primitive +and Modern Diets and Their Effects (EnCognitive.com, 2016). +31. J.-J. Rousseau, Discourse on the Origin of Inequality (Oxford University Press, 1755). +32. M. D. Gurven, D. E. Lieberman, WEIRD bodies: mismatch, medicine and missing diversity. +Evol. Hum. Behav. 41, 330–340 (2020). +33. T. Vos, et al., Global burden of 369 diseases and injuries in 204 countries and territories, + +19 +1990–2019: a systematic analysis for the Global Burden of Disease Study 2019. Lancet +396, 1204–1222 (2020). +34. H. Kaplan, et al., Coronary atherosclerosis in indigenous South American Tsimane: a +cross-sectional cohort study. Lancet 389, 1730–1739 (2017). +35. D. A. Raichlen, et al., Physical activity patterns and biomarkers of cardiovascular disease +risk in hunter-gatherers. Am. J. Hum. Biol. 29, e22919 (2017). +36. A. S. Truswell, B. M. Kennelly, J. D. Hansen, R. B. Lee, Blood pressures of Kung bushmen +in Northern Botswana. Am. Heart J. 84, 5–12 (1972). +37. R. E. Shave, et al., Selection of endurance capabilities and the trade-off between pressure +and volume in the evolution of the human heart. Proc. Natl. Acad. Sci. U. S. A. 116, 19905– +19910 (2019). +38. M. Gurven, A. D. Blackwell, D. E. Rodríguez, J. Stieglitz, H. Kaplan, Does blood pressure +inevitably rise with age?: longitudinal evidence among forager-horticulturalists. +Hypertension 60, 25–33 (2012). +39. I. J. Wallace, et al., Knee osteoarthritis risk in non-industrial societies undergoing an energy +balance transition: evidence from the indigenous Tarahumara of Mexico. Annals of the +Rheumatic Diseases 78, 1693–1698 (2019). +40. S. S. Urlacher, et al., Childhood Daily Energy Expenditure Does Not Decrease with Market +Integration and Is Not Related to Adiposity in Amazonia. J. Nutr. 151, 695–704 (2021). +41. H. Pontzer, Burn: New Research Blows the Lid Off How We Really Burn Calories, Lose +Weight, and Stay Healthy (Penguin, 2021). +42. D. Lieberman, Exercised: Why Something We Never Evolved to Do Is Healthy and +Rewarding (Pantheon Books, 2021). +43. A. V. Nosarev, L. V. Smagliy, Y. Anfinogenova, S. V. Popov, L. V. Kapilevich, Exercise and +NO production: relevance and implications in the cardiopulmonary system. Front Cell Dev +Biol 2, 73 (2014). +44. D. J. Green, M. T. E. Hopman, J. Padilla, M. H. Laughlin, D. H. J. Thijssen, Vascular +Adaptation to Exercise in Humans: Role of Hemodynamic Stimuli. Physiol. Rev. 97, 495– +528 (2017). +45. A. M. W. Petersen, B. K. Pedersen, The anti-inflammatory effect of exercise. J. Appl. +Physiol. 98, 1154–1162 (2005). +46. W. W. Tigbe, M. H. Granat, N. Sattar, M. E. J. Lean, Time spent in sedentary posture is +associated with waist circumference and cardiovascular risk. Int. J. Obes. 41, 689–696 +(2017). +47. A. V. Jaeggi, et al., Do wealth and inequality associate with health in a small-scale +subsistence society? Elife 10 (2021). +48. Noah Snyder-Mackler, Joseph Robert Burger, Lauren Gaydosh, Dan Belsky, Grace A. +Noppert, Fernando A. Campos, Alessandro Bartolomucci, Yang Claire Yang, Allison E. + +20 +Aiello, Angela O’Rand, Kathleen Mullan Harris, Carol A. Shively, Susan C. Alberts, & Jenny +Tung, Social determinants of health and survival in humans and other animals. Science +(2020). +49. A. J. Lea, et al., Social gradients in health vary between rural and urban Turkana. bioRxiv +(2021) https:/doi.org/10.1101/2021.07.23.21260771. +50. D. P. Strachan, Hay fever, hygiene, and household size. BMJ 299, 1259–1260 (1989). +51. J.-F. Bach, The hygiene hypothesis in autoimmunity: the role of pathogens and +commensals. Nat. Rev. Immunol. 18, 105–120 (2018). +52. G. Rook, Darwinian medicine and the “hygiene”or “old friends” hypothesis Clin. Exp. +Immunol 160, 70–79 (2010). +53. R. H. Glew, et al., Cardiovascular disease risk factors and diet of Fulani pastoralists of +northern Nigeria. Am. J. Clin. Nutr. 74, 730–736 (2001). +54. W. R. Leonard, M. H. Crawford, A. G. Comuzzie, R. I. Sukernik, Correlates of low serum +lipid levels among the Evenki herders of Siberia. Am. J. Hum. Biol. 6, 329–338 (1994). +55. J. A. Mbalilaki, et al., Daily energy expenditure and cardiovascular risk in Masai, rural and +urban Bantu Tanzanians. Br. J. Sports Med. 44, 121–126 (2010). +56. , National Health and Nutrition Examination Survey (NHANES) +https:/doi.org/10.4135/9781412963947.n318 (February 16, 2022). +57. H. Arem, et al., Leisure time physical activity and mortality: a detailed pooled analysis of the +dose-response relationship. JAMA Intern. Med. 175, 959–967 (2015). +58. M. Gurven, A. V. Jaeggi, H. Kaplan, D. Cummings, Physical activity and modernization +among Bolivian Amerindians. PLoS One 8, e55679 (2013). +59. Y. Nédélec, et al., Genetic Ancestry and Natural Selection Drive Population Differences in +Immune Responses to Pathogens. Cell 167, 113 (2016). +60. S. Kim-Hellmuth, et al., Genetic regulatory effects modified by immune activation contribute +to autoimmune disease associations. Nat. Commun. 8 (2017). +61. A. Jelenkovic, et al., Genetic and environmental influences on adult human height across +birth cohorts from 1886 to 1994. Elife 5 (2016). +62. G. Athanasiadis, et al., A comprehensive map of genetic relationships among diagnostic +categories based on 48.6 million relative pairs from the Danish genealogy. Proc. Natl. +Acad. Sci. U. S. A. 119 (2022). +63. G. Genovese, et al., Association of Trypanolytic ApoL1 Variants with Kidney Disease in +African Americans. Science 329, 841–845 (2010). +64. D. J. Friedman, M. R. Pollak, APOL1 and Kidney Disease: From Genetics to Biology. Annu. +Rev. Physiol. 82, 323–342 (2020). +65. R. J. F. Loos, CREBRF variant increases obesity risk and protects against diabetes in + +21 +Samoans. Nat. Genet. 48, 976–978 (2016). +66. M. L. Benton, et al., The influence of evolutionary history on human health and disease. +Nat. Rev. Genet. (2021) https:/doi.org/10.1038/s41576-020-00305-9. +67. S. G. Byars, et al., Genetic loci associated with coronary artery disease harbor evidence of +selection and antagonistic pleiotropy. PLoS Genet. 13, e1006328 (2017). +68. J. F. Brinkworth, L. B. Barreiro, The contribution of natural selection to present-day +susceptibility to chronic inflammatory and autoimmune disease. Curr. Opin. Immunol. 31, +66–78 (2014). +69. T. Raj, et al., Common risk alleles for inflammatory diseases are targets of recent positive +selection. Am. J. Hum. Genet. 92, 517–529 (2013). +70. S. Fan, M. E. B. Hansen, Y. Lo, S. A. Tishkoff, Going global by adapting local: A review of +recent human adaptation. Science 354, 54–59 (2016). +71. M. T. Hamblin, A. Di Rienzo, Detection of the signature of natural selection in humans: +evidence from the Duffy blood group locus. Am. J. Hum. Genet. 66, 1669–1679 (2000). +72. I. Hamid, K. L. Korunes, S. Beleza, A. Goldberg, Rapid adaptation to malaria facilitated by +admixture in the human population of Cabo Verde. Elife 10 (2021). +73. P. C. Sabeti, et al., Positive natural selection in the human lineage. Science 312, 1614– +1620 (2006). +74. R. P. Evershed, et al., Dairying, diseases and the evolution of lactase persistence in +Europe. Nature 608, 336–345 (2022). +75. M. E. Belloy, V. Napolioni, M. D. Greicius, A Quarter Century of APOE and Alzheimer’s +Disease: Progress to Date and the Path Forward. Neuron 101, 820–838 (2019). +76. B. C. Trumble, et al., Apolipoprotein E4 is associated with improved cognitive function in +Amazonian forager-horticulturalists with a high parasite burden. FASEB J. 31, 1508–1515 +(2017). +77. A. R. Garcia, et al., APOE4 is associated with elevated blood lipids and lower levels of +innate immune biomarkers in a tropical Amerindian subsistence population. bioRxiv, +2021.03.19.436070 (2021). +78. W. J. Peyrot, et al., Effect of polygenic risk scores on depression in childhood trauma. Br. J. +Psychiatry 205, 113–119 (2014). +79. N. Mullins, et al., Polygenic interactions with environmental adversity in the aetiology of +major depressive disorder. Psychol. Med. 46, 759–770 (2016). +80. R. E. Peterson, et al., Molecular Genetic Analysis Subdivided by Adversity Exposure +Suggests Etiologic Heterogeneity in Major Depression. Am. J. Psychiatry 175, 545–554 +(2018). +81. S. Nagpal, G. Gibson, U. M. Marigorta, Pervasive Modulation of Obesity Risk by the +Environment and Genomic Background. Genes 9 (2018). + +22 +82. A. Abadi, et al., Penetrance of Polygenic Obesity Susceptibility Loci across the Body Mass +Index Distribution. Am. J. Hum. Genet. 101, 925–938 (2017). +83. M. Rask-Andersen, T. Karlsson, W. E. Ek, Å. Johansson, Gene-environment interaction +study for BMI reveals interactions between genetic factors and physical activity, alcohol +consumption and socioeconomic status. PLoS Genet. 13, e1006977 (2017). +84. B. Sheppard, et al., A model and test for coordinated polygenic epistasis in complex traits. +Proc. Natl. Acad. Sci. U. S. A. 118 (2021). +85. A. Gusev, et al., Integrative approaches for large-scale transcriptome-wide association +studies. Nat. Genet. 48, 245–252 (2016). +86. E. R. Gamazon, et al., A gene-based association method for mapping traits using reference +transcriptome data. Nat. Genet. 47, 1091–1098 (2015). +87. B. P. Fairfax, et al., Innate immune activity conditions the effect of regulatory variants upon +monocyte gene expression. Science 343, 1246949 (2014). +88. M. N. H. M. N. Lee, et al., Common Genetic Variants Modulate Pathogen-Sensing +Responses in Human Dendritic Cells. Science 343, 1246980 (2014). +89. K. M. Garske, et al., Reverse gene-environment interaction approach to identify variants +influencing body-mass index in humans. Nat Metab 1, 630–642 (2019). +90. S. Carding, K. Verbeke, D. T. Vipond, B. M. Corfe, L. J. Owen, Dysbiosis of the gut +microbiota in disease. Microb. Ecol. Health Dis. 26, 26191 (2015). +91. J. Aron-Wisnewsky, K. Clément, M. Nieuwdorp, Fecal Microbiota Transplantation: a Future +Therapeutic Option for Obesity/Diabetes? Curr. Diab. Rep. 19, 51 (2019). +92. T. B. Smallwood, et al., Helminth Immunomodulation in Autoimmune Disease. Front. +Immunol. 8, 453 (2017). +93. C. Ober, D. Vercelli, Gene–environment interactions in human disease: nuisance or +opportunity? Trends Genet. 27, 107–115 (2011). +94. D. J. Hunter, Gene–environment interactions in human diseases. Nat. Rev. Genet. 6, 287– +298 (2005). +95. R. N. Carmody, A. Sarkar, A. T. Reese, Gut microbiota through an evolutionary lens. +Science 372, 462–463 (2021). +96. L. Grieneisen, et al., Gut microbiome heritability is nearly universal but environmentally +contingent. Science 373, 181–186 (2021). +97. M. Groussin, et al., Elevated rates of horizontal gene transfer in the industrialized human +microbiome. Cell 184, 2053–2067.e18 (2021). +98. S. L. Schnorr, et al., Gut microbiome of the Hadza hunter-gatherers. Nat. Commun. 5, 3654 +(2014). +99. P. Vangay, et al., US Immigration Westernizes the Human Gut Microbiome. Cell 175, 962– + +23 +972.e10 (2018). +100. J. J. Snodgrass, et al., The emergence of obesity among indigenous Siberians. J. +Physiol. Anthropol. 25, 75–84 (2006). +101. NCD Risk Factor Collaboration (NCD-RisC), Rising rural body-mass index is the main +driver of the global obesity epidemic in adults. Nature 569, 260–264 (2019). +102. V. J. Felitti, et al., Relationship of childhood abuse and household dysfunction to many of +the leading causes of death in adults. The Adverse Childhood Experiences (ACE) Study. +American Journal of Preventative Medicine 14, 245–258 (1998). +103. M. A. Bellis, et al., Life course health consequences and associated annual costs of +adverse childhood experiences across Europe and North America: a systematic review and +meta-analysis. The Lancet Public Health 4, e517–e528 (2019). +104. D. J. Hoffman, R. M. Reynolds, D. B. Hardy, Developmental origins of health and +disease: current knowledge and potential mechanisms. Nutr. Rev. 75, 951–970 (2017). +105. G. Gibson, Decanalization and the origin of complex disease. Nat. Rev. Genet. 10, 134– +140 (2013). +106. G. Gibson, K. A. Lacek, Canalization and Robustness in Human Genetics and Disease. +Annu. Rev. Genet. 54, 189–211 (2020). +107. K. Fox, The illusion of inclusion - the “all of us” research program and indigenous +peoples’ DNA. N. Engl. J. Med. 383, 411–413 (2020). +108. L. Jackson, C. Kuhlman, F. Jackson, P. Keolu Fox, Including Vulnerable Populations in +the Assessment of Data From Vulnerable Populations. Frontiers in Big Data 2 (2019). +109. K. G. Claw, et al., A framework for enhancing ethical genomic research with Indigenous +communities. Nat. Commun. 9, 2957 (2018). +110. M. B. Dignan, L. Burhansstipanov, L. Bemis, Successful implementation of genetic +Education for Native Americans workshops at national conferences. Genetics 169, 517– +521 (2005). +111. N. Tiffin, Tiered informed consent: respecting autonomy, agency and individuality in +Africa. BMJ Glob Health 3, e001249 (2018). +112. M. Hudson, et al., Rights, interests and expectations: Indigenous perspectives on +unrestricted access to genomic data. Nat. Rev. Genet. 21, 377–384 (2020). +113. M. Hernandez, G. H. Perry, Scanning the human genome for “signatures” of positive +selection: Transformative opportunities and ethical obligations. Evol. Anthropol. 30, 113– +121 (2021). +114. N. A. Garrison, et al., Genomic Research Through an Indigenous Lens: Understanding +the Expectations. Annu. Rev. Genomics Hum. Genet. 20, 495–517 (2019). +115. G. Guglielmi, Facing up to injustice in genome science. Nature 568, 290–293 (2019). + +24 +116. S. Cohen, D. Janicki-Deverts, E. Chen, K. Matthews, Childhood socioeconomic status +and adult health. Ann. N. Y. Acad. Sci. 1186, 37–55 (2010). +117. P. Monaghan, Early growth conditions, phenotypic development and environmental +change. Philos. Trans. R. Soc. Lond. B Biol. Sci. 363, 1635–1645 (2008). +118. A. J. Lea, E. A. Archie, J. Tung, S. C. Alberts, Developmental plasticity: bridging +research in evolution and human health. Evolution, Medicine, and Public Health, in press +(2018). +119. A. J. Lea, S. Rosebaum, Understanding how early life effects evolve: progress, gaps, +and future directions. Current Opinion in Behavioral Sciences 36, 29–35 (2020). +120. A. D. Hayward, I. J. Rickard, V. Lummaa, Influence of early-life nutrition on mortality and +reproductive success during a subsequent famine in a preindustrial population. Proc. Natl. +Acad. Sci. U. S. A. 110, 13886–13891 (2013). +121. A. D. Hayward, V. Lummaa, Testing the evolutionary basis of the predictive adaptive +response hypothesis in a preindustrial human population. Evolution, Medicine, and Public +Health 2013, 106–117 (2013). +122. M. P. Muehlenbein, Human Evolutionary Biology (Cambridge University Press, 2010). +123. J. C. N. Chan, et al., Diabetes in Asia: epidemiology, risk factors, and pathophysiology. +JAMA 301, 2129–2140 (2009). +124. R. Huxley, et al., Ethnic comparisons of the cross-sectional relationships between +measures of body size with diabetes and hypertension. Obes. Rev. 9 Suppl 1, 53–61 +(2008). +125. N. R. Nugent, A. R. Tyrka, L. L. Carpenter, L. H. Price, Gene-environment interactions: +early life stress and risk for depressive and anxiety disorders. Psychopharmacology 214, +175–196 (2011). +126. A. Caspi, et al., Influence of life stress on depression: moderation by a polymorphism in +the 5-HTT gene. Science 301, 386–389 (2003). +127. A. Caspi, T. E. Moffitt, Gene–environment interactions in psychiatry: joining forces with +neuroscience. Nat. Rev. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA YL - Department of Parasitology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Faculty of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Universiti Malaya,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kuala Lumpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Malaysia DM1 - Turkana Basin Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Turkana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kenya DM1 - Department of Ecology and Evolution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' NJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA DM2 - Director at County Government of Laikipia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nanyuki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kenya PP - Department of Human Genetics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' David Geffen School of Medicine at UCLA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Los Angeles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA PP - Institute for Precision Health,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' David Geffen School of Medicine at UCLA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Los Angeles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA GP - Department of Anthropology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA GP - Department of Biology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA GP - Huck Institutes of the Life Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA HP - Evolutionary Anthropology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Duke University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Durham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA HP - Duke Global Health Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Duke University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Durham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA BT - School of Human Evolution and Social Change,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Arizona State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Tempe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' US BT - Center for Evolution and Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Arizona State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Tempe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' United States SU - Department of Anthropology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Baylor University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Waco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA SU - Child and Brain Development,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Canadian Institute for Advanced Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Toronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Canada VV - Department of Anthropology and Archaeology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' University of Calgary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Calgary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Canada IW - Department of Anthropology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' University of New Mexico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Albuquerque,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA MG - Department of Anthropology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' University of California: Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA DL - Department of Human Evolutionary Biology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Harvard University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA JA - Department of Ecology and Evolution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' NJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA JA - Lewis Sigler Institute for Integrative Genomics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' NJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' USA 2 Abstract Globally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' we are witnessing the rise of complex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' non-communicable diseases (NCDs) related to changes in our daily environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Obesity, asthma, cardiovascular disease, and type 2 diabetes are part of a long list of “lifestyle” diseases that were rare throughout human history but are now common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A key idea from anthropology and evolutionary biology—the evolutionary mismatch hypothesis—seeks to explain this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' It posits that humans evolved in environments that radically differ from the ones experienced by most people today, and thus traits that were advantageous in past environments may now be “mismatched” and disease-causing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' This hypothesis is, at its core, a genetic one: it predicts that loci with a history of selection will exhibit “genotype by environment” (GxE) interactions and have differential health effects in ancestral versus modern environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Here, we discuss how this concept could be leveraged to uncover the genetic architecture of NCDs in a principled way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Specifically, we advocate for partnering with small-scale, subsistence-level groups that are currently transitioning from environments that are arguably more “matched” with their recent evolutionary history to those that are more “mismatched”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' These populations provide diverse genetic backgrounds as well as the needed levels and types of environmental variation necessary for mapping GxE interactions in an explicit mismatch framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Such work would make important contributions to our understanding of environmental and genetic risk factors for NCDs across diverse ancestries and sociocultural contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 3 Introduction Non-communicable diseases (NCDs) such as cardiovascular disease (CVD), type II diabetes, and Alzheimer’s are among the leading causes of death worldwide (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' NCDs are often difficult to prevent and treat, because they result from complex and poorly understood interactions between a person’s genetic makeup and their environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' For example, cardiovascular disease (CVD) has a heritability of 40-50%, with dozens of loci now mapped through genome-wide association studies (1–3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' However, when tallied together in an additive framework, these loci explain only a small fraction of the heritable genetic effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' This has led many to conclude that environmental risk factors—such as a diet high in processed foods and low levels of physical activity—interact with genetic variation to shape NCD risk (4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In other words, genetic variation may predispose individuals toward physiological sensitivity or resilience in the face of environmental perturbations, a phenomenon known as “genotype x environment” (GxE) interactions (Box 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Despite major interest in GxE interactions in the context of NCDs, scientists have struggled in practice to identify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' There are many complex reasons for this, including that the relevant environmental factors are often unknown, difficult to measure, or minimally variable within the study population (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', most individuals in high income countries (HICs) consume processed foods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Further, large sample sizes are needed to test for interaction effects, and even more so to overcome the multiple testing burden incurred by testing for interactions between many genetic variants and many environments (6, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' To overcome power issues, current state-of-the-art approaches have leveraged very large studies such as the UK Biobank to scan for interactions between genome-wide genetic variation and key lifestyle factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', smoking, diet, or physical activity) (8–11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' However, these studies have not delivered as expected, and have only uncovered a handful of GxE interactions for NCDs like obesity, type II diabetes, and depression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Here, we argue for a complementary approach informed by anthropological traditions, genomic tools, and evolutionary theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In particular, we believe there is much to learn by 1) viewing GxE interactions through the lens of the “evolutionary mismatch” hypothesis and 2) partnering with genetically and environmentally diverse small-scale, subsistence-level populations to map them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' The evolutionary mismatch hypothesis posits that traits that evolved under past selection regimes are often imperfectly or inadequately suited to modern environments, leading to “mismatches” in the form of NCDs (12–15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' At the genetic level, we would thus expect that previously neutral or beneficial alleles are now disease-causing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' While we cannot go back in time to evaluate genotype-phenotype relationships in past environments, we can collaborate with populations that practice non-industrial, subsistence-level 4 lifestyles that are arguably more “matched” to their recent evolutionary history (though we caution that, of course, no modern population is perfectly representative of ancestral conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Importantly, many subsistence-level populations are currently exposed to globalizing forces causing rapid environmental shifts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' this situation creates a quasi-natural experiment for studying the transition from traditional to modern lifeways within a single group (16) (Figure 2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Additionally, many subsistence-level groups have already been well-characterized ecologically and phenotypically through long-term work with anthropologists (Figure 2B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Box 2), setting the stage for integration of genomic studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In this Consensus, we argue that uniting an evolutionary mismatch framework, long-term anthropological work with subsistence-level groups, and cutting-edge genomic tools can increase our power to identify and understand GxE interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Specifically, because the mismatch framework provides clear expectations for the types of loci and environments we expect to affect NCDs, we can narrow the search space considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Further, by focusing on populations where Western diets and lifestyles are the exception rather than the norm, we can design studies that explicitly sample environmental extremes, thereby boosting power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Finally, by studying many genetically distinct populations under a uniting intellectual framework, we can identify new loci that have so far been invisible to studies focused on individuals of European descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' With these goals in mind, we first review the evolutionary mismatch hypothesis and discuss its current support at the phenotypic and genetic levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Second, we propose consensus recommendations for integrating mismatch principles with molecular and genomic techniques, focusing on collaborations with subsistence-level groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Third, we discuss the payoffs for scientists and study communities that would come from implementing these partnerships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Non-communicable diseases are the leading cause of death worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A) Proportion of worldwide deaths attributable to non-communicable diseases, communicable (infectious) diseases, and injuries through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B) Proportion of deaths within the US in 2019, broken down by the top 10 causes of death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' NCDs are highlighted in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' For both panels, data were sourced from ourworldindata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='org and represent all ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A Cause Infectiousdiseases Injuries NCDs B Cause Other NCDs Proportion of total deaths 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='00 CVD - Cancers 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content="75 Chronicrespiratorydisease Alzheimer's and dementia 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='50 Digestive diseases Chronic kidney disease Respiratory infections 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='25 Diabetesmellitus Chronic liver disease 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='00 Drug use disorders 1990 2000 2010 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='3 Year Proportionoftotal deaths(2019)5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Subsistence-level groups experiencing lifestyle change are a potential model for uncovering GxE interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A) Subsistence-level groups faced with urbanization, market-integration, and modernization experience extreme variation in diet and physical activity levels, pathogen and toxin exposures, and social conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' This list of environmental components for which there is extreme variation is not exhaustive, and in many cases will also be population specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' We highlight a few broad categories that tend to change consistently during lifestyle transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B) Studies such as The Turkana Health and Genomics Project (17, 18), The Orang Asli Health and Lifeways Project (19), The Pacific Planetary Health Initiative, Madagascar Health and Environmental Research (20–22), The Tsimane Health and Life History Project (23), and The Shuar Health and Life History Project (24, 25) all combine anthropological and biomedical data collection in transitioning societies, and are thus poised to uncover GxE interactions in the context of evolutionary mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' We note that this list is meant to be illustrative and only includes projects directed by authors of this Consensus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' it does not by any means cover all ongoing projects of small-scale, subsistence-level groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Overview of the evolutionary mismatch hypothesis An evolutionary mismatch is a condition that is more common or severe in an organism because it is imperfectly or inadequately adapted to a novel environment (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' While mismatches are not unique to humans, their frequency may be unusually high in our species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' This is because human culture can generate rapid and profound environmental change: in just a few generations, industrialization has transformed human diets, physical activity patterns, and toxin exposure landscapes, especially in HICs, and these changes presumably contribute to the long list of NCDs that used to be rare or nonexistent (27–29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' For at least a century,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' a wide range of conditions have been assumed to be “diseases of civilization” or “lifestyle diseases” (30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 31),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' but mismatches need to be explicitly and rigorously ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='Subsistence-level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='Urban ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='+processed foods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='Iprocessed foods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='Tphysical activity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='tphysical activity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='tpathogen exposure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='tpathogen exposure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='↓toxin exposure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='↓toxin exposure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='ftl socioeconomic inequality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='↑l socioeconomic inequality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='t social support ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='social support ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='TheTurkana Health ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='and Genomics Project ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='The Orang Asli Health ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='and Lifeways Project ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='TheShuar Health and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='Life History Project ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='ThePacificPlanetary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='The Tsimane Health ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='Health Initiative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='and Life History Project,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Madagascar Health and Environmental Research6 tested according to three criteria (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' First, a mismatch condition should be more common or severe in the “novel” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', post-industrial, HICs) relative to the “ancestral” environment (Figure 3A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Small-scale, subsistence-level societies typically stand in as the best available, though often imperfect, proxy for the “ancestral” condition in humans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' this is because they experience a closer “match” between their recent evolutionary history and their current environments relative to individuals in HICs, though we caution they are not themselves “ancestral” populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In addition to the hypothesized mismatch condition being more prevalent in post-industrial versus subsistence-level groups, the second criteria is that it should also be tied to some environmental variable that differs between these groups (Figure 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' One complication for achieving this is that NCDs arise from complex multifactorial causes, and thus, while between- population comparisons are necessary, they can be confounded by many covariates that must also be taken into account (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', sanitation, access to medical care, age structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' The third criteria is that it is necessary to establish a molecular or physiological mechanism by which the environmental shift generates the proposed mismatch condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' At the genetic level, this should manifest as a locus for which 1) a variant exhibits a past history of positive selection and is associated with health benefits in the ancestral environment but health detriments in the novel environment or 2) past stabilizing selection has created a situation where two intermediate alleles have similar fitness in the ancestral environment, but one allele becomes associated with health detriments in the novel environment (Figure 3C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' see also Box 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Mismatch diseases must be tested according to three criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A) Health phenotypes related to the hypothesized mismatch disease must be more common or severe in the novel versus ancestral environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B) A B Health phenotype Health phenotype Novel Ancestral Environmentalvariable c Ancestral Health phenotype Before After AA AT TT selection selection Genotype7 These health phenotypes must be attributable to an environmental variable, which will most often differ in mean and range between groups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', physical activity influences cardiovascular health and is consistently higher in subsistence-level groups relative to HICs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' C) It is necessary to establish a mechanism by which an environmental shift generates health issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' At the genetic level, this could manifest as a locus for which a variant exhibits a past history of positive selection and is associated with health benefits in the ancestral environment but health detriments in the novel environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In panel C, horizontal lines represent haplotypes and the dark orange circle represents the selected variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In all panels, dark blue represents the novel environment and light blue represents the ancestral environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Current evidence for evolutionary mismatch at the phenotypic level Scientists have been relatively successful at testing the first two criteria for mismatch, especially in the context of CVD, the single largest cause of mortality worldwide (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In support of the first criteria, subsistence-level groups experience remarkably low rates of CVD (29, 34, 35) relative to HICs, as well as minimal age-associated increases in CVD or its biomarkers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', hypertension, cholesterol) (36–38) (Figure 4A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Studies of small-scale societies in the midst of socioeconomic transition have demonstrated within-population effects of industrialization (17, 39, 40), strengthening the findings from between-population comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In support of the second criteria, recent work has also isolated salient environmental changes by which industrialization promotes CVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' People in subsistence-level communities are generally very physically active, accruing 5-10 times more daily physical activity than adults in Europe, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', and other HICs (41, 42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Moderate to vigorous physical activity increases cardiac output promoting nitric oxide production and arterial elasticity (43, 44), it also decreases baseline levels of inflammation, which plays a critical role in all aspects of CVD (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Within industrialized populations, individuals accruing daily physical activity similar to those of subsistence-level individuals experience similarly low rates of CVD as well as NCD-related mortality (46) (Figure 4B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' However, while physical activity plays a critical role in averting CVD, it is not a panacea and several other factors are surely important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' For example, relative to HICs, subsistence-level groups subsist on diets dominated by unprocessed or minimally processed foods and experience different types and degrees of social integration and inequality—all of which impact CVD risk (47–49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Finally, we note that while we have focused this section on CVD as an illustrative example of the type of comprehensive evidence required for diagnosing a mismatch disease, several other conditions also have relatively clear evidence for the first two criteria for mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' For example, inflammatory and autoimmune disorders have increased during the twentieth century, which has been linked to a reduced exposure to parasites and microorganisms (a phenomenon attributed to the “hygiene hypothesis” or “old friends hypothesis”) (50–52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 8 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Evidence for evolutionary mismatch at the phenotypic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A) Mean levels of total cholesterol are much lower in select subsistence-level populations relative to US adults (>18 years old) profiled as part of the National Health and Nutrition Examination Survey (NHANES) (56) (subsistence-level data from (17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B) Evidence that, within industrialized populations, individuals accruing daily physical activity similar to those of men and women in subsistence-level societies experience similarly low rates of CVD as well as all-cause mortality from NCDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Dose response relationship between minutes/week of moderate to vigorous leisure time physical activity and age-adjusted relative risk of death from a sample of 661,137 adult Americans and Europeans (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' The arrow for physical activity estimates in subsistence-level groups is based on studies of the Hadza (estimated at x=944 minutes (35)) and the Tsimane (x=924 minutes (58)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Current evidence for evolutionary mismatch at the genetic level As mentioned above, to fulfill the third criteria for mismatch, we would need to identify a locus for which 1) there is evidence of past selection and 2) performance of at least one allele varies across environments and confers inflated risk of an NCD in the novel environment (see also Figure 1B and Box 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' One would think this would be easy to find, but in fact there are only a handful of clear cases, despite good evidence for the existence of GxE interactions in general (59–62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' One clear example of mismatch involves variants in the APOL1 gene, which provides resistance to trypanosome infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Given the prevalence of trypanosomes across Africa, beneficial alleles are found at high frequency in African populations as well as African Americans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' However, these same variants confer elevated kidney disease risk in African Americans living in the US (63, 64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Another example is related to the “thrifty genotype” hypothesis (14), which suggests that individuals living in environments where food is unpredictably and periodically scarce should experience selection to store body fat in times of plenty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Recently, an intriguing variant was found in Samoans, who are also susceptible to extreme obesity when eating a Western diet: a single amino acid variant (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='Arg475Gln) in the CREBRF gene exhibits signatures of past selection and is currently associated with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='3-fold increased risk of obesity (though puzzlingly, also a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='6-fold decreased risk of type 2 diabetes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Subsequent functional work in cell culture models Sex Both Female Male 250 B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='0 Age-adjusted relative risk of all-cause death 200 I cholesterol levels (mg/dL) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='8 WHOminimum recommendation 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='6 Total 100 Subsistence- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='4 levelgroups 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='2 0 NHANES Bantu Evenki - Hadza Maasai- Shuar Tsimane Turkana 100 300 500 700 900 1100 1300 1500 1700 1900 Minutes perweek of moderate to vigorous physical activity Subsistence-levelgroups9 demonstrated that p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='Arg475Gln has direct effects on metabolism, reducing energy use while increasing lipid storage (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In addition to these well-characterized examples (see also Figure 2 of (66)), recent genomic work has shown that, in aggregate, variants that serve as modern-day risk alleles for particular NCDs (namely CVD and autoimmune diseases) are more likely to show signatures of past selection relative to non-risk alleles (67–69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' More broadly, there is now ample evidence that human populations can adapt to their unique ecologies quite quickly (70), setting the stage for mismatches when local conditions shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' For example, within the last 10,000 years, the high P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' vivax malaria risk experienced by West Africans has selected for changes to a key chemokine receptor encoded by the DARC gene (71, 72), while the spread of dairying in Europe has selected for lactase persistence through changes in the regulation of the LCT gene (73, 74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' As pathogen environments and diets inevitably change, local adaptation sets the stage for mismatches to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Consensus recommendations for a new path forward: integrating genomic tools and partnerships with transitioning populations In principle, GxE interactions are most simply identifiable using a mismatch framework by testing for environmentally-dependent genetic effects in transitioning populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' However, in practice, this would be difficult because most NCDs arise from many small genetic effects distributed across the genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Further, the standard approach to resolve this needle-in-a- haystack problem—using a massive sample size—is difficult in small-scale groups who typically have modest population sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Instead, we discuss how advanced genomic methods can be combined with the mismatch framework in a principled way to quantify the role of GxE interactions in NCDs in subsistence-level settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' First, we can improve GxE test power in transitioning populations by focusing on genetic loci with already demonstrated evidence for phenotypic relevance, for example, 1) those with evidence for recent selection in the study group or 2) those that have already been discovered in urban/industrialized environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' For example, recent work on the APOE locus found that the E4 variant—a well-known risk factor for CVD and Alzheimer’s disease in HICs—is associated with lower innate inflammation and may have beneficial effects on lipid moderation and cognition in a high pathogen/low obesity environment (75–77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' We might expect similar successes in elucidating GxE mismatches at other well-known risk loci that replicate across HICs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', FTO, ADCY3, BRCA1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A related approach is to test for GxE enrichment at the level of known genes or pathways, generalizing single SNP tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' These set-based approaches (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', that target predefined 10 sets of loci) may also perform well in transitioning populations, even if the specific causal variants are not shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Second, polygenic approaches that integrate GxE signals across the genome can improve power when studying complex traits like NCDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' For example, recent methodological developments have extended the popular polygenic risk score (PRS) framework to allow for PRS- environment interaction tests, thus providing a polygenic GxE test (78–80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' This approach has so far been used to show how diet and other lifestyle factors modulate the genetic risk of obesity (81–83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' While polygenic approaches such as PRS sacrifice variant-level resolution, they yield much greater power to detect GxE interactions, an invaluable exchange for quantifying evolutionary mismatch in transitioning populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Three downsides however are that: 1) compared to single, large-effect allele results, one can be left with no suggestion of underlying mechanism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 2) for PRS-environment interaction tests, power unavoidably depends on the predictive power of the PRS as well as its portability across contexts and ancestries, which is a clear problem given that most PRS work has focused on European ancestry individuals in HICs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' and 3) again for PRS-environment interaction tests, an underlying assumption is that risk effects are systematically stronger in one environment than another (84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Finally, we can add power and interpretability for GxE interactions using intermediate phenotypes like gene expression, DNA methylation, and chromatin accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' One approach is to impute these functional genomic features from genotype data and then test them for environmental interaction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', akin to a GxE version of transcriptome-wide association studies (TWAS) (85, 86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' The imputation step can use large, publicly available functional genomic datasets from HICs, but will improve when similar datasets are available for the study populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A second approach is to test GxE in the map from genotype to functional genomic feature by identifying environmentally-sensitive variants that impact nearby gene expression, DNA methylation, chromatin accessibility, etc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' this “molecular QTL” framework has so far proven very powerful and could be extended to transitioning populations (59, 87, 88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Moreover, GxE molecular QTLs can be validated experimentally by exposing cell lines or model organisms to stimuli that mimic aspects of the environmental gradients experienced by transitioning populations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' indeed, this can pinpoint key components of the incredibly complex environmental shifts that drive GxE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Finally, a third option is to use functional genomic experiments to narrow the search space, by first identifying regulatory elements that respond to mismatch-relevant environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' For example, Garske and colleagues recently identified chromatin elements that respond to dietary fatty acids in adipocytes and then focused GxE follow up work on variants in these responsive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' By doing so, they were able to gain power to search for interaction 11 effects between genotype and dietary saturated fat intake on BMI (89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Similar in vitro functional genomic experiments (using field-collected samples) could be leveraged to target regions of the genome that may be most important for responding to key aspects of lifestyle transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Payoffs for NCD prevention and treatment Testing the degree to which GxE interactions arise from evolutionary mismatch would answer mechanistic questions about how GxE interactions manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' For example, are loci that were involved in adaptation to a population’s past environment more likely to exhibit GxE effects when the environment shifts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' To what degree does the nature of GxE interactions vary across ancestries with distinct evolutionary histories?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' What is the envelope of “optimal” human environmental conditions that do not provoke mismatch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Molecular insights into evolutionary mismatch would allow us to prioritize the study of genetic variants that may adversely affect health outcomes in novel environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' It would also enable prediction of potential future adverse environments that could accelerate the onset of disease, and it could help us refine explanations for already observed ancestry-related differences in disease susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' The studies we recommend would also advance our understanding of health issues in minority, indigenous, and other underrepresented groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Most subsistence-level populations in low- and middle-income countries (LMICs) are facing rapid rises in NCD risk, and the limited reports from these counties suggest that population responses to urbanization and market- integration are highly variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Studies of European ancestry individuals in HICs are not well- suited to explain why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Partnering with transitioning groups to conduct evolutionarily and culturally informed studies is needed to better serve their health concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Conclusions and future directions The basic argument of this review is that we can further our understanding of evolution as well as the genetic architecture of human disease by combining genomic tools with studies of transitioning populations (as has been discussed previously (16), though not in the context of genomics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' This recommended path improves on current approaches, which typically rely on “brute forcing” GxE scans across many SNPs and many environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Instead, we advocate for using evolutionary theory to parse a priori which G and E we expect to interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Doing so would boost power, better position us to understand and predict GxE interactions in the etiology of NCDs, and provide much needed insight into urgent health issues affecting vulnerable populations around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 12 Because the interdisciplinary perspective we take here necessarily touches on several fields, we did not attempt an exhaustive review of research on either evolutionary mismatch or GxE interactions (instead, we refer readers to excellent existing work (6, 12, 13, 15, 93, 94)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' However, there are several interesting new directions in these fields that are worth highlighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' For example, a growing body of work has begun to conceptualize the human microbiome as an evolved trait that is currently “mismatched” to its environment, often with serious health implications (95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Given that 1) the microbiome is under host genetic control and can therefore be a target of natural selection (96), and 2) industrialization can induce large scale changes in gut microbial communities (97–99), this is an exciting area in which to investigate GxE interactions that generate mismatch diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Another emerging research topic is sex differences in the response to lifestyle change: several recent studies have found that women experience greater NCD risk following economic and nutritional transitions than men (17, 24, 100, 101), yet how sex- specific genetic, physiological, or environmental variation interact to produce this phenomenon is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Finally, it is well-established that early life experiences are important for predicting NCD risk later in life (102–104), and the timing of lifestyle change, as well as the degree to which individuals experience environmental mismatches within their lifetimes, may therefore be important to consider and to intersect with GxE frameworks (Box 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In many cases, long-term partnerships with focal communities have already led to the creation of longitudinal datasets well positioned to take a lifecourse approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Moving forward, we expect that longitudinal perspectives on environmental change, NCD risk, and GxE interactions will be especially fruitful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Acknowledgments This work was supported by a postdoctoral research fellowship to AJL from the Helen Hay Whitney Foundation as well as grants from the Searle Scholars Program, Canadian Institute for Advanced Research, and the National Institutes of Health (R35-GM147267).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' This work was also supported by grants from the National Institutes of Health to JFA (R01-ES029929 and R35- GM124881).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' We thank all participants from the “Evolutionary Mismatch Hypothesis in the Genomics Era” symposium, which generated many of the ideas discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Author contributions AJL and JFA conceived the idea for the review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' All authors drafted and edited the review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 13 Competing interests The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 14 Boxes Box 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' GxE interactions in population genetics: definitions and related concepts In population genetics, the simplest conceptualization of a GxE interaction involves three genotypes for a single bi-allelic locus, with each of the three genotypes found in two different environments and with fitnesses varying across these six conditions (Figure 3C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' At equilibrium, this population will harbor, among other types of genetic variation, 1) alleles that have been selected to high frequency as a consequence of positive selection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', selection on a trait value in a particular direction) and 2) alleles that are at intermediate frequency as a consequence of stabilizing selection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', selection to keep trait values near an optimum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Now let’s suppose the environment changes quickly: previously selected alleles may now be associated with a trait that is no longer beneficial, and even disease-causing, but they will remain at high frequency for some time before selection is able to purge them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Note that loci with no genetic variation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', fixed beneficial mutations) could still be involved in mismatches in the new environment, but in the absence of genetic variation we will be unable to identify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In addition to GxE interactions, another population genetic concept relating to evolutionary mismatch and the modern increase in NCDs is decanalization (105).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Canalization refers to the process of stabilizing selection that minimizes genetic variation associated with fitness-related traits in a given environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Decanalization, then, is a perturbation from this state that reveals genetic variation for health- or disease-associated phenotypes (106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Though similar, evolutionary mismatch is more specific than decanalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Evolutionary mismatch can occur without having a previously canalized trait, and is a more general term not necessarily linked to stabilizing selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Decanalization is always a form of evolutionary mismatch, but not the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A final term that is distinct from all of these is robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Robustness refers to a property of individual genotypes, wherein they are able to retain an advantageous phenotype despite genetic or environmental hazards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In contrast, evolutionary mismatch and decanalization are population- level phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Box 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Ethical considerations of conducting genomic work in diverse populations Community engagement and ethical research is fundamental to achieving the broader vision of this Consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' There is widespread consensus that broader population representation in biomedical research is critical for reducing health disparities (107), but moving forward on this agenda requires that we simultaneously acknowledge and learn from past mistakes and abuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 15 At the heart of ethical considerations in genetics research is a situation in which diverse populations are dually under-represented and under-consulted (108).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Recent work has outlined best practices for overcoming these issues (108–115).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' For example, Claw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' (109) suggest six principles of research ethics: 1) understand community sovereignty and research regulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 2) engage and collaborate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 3) build cultural competencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 4) improve transparency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 5) build local research capacity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' and 6) disseminate research in accessible formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' The common thread behind these principles is the importance of building trustful and long-term relationships based on principles of dynamic consent, reciprocity, beneficence, and sovereignty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In our own experience, building these sorts of relationships takes time (typically years) but is essential to do before engaging in research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Basic research with populations in LMICs can lead to important insights, yet the value- added benefits from basic research (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', shaping health policy based on epidemiological trends, and/or the development of novel treatment strategies) often can take decades to materialize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Mechanisms for participant community involvement in these longer-term benefits should be explicitly embedded in initial plans (107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' It is also important to recognize that community benefits can extend beyond the research itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' The needs and desires of local communities will vary widely, but populations in LMICs may face problems that are deeply inter-connected and often stem from systemic discrimination: poor nutrition and sanitation (often due to environmental degradation), minimal access to education, few economic opportunities, and loss of land rights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' The priorities of communities will seldom match perfectly with the aims of scientists, especially when participant communities lack basic infrastructure and face discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Prioritizing solutions to these problems is an opportunity to have great impact that will require cooperation between researchers, study participants, universities, NGOs, governments, and funding bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Box 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Life course perspectives on NCD risk Development is a period of heightened environmental sensitivity, and challenging experiences early in life increase lifelong risk of most NCDs (102, 104, 116).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Subsistence-level societies are an under-utilized yet potentially powerful model for studying early life influences on NCD risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Many of these groups are currently experiencing rapid lifestyle changes leading to 1) extreme variation in early life conditions within a single population and 2) frequent mismatch between early life and adult environments—a situation that is thought to put individuals at risk for later life health issues (117-119).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Point #1 provides a clear opportunity to leverage the distributional extremes to study early life effects on health (25, 122).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Further, point #2 affords us 16 the opportunity to compare outcomes when individuals experience within-lifetime environmental “matches” versus “mismatches”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' To date, studies of industrial transitions have come to mixed conclusions about the importance of within-lifetime mismatches (17, 39, 123, 124).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' More work in this area is urgently needed to understand when, why, and how early life experiences shape adult health in these groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genomic tools applied to populations undergoing lifestyle change could also provide valuable insight into how early life experiences become “embedded” into lifelong physiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' At the molecular level, this process is thought to be mediated by stable changes in gene regulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', DNA methylation, chromatin accessibility, and gene expression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' However, many gene regulatory elements are also dynamic and responsive to environmental perturbations throughout life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' This fact leads to challenges in disentangling the effects of early versus later life environments, especially when the two are highly correlated (as is common in HICs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' In contrast, subsistence-level groups in transition often experience decoupled early life and adult experiences, which could be leveraged to disentangle early versus later life influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genotype data collected for the same individuals could also be used to identify rarely studied GxE interactions where the “E” encompasses early life experiences (125–127).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Overall, integrative studies of transitioning populations are primed to reveal which individuals will be most susceptible to NCDs during lifestyle transitions as well as when in the life course these exposures matter most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 17 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kathiresan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Srivastava, Genetics of human cardiovascular disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Cell 148, 1242– 1257 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Khera, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kathiresan, Genetics of coronary artery disease: discovery, biology and clinical translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 18, 331–344 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' McPherson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Tybjaerg-Hansen, Genetics of Coronary Artery Disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Circ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 118, 564–578 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Manolio, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Finding the missing heritability of complex diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nature 461, 747– 753 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Visscher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Brown, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' McCarthy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Yang, Five years of GWAS discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 90, 7–24 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' McAllister, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Current Challenges and New Opportunities for Gene-Environment Interaction Studies of Complex Diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Epidemiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 186, 753–761 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Thomas, Gene–environment-wide association studies: emerging approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 11, 259–272 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Genotype-by-environment interactions inferred from genetic effects on phenotypic variability in the UK Biobank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Science Advances 5, eaaw3538 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Tyrrell, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Gene–obesogenic environment interactions in the UK Biobank study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' International Journal of Epidemiology, dyw337 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nag, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Mahajan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Mccarthy, 306-OR: Gene-Environment Interactions for Type 2 Diabetes in UK Biobank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Diabetes 68, 306–OR (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Arnau-Soler, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Genome-wide by environment interaction studies of depressive symptoms and psychosocial stress in UK Biobank and Generation Scotland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Transl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Psychiatry 9, 14 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lieberman, The Story of the Human Body: Evolution, Health, and Disease (Vintage, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nesse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Williams, Why We Get Sick: The New Science of Darwinian Medicine (Vintage, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Neel, Diabetes Mellitus: A “Thrifty” Genotype Rendered Detrimental by “Progress”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', 353–362 (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Corbett, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Courtiol, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lummaa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Moorad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Stearns, The transition to modernity and chronic disease: mismatch and natural selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 19, 419–430 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Garruto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Little, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' James, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Brown, Natural experimental models: the global search for biomedical paradigms among traditional, modernizing, and modern populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 96, 10536–10543 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lea, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Martins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kamau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Gurven, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Ayroles, Urbanization and market 18 integration have strong, nonlinear effects on cardiometabolic health in the Turkana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sci Adv 6 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lea, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Socioeconomic status effects on health vary between rural and urban Turkana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Evol Med Public Health 9, 406–419 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Wallace, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Orang Asli Health and Lifeways Project (OA HeLP): Study Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' bioRxiv (2021) https:/doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='1101/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='21265442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Golden, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Cohort Profile: The Madagascar Health and Environmental Research (MAHERY) study in north-eastern Madagascar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Epidemiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 46, 1747–1748d (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Golden, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Cohort Description of the Madagascar Health and Environmental Research–Antongil (MAHERY–Antongil) Study in Madagascar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Frontiers in Nutrition 6 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Golden, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Study Protocol: A Cross-Sectional Examination of Socio-Demographic and Ecological Determinants of Nutrition and Disease Across Madagascar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Front Public Health 8, 500 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Gurven, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', The Tsimane Health and Life History Project: Integrating anthropology and biomedicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Evol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Anthropol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 26, 54–73 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Liebert, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Implications of market integration for cardiovascular and metabolic health among an indigenous Amazonian Ecuadorian population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 40, 228– 242 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Urlacher, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Heterogeneous effects of market integration on sub-adult body size and nutritional status among the Shuar of Amazonian Ecuador.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 43, 316– 329 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lieberman, The Story of the Human Body: Evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Health, and Disease (Pantheon, New York) (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Holowka, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Wallace, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lieberman, Foot strength and stiffness are related to footwear use in a comparison of minimally- vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' conventionally-shod populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 8, 3679 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Trumble, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Finch, THE EXPOSOME IN HUMAN EVOLUTION: FROM DUST TO DIESEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 94, 333–394 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Pontzer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Wood, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Raichlen, Hunter-gatherers as models in public health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Obes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 19 Suppl 1, 24–35 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Price, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nguyen, Nutrition and Physical Degeneration: A Comparison of Primitive and Modern Diets and Their Effects (EnCognitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='com, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rousseau, Discourse on the Origin of Inequality (Oxford University Press, 1755).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Gurven, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lieberman, WEIRD bodies: mismatch, medicine and missing diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Evol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Behav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 41, 330–340 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Vos, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Global burden of 369 diseases and injuries in 204 countries and territories, 19 1990–2019: a systematic analysis for the Global Burden of Disease Study 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lancet 396, 1204–1222 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kaplan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Coronary atherosclerosis in indigenous South American Tsimane: a cross-sectional cohort study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lancet 389, 1730–1739 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Raichlen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Physical activity patterns and biomarkers of cardiovascular disease risk in hunter-gatherers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 29, e22919 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Truswell, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kennelly, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hansen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lee, Blood pressures of Kung bushmen in Northern Botswana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Heart J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 84, 5–12 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Shave, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Selection of endurance capabilities and the trade-off between pressure and volume in the evolution of the human heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 116, 19905– 19910 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Gurven, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Blackwell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rodríguez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Stieglitz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kaplan, Does blood pressure inevitably rise with age?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' : longitudinal evidence among forager-horticulturalists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hypertension 60, 25–33 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Wallace, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Knee osteoarthritis risk in non-industrial societies undergoing an energy balance transition: evidence from the indigenous Tarahumara of Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Annals of the Rheumatic Diseases 78, 1693–1698 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Urlacher, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Childhood Daily Energy Expenditure Does Not Decrease with Market Integration and Is Not Related to Adiposity in Amazonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nutr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 151, 695–704 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Pontzer, Burn: New Research Blows the Lid Off How We Really Burn Calories, Lose Weight, and Stay Healthy (Penguin, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lieberman, Exercised: Why Something We Never Evolved to Do Is Healthy and Rewarding (Pantheon Books, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nosarev, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Smagliy, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Anfinogenova, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Popov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kapilevich, Exercise and NO production: relevance and implications in the cardiopulmonary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Front Cell Dev Biol 2, 73 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Green, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hopman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Padilla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Laughlin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Thijssen, Vascular Adaptation to Exercise in Humans: Role of Hemodynamic Stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Physiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 97, 495– 528 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Petersen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Pedersen, The anti-inflammatory effect of exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Physiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 98, 1154–1162 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Tigbe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Granat, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sattar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lean, Time spent in sedentary posture is associated with waist circumference and cardiovascular risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Obes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 41, 689–696 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Jaeggi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Do wealth and inequality associate with health in a small-scale subsistence society?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Elife 10 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Noah Snyder-Mackler, Joseph Robert Burger, Lauren Gaydosh, Dan Belsky, Grace A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Noppert, Fernando A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Campos, Alessandro Bartolomucci, Yang Claire Yang, Allison E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 20 Aiello, Angela O’Rand, Kathleen Mullan Harris, Carol A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Shively, Susan C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Alberts, & Jenny Tung, Social determinants of health and survival in humans and other animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Science (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lea, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Social gradients in health vary between rural and urban Turkana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' bioRxiv (2021) https:/doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='1101/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='21260771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Strachan, Hay fever, hygiene, and household size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' BMJ 299, 1259–1260 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Bach, The hygiene hypothesis in autoimmunity: the role of pathogens and commensals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Immunol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 18, 105–120 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rook, Darwinian medicine and the “hygiene”or “old friends” hypothesis Clin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Immunol 160, 70–79 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Glew, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Cardiovascular disease risk factors and diet of Fulani pastoralists of northern Nigeria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Clin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nutr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 74, 730–736 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Leonard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Crawford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Comuzzie, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sukernik, Correlates of low serum lipid levels among the Evenki herders of Siberia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 6, 329–338 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Mbalilaki, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Daily energy expenditure and cardiovascular risk in Masai, rural and urban Bantu Tanzanians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sports Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 44, 121–126 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' , National Health and Nutrition Examination Survey (NHANES) https:/doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='4135/9781412963947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='n318 (February 16, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Arem, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Leisure time physical activity and mortality: a detailed pooled analysis of the dose-response relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' JAMA Intern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 175, 959–967 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Gurven, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Jaeggi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kaplan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Cummings, Physical activity and modernization among Bolivian Amerindians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' PLoS One 8, e55679 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nédélec, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Genetic Ancestry and Natural Selection Drive Population Differences in Immune Responses to Pathogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Cell 167, 113 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kim-Hellmuth, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Genetic regulatory effects modified by immune activation contribute to autoimmune disease associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 8 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Jelenkovic, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Genetic and environmental influences on adult human height across birth cohorts from 1886 to 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Elife 5 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Athanasiadis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', A comprehensive map of genetic relationships among diagnostic categories based on 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='6 million relative pairs from the Danish genealogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 119 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genovese, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Association of Trypanolytic ApoL1 Variants with Kidney Disease in African Americans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Science 329, 841–845 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Friedman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Pollak, APOL1 and Kidney Disease: From Genetics to Biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Physiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 82, 323–342 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Loos, CREBRF variant increases obesity risk and protects against diabetes in 21 Samoans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 48, 976–978 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Benton, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', The influence of evolutionary history on human health and disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' (2021) https:/doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='1038/s41576-020-00305-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Byars, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Genetic loci associated with coronary artery disease harbor evidence of selection and antagonistic pleiotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' PLoS Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 13, e1006328 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Brinkworth, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Barreiro, The contribution of natural selection to present-day susceptibility to chronic inflammatory and autoimmune disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Curr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Opin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Immunol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 31, 66–78 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Raj, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Common risk alleles for inflammatory diseases are targets of recent positive selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 92, 517–529 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Fan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hansen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Tishkoff, Going global by adapting local: A review of recent human adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Science 354, 54–59 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hamblin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Di Rienzo, Detection of the signature of natural selection in humans: evidence from the Duffy blood group locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 66, 1669–1679 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hamid, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Korunes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Beleza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Goldberg, Rapid adaptation to malaria facilitated by admixture in the human population of Cabo Verde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Elife 10 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sabeti, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Positive natural selection in the human lineage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Science 312, 1614– 1620 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Evershed, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Dairying, diseases and the evolution of lactase persistence in Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nature 608, 336–345 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Belloy, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Napolioni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Greicius, A Quarter Century of APOE and Alzheimer’s Disease: Progress to Date and the Path Forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Neuron 101, 820–838 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Trumble, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Apolipoprotein E4 is associated with improved cognitive function in Amazonian forager-horticulturalists with a high parasite burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' FASEB J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 31, 1508–1515 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Garcia, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', APOE4 is associated with elevated blood lipids and lower levels of innate immune biomarkers in a tropical Amerindian subsistence population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' bioRxiv, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='436070 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Peyrot, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Effect of polygenic risk scores on depression in childhood trauma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Psychiatry 205, 113–119 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Mullins, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Polygenic interactions with environmental adversity in the aetiology of major depressive disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Psychol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 46, 759–770 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Peterson, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Molecular Genetic Analysis Subdivided by Adversity Exposure Suggests Etiologic Heterogeneity in Major Depression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Psychiatry 175, 545–554 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nagpal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Gibson, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Marigorta, Pervasive Modulation of Obesity Risk by the Environment and Genomic Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genes 9 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 22 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Abadi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Penetrance of Polygenic Obesity Susceptibility Loci across the Body Mass Index Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 101, 925–938 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rask-Andersen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Karlsson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Ek, Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Johansson, Gene-environment interaction study for BMI reveals interactions between genetic factors and physical activity, alcohol consumption and socioeconomic status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' PLoS Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 13, e1006977 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sheppard, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', A model and test for coordinated polygenic epistasis in complex traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 118 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Gusev, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Integrative approaches for large-scale transcriptome-wide association studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 48, 245–252 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Gamazon, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', A gene-based association method for mapping traits using reference transcriptome data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 47, 1091–1098 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Fairfax, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Innate immune activity conditions the effect of regulatory variants upon monocyte gene expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Science 343, 1246949 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lee, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Common Genetic Variants Modulate Pathogen-Sensing Responses in Human Dendritic Cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Science 343, 1246980 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Garske, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Reverse gene-environment interaction approach to identify variants influencing body-mass index in humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat Metab 1, 630–642 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Carding, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Verbeke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Vipond, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Corfe, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Owen, Dysbiosis of the gut microbiota in disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Microb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Ecol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Health Dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 26, 26191 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Aron-Wisnewsky, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Clément, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nieuwdorp, Fecal Microbiota Transplantation: a Future Therapeutic Option for Obesity/Diabetes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Curr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Diab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 19, 51 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Smallwood, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Helminth Immunomodulation in Autoimmune Disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Immunol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 8, 453 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Ober, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Vercelli, Gene–environment interactions in human disease: nuisance or opportunity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Trends Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 27, 107–115 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hunter, Gene–environment interactions in human diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 6, 287– 298 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Carmody, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sarkar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Reese, Gut microbiota through an evolutionary lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Science 372, 462–463 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Grieneisen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Gut microbiome heritability is nearly universal but environmentally contingent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Science 373, 181–186 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Groussin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Elevated rates of horizontal gene transfer in the industrialized human microbiome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Cell 184, 2053–2067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='e18 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Schnorr, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Gut microbiome of the Hadza hunter-gatherers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 5, 3654 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Vangay, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', US Immigration Westernizes the Human Gut Microbiome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Cell 175, 962– 23 972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content='e10 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Snodgrass, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', The emergence of obesity among indigenous Siberians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Physiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Anthropol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 25, 75–84 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' NCD Risk Factor Collaboration (NCD-RisC), Rising rural body-mass index is the main driver of the global obesity epidemic in adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nature 569, 260–264 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Felitti, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Relationship of childhood abuse and household dysfunction to many of the leading causes of death in adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' The Adverse Childhood Experiences (ACE) Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' American Journal of Preventative Medicine 14, 245–258 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Bellis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Life course health consequences and associated annual costs of adverse childhood experiences across Europe and North America: a systematic review and meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' The Lancet Public Health 4, e517–e528 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hoffman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Reynolds, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hardy, Developmental origins of health and disease: current knowledge and potential mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nutr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 75, 951–970 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Gibson, Decanalization and the origin of complex disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 10, 134– 140 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Gibson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lacek, Canalization and Robustness in Human Genetics and Disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 54, 189–211 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Fox, The illusion of inclusion - the “all of us” research program and indigenous peoples’ DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Engl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 383, 411–413 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Jackson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Kuhlman, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Jackson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Keolu Fox, Including Vulnerable Populations in the Assessment of Data From Vulnerable Populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Frontiers in Big Data 2 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Claw, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', A framework for enhancing ethical genomic research with Indigenous communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 9, 2957 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Dignan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Burhansstipanov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Bemis, Successful implementation of genetic Education for Native Americans workshops at national conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genetics 169, 517– 521 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Tiffin, Tiered informed consent: respecting autonomy, agency and individuality in Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' BMJ Glob Health 3, e001249 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hudson, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Rights, interests and expectations: Indigenous perspectives on unrestricted access to genomic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 21, 377–384 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hernandez, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Perry, Scanning the human genome for “signatures” of positive selection: Transformative opportunities and ethical obligations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Evol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Anthropol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 30, 113– 121 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Garrison, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Genomic Research Through an Indigenous Lens: Understanding the Expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genomics Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 20, 495–517 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Guglielmi, Facing up to injustice in genome science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nature 568, 290–293 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 24 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Cohen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Janicki-Deverts, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Matthews, Childhood socioeconomic status and adult health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 1186, 37–55 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Monaghan, Early growth conditions, phenotypic development and environmental change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' B Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 363, 1635–1645 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lea, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Archie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Tung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Alberts, Developmental plasticity: bridging research in evolution and human health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Evolution, Medicine, and Public Health, in press (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lea, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rosebaum, Understanding how early life effects evolve: progress, gaps, and future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Current Opinion in Behavioral Sciences 36, 29–35 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hayward, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rickard, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lummaa, Influence of early-life nutrition on mortality and reproductive success during a subsequent famine in a preindustrial population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 110, 13886–13891 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Hayward, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Lummaa, Testing the evolutionary basis of the predictive adaptive response hypothesis in a preindustrial human population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Evolution, Medicine, and Public Health 2013, 106–117 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Muehlenbein, Human Evolutionary Biology (Cambridge University Press, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Chan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Diabetes in Asia: epidemiology, risk factors, and pathophysiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' JAMA 301, 2129–2140 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Huxley, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Ethnic comparisons of the cross-sectional relationships between measures of body size with diabetes and hypertension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Obes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 9 Suppl 1, 53–61 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nugent, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Tyrka, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Carpenter, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Price, Gene-environment interactions: early life stress and risk for depressive and anxiety disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Psychopharmacology 214, 175–196 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Caspi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=', Influence of life stress on depression: moderation by a polymorphism in the 5-HTT gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Science 301, 386–389 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Caspi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Moffitt, Gene–environment interactions in psychiatry: joining forces with neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' Neurosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} +page_content=' 7, 583–590 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfxA2-/content/2301.05255v1.pdf'} diff --git a/udAzT4oBgHgl3EQfdPwF/content/tmp_files/2301.01415v1.pdf.txt b/udAzT4oBgHgl3EQfdPwF/content/tmp_files/2301.01415v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..aac6b0e131b61d956bd70136a6430f35edfe4c9d --- /dev/null +++ b/udAzT4oBgHgl3EQfdPwF/content/tmp_files/2301.01415v1.pdf.txt @@ -0,0 +1,2012 @@ +Machine Learning technique for isotopic determination of radioisotopes using +HPGe γ-ray spectra +Ajeeta Khatiwada, Marc Klasky, Marcie Lombardi, Jason Matheny, Arvind Mohan +Los Alamos National Laboratory, Los Alamos, NM 87545, USA +Abstract +γ-ray spectroscopy is a quantitative, non-destructive technique that may be utilized for the identification and quan- +titative isotopic estimation of radionuclides. Traditional methods of isotopic determination have various challenges +that contribute to statistical and systematic uncertainties in the estimated isotopics. Furthermore, these methods typi- +cally require numerous pre-processing steps, and have only been rigorously tested in laboratory settings with limited +shielding. In this work, we examine the application of a number of machine learning based regression algorithms +as alternatives to conventional approaches for analyzing γ-ray spectroscopy data in the Emergency Response arena. +This approach not only eliminates many steps in the analysis procedure, and therefore offers potential to reduce this +source of systematic uncertainty, but is also shown to offer comparable performance to conventional approaches in the +Emergency Response Application. +Keywords: Radionuclides, γ-ray spectroscopy, Isotopic determination, enrichment determination, Machine +Learning, Nuclear safeguards, Nuclear Threat Detection +1. Introduction +The identification and quantitative determination of +the isotopic content of samples/objects potentially con- +taining uranium and/or plutonium is of paramount im- +portance to the nuclear materials safeguards, arms con- +trol verification, nuclear security, Emergency Response +(ER), as well in nuclear remediation arenas [1, 2, 3, 4, +5, 6, 7, 8, 9, 10]. Conventional methods for determin- +ing the isotopics/enrichment using γ-ray spectroscopy +require many time consuming steps +1. photo-peak identification, +2. background and continuum subtraction, +3. feature extraction, +4. estimation of the relative efficiency curve, and +5. matching of the extracted features with those of +known nuclides to estimate the fraction of iso- +topes [11]. +In many of these application areas, it is imperative +to rapidly determine the isotopic fractions using remote +detection techniques. These constraints necessitate the +∗Corresponding Author +Email address: ajeeta@lanl.gov (Ajeeta Khatiwada) +use of non-destructive assay methods (NDA) and ac- +companying automated algorithms to perform quanti- +tative analysis. +In some applications, details regard- +ing the physical arrangement of the nuclear materials +cannot be revealed due to security concerns, e.g. +in +treaty verification activities, or are unknown e.g. in nu- +clear security and ER activities in which the shielding +and other aspects of the physical configuration are un- +known. In this work, we examine the ability of numer- +ous machine learning (ML) techniques to address the +automated identification and quantification of uranium +and plutonium isotopics for ER applications. +2. Organization of Paper +In this work, we investigate the application of a vari- +ety of machine learning algorithms to perform uranium +and plutonium isotopic estimation for Emergency Re- +sponse applications. Before discussing the ML algo- +rithms utilized in these investigations, we present a re- +view of both the traditional as well as the ML methods +to perform quantitative isotopic identification in Sec- +tion 3. The machine algorithms utilized in this investi- +gation are presented in Section 4. In Section 5, the gen- +eration of ML training data is discussed along with an +Preprint submitted to Elsevier +January 5, 2023 +arXiv:2301.01415v1 [physics.data-an] 4 Jan 2023 + +investigation of the accuracy of these simulations to em- +ulate experimental data. Details of the pre-processing of +the spectral data including background, continuum sub- +traction, and feature extraction are given in Section 6. +ML results using simulations are presented in Section 7. +Hyper-parameter investigations are presented in Sec- +tion 8. Investigations using experimental data and dis- +cussions of the results are presented in Section 9. Lastly, +summary and conclusions are provided in Section 10. +3. Background +3.1. Traditional Methods +Starting in the early 1970s, researchers developed +several approaches to perform quantitative NDA spec- +troscopic analysis for both uranium and plutonium iso- +topics [12]. Today there are three general variations of +the NDA method that have been utilized to infer the +isotopic content of 235U. The first method, currently +utilized by the International Atomic Energy Agency +(IAEA), is based on the measurement of the 186 keV +line of 235U in the spectra obtained using either germa- +nium or sodium iodide spectrometer systems [13], and +requires a calibration with a known enrichment stan- +dard. Provided that the sample measured is similar to +the reference i.e. has the same geometry and thickness +and the measurement conditions are constant, the count- +ing rate for the 185.7 keV peak is proportional to the +enrichment. While this approach has been utilized to +successfully infer the content of 235U, there are several +limitations: the samples must satisfy the infinite thick- +ness criterion [14], calibrations need to be performed +for samples with different containers, and wall thick- +nesses need to be determined prior to the enrichment +measurement [15]. In practice, this constraint limits the +applicability of the the enrichment of an object’s sur- +face to a depth of 0.26 cm and 0.74 cm for uranium +metal and U3O8 powder, respectively [14]. An auto- +mated version of this method, called NaIGEM (NaI(Tl) +Gamma Enrichment Measurements), is included in the +HM-5 instrument used by the IAEA [16]. Enrichment +measurements of uranium without contaminants using +low-resolution detectors can achieve 1% precision for +arbitrary enrichment while contamination by minor ura- +nium isotopes has a biasing effect of 5–10% [17]. +Methods employing multi-peak self calibration were +proposed to overcome the drawbacks of the enrich- +ment meter principle. +The first variation, Peak Area +(PA), utilizes the spectral lines in the range 89 to 120 +keV [3]. The relative efficiency curves of different ura- +nium isotopes or their daughters are estimated from +a limited number of peaks in the spectrum. +Sophis- +ticated codes such as MGAU (Multi-Group Analysis +for uranium) are based on this principle. +The preci- +sion of the estimated efficiency response depends on the +the number and intensity of the isotope peaks. How- +ever, these methods still experience performance issues +when measuring uranium through thick walled contain- +ers [18, 19, 2, 20, 21]. +To overcome the limitation of the finite thickness of +shielding, the relative-efficiency (RE) method was pro- +posed [12]. +The RE method computes the uranium +enrichment using the relative efficiency obtained from +the peaks expressed in the measured spectra using an +energy range from 144 to 1001 keV. Several software +packages, including FRAM and MGA++1, have imple- +mented this approach [8, 22, 23]. Both MGA++ and +FRAM may be utilized to perform Pu isotopics analysis +using the low-energy γ-ray spectrum, along with higher +energy γ-rays [3, 22]. +A comparison of three implementations of the RE +method concluded that the performance and applica- +bility with increasing wall thickness at low enrich- +ment grades was in the order PC/FRAM, MGA++, and +MGAU. Therefore, in shielded conditions, it was rec- +ommended that PC/FRAM for γ-rays above 200 keV +using the coaxial detector spectrum be utilized. +Be- +fore concluding, it should be noted that the shielding +thicknesses that were evaluated are significantly below +those that might be encountered in ER scenarios [23], +i.e., the shielding thickness may be significantly greater +than those analyzed with the traditional approaches for +determining uranium enrichment and or plutonium iso- +topics. +3.2. Machine Learning Methods +The traditional methods utilized to perform NDA +of uranium enrichment and plutonium isotopic quan- +tification require numerous pre-processing steps, and +also have difficulty in treating environments in which +unknown shielding, overlapping peaks, and or thick +shielding is present. These issues, in conjunction with +the success in the development and application of ma- +chine learning (ML) techniques in the last decade, have +motivated the examination of machine learning tech- +niques to address these shortcomings. Indeed, the appli- +cation of ML techniques to address both classification +and regression problems in radiation detection, source +1A suite of three software programs (MGA, U235, and MGAHI, a +Pu isotopic analysis code that uses the 200 keV -1 MeV energy region) +for the analysis of actinide spectra acquired by Ge detectors. +2 + +identification, and quantitative assessment of radionu- +clides applications have become increasingly popular. +One of the more prominent applications of ML in +addressing radioisotopes has been in the detection and +identification arena for nuclear safeguards and arms +control applications. To that effect, one of the first ap- +plications of a neural network to identify radioisotopes +was performed by Olmos using a low resolution NaI de- +tector [24]. Additional early work by Yoshida utilized +a multilayer perceptron (MLP) network with a HPGe +spectra to identify radioisotopes in samples with mixed +radioisotopes [25]. Kangas also developed a neural net- +work to analyze very low resolution Polyvinyl toluene +(PVT) spectra for use in the identification of radioac- +tive materials at international border crossings [26]. +More recently, Liang has demonstrated that a Convo- +lutional Neural Network (CNN) algorithm trained us- +ing Monte Carlo N-Particle Transport (MCNP) [27] +simulations with a NaI detector could, in a low count +rate regime, identify radioisotopes that are nominally +difficult to identify, eliminating the necessity to per- +form spectra pre-processing such as background sub- +traction and spectrum smoothing [28]. Bobin utilized a +Bayesian sequential approach combined with a spiking +neural network to enable the real-time processing of sig- +nals detected from a mixture of γ-emitting radionuclides +in spectroscopic portal systems [29]. Finally, Sharma et +al. implemented machine learning techniques to reduce +false alarm rates when using γ-ray spectrometers for the +identification of persons concealing radioactive materi- +als [30]. +Additional investigations have been performed in the +application of neural networks for radioisotope identi- +fication [31, 32, 33, 34]. In general, these investiga- +tions utilized either MLP or CNNs with a number of +different methods for feature extraction, including the +Discrete Cosine Transform (DCT) and the Karhunen- +Lo`eve Transform (KLT). A more advanced neural net- +work architecture employing an autoencoder with a low +resolution NaI detector was shown to improve anomaly +detection relative to traditional techniques [35]. +Another application of ML is in the area of identifi- +cation of radioisotopics in environmental samples [36, +37]. Hata investigated the feasibility of using a support +vector machine (SVM) to classify uranium waste drums +as natural uranium or reprocessed uranium using NaI +detectors [37]. Wei applied a radial basis neural net- +work algorithm for environmental and treatment evalu- +ation of decommissioned uranium tailing ponds [4]. Fi- +nally, Chen used a KLT and an artificial neural network +in conjunction with NaI [38]. +Additional application areas of ML have been investi- +gated including the analysis of complex spectra (fission +and activation products). In these applications, it was +shown that the application of feed forward neural net- +works in conjunction with the Singular Value Decom- +position (SVD) can significantly improve performance +and reduce the required analysis time once the neural +networks have been trained [39]. +In some applications, the objective is to determine the +isotopic content of the radioisotopes. In particular, in +the ER application the objective is to determine the ura- +nium enrichment and or plutonium isotopics in objects +containing nuclear material. In this scenario, HPGe de- +tectors are typically utilized, and the geometry of the +object containing the nuclear material along with the +characteristics of the intervening shielding materials, +i.e. material composition and thicknesses of the compo- +nents containing the nuclear materials, are not known. +Although many investigations have been performed us- +ing ML algorithms to determine isotopic content, al- +most all of these have been conducted in applications +related to Nuclear Material Safeguards and other appli- +cation areas in which either the configuration is known +and or the shielding materials are both known and/or +relatively thin i.e. less than 1 cm. Notably, Shaban uti- +lized a feed forward neural network to predict uranium +enrichment in laboratory size samples [40]. +Early work by Vigneron demonstrated that HPGe +spectroscopic measurements in conjunction with Prin- +ciple Component Analysis (PCA), to reduce the dimen- +sionality of the spectra, could be successfully utilized to +determine the enrichment of laboratory samples using +the low energy range 83 to 103 KeV using a MLP [41]. +Ryu investigated the use of a neural network model +using low resolution NaI spectra to analyze uranium en- +richment, from depleted to low enrichment, from very +low radioactivity samples present in small beakers with +very short count times [42]. Elmaghraby also utilized a +neural network architecture to determine the uranium +isotopics using a HPGe detector on laboratory sam- +ples [43]. +Lastly, Aitkenhead using simulated data, evaluated +the spectra of shielded plutonium using ANNs to detect +the presence or absence of plutonium, estimate 239Pu +content, as well as distinguish material age of shielded +plutonium [10]. +4. Machine Learning algorithms +While a great deal of work has been performed to +investigate the use of machine learning in the areas of +radioisotope detection and identification as well in the +quantification of radioisotopes, almost all of this work +3 + +has been conducted under conditions that are not di- +rectly relevant to the ER community. Accordingly, in +this work we examine the application of machine learn- +ing algorithms (MLP and Convolutional) Neural Net- +works, Gaussian Processes, Decision Tees and their +variants i.e. Gradient Boosted Decision Trees and Ran- +dom forest, as well as Nearest-Neighbors to 1) study +if ML based regression algorithms are a reasonable al- +ternative to the conventional methods and 2) to iden- +tify a general class of ML algorithms that are robust +to achieving the aforementioned goal without excessive +fine-tuning of the hyper parameters to enable the deter- +mination of the isotopic content of uranium and pluto- +nium under conditions more consistent with ER appli- +cation. +4.1. Methods +Examinations in this paper are performed based on +supervised learning of training datasets using regression +algorithms that are integrated into the Scikit-learn [44] +package in Python as well as ML algorithms available +in Mathematica [45]. The results from Mathematica are +labeled with ‘*’ next to the algorithm names in the ta- +bles. +4.1.1. Decision Tree +Decision trees are one of the most commonly used, +practical approaches for supervised learning. They can +be used to solve both regression and classification tasks. +A decision tree builds regression or classification mod- +els in the form of a tree structure. They break down +a dataset into smaller and smaller subsets while at the +same time an associated decision tree is incrementally +developed. The final result is a tree with decision nodes +and leaf nodes [46]. Each tree is composed of nodes, +which are chosen by looking for the optimum split of +the features. The split of features is determined utilizing +an impurity measure. For regression trees, two common +impurity measures are least squares and least absolute +deviations. In the former, the method is similar to min- +imizing least squares in a linear model. The splits are +chosen to minimize the residual sum of squares between +the observation and the mean in each node. In the latter +method, a minimization of the mean absolute deviation +from the median within a node is performed. +Two popular techniques to improve the robustness of +a decision tree are ensemble methods such as Random +Forest methods and Boosting methods. These methods +are described below. +4.1.2. Random Forest +Random forests are a popular technique in classical +machine learning, due to their predictive ability at a +lower computational burden than neural networks [47]. +At their core, random forests are an “ensemble” learn- +ing technique based on decision trees. Ensemble learn- +ing is the strategy of averaging predictions from mul- +tiple individual models or estimators, leading to more +robust and accurate predictions. +The random forests +can be configured to train a predefined number of de- +cision tree estimators for the same training data. Each +decision tree makes a target prediction based on train- +ing data. Each tree also has a user-specified depth pa- +rameter. The depth parameter denotes the number of +branches the tree is allowed to create, when fitting to +the training data. Typically, increasing depth can in- +crease the predictive capability of the decision tree, as it +can learn more intricate features in the data. However, +increasing depth beyond a certain limit can also cause +over-fitting and reduce accuracy. The precise limit is +dependent on the data, and is discovered by trial-and- +error. The random forest aggregates the model from all +these individual trees, to create an ensemble model. +4.1.3. Gradient Boosted Trees +Gradient boosting is another family of ensemble +methods fitting a sequence of weak learners (estima- +tor that gives a prediction slightly better than a random +guess) on modified versions of the dataset [48]. In the +Gradient Boosted Tree algorithm, the convergence of +the boosting algorithm is improved by computing the +gradient of a differentiable loss functions. In Gradient +Boosting the base estimator is the Decision Tree estima- +tor and the hyper-parameters in the tuning phase are the +number of estimator and the learning rate. +4.1.4. K-Nearest Neighbors Regression +The central idea behind the K-nearest neighbors +(KNN) is based on the nearest neighbors to query a data +point, where k is an integer algorithm parameter. There- +fore, the value of a quantity at a point is a weighted +average of the k points closest to it [49, 50]. The user +specifies the distance metric for computing the weights. +There are multiple choices: Uniform, Euclidean, Man- +hattan, Minkowski etc. +With Uniform weights, each +neighbor is provided the same weight irrespective of its +distance from the query point. In the other distance met- +rics, the neighbors closer to the query point in that par- +ticular space are assigned higher weights than the those +further away. Therefore, this metric acts as a weighted +average. +4 + +4.1.5. Gaussian Process Regression +Gaussian process regression is a non-parametric +Bayesian approach towards regression problems. It can +capture a wide variety of relations between inputs and +outputs by utilizing a theoretically infinite number of +parameters and letting the data determine the level of +complexity through the means of Bayesian inference +[51, 52, 53]. +4.1.6. Multi-layer Perceptron (MLP) Regression +MLPs are a type of neural network consisting of mul- +tiple layers: an input layer, one or more hidden layers, +and an output layer. Each layer is fully connected to +the next one via non-linear activation functions. Train- +ing a neural network on a simulation such that it can +be generalized to apply to an experimental dataset that +differs from the simulation model in many ways is of- +ten challenging. MLPs are particularly susceptible to +over-fitting, although there are regularization methods +available to counter the problem of over-fitting. Tun- +ing of hyper parameters, such as the activation func- +tion, number of hidden layers, number of nodes in +each hidden layers, amount of regularization, dropout, +enabling/disabling early stopping, and choosing learn- +ing rates and optimization strategies are necessary to +achieve the best possible performance. +4.1.7. Convolutional Neural Networks (CNN) +CNNs are a form of neural network in which the lin- +ear layers take the form of a set of convolutions [54, +55, 56]. This greatly reduces the number of trainable +weights, thereby decreasing the risk of over-fitting, and +also allows for computationally efficient implementa- +tion. These methods are typically only suitable, how- +ever, when the input data has the shift-invariance prop- +erties implied by the use of convolutions. +5. Training Data Generation +The training data for the ML algorithms was gen- +erated utilizing GADRAS [57], incorporating a 145% +relative efficiency HPGe detector with a bismuth side +shield and tin filter using either Pu or U sources in ei- +ther metal or oxide forms. The sources were contained +in one of three geometries, i.e. +shells, cylinders, or +spheres. Since the “self-shielding” is dependent on the +source geometry, the γ spectra are not identical for two +identical sources that differ only in geometry. There- +fore, an ensemble of training data for each of the re- +spective geometries was generated using a variety of +Geometry +No. of +Enrichment +Shielding +Decks +fraction (235U ) +present +Shell +1800 +0.000 – 0.989 +No +Shell +15839 +0.003 – 1.000 +Yes +Sphere +1800 +0.000 – 0.995 +No +Cylinder +7000 +0.000 – 1.000 +No +Cylinder +20000 +0.000 – 1.000 +Yes +Table 1: Summary of training data simulations used for uranium in +various configurations. +Geometry +No. of +Isotopics +Shielding +Decks +fraction (239Pu ) +present +Shell +1800 +0.000 – 0.995 +No +Shell +7920 +0.230 – 1.000 +Yes +Sphere +1800 +0.000 – 0.995 +No +Cylinder +5000 +0.560 – 1.000 +No +Cylinder +20000 +0.560 – 1.000 +Yes +Table 2: Summary of training data simulations used for plutonium in +various configurations. +235U enrichment/239Pu isotopic fractions, source thick- +ness, and shielding materials with accompanying thick- +nesses. Characteristics of the training data are summa- +rized in Tables 1 and 2. +For the dataset created with a shell configuration, the +thickness of the source shells was between 0.02 and 4 +cm. The interior of the shell had a void of radius 1.6 cm +for Pu sources with source thickness greater than 2 cm, +and for all other cases the outer surface of the source +was 6 cm. Spectra generated for sources with spherical +geometry had radii ranging from 0.02 to 4 cm. Cylin- +drical plutonium sources were generated with heights +ranging from 0.35 to 1.57 cm and 0.142 to 0.59 cm +with corresponding radii ranging from 0.4 to 1.1 cm and +0.353 to 0.931 cm in bare and shielded configurations, +respectively. Cylindrical uranium sources were gener- +ated with heights ranging from 3.9 to 7.7 cm and 2.45 +to 9.76 cm with corresponding radii ranging from 5.35 +to 5.44 cm and with 5 to 6 cm radii in bare and shielded +configurations, respectively. +For the Shell and Sphere configuration simulations +with the shielding material present, iron (Fe), Tanta- +lum (Ta), Polypropylene, or some combination of the +aforementioned materials was utilized. The thickness +of shielding materials ranged from 1–10, 1–6, and 1–64 +cm respectively for the aforementioned materials. The +Cylindrical geometry dataset utilized various combina- +tion of aluminum (Al), Tantalum (Ta), Iron (Fe), Lead +(Pb), and Polypropylene for shielding, while the shield- +ing thickness ranged from 0.05 – 2 cm. +5 + +Figure 1: Comparison of 8x binned simulated spectra and background +subtracted spectra for a U source, UISO17. +5.1. Comparison of experimental data with GADRAS +simulations +In machine learning, the ability of the simulations to +replicate the experimental data is a fundamental issue +that must be addressed when simulations are utilized for +training and the testing is performed using experimental +measurements. To this end, we performed experiments +with a HPGe detector using both uranium as well as plu- +tonium sources with and without accompanying shield- +ing and compared these with GADRAS simulated spec- +tra. The simulations were generated using source data +sheets for the primary isotopics, geometry, reported age +of the material, and dimensions/configurations to model +the experimental data. Furthermore, for the GADRAS +simulation of U3O8 and PuO2 sources, the mass frac- +tions of uranium and plutonium were adjusted to ac- +count for the oxide forms utilized in the experimen- +tal data. The isotopic fractions of U/Pu isotopes other +than 235U, 238U, 239Pu, and 240Pu were chosen based +on the certification sheets for the sources. For the de- +pleted uranium shell simulations, generic values of en- +richment and miscellaneous isotopic fractions were uti- +lized, while the void/shell thickness were matched to +those in the experimental setup. Additionally, for the +U3O8 simulations, 40K and 232Th contents were adjusted +to match the background data. The 232U content in the +simulation was also adjusted based on the height of the +2614.5 keV photopeak. Some fine tuning in the normal- +ization was performed to match the container material +and thickness, where appropriate. The simulation mod- +els were run with Poission statistics, and compared to +data with terrestrial background contribution subtracted +from the spectra. Comparisons of the simulated spec- +tra with the experimental spectra are presented in Fig- +ures 1–6. +Figure 2: Comparison of 2x binned simulated spectra vs background +subtracted spectra for a U source, UISO17. +Figure 3: Comparison of 8x binned simulated spectra and background +subtracted spectra for a U source, A1127. +Figure 4: Comparison of 2x binned simulated spectra vs background +subtracted spectra for a U source, A1127. +6 + +GADRAS +$106 +uno +Data +0105 +104 +103 +102 +10 +500 +1000 +1500 +2000 +2500 +Energy(keV)GADRAS +$105 +Counts +Data +104 +103 +102 +50 +100 +150 +200 +250 +300 +350 +400 +450 +Energy (keV)$106 +GADRAS +uno +Data +0105 +104 +103 +102 +10 +500 +1000 +1500 +2000 +2500 +Energy (keV)GADRAS +$105 +Counts +Data +104 +103 +102 +山 +50 +100 +150 +200 +250 +300 +350 +400 +450 +Energy (keV)Figure 5: Comparison of 8x binned simulated spectra vs background +subtracted spectra for a Pu source, CBNMPu84. +Figure 6: Comparison of 8x binned simulated spectra vs background +subtracted spectra for a Du-Shell. +6. Pre-processing and feature extraction +The experimental analog data pulses obtained from +the HPGe detector system, after being converted into +digital pulses, are recorded in the units of count per dis- +crete channels. γ spectra obtained from the GADRAS +simulation are also obtained in the units of counts per +channel. In either case the counts may be, optionally, +pre-processed to remove the continuum background. +Detailed discussion on continuum subtraction is pro- +vided in subsection 6.1. +Additionally, for the exper- +imental data, where terrestrial background is present, +contributions from such background sources are esti- +mated and subtracted from the foreground counts. The +net-counts, after optional continuum subtraction and +terrestrial background subtraction, are then integrated +in a region of interest around photo-peaks of interest +to estimate counts associated with each of the photo- +peaks. +The regions of interests are chosen based on +the expected photo-peaks for the two isotopes of U and +Pu examined in this study. The mean value of the en- +ergy associated with these photo-peaks and their associ- +ated net-counts constitute the features for ML training. +The impact of the number of features, and the means of +reducing the dimensionality of the features during the +supervised ML training is investigated in Section 9.2. +For training samples, the features are accompanied by +answer “keys”, which are the relative fraction of 235U +(239Pu) with respect to the total fraction of 235U and 238U +(239Pu and 240Pu). Here onward, for simplicity, these +quantities will be together referred as isotopic ratios or +as 235U frac and 239Pu frac individually. +6.1. Continuum Subtraction +Subtraction of the continuum background produced +from scattering were examined to understand the im- +pact on the isotopic determination. +As such, a Sen- +sitive Nonlinear Iterative Peak (SNIP) clipping algo- +rithm implemented in TSpectrum class of ROOT frame- +work [58, 59, 60] was utilized for one-dimensional +background estimation. The number of iterations was +examined in estimating the continuum. +The optimal +number of iterations was chosen to be 20 based on the +ability to remove adequate amount of continuum with- +out resulting in negative counts in the subtracted spec- +tra. An example spectra with the continuum background +estimate with this method is provided in Figure 7. +7. Investigations using simulated data +The determination of the isotopic content of ura- +nium or plutonium is a complex function of numerous +7 + +$106 +GADRAS +uno +Data +0105 +104 +103 +102 +10 +500 +1000 +1500 +2000 +2500 +Energy(keV)$106 +GADRAS +uno +Data +0105 +104 +103 +102 +10 +500 +1000 +1500 +2000 +2500 +Energy(keV)Figure 7: An example of a continuum estimate performed on a γ spec- +tra for a Pu source. +factors including: the source geometry, source thick- +ness, shielding material composition, shielding thick- +ness, possible inherent impurities e.g. 232U along with +the isotopic ratios of the isotopes in question i.e. 235U +or 239Pu . +Furthermore, the accuracy in determining +the isotopic fraction is determined by the ability to ad- +equately sample these variables in the training set as +well as the representativeness of the training data to the +testing data, the quantity of training data from which to +learn, and the ability to adequately train the given ML +algorithm. +Since the final goal of this study is to apply the ML +algorithm to experimental data that may differ signif- +icantly from the training sample in multiple different +ways, i.e. amount of shielding present, source geom- +etry, background spectra, etc., the algorithm needs to be +robust against over-fitting. +7.1. Bare: No Shielding +As an initial test, two simulated datasets (Spheres and +Cylinders) were utilized to examined the ability of dif- +ferent ML algorithms to predict the isotopic ratios for +both Pu and U with no shielding materials present. This +test represents the most simplistic mapping from the +spectra to isotopic ratios that can be learnt. That is, +no alteration of the line intensities due to the shielding +needs to be learned. A sample result showing the abso- +lute mean error ( +���true − predicted +���) and the standard de- +viation of the error for training and testing with a dataset +generated using cylindrical geometry is presented in Ta- +ble 3. +Examination of results of the bare geometries, ex- +ample shown in Table 3, indicates that all of the ML +methods with the chosen parameter settings perform an +excellent job at predicting the isotopics. This is to be +Method +235U frac +239Pu frac +Nearest +0.0012 ± 0.0041 +0.0052 ± 0.0044 +Decision +0.0014 ± 0.0016 +0.0021 ± 0.0027 +Random +0.0009 ± 0.0009 +0.0012 ± 0.0016 +GB +0.0023 ± 0.0018 +0.0027 ± 0.0025 +Gaussian* +0.0007 ± 0.0007 +0.0009 ± 0.0007 +FCNN* +0.0007 ± 0.0006 +0.0006 ± 0.0006 +Table 3: Mean error and the standard deviation of error in the isotopic +ratios using cylinder simulations with no shielding materials. Results +marked as ‘*’ were produced using algorithm implemented in Mathe- +matica. +Method +235U frac +239Pu frac +Nearest +0.0013 ± 0.0042 +0.0051 ± 0.0045 +Decision +0.0020 ± 0.0028 +0.0031 ± 0.0030 +Random +0.0011 ± 0.0012 +0.0017 ± 0.0016 +GB +0.0024 ± 0.0022 +0.0033 ± 0.0036 +Gaussian* +0.0010 ± 0.0016 +0.0020 ± 0.0019 +FCNN* +0.0011 ± 0.0009 +0.0012 ± 0.0013 +Table 4: Mean error and the standard deviation of error in the isotopic +ratios using simulations with no shielding materials after the simula- +tion was pre-processed to subtract the continuum. Results marked as +‘*’ were produced using algorithm implemented in Mathematica. +expected since the ratio of the line intensities is solely a +function of the thickness of the radioisotopes. One addi- +tional finding from the analysis of these datasets is that +the ML algorithms are able to adequately treat the con- +tinuum and therefore remove the time consuming con- +tinuum subtraction step. However, to quantify the abil- +ity of the ML algorithms to perform this function we +utilized a continuum subtraction algorithm, as outlined +in Section 6.1. An example result is provided in Table 4. +More information on the hyperparameter examination is +provided in Section 8. +Examination of Table 3 and 4 indicates that the ML +algorithms indeed perform well in removing the contin- +uum. The slight decrease in performance upon separate +continuum subtraction may be attributed to the decrease +in statistics, and the uncertainty in continuum subtrac- +tion procedure. +7.2. Testing the impact of shielding +The previous investigations did not include any +shielding. +It is instructive to examine the ability of +the learning algorithms to learn a much more complex +multi-dimensional function i.e. determine the isotopic +ratios of 235U and 239Pu when different shielding mate- +rials with different thicknesses are present. Indeed, as +8 + +107 +Counts +. Total +106 +Background +105 +104 +103 +102 +0 +500 +1000 +1500 +2000 +Energy (keV)Method +235U frac +239Pu frac +Nearest +0.0636 ± 0.0684 +0.0253 ± 0.0256 +Decision +0.0370 ± 0.0492 +0.0060 ± 0.0078 +Random +0.0248 ± 0.0322 +0.0032 ± 0.0040 +GB +0.0264 ± 0.0317 +0.0054 ± 0.0056 +Gaussian* +0.0290 ± 0.0290 +0.0012 ± 0.0012 +FCNN* +0.0310 ± 0.0290 +0.0083 ± 0.0084 +Table 5: Mean error and the standard deviation of error in the isotopic +ratios using simulations with shielding materials. Results marked as +‘*’ were produced using algorithm implemented in Mathematica. +Method +235U frac +239Pu frac +Nearest +0.0503 ± 0.0539 +0.0237 ± 0.0255 +Decision +0.0200 ± 0.0222 +0.0075 ± 0.0082 +Random +0.0134 ± 0.0161 +0.0039 ± 0.0043 +GB +0.0148 ± 0.0141 +0.0064 ± 0.0052 +Gaussian* +0.0148 ± 0.0145 +0.0026 ± 0.0023 +FCNN* +0.0120 ± 0.0110 +0.0035 ± 0.0032 +Table 6: Mean error and the standard deviation of error in the isotopic +ratios using simulations with shielding materials after the simulation +was pre-processed to remove continuum background. Results marked +as ‘*’ were produced using algorithm implemented in Mathematica. +may be observed from examination of Table 5, 6 errors +increase relative to those obtained without shielding. +Examination of Table 5 and 6 also reveals that the +impact of background subtraction has a minimal im- +pact on the errors. All ML methods, with the possi- +ble exception of the nearest neighbor, appear to offer +comparable performance. Finally, we observe that in +the dataset with shielding applied, the plutonium pre- +dictions are significantly better than those for uranium. +Shielding adds extra scattering background to the ob- +served spectra, which makes the ratio of the photo-peak +counts to the scatter background smaller. Most of the +photo-peaks features that are useful for the uranium en- +richment determination are far apart in energy, with dif- +ferent amount of scatter present under the peaks. Fur- +thermore, these photo-peaks are often also in the low +energy region, where photo-peak to continuum back- +ground ratio is already smaller than for photo-peaks in +medium energy range, which are more useful for plu- +tonium isotopic determination. Therefore, accuracy in +the continuum background determination, whether it is +through a separate step applied during pre-processing +or one done automatically by the ML algorithm, im- +pacts the uranium enrichment estimate asymmetrically +as compared to the plutonium isotopics determination. +7.3. Generalization of ML Algorithms +In nuclear safeguards applications, many field param- +eters, such as source geometry and shielding material +properties, are unknown. The previous investigations +reported results for cases in which the training and test- +ing datasets were drawn from the same general popula- +tion e.g. training and testing on cylinders or other com- +mon geometries; or with common shielding materials +and thickness and fixed geometries. A common issue in +ML is the ability of a given ML algorithm, with a given +training set, to generalize e.g. to make predictions using +testing data that may be different in either a known or +unknown manner from the training data. Testing the va- +lidity of the ML algorithm’s performance with data that +differs from the training sample in either source geom- +etry or shielding materials allows for identification and +quantification of possible sources of uncertainty. In the +first investigations, training with one geometry and test- +ing on another was examined. It was observed that the +training with bare spheres and testing on bare cylinders +resulted in significantly worse performance than those +results obtained above, with mean errors on the order of +0.10–0.15 for most algorithms. An additional investiga- +tion in which training with shielded shells and testing +with shielded cylinders revealed even higher degrada- +tion in the performance for all of the ML algorithms ow- +ing to the increase in complexity and difference between +the phase space of the training and testing samples. To +illustrate the second issue, ML algorithms were initially +trained on simulations with cylindrical geometry gen- +erated without shielding materials and tested on simu- +lations with shielding materials. The predictive ability +and generalization ability of the ML algorithms was de- +graded as reflected by mean error values in the range of +0.05–0.10 for Pu and 0.10–0.15 for U. The process was +later repeated with the training and testing populations +swapped. The mean absolute errors obtained were < +0.01 for most ML algorithms. The lower value of mean +absolute error when training on a sample that was pro- +duced with shielding materials ranging in material type +and thickness implies that increasing the heterogeneity +in the training sample to widen the physics phase space +increases the overall generalization ability of the algo- +rithm, as predicted. +To address the degradation in the performance when +a variety of geometries may be present all of the training +data was combined. The results of these investigations +are provided in Tables 7 and 8 with and without contin- +uum subtraction, respectively. +Examination of Table 7 and Table 8 reveals excel- +lent performance of the ML algorithms, without contin- +uum subtraction, in determining the Pu isotopic content. +9 + +Method +235U frac +239Pu frac +Nearest +0.0120 ± 0.0310 +0.0190 ± 0.0270 +Decision +0.0073 ± 0.0140 +0.0087 ± 0.0240 +Random +0.0038 ± 0.0080 +0.0056 ± 0.0160 +GB +0.0170 ± 0.0160 +0.0150 ± 0.0200 +Gaussian* +0.1300 ± 0.1100 +0.0093 ± 0.0095 +FCNN* +0.1500 ± 0.1300 +0.0350 ± 0.0250 +CNN* +0.2300 ± 0.1600 +0.0300 ± 0.0230 +Table 7: Mean error and the standard deviation of error in the isotopic +ratios using all simulations. Results marked as ‘*’ were produced +using algorithm implemented in Mathematica. +Method +235U frac +239Pu frac +Nearest +0.0130 ± 0.0330 +0.0190 ± 0.0290 +Decision +0.0076 ± 0.0190 +0.0085 ± 0.0210 +Random +0.0041 ± 0.0081 +0.0048 ± 0.0150 +GB +0.0180 ± 0.0170 +0.0130 ± 0.0180 +Gaussian* +0.0050 ± 0.0060 +0.0050 ± 0.0090 +FCNN* +0.0270 ± 0.0210 +0.0090 ± 0.0070 +CNN* +0.0540 ± 0.0460 +0.0140 ± 0.0110 +Table 8: Mean error and the standard deviation of error in the iso- +topic ratios using all simulations after subtracting contributions from +continuum. Results marked as ‘*’ were produced using algorithm im- +plemented in Mathematica. +However, for the uranium isotopic content, it was found +that the Gaussian processes and the neural networks did +not perform adequately. Examinations of the isotopic +content predictions using all ML algorithms revealed +excellent performance when a separate continuum sub- +traction was done during pre-processing. +Finally, the isotopics for all of the simulated data for +plutonium and uranium were evaluated using FRAM. +The results were significantly worse 0.074 for pluto- +nium and 0.11 for uranium than those obtained via the +machine learning algorithms. +8. Hyper-parameter Examinations +The parameters that need to be defined prior to train- +ing a ML algorithm are commonly termed as hyperpa- +rameters. There is currently no known method to deter- +mine which hyperparameters have an impact on model +performance before training. +Consequently, for each +of the respective methods a range of hyperparameters +was explored. Furthermore, because the objective of +this work is to train models using simulation data and +test using experimental data, pre-cautions were taken to +avoid over-fitting. A summary of the hyper-parameters +examined, and the parameters utilized for subsequent +Method +Parameters +Range/ +Value/ +Methods +Range +Explored +Selected +Nearest +Neighbors: +1–32000 +1 +Neighbors +Methods: +KDtree/ +Auto +Brute/Auto +Distance: +Uniform/ +Minkowski +Chebyshev/ +(Euclidean) +Euclidean/ +Manhattan/ +Minkowski +Decision +Max Depth: +10–100 +50 +Tree +Splitter: +Best +Best +Loss: +MSE +MSE +Feature +Fraction: +0.05–1 +1 +Random +No. of trees: +10–100 +100 +Forest +Leaf Size: +Unlimited +Unlimited +Max Depth: +10–100/None +50 +Loss: +MSE +MSE +Feature +Fraction: +0.05–1 +1 +Gradient +No. of trees: +10–300 +200 +Boosted +Leaf Number: +5–50/None +None +Trees +Max Depth: +2–25 +4 +Min samples +for split: +2–10 +5 +Loss: +MSE +MSE +Learning +Rate: +0.01–0.4 +0.1 +Feature +Fraction: +0.1–1 +1 +Fully +Layers: +2–10 +2 +Connected +Activation: +SELU/Tanh +Tanh +Neural +No of params: +15250–100000 +10250 +Network +DropOut: +0–0.1 +0.01 +Epochs: +100–1000 +100 +Optimization +MSE +Method: +ADAM/SGD +ADAM +Learning +Rate: +0.001–0.1 +0.01 +Convol- +Layers: +2–10 +2 +utional +No of params: +10250–100000 +35324 +Neural +DropOut: +0–0.1 +0.1937 +Network +Activation: +SELU/ Tanh +SELU +Epochs: +100–1000 +100 +Optimization +Method: +ADAM/SGD/ +Logistic +LogisticSigmoid +Sigmoid +L2: +0–0.1 +0.01 +Learning +Rate: +0.001–0.1 +0.001 +Table 9: Hyperparameters tested and selected for different ML algo- +rithms. +10 + +investigations, for each of the respective methods is pro- +vided in Table 9. +9. Experimental Data and Results +The previous analyses were performed using simu- +lated spectra generated using GADRAS for both train- +ing as well as testing. In this Section, we explore the use +of the simulations for training and experimental data for +testing. The details of the experimental configurations +are outlined in Section 9.1 and the application of ML +algorithms are presented in Section 9.3. However, be- +fore presenting these details, we note that at the time +of experimental data collection, terrestrial background +data is taken with identical settings to the experimen- +tal data. Often the background files are generated with +longer collection times than the experimental data so as +to minimize the effects of statistical fluctuation when +subtracting the terrestrial background counts from the +foreground counts. Prior to subtraction from the fore- +ground counts, the background counts are scaled ac- +cordingly based on the relative count time for the back- +ground file with respect to the count time for the exper- +imental data. +We note that in the GADRAS simulations, the spec- +tra do not include contributions from terrestrial back- +ground; hence, this process is not applicable for simula- +tion. +9.1. Experimental Description +Experimental dataset with multiple source and shield- +ing configurations, source geometries, and source forms +were utilized to enable the testing of the ML algo- +rithms. Configurations included bare and shielded cans +of uranium and plutonium oxide with a wide range of +isotopics, depleted uranium spheres and shells (both +bare and shielded), and plutonium spheres with various +shielding materials and thicknesses. +9.1.1. Uranium and plutonium oxide sources +The uranium (U3O8) and plutonium (PuO2) oxide +dataset analyzed were collected with an ORTEC Detec- +tive X and LANL Detector S respectively. The Detec- +tive X is a handheld, mechanically cooled HPGe detec- +tor with 50% relative efficiency. The Detective X has a +range of 8 MeV with 214 channels. The LANL Detector +S is an ORTEC poptop liquid-nitrogen cooled HPGe de- +tector with a relative efficiency of roughly 140%. This +detector has a range of 12 MeV with 215 channels. The +Detector S response function was incorporated into the +GADRAS simulations as detailed in Section 5. At the +%U234 +%U235 +%U236 +%U238 +0.005–0.910 +0.716–91.340 +0.002–0.335 +7.417–99.277 +Table 10: Range in weight % of reported isotopes relative to total U +for U3O8 cylindrical sources dated 9/6/1988. +%Pu238 +%Pu241 +%Pu242 +%Am241 +0.002–1.177 +0.014–5.693 +0.003–4.239 +0.009–2.510 +Table 11: Range in weight % of miscellaneous isotopes relative to +total Pu for PuO2 cylindrical sources dated 1/1/1990. +time of data collection, Detector S incorporated a bis- +muth side shield to reduce the background radiation +contribution to the measured spectra. Additionally, a +thin front filter made of tin was also present to filter out +low energy photons. +Fourteen data sets, seven without shielding and seven +with shielding material present, were collected for both +the uranium and plutonium oxide sources. Uranium en- +richment and plutonium isotopics were in the range of +0.7–91.3% and 63.2(25.7) – 98.0(2.0)% 239Pu (240Pu) +respectively. Additional reported isotopes are listed in +Tables 10 and 11. The uranium oxide samples were ap- +proximately 1 kg whereas the plutonium samples had +mass between 1.6–5.8 g. +Thin sheets of lead were +used for the shielded measurements, with a thickness +of 3.175 mm for the uranium and 4 mm for the pluto- +nium. A complete set of plutonium oxide measurements +were collected for 300 seconds at a source-to-detector +distance of 50 cm. The uranium oxide measurements +was taken with better counting statistics: 600 seconds at +a source-to-detector distance of 25 cm, with an excep- +tion of an unshielded 91% enriched oxide measurement, +which performed at a source-to-detector distance of 50 +cm to ensure an acceptable dead time in the detector. +9.1.2. Depleted uranium shell data +The depleted uranium (DU) measurements were per- +formed using a LANL Detector S, described above, with +a bismuth collimator and tin front filter. Fifty-four data +sets were taken, six in nine different configurations at +a source to detector distance of 1m. The configurations +utilized various combinations of three stacked DU shells +of 6.35 mm thickness, while keeping the outer diameter +of the DU shells at 15.24 cm. Some configurations were +taken without any shielding, and some utilized shielding +from either one or two stacked aluminum shells. The +aluminum shells were 1.27 cm in thickness. +9.1.3. BeRP ball data +The BeRP (Beryllium Reflected Plutonium) ball [61] +data was collected with Detector K, a 140% relative ef- +11 + +ficiency liquid-nitrogen-cooled HPGe detector that is +similar to Detector S, at the Nevada Nuclear Secu- +rity Site. +The BeRP ball is a sphere of 7.59 cm di- +ameter alpha-phase plutonium clad with a 0.3 mm of +SS304, and weighs 4.48 kg [62]. Although present in +the name of the object, the original beryllium reflector +was not used in these configurations. The dataset col- +lected was taken at a source-to-detector distance of 50 +cm, both unshielded and with shielding (polyethylene) +of thicknesses between 2.54-10.16 cm in conjunction +with other combinations of shielding materials such as +nickel, steel, mock high explosives, and aluminum rang- +ing in total thickness from 1.27-7.62 cm. +9.2. Dimensionality Reduction +The HPGe detectors utilized in these investigations +have 16384 (Detector X) and 32768 (Detector (S) chan- +nels. To reduce the dimensionality of the features for +which the ML algorithms were trained, we selected a +total of 172 features based on the emission lines of the +isotopes under investigation. +This dimensionality re- +duction was performed due to established observation +that when training a ML algorithm in a large multi- +dimensional space, there are often redundant features +that add noise to the dataset, without improving the per- +formance of the algorithm. Further investigations into +improving ML algorithm performance were performed +by applying additional dimensionality reduction using +two approaches: 1) physics based feature reduction, and +2) Principle component analysis (PCA). +For the physics based feature reduction, we se- +lect 9 and 10 prominent γ peaks for U and Pu +data/simulations, respectively. +The selections were +made based on the most commonly used photo-peaks +in γ spectroscopy for plutonium and uranium. Compar- +ison of the results from this method of dimensionality +reduction did not improve the mean absolute error and +the standard deviation in simulation test dataset. Sim- +ilarly, the PCA based method also did not reduced the +absolute error in a systematic way. +As discussed in Section 5.1, since we utilized a simu- +lation model for constructing a training dataset, there +is a potential for biased results due to model depen- +dence. This potential bias was examined by inspect- +ing the spectra generated with a GADRAS model for +a sampling of the experimental dataset with known pa- +rameters. After observing larger disagreements in the +lower and higher energy ranges, the number of features +was reduced to include features only in the 100–1000 +keV range. Although this improved the mean absolute +error, this type of ad hoc dimensionality reduction can- +not be generalized without the knowledge of the source +of data-simulation discrepancy. +9.3. Results +To investigate the performance of ML algorithms us- +ing the experimental data discussed in Section 9.1, five +algorithms: Decision Trees, MLP, Gradient Boosted +Trees, Nearest Neighbors and Random Forests, were +considered. The results were compared with results ob- +tained using FRAM software. For the uranium dataset, +the comparisons were performed using both the ‘HEU’ +and ‘LEU’ models. +The results obtained using the small scale plutonium +oxide sources are presented in Figure 8. The error bars +for results, provided in Figure 8, include combined sta- +tistical and systematic uncertainties for all the methods +except for Decision Trees and Nearest Neighbors (these +methods were found to have very low errors due to the +lack of systematic uncertainties which were found to +be the dominant source of error). The statistical un- +certainties were estimated by varying the photo-peak +counts with a poisson model and repeating the ML al- +gorithm implementation for each instance of the varia- +tion. The systematic uncertainties account for the vari- +ation in the ML results when repeating the training and +testing with identical conditions and parameters and in- +corporating a different random seed for algorithm ini- +tialization. Figure 8 (top) shows that no single ML al- +gorithm outperforms the others for all 14 experiments +considered; however, the MLP and Nearest Neighbor +methods were found to perform better than the conven- +tional method in a few of the experimental cases. Once +the data is pre-processed to remove the continuum, in +general the MLP algorithm performs comparable to the +conventional method, within the uncertainties of both +methods, as shown in Figure 8 (bottom). +The uranium oxide results, as provided in Figure 9 +(top), show consistently smaller absolute deviations for +the MLP as compared to other ML algorithms. The bot- +tom Figure 9 shows that although the results from MLP +method are comparable to FRAM results, the estimated +uncertainties in some cases (experiment numbers: 6, 8, +10, 12 and 13) are much smaller for the ML method than +for the conventional approach. +The results for depleted uranium shell in Figure 10 +(top) show that decision tree based methods do not +perform as well as the Nearest Neighbor and MLP +methods. +Absolute deviation for the Nearest Neigh- +bor method are comparable to the conventional method +for most experiments. The good performance of Near- +est Neighbor method is perhaps due to the inclusion of +a large number of simulations with enrichment close +12 + +Figure 8: Comparison of absolute deviation from true isotopics ratio +for Pu oxide data for various ML algorithms and FRAM (top), and +MLP with and without continuum subtraction and FRAM (bottom). +Combined statistical and systematic uncertainties are reported at 1 σ +for all the algorithms except for Random Forest and Nearest Neigh- +bors. Error bars in FRAM results are the ’sigma’ values returned by +the FRAM software. +Figure 9: Comparison of absolute deviation from true isotopics ratio +for U oxide data for various ML algorithms and FRAM HEU model +(top), MLP vs FRAM LEU model (bottom, red) and FRAM HEU +model (bottom, blue). Combined statistical and systematic uncertain- +ties are reported at 1 σ for all the algorithms except for Random Forest +and Nearest Neighbors. Error bars in FRAM results are the ’sigma’ +values returned by the FRAM software. +13 + +UO2 TA66 +1.2 +DT +MLP +1.0 +GB +NN +0.8 +RF +Absolute Deviation +FRAM +0.6 +0.4 +0.2 +0.0 +0.2 +2 +4 +6 +10 +12 +14 +Experiment#MLPUO2TA66 +1.2 +MLP. No Cont. Subt. +1.0 +FRAM LEU +FRAM +0.8 +AbsoluteDeviation +0.6 +0.4 +0.2 +0.0 +0.2 +2 +4 +6 +8 +10 +12 +14 +Experiment#PuO2TA66 +0.8 +DT +0.7 +MLP +GB +0.6 +NN +RF +Absolute Deviation +0.5 +FH +FRAM +0.4 +0.3 +0.2 +0.1 +0.0 +0.1 +2 +4 +6 +8 +10 +12 +14 +Experiment#MLPPuO2TA66 +0.8 +MLP, No Cont. Subt. +0.7 +MLP,Cont.Subt. +FRAM +0.6 +Absolute Deviation +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +0.1 +2 +4 +6 +8 +10 +12 +14 +Experiment#Figure 10: Comparison of absolute deviation from true enrichment +value for depleted uranium data for various ML algorithms and FRAM +HEU model (top), MLP vs FRAM LEU model (bottom, red) and +FRAM HEU model (bottom, blue). Combined statistical and system- +atic uncertainties are reported at 1 σ for MLP, Decision Trees, and +Gradient Boosted Trees. Error bars in FRAM results are the ’sigma’ +values returned by the FRAM software. +to that of the depleted uranium in shell configurations. +Since this method relies on finding sets of training data +points closest in distance to the query, and taking an av- +erage of the closest solutions, having a well represented +training sample is expected to enhance the performance. +The MLP method, despite having larger uncertainty in +the estimate as compared to the traditional approach, +shows mostly small mean absolute deviation of < 0.05 +for most experiments as shown in Figure 10 (bottom). +The BeRP ball results are presented in Figures 11 and +12. The former figure presents absolute deviation from +the true isotopics ratio for experiments with different +shielding material combinations and thicknesses, while +the latter presents analogous results with a polyethylene +shielding of 2.52 cm. As observed previously, the MLP +results are comparable to the FRAM results. The bot- +tom figures show improved performance for the MLP +Figure 11: Comparison of absolute deviation from true isotopics ratio +for BeRP ball data for various ML algorithms and FRAM (top), and +MLP with and without continuum subtraction and FRAM (bottom). +Combined statistical and systematic uncertainties are reported at 1 σ +for all the algorithms except for Decision Trees and Nearest Neigh- +bors. Error bars in FRAM results are the ’sigma’ values returned by +the FRAM software. +method when continuum subtraction is performed in +line with the previous observation in the plutonium ox- +ide results. +Upon considering all of the analyzed experimental +data, the MLP algorithm performed better than the other +ML algorithms evaluated. The better performance of +MLP as compared to the other ML algorithms may be +attributed to the large interconnections of the fully con- +nected neural network enabling highly non-linear be- +havior to be learned more readily. +Improvements in +the predictions of the MLP was observed for the plu- +tonium data set when continuum subtraction was per- +formed prior to the ML application. +Although, the +amount of improvement varied experiment to experi- +ment, the largest improvement in absolute deviation was +seen for Pu oxide data at a value of roughly 0.3. +14 + +DU3Shell +0.8 +DT +0.7 +MLP +GB +0.6 +NN +H +RF +AbsoluteDeviation +0.5 +FRAM +0.4 +0.3 +0.2 +0.1 +0.0 +0.1 +0 +10 +20 +30 +40 +50 +Experiment #MLPDU3Shell +0.200 +MLP, No Cont. Subt. +0.175 +FRAMLEU +FRAM +0.150 +eviation +0.125 +0.100 +D +Absolute +0.075 +0.050 +0.025 +0.000 +0 +10 +20 +30 +40 +50 +Experiment#BeRP ball +0.8 +H +DT +0.7 +MLP +GB +0.6 +NN +RF +Absolute Deviation +0.5 +FRAM +0.4 +0.3 +0.2 +0.1 +0.0 +0.1 +1 +2 +3 +4 +5 +6 +7 +8 +Experiment#MLPBeRPball +0.200 +MLP,No Cont.Subt. +0.175 +MLP,Cont.Subt. +FRAM +0.150 +AbsoluteDeviation +0.125 +0.100 +0.075 +0.050 +0.025 +0.000 +1 +2 +3 +4 +5 +6 +7 +8 +Experiment#Figure 12: Comparison of absolute deviation from true isotopics ratio +for BeRP ball data for various ML algorithms and FRAM (top), and +MLP with and without continuum subtraction and FRAM (bottom) +as a function of polyethylene (shielding material) thickness in inch. +Combined statistical and systematic uncertainties are reported at 1 σ +for all the algorithms except for Decision Trees and Nearest Neigh- +bors. Error bars in FRAM results are the ’sigma’ values returned by +the FRAM software. +10. Conclusions +Several machine learning (ML) based regression al- +gorithms were investigated to perform quantitative de- +termination of uranium and plutonium isotopics using +γ-ray spectroscopy data collected with HPGe detectors. +The algorithms were trained using GADRAS simula- +tions with different source geometries and thicknesses +as well as shielding material types and thicknesses to +address the needs of the Emergency Response commu- +nity. Performance of the algorithms was examined using +both simulations as well as experimental datasets incor- +porating both uranium and plutonium sources in oxide +and metal forms. +Without time-consuming pre-processing that is of- +ten required using conventional methods, all the inves- +tigated algorithms were found to offer excellent per- +formance when simulation data was utilized. A slight +decrease in performance was observed with increasing +complexity, i.e. +wider ranges in source thicknesses, +shielding conditions, etc. Additional subtraction of the +continuum background in the pre-processing stage had +a minimum impact in the performance, indicating that +ML algorithms were able to adequately learn the fea- +ture relationships in the presence of a large continuum +background. +For the experimental dataset, the results were found +to be consistently better using a fully connected neural +network (or MLP) algorithm as compared to other al- +gorithms that were investigated. Comparison of these +results with results obtained from conventional meth- +ods (FRAM software) showed comparable error in the +isotopic ratio estimate. Finally, our results demonstrate +that with minimum pre-processing, ML algorithms are a +good alternative to conventional methods of isotopic de- +termination. The performance of ML algorithms may be +enhanced by substantially increasing the training data +volume and the physics phase space it covers for im- +proved machine learning interpolation at unknown con- +figurations. +References +[1] R. Walton, T. Reilly, J. Parker, J. Menzel, E. Marshall, L. Fields, +Measurements of UF6 cylinders with portable instruments, Nu- +clear Technology 21 (2) (1974) 133–148. +[2] R. Gunnink, MGAU: a new analysis code for measuring u-235 +enrichments in arbitrary samples, lawrence livermore national +laboratory, Tech. rep., UCRL-LR-114713 (1994). +[3] R. Gunnink, MGA (multi-group analysis): a gamma-ray spec- +trum analysis code for determining plutonium isotopic abun- +dances, Tech. rep., Lawrence Livermore National Lab., CA +(USA) (1990). +15 + +BeRP ball +0.8 +DT +0.7 +MLP +GB +0.6 +NN +RF +Absolute Deviation +0.5 +FRAM +0.4 +0.3 +0.2 +0.1 +0.0 +0.1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Poly thickness (inch)MLPBeRPball +0.5 +MLP. No Cont. Subt. +MLP, Cont. Subt. +FRAM +0.4 +AbsoluteDeviation +0.3 +0.2 +0.1 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Poly thickness (inch)[4] K. Wei, G. Xiong, Z. Xu, Y. Liu, Application analysis of ra- +dial basis function neural network algorithm of genetic algo- +rithm for environmental restoration and treatment effect evalua- +tion of decommissioned uranium tailings ponds, Computational +Intelligence and Neuroscience 2021. +[5] R. G. Peyvandi, et al., Determination of uranium and thorium +concentrations in thorium ore sample using artificial neural net- +work and comparison with net area peak method, Radiochimica +Acta 106 (8) (2018) 669–674. +[6] D. Connor, P. G. Martin, T. B. Scott, Airborne radiation map- +ping: overview and application of current and future aerial sys- +tems, International journal of remote sensing 37 (24) (2016) +5953–5987. +[7] M. Hosoda, K. Inoue, M. Oka, Y. Omori, K. Iwaoka, S. Tokon- +ami, Environmental radiation monitoring and external dose esti- +mation in aomori prefecture after the Fukushima Daiichi nuclear +power plant accident, Health Physics 51 (1) (2016) 41–50. +[8] R. Korob, G. B. Nu˜no, A simple method for the absolute deter- +mination of uranium enrichment by high-resolution γ spectrom- +etry, Applied radiation and isotopes 64 (5) (2006) 525–531. +[9] A. Apostol, A. Pantelica, O. Sima, V. Fugaru, Isotopic composi- +tion analysis and age dating of uranium samples by high resolu- +tion gamma ray spectrometry, Nuclear Instruments and Methods +in Physics Research Section B: Beam Interactions with Materi- +als and Atoms 383 (2016) 103–108. +[10] M. J. Aitkenhead, M. Owen, D. M. Chambers, Use of artificial +neural networks in measuring characteristics of shielded pluto- +nium for arms control, Journal of Analytical Atomic Spectrom- +etry 27 (3) (2012) 432–439. +[11] M. Rawool-Sullivan, J. Bounds, S. Brumby, L. Prasad, J. Sul- +livan, Steps toward automated gamma ray spectroscopy, Tech. +rep., Los Alamos National Laboratory (2010). +[12] J. Parker, T. Reilly, Plutonium isotopic determination by +gamma-ray spectroscopy, Nuclear analysis research and devel- +opment program status report. +[13] T. Reilly, R. Walton, J. Parker, progress report la-4605-ms, Los +Alamos National Laboratory, NNM (197) 19. +[14] D. Reilly, N. Ensslin, H. Smith Jr, S. Kreiner, Passive nonde- +structive assay of nuclear materials, Tech. Rep. LA-UR-90-732, +Office of Nuclear Regulatory Commission, Washington, DC, +USA (1991). +[15] T. E. Sampson, G. W. Nelson, T. A. Kelley, FRAM: A versatile +code for analyzing the isotopic composition of plutonium from +gamma-ray pulse height spectra, Tech. rep., Los Alamos Na- +tional Lab.(LANL), Los Alamos, NM (United States) (1989). +[16] P. Mortreau, R. Berndt, Determination of the uranium enrich- +ment with the NaIGEM code, Nuclear Instruments and Meth- +ods in Physics Research Section A: Accelerators, Spectrome- +ters, Detectors and Associated Equipment 530 (3) (2004) 559– +567. +[17] J. Sprinkle Jr, A. Christiansen, R. Cole, M. Collins, S.-T. Hsue, +P. Knepper, T. McKown, R. Siebelist, Low-resolution gamma- +ray measurements of uranium enrichment, Applied radiation +and isotopes 48 (10-12) (1997) 1525–1528. +[18] S. Abousahl, A. Michiels, M. Bickel, R. Gunnink, J. Ver- +plancke, Applicability and limits of the MGAU code for the +determination of the enrichment of uranium samples, Nuclear +Instruments and Methods in Physics Research Section A: Ac- +celerators, Spectrometers, Detectors and Associated Equipment +368 (2) (1996) 443–448. +[19] R. Gunnink, W. Ruhter, P. Miller, J. Goerten, M. Swinhoe, +H. Wagner, J. Verplancke, M. Bickel, S. Abousahl, MGAU: +A new analysis code for measuring 235U enrichments in arbi- +trary samples, Tech. rep., Lawrence Livermore National Lab., +CA (United States) (1994). +[20] J. Morel, H. Goenvec, J. Dalmazzone, G. Malet, R´ef´erences +pour la d´etermination de l’uranium 235 dans les combustibles +nucl´eaires, IAEA Nuclear Safeguard Technology. +[21] J. Morel, C. Hill, M. Bickel, A. Alonso-Munoz, S. Napier, +B. Thaurel, E. S. Research, D. A. N.-W. Members, Results from +the international evaluation exercise for uranium enrichment +measurements, Applied Radiation and Isotopes 52 (3) (2000) +509–522. +[22] T. E. Sampson, T. A. Kelley, D. T. Vo, Application guide to +gamma-ray isotopic analysis using the FRAM software, LA- +14018. +[23] M. Darweesh, S. Shawky, Study on the performance of differ- +ent uranium isotopic codes used in nuclear safeguards activities, +Heliyon 5 (4) (2019) e01470. +[24] P. Olmos, J. Diaz, J. Perez, G. Garcia-Belmonte, P. Gomez, +V. Rodellar, Application of neural network techniques in gamma +spectroscopy, Nuclear Instruments and Methods in Physics Re- +search Section A: Accelerators, Spectrometers, Detectors and +Associated Equipment 312 (1-2) (1992) 167–173. +[25] E. Yoshida, K. Shizuma, S. Endo, T. Oka, Application of neural +networks for the analysis of gamma-ray spectra measured with +a Ge spectrometer, Nuclear Instruments and Methods in Physics +Research Section A: Accelerators, Spectrometers, Detectors and +Associated Equipment 484 (1-3) (2002) 557–563. +[26] L. J. Kangas, P. E. Keller, E. R. Siciliano, R. T. Kouzes, J. H. +Ely, The use of artificial neural networks in PVT-based radiation +portal monitors, Nuclear Instruments and Methods in Physics +Research Section A: Accelerators, Spectrometers, Detectors and +Associated Equipment 587 (2-3) (2008) 398–412. +[27] J. F. Briesmeister, MCNP-a general Monte Carlo n-particle +transport code, Tech. Rep. LA–13709–M, Los Alamos National +Laboratory (1993). +[28] D. Liang, P. Gong, X. Tang, P. Wang, L. Gao, Z. Wang, +R. Zhang, Rapid nuclide identification algorithm based on con- +volutional neural network, Annals of Nuclear Energy 133 (2019) +483–490. +[29] C. Bobin, O. Bichler, V. Lourenc¸o, C. Thiam, M. Th´evenin, +Real-time radionuclide identification in γ-emitter mixtures +based on spiking neural network, Applied Radiation and Iso- +topes 109 (2016) 405–409. +[30] S. Sharma, C. Bellinger, N. Japkowicz, R. Berg, K. Ungar, +Anomaly detection in gamma ray spectra: A machine learning +perspective, in: 2012 IEEE Symposium on Computational Intel- +ligence for Security and Defence Applications, IEEE, 2012, pp. +1–8. +[31] J. He, X. Tang, P. Gong, P. Wang, L. Wen, X. Huang, Z. Han, +W. Yan, L. Gao, Rapid radionuclide identification algorithm +based on the discrete cosine transform and bp neural network, +Annals of Nuclear Energy 112 (2018) 1–8. +[32] E. J. Hague, M. Kamuda, W. P. Ford, E. T. Moore, J. Turk, A +comparison of adaptive and template matching techniques for +radio-isotope identification, in: Algorithms, Technologies, and +Applications for Multispectral and Hyperspectral Imagery XXV, +Vol. 10986, SPIE, 2019, pp. 62–73. +[33] M. M. Kamuda, Automated isotope identification and quantifi- +cation using artificial neural networks, Ph.D. thesis, University +of Illinois at Urbana-Champaign (2019). +[34] C. Zhang, G. Hu, F. Luo, Y. Xiang, G. Ding, C. Chu, J. Zeng, +R. Ze, Q. Xiang, Identification of SNM based on low-resolution +gamma-ray characteristics and neural network, Nuclear Instru- +ments and Methods in Physics Research Section A: Accelera- +tors, Spectrometers, Detectors and Associated Equipment 927 +(2019) 155–160. +[35] K. J. Bilton, T. H. Joshi, M. S. Bandstra, J. C. Curtis, D. Hellfeld, +K. Vetter, Neural network approaches for mobile spectroscopic +16 + +gamma-ray source detection, Journal of Nuclear Engineering +2 (2) (2021) 190–206. +[36] M. Gomez-Fernandez, K. Higley, A. Tokuhiro, K. Welter, W.-K. +Wong, H. Yang, Status of research and development of learning- +based approaches in nuclear science and engineering: A review, +Nuclear Engineering and Design 359 (2020) 110479. +[37] H. Hata, K. Yokoyama, Y. Ishimori, Y. Ohara, Y. Tanaka, +N. Sugitsue, Application of support vector machine to rapid +classification of uranium waste drums using low-resolution γ- +ray spectra, Applied Radiation and Isotopes 104 (2015) 143– +146. +[38] L. Chen, Y.-X. Wei, Nuclide identification algorithm based on +K–L transform and neural networks, Nuclear Instruments and +Methods in Physics Research Section A: Accelerators, Spec- +trometers, Detectors and Associated Equipment 598 (2) (2009) +450–453. +[39] V. Pilato, F. Tola, J. Martinez, M. Huver, Application of neu- +ral networks to quantitative spectrometry analysis, Nuclear In- +struments and Methods in Physics Research Section A: Ac- +celerators, Spectrometers, Detectors and Associated Equipment +422 (1-3) (1999) 423–427. +[40] S. E. Shaban, M. Hazzaa, R. El-Tayebany, Applying Monte +Carlo and artificial intelligence techniques for 235u mass pre- +diction in samples with different enrichments, Nuclear Instru- +ments and Methods in Physics Research Section A: Accelera- +tors, Spectrometers, Detectors and Associated Equipment 916 +(2019) 322–326. +[41] V. Vigneron, J. Morel, M.-C. Lepy, J.-M. Martinez, Statisti- +cal modelling of neural networks in γ-spectrometry, Nuclear +Instruments and Methods in Physics Research Section A: Ac- +celerators, Spectrometers, Detectors and Associated Equipment +369 (2-3) (1996) 642–647. +[42] J. Ryu, C. Park, J. Park, N. Cho, J. Park, G. Cho, Develop- +ment of neural network model with explainable AI for measur- +ing uranium enrichment, IEEE Transactions on Nuclear Science +68 (11) (2021) 2670–2681. +[43] E. K. Elmaghraby, M. Tohamy, M. Comsan, Determination +of isotopes activity ratio using gamma ray spectroscopy based +on neural network model, Applied Radiation and Isotopes 148 +(2019) 19–26. +[44] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, +O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, +J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Per- +rot, E. Duchesnay, Scikit-learn: Machine learning in Python, +Journal of Machine Learning Research 12 (2011) 2825–2830. +[45] S. Wolfram, The Mathematica book, Vol. 1, Wolfram Research, +Inc., 2003. +[46] L. Breiman, J. H. Friedman, C. J. Stone, R. A. Olshen, Classifi- +cation and Regression Trees, Taylor & Francis, 1984. +[47] G. Biau, E. Scornet, A random forest guided tour, TEST 25 (2) +(2016) 197–227. doi:10.1007/s11749-016-0481-7. +[48] A. Natekin, A. Knoll, Gradient boosting machines, a tutorial, +Frontiers in neurorobotics 7 (2013) 21. +[49] E. Fix, J. L. Hodges, Discriminatory analysis. nonparametric +discrimination: Consistency properties, International Statistical +Review/Revue Internationale de Statistique 57 (3) (1989) 238– +247. +[50] N. S. Altman, An introduction to kernel and nearest-neighbor +nonparametric regression, The American Statistician 46 (3) +(1992) 175–185. +[51] S. J. Gershman, D. M. Blei, A tutorial on Bayesian nonparamet- +ric models, Journal of Mathematical Psychology 56 (1) (2012) +1–12. +[52] C. K. Williams, D. Barber, Bayesian classification with Gaus- +sian processes, IEEE Transactions on Pattern Analysis and Ma- +chine Intelligence 20 (12) (1998) 1342–1351. +[53] C. Williams, C. Rasmussen, Gaussian processes for regression, +Advances in Neural Information Processing Systems 8. +[54] K. Fukushima, Neocognitron: A self-organizing neural network +model for a mechanism of pattern recognition unaffected by +shift in position, Biological Cybernetics 36 (1980) 193–202. +doi:10.1007/BF00344251. +[55] Y. LeCun, Y. Bengio, G. Hinton, Deep learning, Nature +521 (7553) (2015) 436–444. doi:10.1038/nature14539. +[56] J. Schmidhuber, +Deep learning in neural networks: +An +overview, Neural Networks 61 (2015) 85–117. doi:10.1016/ +j.neunet.2014.09.003. +[57] D. J. Mitchell, L. T. Harding, GADRAS isotope id users manual +for analysis of gamma-ray measurements and API for Linux and +Android, Tech. rep., Sandia National Laboratory (5 2014). doi: +10.2172/1177049. +[58] M. Morh´aˇc, J. Kliman, V. Matouˇsek, M. Veselsk´y, I. Turzo, +Background elimination methods for multidimensional coinci- +dence gamma-ray spectra, Nuclear Instruments and Methods +in Physics Research Section A: Accelerators, Spectrometers, +Detectors and Associated Equipment 401 (1) (1997) 113–132. +doi:10.1016/S0168-9002(97)01023-1. +[59] M. Morh´aˇc, J. Kliman, V. Matouˇsek, M. Veselsk´y, I. Turzo, +Efficient one- and two-dimensional gold deconvolution and its +application to gamma-ray spectra decomposition, Nuclear In- +struments and Methods in Physics Research Section A: Ac- +celerators, Spectrometers, Detectors and Associated Equipment +401 (2) (1997) 385–408. +doi:10.1016/S0168-9002(97) +01058-9. +[60] M. Morh´aˇc, J. Kliman, V. Matouˇsek, M. Veselsk´y, I. Turzo, +Identification of peaks in multidimensional coincidence γ-ray +spectra, Nuclear Instruments and Methods in Physics Research +A 443 (1) (2000) 108–125. doi:10.1016/S0168-9002(99) +01005-0. +[61] J. Hutchinson, T. Valentine, Subcritical measurements of a plu- +tonium sphere reflected by polyethylene and acrylic, Nuclear +Science and Engineering 161 (3) (2009) 357–362. +doi:10. +13182/NSE161-357. +[62] Nuclear Energy Agency, ICSBEP handbook 2019, interna- +tional Criticality Safety Benchmark Evaluation Project Hand- +book (database) (2019). doi:10.1787/e2703cd5-en. +17 + diff --git a/udAzT4oBgHgl3EQfdPwF/content/tmp_files/load_file.txt b/udAzT4oBgHgl3EQfdPwF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..68cec17e77fc2e052ca067ba40c624afb4d9b9d4 --- /dev/null +++ b/udAzT4oBgHgl3EQfdPwF/content/tmp_files/load_file.txt @@ -0,0 +1,1234 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf,len=1233 +page_content='Machine Learning technique for isotopic determination of radioisotopes using HPGe γ-ray spectra Ajeeta Khatiwada, Marc Klasky, Marcie Lombardi, Jason Matheny, Arvind Mohan Los Alamos National Laboratory, Los Alamos, NM 87545, USA Abstract γ-ray spectroscopy is a quantitative, non-destructive technique that may be utilized for the identification and quan- titative isotopic estimation of radionuclides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Traditional methods of isotopic determination have various challenges that contribute to statistical and systematic uncertainties in the estimated isotopics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Furthermore, these methods typi- cally require numerous pre-processing steps, and have only been rigorously tested in laboratory settings with limited shielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In this work, we examine the application of a number of machine learning based regression algorithms as alternatives to conventional approaches for analyzing γ-ray spectroscopy data in the Emergency Response arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' This approach not only eliminates many steps in the analysis procedure, and therefore offers potential to reduce this source of systematic uncertainty, but is also shown to offer comparable performance to conventional approaches in the Emergency Response Application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Keywords: Radionuclides, γ-ray spectroscopy, Isotopic determination, enrichment determination, Machine Learning, Nuclear safeguards, Nuclear Threat Detection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Introduction The identification and quantitative determination of the isotopic content of samples/objects potentially con- taining uranium and/or plutonium is of paramount im- portance to the nuclear materials safeguards, arms con- trol verification, nuclear security, Emergency Response (ER), as well in nuclear remediation arenas [1, 2, 3, 4, 5, 6, 7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Conventional methods for determin- ing the isotopics/enrichment using γ-ray spectroscopy require many time consuming steps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' photo-peak identification, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' background and continuum subtraction, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' feature extraction, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' estimation of the relative efficiency curve, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' matching of the extracted features with those of known nuclides to estimate the fraction of iso- topes [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In many of these application areas, it is imperative to rapidly determine the isotopic fractions using remote detection techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' These constraints necessitate the ∗Corresponding Author Email address: ajeeta@lanl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='gov (Ajeeta Khatiwada) use of non-destructive assay methods (NDA) and ac- companying automated algorithms to perform quanti- tative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In some applications, details regard- ing the physical arrangement of the nuclear materials cannot be revealed due to security concerns, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' in treaty verification activities, or are unknown e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' in nu- clear security and ER activities in which the shielding and other aspects of the physical configuration are un- known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In this work, we examine the ability of numer- ous machine learning (ML) techniques to address the automated identification and quantification of uranium and plutonium isotopics for ER applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Organization of Paper In this work, we investigate the application of a vari- ety of machine learning algorithms to perform uranium and plutonium isotopic estimation for Emergency Re- sponse applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Before discussing the ML algo- rithms utilized in these investigations, we present a re- view of both the traditional as well as the ML methods to perform quantitative isotopic identification in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The machine algorithms utilized in this investi- gation are presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In Section 5, the gen- eration of ML training data is discussed along with an Preprint submitted to Elsevier January 5, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='01415v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='data-an] 4 Jan 2023 investigation of the accuracy of these simulations to em- ulate experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Details of the pre-processing of the spectral data including background, continuum sub- traction, and feature extraction are given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' ML results using simulations are presented in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hyper-parameter investigations are presented in Sec- tion 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Investigations using experimental data and dis- cussions of the results are presented in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Lastly, summary and conclusions are provided in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Background 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Traditional Methods Starting in the early 1970s, researchers developed several approaches to perform quantitative NDA spec- troscopic analysis for both uranium and plutonium iso- topics [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Today there are three general variations of the NDA method that have been utilized to infer the isotopic content of 235U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The first method, currently utilized by the International Atomic Energy Agency (IAEA), is based on the measurement of the 186 keV line of 235U in the spectra obtained using either germa- nium or sodium iodide spectrometer systems [13], and requires a calibration with a known enrichment stan- dard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Provided that the sample measured is similar to the reference i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' has the same geometry and thickness and the measurement conditions are constant, the count- ing rate for the 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='7 keV peak is proportional to the enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' While this approach has been utilized to successfully infer the content of 235U, there are several limitations: the samples must satisfy the infinite thick- ness criterion [14], calibrations need to be performed for samples with different containers, and wall thick- nesses need to be determined prior to the enrichment measurement [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In practice, this constraint limits the applicability of the the enrichment of an object’s sur- face to a depth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='26 cm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='74 cm for uranium metal and U3O8 powder, respectively [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' An auto- mated version of this method, called NaIGEM (NaI(Tl) Gamma Enrichment Measurements), is included in the HM-5 instrument used by the IAEA [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Enrichment measurements of uranium without contaminants using low-resolution detectors can achieve 1% precision for arbitrary enrichment while contamination by minor ura- nium isotopes has a biasing effect of 5–10% [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Methods employing multi-peak self calibration were proposed to overcome the drawbacks of the enrich- ment meter principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The first variation, Peak Area (PA), utilizes the spectral lines in the range 89 to 120 keV [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The relative efficiency curves of different ura- nium isotopes or their daughters are estimated from a limited number of peaks in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Sophis- ticated codes such as MGAU (Multi-Group Analysis for uranium) are based on this principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The preci- sion of the estimated efficiency response depends on the the number and intensity of the isotope peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' How- ever, these methods still experience performance issues when measuring uranium through thick walled contain- ers [18, 19, 2, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' To overcome the limitation of the finite thickness of shielding, the relative-efficiency (RE) method was pro- posed [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The RE method computes the uranium enrichment using the relative efficiency obtained from the peaks expressed in the measured spectra using an energy range from 144 to 1001 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Several software packages, including FRAM and MGA++1, have imple- mented this approach [8, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Both MGA++ and FRAM may be utilized to perform Pu isotopics analysis using the low-energy γ-ray spectrum, along with higher energy γ-rays [3, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' A comparison of three implementations of the RE method concluded that the performance and applica- bility with increasing wall thickness at low enrich- ment grades was in the order PC/FRAM, MGA++, and MGAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Therefore, in shielded conditions, it was rec- ommended that PC/FRAM for γ-rays above 200 keV using the coaxial detector spectrum be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Be- fore concluding, it should be noted that the shielding thicknesses that were evaluated are significantly below those that might be encountered in ER scenarios [23], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=', the shielding thickness may be significantly greater than those analyzed with the traditional approaches for determining uranium enrichment and or plutonium iso- topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Machine Learning Methods The traditional methods utilized to perform NDA of uranium enrichment and plutonium isotopic quan- tification require numerous pre-processing steps, and also have difficulty in treating environments in which unknown shielding, overlapping peaks, and or thick shielding is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' These issues, in conjunction with the success in the development and application of ma- chine learning (ML) techniques in the last decade, have motivated the examination of machine learning tech- niques to address these shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Indeed, the appli- cation of ML techniques to address both classification and regression problems in radiation detection, source 1A suite of three software programs (MGA, U235, and MGAHI, a Pu isotopic analysis code that uses the 200 keV -1 MeV energy region) for the analysis of actinide spectra acquired by Ge detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 2 identification, and quantitative assessment of radionu- clides applications have become increasingly popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' One of the more prominent applications of ML in addressing radioisotopes has been in the detection and identification arena for nuclear safeguards and arms control applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' To that effect, one of the first ap- plications of a neural network to identify radioisotopes was performed by Olmos using a low resolution NaI de- tector [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Additional early work by Yoshida utilized a multilayer perceptron (MLP) network with a HPGe spectra to identify radioisotopes in samples with mixed radioisotopes [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Kangas also developed a neural net- work to analyze very low resolution Polyvinyl toluene (PVT) spectra for use in the identification of radioac- tive materials at international border crossings [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' More recently, Liang has demonstrated that a Convo- lutional Neural Network (CNN) algorithm trained us- ing Monte Carlo N-Particle Transport (MCNP) [27] simulations with a NaI detector could, in a low count rate regime, identify radioisotopes that are nominally difficult to identify, eliminating the necessity to per- form spectra pre-processing such as background sub- traction and spectrum smoothing [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Bobin utilized a Bayesian sequential approach combined with a spiking neural network to enable the real-time processing of sig- nals detected from a mixture of γ-emitting radionuclides in spectroscopic portal systems [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Finally, Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' implemented machine learning techniques to reduce false alarm rates when using γ-ray spectrometers for the identification of persons concealing radioactive materi- als [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Additional investigations have been performed in the application of neural networks for radioisotope identi- fication [31, 32, 33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In general, these investiga- tions utilized either MLP or CNNs with a number of different methods for feature extraction, including the Discrete Cosine Transform (DCT) and the Karhunen- Lo`eve Transform (KLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' A more advanced neural net- work architecture employing an autoencoder with a low resolution NaI detector was shown to improve anomaly detection relative to traditional techniques [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Another application of ML is in the area of identifi- cation of radioisotopics in environmental samples [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hata investigated the feasibility of using a support vector machine (SVM) to classify uranium waste drums as natural uranium or reprocessed uranium using NaI detectors [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Wei applied a radial basis neural net- work algorithm for environmental and treatment evalu- ation of decommissioned uranium tailing ponds [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Fi- nally, Chen used a KLT and an artificial neural network in conjunction with NaI [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Additional application areas of ML have been investi- gated including the analysis of complex spectra (fission and activation products).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In these applications, it was shown that the application of feed forward neural net- works in conjunction with the Singular Value Decom- position (SVD) can significantly improve performance and reduce the required analysis time once the neural networks have been trained [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In some applications, the objective is to determine the isotopic content of the radioisotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In particular, in the ER application the objective is to determine the ura- nium enrichment and or plutonium isotopics in objects containing nuclear material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In this scenario, HPGe de- tectors are typically utilized, and the geometry of the object containing the nuclear material along with the characteristics of the intervening shielding materials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' material composition and thicknesses of the compo- nents containing the nuclear materials, are not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Although many investigations have been performed us- ing ML algorithms to determine isotopic content, al- most all of these have been conducted in applications related to Nuclear Material Safeguards and other appli- cation areas in which either the configuration is known and or the shielding materials are both known and/or relatively thin i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' less than 1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Notably, Shaban uti- lized a feed forward neural network to predict uranium enrichment in laboratory size samples [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Early work by Vigneron demonstrated that HPGe spectroscopic measurements in conjunction with Prin- ciple Component Analysis (PCA), to reduce the dimen- sionality of the spectra, could be successfully utilized to determine the enrichment of laboratory samples using the low energy range 83 to 103 KeV using a MLP [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ryu investigated the use of a neural network model using low resolution NaI spectra to analyze uranium en- richment, from depleted to low enrichment, from very low radioactivity samples present in small beakers with very short count times [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Elmaghraby also utilized a neural network architecture to determine the uranium isotopics using a HPGe detector on laboratory sam- ples [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Lastly, Aitkenhead using simulated data, evaluated the spectra of shielded plutonium using ANNs to detect the presence or absence of plutonium, estimate 239Pu content, as well as distinguish material age of shielded plutonium [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Machine Learning algorithms While a great deal of work has been performed to investigate the use of machine learning in the areas of radioisotope detection and identification as well in the quantification of radioisotopes, almost all of this work 3 has been conducted under conditions that are not di- rectly relevant to the ER community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Accordingly, in this work we examine the application of machine learn- ing algorithms (MLP and Convolutional) Neural Net- works, Gaussian Processes, Decision Tees and their variants i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gradient Boosted Decision Trees and Ran- dom forest, as well as Nearest-Neighbors to 1) study if ML based regression algorithms are a reasonable al- ternative to the conventional methods and 2) to iden- tify a general class of ML algorithms that are robust to achieving the aforementioned goal without excessive fine-tuning of the hyper parameters to enable the deter- mination of the isotopic content of uranium and pluto- nium under conditions more consistent with ER appli- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Methods Examinations in this paper are performed based on supervised learning of training datasets using regression algorithms that are integrated into the Scikit-learn [44] package in Python as well as ML algorithms available in Mathematica [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The results from Mathematica are labeled with ‘*’ next to the algorithm names in the ta- bles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Decision Tree Decision trees are one of the most commonly used, practical approaches for supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' They can be used to solve both regression and classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' A decision tree builds regression or classification mod- els in the form of a tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' They break down a dataset into smaller and smaller subsets while at the same time an associated decision tree is incrementally developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The final result is a tree with decision nodes and leaf nodes [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Each tree is composed of nodes, which are chosen by looking for the optimum split of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The split of features is determined utilizing an impurity measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' For regression trees, two common impurity measures are least squares and least absolute deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In the former, the method is similar to min- imizing least squares in a linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The splits are chosen to minimize the residual sum of squares between the observation and the mean in each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In the latter method, a minimization of the mean absolute deviation from the median within a node is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Two popular techniques to improve the robustness of a decision tree are ensemble methods such as Random Forest methods and Boosting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' These methods are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Random Forest Random forests are a popular technique in classical machine learning, due to their predictive ability at a lower computational burden than neural networks [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' At their core, random forests are an “ensemble” learn- ing technique based on decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ensemble learn- ing is the strategy of averaging predictions from mul- tiple individual models or estimators, leading to more robust and accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The random forests can be configured to train a predefined number of de- cision tree estimators for the same training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Each decision tree makes a target prediction based on train- ing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Each tree also has a user-specified depth pa- rameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The depth parameter denotes the number of branches the tree is allowed to create, when fitting to the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Typically, increasing depth can in- crease the predictive capability of the decision tree, as it can learn more intricate features in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' However, increasing depth beyond a certain limit can also cause over-fitting and reduce accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The precise limit is dependent on the data, and is discovered by trial-and- error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The random forest aggregates the model from all these individual trees, to create an ensemble model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gradient Boosted Trees Gradient boosting is another family of ensemble methods fitting a sequence of weak learners (estima- tor that gives a prediction slightly better than a random guess) on modified versions of the dataset [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In the Gradient Boosted Tree algorithm, the convergence of the boosting algorithm is improved by computing the gradient of a differentiable loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In Gradient Boosting the base estimator is the Decision Tree estima- tor and the hyper-parameters in the tuning phase are the number of estimator and the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' K-Nearest Neighbors Regression The central idea behind the K-nearest neighbors (KNN) is based on the nearest neighbors to query a data point, where k is an integer algorithm parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' There- fore, the value of a quantity at a point is a weighted average of the k points closest to it [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The user specifies the distance metric for computing the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' There are multiple choices: Uniform, Euclidean, Man- hattan, Minkowski etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' With Uniform weights, each neighbor is provided the same weight irrespective of its distance from the query point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In the other distance met- rics, the neighbors closer to the query point in that par- ticular space are assigned higher weights than the those further away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Therefore, this metric acts as a weighted average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gaussian Process Regression Gaussian process regression is a non-parametric Bayesian approach towards regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' It can capture a wide variety of relations between inputs and outputs by utilizing a theoretically infinite number of parameters and letting the data determine the level of complexity through the means of Bayesian inference [51, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Multi-layer Perceptron (MLP) Regression MLPs are a type of neural network consisting of mul- tiple layers: an input layer, one or more hidden layers, and an output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Each layer is fully connected to the next one via non-linear activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Train- ing a neural network on a simulation such that it can be generalized to apply to an experimental dataset that differs from the simulation model in many ways is of- ten challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' MLPs are particularly susceptible to over-fitting, although there are regularization methods available to counter the problem of over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Tun- ing of hyper parameters, such as the activation func- tion, number of hidden layers, number of nodes in each hidden layers, amount of regularization, dropout, enabling/disabling early stopping, and choosing learn- ing rates and optimization strategies are necessary to achieve the best possible performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Convolutional Neural Networks (CNN) CNNs are a form of neural network in which the lin- ear layers take the form of a set of convolutions [54, 55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' This greatly reduces the number of trainable weights, thereby decreasing the risk of over-fitting, and also allows for computationally efficient implementa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' These methods are typically only suitable, how- ever, when the input data has the shift-invariance prop- erties implied by the use of convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Training Data Generation The training data for the ML algorithms was gen- erated utilizing GADRAS [57], incorporating a 145% relative efficiency HPGe detector with a bismuth side shield and tin filter using either Pu or U sources in ei- ther metal or oxide forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The sources were contained in one of three geometries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' shells, cylinders, or spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Since the “self-shielding” is dependent on the source geometry, the γ spectra are not identical for two identical sources that differ only in geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' There- fore, an ensemble of training data for each of the re- spective geometries was generated using a variety of Geometry No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' of Enrichment Shielding Decks fraction (235U ) present Shell 1800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='989 No Shell 15839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='003 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 Yes Sphere 1800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='995 No Cylinder 7000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 No Cylinder 20000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 Yes Table 1: Summary of training data simulations used for uranium in various configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Geometry No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' of Isotopics Shielding Decks fraction (239Pu ) present Shell 1800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='995 No Shell 7920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='230 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 Yes Sphere 1800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='995 No Cylinder 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='560 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 No Cylinder 20000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='560 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 Yes Table 2: Summary of training data simulations used for plutonium in various configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 235U enrichment/239Pu isotopic fractions, source thick- ness, and shielding materials with accompanying thick- nesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Characteristics of the training data are summa- rized in Tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' For the dataset created with a shell configuration, the thickness of the source shells was between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='02 and 4 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The interior of the shell had a void of radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='6 cm for Pu sources with source thickness greater than 2 cm, and for all other cases the outer surface of the source was 6 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Spectra generated for sources with spherical geometry had radii ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='02 to 4 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Cylin- drical plutonium sources were generated with heights ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='35 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='57 cm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='142 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='59 cm with corresponding radii ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='4 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 cm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='353 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='931 cm in bare and shielded configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Cylindrical uranium sources were gener- ated with heights ranging from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='9 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='7 cm and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='45 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='76 cm with corresponding radii ranging from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='35 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='44 cm and with 5 to 6 cm radii in bare and shielded configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' For the Shell and Sphere configuration simulations with the shielding material present, iron (Fe), Tanta- lum (Ta), Polypropylene, or some combination of the aforementioned materials was utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The thickness of shielding materials ranged from 1–10, 1–6, and 1–64 cm respectively for the aforementioned materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The Cylindrical geometry dataset utilized various combina- tion of aluminum (Al), Tantalum (Ta), Iron (Fe), Lead (Pb), and Polypropylene for shielding, while the shield- ing thickness ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='05 – 2 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 5 Figure 1: Comparison of 8x binned simulated spectra and background subtracted spectra for a U source, UISO17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Comparison of experimental data with GADRAS simulations In machine learning, the ability of the simulations to replicate the experimental data is a fundamental issue that must be addressed when simulations are utilized for training and the testing is performed using experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' To this end, we performed experiments with a HPGe detector using both uranium as well as plu- tonium sources with and without accompanying shield- ing and compared these with GADRAS simulated spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The simulations were generated using source data sheets for the primary isotopics, geometry, reported age of the material, and dimensions/configurations to model the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Furthermore, for the GADRAS simulation of U3O8 and PuO2 sources, the mass frac- tions of uranium and plutonium were adjusted to ac- count for the oxide forms utilized in the experimen- tal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The isotopic fractions of U/Pu isotopes other than 235U, 238U, 239Pu, and 240Pu were chosen based on the certification sheets for the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' For the de- pleted uranium shell simulations, generic values of en- richment and miscellaneous isotopic fractions were uti- lized, while the void/shell thickness were matched to those in the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Additionally, for the U3O8 simulations, 40K and 232Th contents were adjusted to match the background data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The 232U content in the simulation was also adjusted based on the height of the 2614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='5 keV photopeak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Some fine tuning in the normal- ization was performed to match the container material and thickness, where appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The simulation mod- els were run with Poission statistics, and compared to data with terrestrial background contribution subtracted from the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Comparisons of the simulated spec- tra with the experimental spectra are presented in Fig- ures 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Figure 2: Comparison of 2x binned simulated spectra vs background subtracted spectra for a U source, UISO17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Figure 3: Comparison of 8x binned simulated spectra and background subtracted spectra for a U source, A1127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Figure 4: Comparison of 2x binned simulated spectra vs background subtracted spectra for a U source, A1127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 6 GADRAS $106 uno Data 0105 104 103 102 10 500 1000 1500 2000 2500 Energy(keV)GADRAS $105 Counts Data 104 103 102 50 100 150 200 250 300 350 400 450 Energy (keV)$106 GADRAS uno Data 0105 104 103 102 10 500 1000 1500 2000 2500 Energy (keV)GADRAS $105 Counts Data 104 103 102 山 50 100 150 200 250 300 350 400 450 Energy (keV)Figure 5: Comparison of 8x binned simulated spectra vs background subtracted spectra for a Pu source, CBNMPu84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Figure 6: Comparison of 8x binned simulated spectra vs background subtracted spectra for a Du-Shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Pre-processing and feature extraction The experimental analog data pulses obtained from the HPGe detector system, after being converted into digital pulses, are recorded in the units of count per dis- crete channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' γ spectra obtained from the GADRAS simulation are also obtained in the units of counts per channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In either case the counts may be, optionally, pre-processed to remove the continuum background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Detailed discussion on continuum subtraction is pro- vided in subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Additionally, for the exper- imental data, where terrestrial background is present, contributions from such background sources are esti- mated and subtracted from the foreground counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The net-counts, after optional continuum subtraction and terrestrial background subtraction, are then integrated in a region of interest around photo-peaks of interest to estimate counts associated with each of the photo- peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The regions of interests are chosen based on the expected photo-peaks for the two isotopes of U and Pu examined in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The mean value of the en- ergy associated with these photo-peaks and their associ- ated net-counts constitute the features for ML training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The impact of the number of features, and the means of reducing the dimensionality of the features during the supervised ML training is investigated in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' For training samples, the features are accompanied by answer “keys”, which are the relative fraction of 235U (239Pu) with respect to the total fraction of 235U and 238U (239Pu and 240Pu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Here onward, for simplicity, these quantities will be together referred as isotopic ratios or as 235U frac and 239Pu frac individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Continuum Subtraction Subtraction of the continuum background produced from scattering were examined to understand the im- pact on the isotopic determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' As such, a Sen- sitive Nonlinear Iterative Peak (SNIP) clipping algo- rithm implemented in TSpectrum class of ROOT frame- work [58, 59, 60] was utilized for one-dimensional background estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The number of iterations was examined in estimating the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The optimal number of iterations was chosen to be 20 based on the ability to remove adequate amount of continuum with- out resulting in negative counts in the subtracted spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' An example spectra with the continuum background estimate with this method is provided in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Investigations using simulated data The determination of the isotopic content of ura- nium or plutonium is a complex function of numerous 7 $106 GADRAS uno Data 0105 104 103 102 10 500 1000 1500 2000 2500 Energy(keV)$106 GADRAS uno Data 0105 104 103 102 10 500 1000 1500 2000 2500 Energy(keV)Figure 7: An example of a continuum estimate performed on a γ spec- tra for a Pu source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' factors including: the source geometry, source thick- ness, shielding material composition, shielding thick- ness, possible inherent impurities e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 232U along with the isotopic ratios of the isotopes in question i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 235U or 239Pu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Furthermore, the accuracy in determining the isotopic fraction is determined by the ability to ad- equately sample these variables in the training set as well as the representativeness of the training data to the testing data, the quantity of training data from which to learn, and the ability to adequately train the given ML algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Since the final goal of this study is to apply the ML algorithm to experimental data that may differ signif- icantly from the training sample in multiple different ways, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' amount of shielding present, source geom- etry, background spectra, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=', the algorithm needs to be robust against over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Bare: No Shielding As an initial test, two simulated datasets (Spheres and Cylinders) were utilized to examined the ability of dif- ferent ML algorithms to predict the isotopic ratios for both Pu and U with no shielding materials present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' This test represents the most simplistic mapping from the spectra to isotopic ratios that can be learnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' That is, no alteration of the line intensities due to the shielding needs to be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' A sample result showing the abso- lute mean error ( ���true − predicted ���) and the standard de- viation of the error for training and testing with a dataset generated using cylindrical geometry is presented in Ta- ble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Examination of results of the bare geometries, ex- ample shown in Table 3, indicates that all of the ML methods with the chosen parameter settings perform an excellent job at predicting the isotopics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' This is to be Method 235U frac 239Pu frac Nearest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0012 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0052 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0044 Decision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0021 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0027 Random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0009 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0012 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0016 GB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0023 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0027 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0025 Gaussian* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0007 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0009 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0007 FCNN* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0007 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0006 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0006 Table 3: Mean error and the standard deviation of error in the isotopic ratios using cylinder simulations with no shielding materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Results marked as ‘*’ were produced using algorithm implemented in Mathe- matica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Method 235U frac 239Pu frac Nearest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0013 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0051 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0045 Decision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0020 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0031 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0030 Random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0011 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0017 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0016 GB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0024 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0033 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0036 Gaussian* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0010 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0020 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0019 FCNN* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0011 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0012 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0013 Table 4: Mean error and the standard deviation of error in the isotopic ratios using simulations with no shielding materials after the simula- tion was pre-processed to subtract the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Results marked as ‘*’ were produced using algorithm implemented in Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' expected since the ratio of the line intensities is solely a function of the thickness of the radioisotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' One addi- tional finding from the analysis of these datasets is that the ML algorithms are able to adequately treat the con- tinuum and therefore remove the time consuming con- tinuum subtraction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' However, to quantify the abil- ity of the ML algorithms to perform this function we utilized a continuum subtraction algorithm, as outlined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' An example result is provided in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' More information on the hyperparameter examination is provided in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Examination of Table 3 and 4 indicates that the ML algorithms indeed perform well in removing the contin- uum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The slight decrease in performance upon separate continuum subtraction may be attributed to the decrease in statistics, and the uncertainty in continuum subtrac- tion procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Testing the impact of shielding The previous investigations did not include any shielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' It is instructive to examine the ability of the learning algorithms to learn a much more complex multi-dimensional function i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' determine the isotopic ratios of 235U and 239Pu when different shielding mate- rials with different thicknesses are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Indeed, as 8 107 Counts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Total 106 Background 105 104 103 102 0 500 1000 1500 2000 Energy (keV)Method 235U frac 239Pu frac Nearest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0636 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0253 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0256 Decision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0370 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0492 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0060 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0078 Random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0248 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0032 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0040 GB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0264 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0054 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0056 Gaussian* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0290 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0290 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0012 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0012 FCNN* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0310 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0290 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0083 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0084 Table 5: Mean error and the standard deviation of error in the isotopic ratios using simulations with shielding materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Results marked as ‘*’ were produced using algorithm implemented in Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Method 235U frac 239Pu frac Nearest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0503 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0539 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0237 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0255 Decision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0200 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0075 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0082 Random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0134 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0039 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0043 GB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0148 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0064 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0052 Gaussian* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0148 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0026 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0023 FCNN* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0120 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0035 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0032 Table 6: Mean error and the standard deviation of error in the isotopic ratios using simulations with shielding materials after the simulation was pre-processed to remove continuum background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Results marked as ‘*’ were produced using algorithm implemented in Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' may be observed from examination of Table 5, 6 errors increase relative to those obtained without shielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Examination of Table 5 and 6 also reveals that the impact of background subtraction has a minimal im- pact on the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' All ML methods, with the possi- ble exception of the nearest neighbor, appear to offer comparable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Finally, we observe that in the dataset with shielding applied, the plutonium pre- dictions are significantly better than those for uranium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Shielding adds extra scattering background to the ob- served spectra, which makes the ratio of the photo-peak counts to the scatter background smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Most of the photo-peaks features that are useful for the uranium en- richment determination are far apart in energy, with dif- ferent amount of scatter present under the peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Fur- thermore, these photo-peaks are often also in the low energy region, where photo-peak to continuum back- ground ratio is already smaller than for photo-peaks in medium energy range, which are more useful for plu- tonium isotopic determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Therefore, accuracy in the continuum background determination, whether it is through a separate step applied during pre-processing or one done automatically by the ML algorithm, im- pacts the uranium enrichment estimate asymmetrically as compared to the plutonium isotopics determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Generalization of ML Algorithms In nuclear safeguards applications, many field param- eters, such as source geometry and shielding material properties, are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The previous investigations reported results for cases in which the training and test- ing datasets were drawn from the same general popula- tion e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' training and testing on cylinders or other com- mon geometries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' or with common shielding materials and thickness and fixed geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' A common issue in ML is the ability of a given ML algorithm, with a given training set, to generalize e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' to make predictions using testing data that may be different in either a known or unknown manner from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Testing the va- lidity of the ML algorithm’s performance with data that differs from the training sample in either source geom- etry or shielding materials allows for identification and quantification of possible sources of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In the first investigations, training with one geometry and test- ing on another was examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' It was observed that the training with bare spheres and testing on bare cylinders resulted in significantly worse performance than those results obtained above, with mean errors on the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='10–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='15 for most algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' An additional investiga- tion in which training with shielded shells and testing with shielded cylinders revealed even higher degrada- tion in the performance for all of the ML algorithms ow- ing to the increase in complexity and difference between the phase space of the training and testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' To illustrate the second issue, ML algorithms were initially trained on simulations with cylindrical geometry gen- erated without shielding materials and tested on simu- lations with shielding materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The predictive ability and generalization ability of the ML algorithms was de- graded as reflected by mean error values in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='05–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='10 for Pu and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='10–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='15 for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The process was later repeated with the training and testing populations swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The mean absolute errors obtained were < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='01 for most ML algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The lower value of mean absolute error when training on a sample that was pro- duced with shielding materials ranging in material type and thickness implies that increasing the heterogeneity in the training sample to widen the physics phase space increases the overall generalization ability of the algo- rithm, as predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' To address the degradation in the performance when a variety of geometries may be present all of the training data was combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The results of these investigations are provided in Tables 7 and 8 with and without contin- uum subtraction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Examination of Table 7 and Table 8 reveals excel- lent performance of the ML algorithms, without contin- uum subtraction, in determining the Pu isotopic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 9 Method 235U frac 239Pu frac Nearest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0120 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0190 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0270 Decision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0073 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0087 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0240 Random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0038 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0056 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0160 GB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0170 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0150 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0200 Gaussian* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1300 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0093 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0095 FCNN* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1500 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0350 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0250 CNN* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2300 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0300 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0230 Table 7: Mean error and the standard deviation of error in the isotopic ratios using all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Results marked as ‘*’ were produced using algorithm implemented in Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Method 235U frac 239Pu frac Nearest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0130 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0190 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0290 Decision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0076 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0085 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0210 Random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0041 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0048 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0150 GB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0180 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0130 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0180 Gaussian* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0050 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0050 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0090 FCNN* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0270 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0090 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0070 CNN* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0540 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0140 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0110 Table 8: Mean error and the standard deviation of error in the iso- topic ratios using all simulations after subtracting contributions from continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Results marked as ‘*’ were produced using algorithm im- plemented in Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' However, for the uranium isotopic content, it was found that the Gaussian processes and the neural networks did not perform adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Examinations of the isotopic content predictions using all ML algorithms revealed excellent performance when a separate continuum sub- traction was done during pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Finally, the isotopics for all of the simulated data for plutonium and uranium were evaluated using FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The results were significantly worse 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='074 for pluto- nium and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='11 for uranium than those obtained via the machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hyper-parameter Examinations The parameters that need to be defined prior to train- ing a ML algorithm are commonly termed as hyperpa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' There is currently no known method to deter- mine which hyperparameters have an impact on model performance before training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Consequently, for each of the respective methods a range of hyperparameters was explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Furthermore, because the objective of this work is to train models using simulation data and test using experimental data, pre-cautions were taken to avoid over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' A summary of the hyper-parameters examined, and the parameters utilized for subsequent Method Parameters Range/ Value/ Methods Range Explored Selected Nearest Neighbors: 1–32000 1 Neighbors Methods: KDtree/ Auto Brute/Auto Distance: Uniform/ Minkowski Chebyshev/ (Euclidean) Euclidean/ Manhattan/ Minkowski Decision Max Depth: 10–100 50 Tree Splitter: Best Best Loss: MSE MSE Feature Fraction: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='05–1 1 Random No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' of trees: 10–100 100 Forest Leaf Size: Unlimited Unlimited Max Depth: 10–100/None 50 Loss: MSE MSE Feature Fraction: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='05–1 1 Gradient No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' of trees: 10–300 200 Boosted Leaf Number: 5–50/None None Trees Max Depth: 2–25 4 Min samples for split: 2–10 5 Loss: MSE MSE Learning Rate: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='01–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 Feature Fraction: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1–1 1 Fully Layers: 2–10 2 Connected Activation: SELU/Tanh Tanh Neural No of params: 15250–100000 10250 Network DropOut: 0–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='01 Epochs: 100–1000 100 Optimization MSE Method: ADAM/SGD ADAM Learning Rate: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='001–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='01 Convol- Layers: 2–10 2 utional No of params: 10250–100000 35324 Neural DropOut: 0–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1937 Network Activation: SELU/ Tanh SELU Epochs: 100–1000 100 Optimization Method: ADAM/SGD/ Logistic LogisticSigmoid Sigmoid L2: 0–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='01 Learning Rate: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='001–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='001 Table 9: Hyperparameters tested and selected for different ML algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 10 investigations, for each of the respective methods is pro- vided in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Experimental Data and Results The previous analyses were performed using simu- lated spectra generated using GADRAS for both train- ing as well as testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' In this Section, we explore the use of the simulations for training and experimental data for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The details of the experimental configurations are outlined in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 and the application of ML algorithms are presented in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' However, be- fore presenting these details, we note that at the time of experimental data collection, terrestrial background data is taken with identical settings to the experimen- tal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Often the background files are generated with longer collection times than the experimental data so as to minimize the effects of statistical fluctuation when subtracting the terrestrial background counts from the foreground counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Prior to subtraction from the fore- ground counts, the background counts are scaled ac- cordingly based on the relative count time for the back- ground file with respect to the count time for the exper- imental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' We note that in the GADRAS simulations, the spec- tra do not include contributions from terrestrial back- ground;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' hence, this process is not applicable for simula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Experimental Description Experimental dataset with multiple source and shield- ing configurations, source geometries, and source forms were utilized to enable the testing of the ML algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Configurations included bare and shielded cans of uranium and plutonium oxide with a wide range of isotopics, depleted uranium spheres and shells (both bare and shielded), and plutonium spheres with various shielding materials and thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Uranium and plutonium oxide sources The uranium (U3O8) and plutonium (PuO2) oxide dataset analyzed were collected with an ORTEC Detec- tive X and LANL Detector S respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The Detec- tive X is a handheld, mechanically cooled HPGe detec- tor with 50% relative efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The Detective X has a range of 8 MeV with 214 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The LANL Detector S is an ORTEC poptop liquid-nitrogen cooled HPGe de- tector with a relative efficiency of roughly 140%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' This detector has a range of 12 MeV with 215 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The Detector S response function was incorporated into the GADRAS simulations as detailed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' At the %U234 %U235 %U236 %U238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='005–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='910 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='716–91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='340 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='002–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='335 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='417–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='277 Table 10: Range in weight % of reported isotopes relative to total U for U3O8 cylindrical sources dated 9/6/1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' %Pu238 %Pu241 %Pu242 %Am241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='002–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='014–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='693 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='003–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='239 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='009–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='510 Table 11: Range in weight % of miscellaneous isotopes relative to total Pu for PuO2 cylindrical sources dated 1/1/1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' time of data collection, Detector S incorporated a bis- muth side shield to reduce the background radiation contribution to the measured spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Additionally, a thin front filter made of tin was also present to filter out low energy photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Fourteen data sets, seven without shielding and seven with shielding material present, were collected for both the uranium and plutonium oxide sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Uranium en- richment and plutonium isotopics were in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='7–91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='3% and 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='7) – 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0)% 239Pu (240Pu) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Additional reported isotopes are listed in Tables 10 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The uranium oxide samples were ap- proximately 1 kg whereas the plutonium samples had mass between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='6–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='8 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Thin sheets of lead were used for the shielded measurements, with a thickness of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='175 mm for the uranium and 4 mm for the pluto- nium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' A complete set of plutonium oxide measurements were collected for 300 seconds at a source-to-detector distance of 50 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The uranium oxide measurements was taken with better counting statistics: 600 seconds at a source-to-detector distance of 25 cm, with an excep- tion of an unshielded 91% enriched oxide measurement, which performed at a source-to-detector distance of 50 cm to ensure an acceptable dead time in the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Depleted uranium shell data The depleted uranium (DU) measurements were per- formed using a LANL Detector S, described above, with a bismuth collimator and tin front filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Fifty-four data sets were taken, six in nine different configurations at a source to detector distance of 1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The configurations utilized various combinations of three stacked DU shells of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='35 mm thickness, while keeping the outer diameter of the DU shells at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='24 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Some configurations were taken without any shielding, and some utilized shielding from either one or two stacked aluminum shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The aluminum shells were 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='27 cm in thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' BeRP ball data The BeRP (Beryllium Reflected Plutonium) ball [61] data was collected with Detector K, a 140% relative ef- 11 ficiency liquid-nitrogen-cooled HPGe detector that is similar to Detector S, at the Nevada Nuclear Secu- rity Site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The BeRP ball is a sphere of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='59 cm di- ameter alpha-phase plutonium clad with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='3 mm of SS304, and weighs 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='48 kg [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Although present in the name of the object, the original beryllium reflector was not used in these configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The dataset col- lected was taken at a source-to-detector distance of 50 cm, both unshielded and with shielding (polyethylene) of thicknesses between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='54-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='16 cm in conjunction with other combinations of shielding materials such as nickel, steel, mock high explosives, and aluminum rang- ing in total thickness from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='27-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='62 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Dimensionality Reduction The HPGe detectors utilized in these investigations have 16384 (Detector X) and 32768 (Detector (S) chan- nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' To reduce the dimensionality of the features for which the ML algorithms were trained, we selected a total of 172 features based on the emission lines of the isotopes under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' This dimensionality re- duction was performed due to established observation that when training a ML algorithm in a large multi- dimensional space, there are often redundant features that add noise to the dataset, without improving the per- formance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Further investigations into improving ML algorithm performance were performed by applying additional dimensionality reduction using two approaches: 1) physics based feature reduction, and 2) Principle component analysis (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' For the physics based feature reduction, we se- lect 9 and 10 prominent γ peaks for U and Pu data/simulations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The selections were made based on the most commonly used photo-peaks in γ spectroscopy for plutonium and uranium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Compar- ison of the results from this method of dimensionality reduction did not improve the mean absolute error and the standard deviation in simulation test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Sim- ilarly, the PCA based method also did not reduced the absolute error in a systematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' As discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1, since we utilized a simu- lation model for constructing a training dataset, there is a potential for biased results due to model depen- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' This potential bias was examined by inspect- ing the spectra generated with a GADRAS model for a sampling of the experimental dataset with known pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' After observing larger disagreements in the lower and higher energy ranges, the number of features was reduced to include features only in the 100–1000 keV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Although this improved the mean absolute error, this type of ad hoc dimensionality reduction can- not be generalized without the knowledge of the source of data-simulation discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Results To investigate the performance of ML algorithms us- ing the experimental data discussed in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1, five algorithms: Decision Trees, MLP, Gradient Boosted Trees, Nearest Neighbors and Random Forests, were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The results were compared with results ob- tained using FRAM software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' For the uranium dataset, the comparisons were performed using both the ‘HEU’ and ‘LEU’ models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The results obtained using the small scale plutonium oxide sources are presented in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The error bars for results, provided in Figure 8, include combined sta- tistical and systematic uncertainties for all the methods except for Decision Trees and Nearest Neighbors (these methods were found to have very low errors due to the lack of systematic uncertainties which were found to be the dominant source of error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The statistical un- certainties were estimated by varying the photo-peak counts with a poisson model and repeating the ML al- gorithm implementation for each instance of the varia- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The systematic uncertainties account for the vari- ation in the ML results when repeating the training and testing with identical conditions and parameters and in- corporating a different random seed for algorithm ini- tialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Figure 8 (top) shows that no single ML al- gorithm outperforms the others for all 14 experiments considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' however, the MLP and Nearest Neighbor methods were found to perform better than the conven- tional method in a few of the experimental cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Once the data is pre-processed to remove the continuum, in general the MLP algorithm performs comparable to the conventional method, within the uncertainties of both methods, as shown in Figure 8 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The uranium oxide results, as provided in Figure 9 (top), show consistently smaller absolute deviations for the MLP as compared to other ML algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The bot- tom Figure 9 shows that although the results from MLP method are comparable to FRAM results, the estimated uncertainties in some cases (experiment numbers: 6, 8, 10, 12 and 13) are much smaller for the ML method than for the conventional approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The results for depleted uranium shell in Figure 10 (top) show that decision tree based methods do not perform as well as the Nearest Neighbor and MLP methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Absolute deviation for the Nearest Neigh- bor method are comparable to the conventional method for most experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The good performance of Near- est Neighbor method is perhaps due to the inclusion of a large number of simulations with enrichment close 12 Figure 8: Comparison of absolute deviation from true isotopics ratio for Pu oxide data for various ML algorithms and FRAM (top), and MLP with and without continuum subtraction and FRAM (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Combined statistical and systematic uncertainties are reported at 1 σ for all the algorithms except for Random Forest and Nearest Neigh- bors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Error bars in FRAM results are the ’sigma’ values returned by the FRAM software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Figure 9: Comparison of absolute deviation from true isotopics ratio for U oxide data for various ML algorithms and FRAM HEU model (top), MLP vs FRAM LEU model (bottom, red) and FRAM HEU model (bottom, blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Combined statistical and systematic uncertain- ties are reported at 1 σ for all the algorithms except for Random Forest and Nearest Neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Error bars in FRAM results are the ’sigma’ values returned by the FRAM software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 13 UO2 TA66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2 DT MLP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 GB NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='8 RF Absolute Deviation FRAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2 2 4 6 10 12 14 Experiment#MLPUO2TA66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2 MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' No Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Subt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 FRAM LEU FRAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='8 AbsoluteDeviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2 2 4 6 8 10 12 14 Experiment#PuO2TA66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='8 DT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='7 MLP GB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='6 NN RF Absolute Deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='5 FH FRAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 2 4 6 8 10 12 14 Experiment#MLPPuO2TA66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='8 MLP, No Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Subt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='7 MLP,Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='Subt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' FRAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='6 Absolute Deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 2 4 6 8 10 12 14 Experiment#Figure 10: Comparison of absolute deviation from true enrichment value for depleted uranium data for various ML algorithms and FRAM HEU model (top), MLP vs FRAM LEU model (bottom, red) and FRAM HEU model (bottom, blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Combined statistical and system- atic uncertainties are reported at 1 σ for MLP, Decision Trees, and Gradient Boosted Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Error bars in FRAM results are the ’sigma’ values returned by the FRAM software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' to that of the depleted uranium in shell configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Since this method relies on finding sets of training data points closest in distance to the query, and taking an av- erage of the closest solutions, having a well represented training sample is expected to enhance the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The MLP method, despite having larger uncertainty in the estimate as compared to the traditional approach, shows mostly small mean absolute deviation of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='05 for most experiments as shown in Figure 10 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The BeRP ball results are presented in Figures 11 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The former figure presents absolute deviation from the true isotopics ratio for experiments with different shielding material combinations and thicknesses, while the latter presents analogous results with a polyethylene shielding of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='52 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' As observed previously, the MLP results are comparable to the FRAM results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The bot- tom figures show improved performance for the MLP Figure 11: Comparison of absolute deviation from true isotopics ratio for BeRP ball data for various ML algorithms and FRAM (top), and MLP with and without continuum subtraction and FRAM (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Combined statistical and systematic uncertainties are reported at 1 σ for all the algorithms except for Decision Trees and Nearest Neigh- bors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Error bars in FRAM results are the ’sigma’ values returned by the FRAM software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' method when continuum subtraction is performed in line with the previous observation in the plutonium ox- ide results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Upon considering all of the analyzed experimental data, the MLP algorithm performed better than the other ML algorithms evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The better performance of MLP as compared to the other ML algorithms may be attributed to the large interconnections of the fully con- nected neural network enabling highly non-linear be- havior to be learned more readily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Improvements in the predictions of the MLP was observed for the plu- tonium data set when continuum subtraction was per- formed prior to the ML application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Although, the amount of improvement varied experiment to experi- ment, the largest improvement in absolute deviation was seen for Pu oxide data at a value of roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 14 DU3Shell 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 0 10 20 30 40 50 Experiment #MLPDU3Shell 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='200 MLP, No Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Subt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='175 FRAMLEU FRAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='150 eviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='100 D Absolute 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 0 10 20 30 40 50 Experiment#BeRP ball 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='8 H DT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='7 MLP GB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='6 NN RF Absolute Deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='5 FRAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='4 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='175 MLP,Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='Subt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' FRAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='150 AbsoluteDeviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='000 1 2 3 4 5 6 7 8 Experiment#Figure 12: Comparison of absolute deviation from true isotopics ratio for BeRP ball data for various ML algorithms and FRAM (top), and MLP with and without continuum subtraction and FRAM (bottom) as a function of polyethylene (shielding material) thickness in inch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Combined statistical and systematic uncertainties are reported at 1 σ for all the algorithms except for Decision Trees and Nearest Neigh- bors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Error bars in FRAM results are the ’sigma’ values returned by the FRAM software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Conclusions Several machine learning (ML) based regression al- gorithms were investigated to perform quantitative de- termination of uranium and plutonium isotopics using γ-ray spectroscopy data collected with HPGe detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The algorithms were trained using GADRAS simula- tions with different source geometries and thicknesses as well as shielding material types and thicknesses to address the needs of the Emergency Response commu- nity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Performance of the algorithms was examined using both simulations as well as experimental datasets incor- porating both uranium and plutonium sources in oxide and metal forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Without time-consuming pre-processing that is of- ten required using conventional methods, all the inves- tigated algorithms were found to offer excellent per- formance when simulation data was utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' A slight decrease in performance was observed with increasing complexity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' wider ranges in source thicknesses, shielding conditions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Additional subtraction of the continuum background in the pre-processing stage had a minimum impact in the performance, indicating that ML algorithms were able to adequately learn the fea- ture relationships in the presence of a large continuum background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' For the experimental dataset, the results were found to be consistently better using a fully connected neural network (or MLP) algorithm as compared to other al- gorithms that were investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Comparison of these results with results obtained from conventional meth- ods (FRAM software) showed comparable error in the isotopic ratio estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Finally, our results demonstrate that with minimum pre-processing, ML algorithms are a good alternative to conventional methods of isotopic de- termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' The performance of ML algorithms may be enhanced by substantially increasing the training data volume and the physics phase space it covers for im- proved machine learning interpolation at unknown con- figurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Walton, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Reilly, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Parker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Menzel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Marshall, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Fields, Measurements of UF6 cylinders with portable instruments, Nu- clear Technology 21 (2) (1974) 133–148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gunnink, MGAU: a new analysis code for measuring u-235 enrichments in arbitrary samples, lawrence livermore national laboratory, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=', UCRL-LR-114713 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gunnink, MGA (multi-group analysis): a gamma-ray spec- trum analysis code for determining plutonium isotopic abun- dances, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=', Lawrence Livermore National Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=', CA (USA) (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 15 BeRP ball 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='8 DT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='7 MLP GB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='6 NN RF Absolute Deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='5 FRAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='5 2.' 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+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='0 Poly thickness (inch)[4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Wei, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Xiong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Liu, Application analysis of ra- dial basis function neural network algorithm of genetic algo- rithm for environmental restoration and treatment effect evalua- tion of decommissioned uranium tailings ponds, Computational Intelligence and Neuroscience 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Peyvandi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=', Determination of uranium and thorium concentrations in thorium ore sample using artificial neural net- work and comparison with net area peak method, Radiochimica Acta 106 (8) (2018) 669–674.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Connor, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Martin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Scott, Airborne radiation map- ping: overview and application of current and future aerial sys- tems, International journal of remote sensing 37 (24) (2016) 5953–5987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hosoda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Inoue, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Oka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Omori, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Iwaoka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Tokon- ami, Environmental radiation monitoring and external dose esti- mation in aomori prefecture after the Fukushima Daiichi nuclear power plant accident, Health Physics 51 (1) (2016) 41–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Korob, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Nu˜no, A simple method for the absolute deter- mination of uranium enrichment by high-resolution γ spectrom- etry, Applied radiation and isotopes 64 (5) (2006) 525–531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Apostol, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Pantelica, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Sima, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Fugaru, Isotopic composi- tion analysis and age dating of uranium samples by high resolu- tion gamma ray spectrometry, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materi- als and Atoms 383 (2016) 103–108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Aitkenhead, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Owen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Chambers, Use of artificial neural networks in measuring characteristics of shielded pluto- nium for arms control, Journal of Analytical Atomic Spectrom- etry 27 (3) (2012) 432–439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Rawool-Sullivan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Bounds, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Brumby, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Prasad, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Sul- livan, Steps toward automated gamma ray spectroscopy, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=', Los Alamos National Laboratory (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Parker, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Reilly, Plutonium isotopic determination by gamma-ray spectroscopy, Nuclear analysis research and devel- opment program status report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [13] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Reilly, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Walton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Parker, progress report la-4605-ms, Los Alamos National Laboratory, NNM (197) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Reilly, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ensslin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Smith Jr, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Kreiner, Passive nonde- structive assay of nuclear materials, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' LA-UR-90-732, Office of Nuclear Regulatory Commission, Washington, DC, USA (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Sampson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Nelson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Kelley, FRAM: A versatile code for analyzing the isotopic composition of plutonium from gamma-ray pulse height spectra, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=', Los Alamos Na- tional Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' (LANL), Los Alamos, NM (United States) (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Mortreau, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Berndt, Determination of the uranium enrich- ment with the NaIGEM code, Nuclear Instruments and Meth- ods in Physics Research Section A: Accelerators, Spectrome- ters, Detectors and Associated Equipment 530 (3) (2004) 559– 567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Sprinkle Jr, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Christiansen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Cole, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Collins, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hsue, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Knepper, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' McKown, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Siebelist, Low-resolution gamma- ray measurements of uranium enrichment, Applied radiation and isotopes 48 (10-12) (1997) 1525–1528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Abousahl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Michiels, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Bickel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gunnink, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ver- plancke, Applicability and limits of the MGAU code for the determination of the enrichment of uranium samples, Nuclear Instruments and Methods in Physics Research Section A: Ac- celerators, Spectrometers, Detectors and Associated Equipment 368 (2) (1996) 443–448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gunnink, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ruhter, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Miller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Goerten, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Swinhoe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Wagner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Verplancke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Bickel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Abousahl, MGAU: A new analysis code for measuring 235U enrichments in arbi- trary samples, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=', Lawrence Livermore National Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=', CA (United States) (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Morel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Goenvec, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Dalmazzone, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Malet, R´ef´erences pour la d´etermination de l’uranium 235 dans les combustibles nucl´eaires, IAEA Nuclear Safeguard Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Morel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hill, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Bickel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Alonso-Munoz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Napier, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Thaurel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Research, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Members, Results from the international evaluation exercise for uranium enrichment measurements, Applied Radiation and Isotopes 52 (3) (2000) 509–522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [22] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Sampson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Kelley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Vo, Application guide to gamma-ray isotopic analysis using the FRAM software, LA- 14018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Darweesh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Shawky, Study on the performance of differ- ent uranium isotopic codes used in nuclear safeguards activities, Heliyon 5 (4) (2019) e01470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [24] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Olmos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Diaz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Perez, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Garcia-Belmonte, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gomez, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Rodellar, Application of neural network techniques in gamma spectroscopy, Nuclear Instruments and Methods in Physics Re- search Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 312 (1-2) (1992) 167–173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [25] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Yoshida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Shizuma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Endo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Oka, Application of neural networks for the analysis of gamma-ray spectra measured with a Ge spectrometer, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 484 (1-3) (2002) 557–563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [26] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Kangas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Keller, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Siciliano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Kouzes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ely, The use of artificial neural networks in PVT-based radiation portal monitors, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 587 (2-3) (2008) 398–412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Briesmeister, MCNP-a general Monte Carlo n-particle transport code, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' LA–13709–M, Los Alamos National Laboratory (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [28] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Liang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Tang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Zhang, Rapid nuclide identification algorithm based on con- volutional neural network, Annals of Nuclear Energy 133 (2019) 483–490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [29] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Bobin, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Bichler, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Lourenc¸o, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Thiam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Th´evenin, Real-time radionuclide identification in γ-emitter mixtures based on spiking neural network, Applied Radiation and Iso- topes 109 (2016) 405–409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Sharma, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Bellinger, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Japkowicz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Berg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ungar, Anomaly detection in gamma ray spectra: A machine learning perspective, in: 2012 IEEE Symposium on Computational Intel- ligence for Security and Defence Applications, IEEE, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Tang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Wen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Han, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Yan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gao, Rapid radionuclide identification algorithm based on the discrete cosine transform and bp neural network, Annals of Nuclear Energy 112 (2018) 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [32] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hague, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Kamuda, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ford, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Moore, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Turk, A comparison of adaptive and template matching techniques for radio-isotope identification, in: Algorithms, Technologies, and Applications for Multispectral and Hyperspectral Imagery XXV, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 10986, SPIE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 62–73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Kamuda, Automated isotope identification and quantifi- cation using artificial neural networks, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' thesis, University of Illinois at Urbana-Champaign (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [34] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Luo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Xiang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ding, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Chu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Zeng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ze, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Xiang, Identification of SNM based on low-resolution gamma-ray characteristics and neural network, Nuclear Instru- ments and Methods in Physics Research Section A: Accelera- tors, Spectrometers, Detectors and Associated Equipment 927 (2019) 155–160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [35] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Bilton, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Joshi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Bandstra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Curtis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hellfeld, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Vetter, Neural network approaches for mobile spectroscopic 16 gamma-ray source detection, Journal of Nuclear Engineering 2 (2) (2021) 190–206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gomez-Fernandez, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Higley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Tokuhiro, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Welter, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Wong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Yang, Status of research and development of learning- based approaches in nuclear science and engineering: A review, Nuclear Engineering and Design 359 (2020) 110479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [37] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Yokoyama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ishimori, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ohara, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Tanaka, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Sugitsue, Application of support vector machine to rapid classification of uranium waste drums using low-resolution γ- ray spectra, Applied Radiation and Isotopes 104 (2015) 143– 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [38] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Wei, Nuclide identification algorithm based on K–L transform and neural networks, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spec- trometers, Detectors and Associated Equipment 598 (2) (2009) 450–453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [39] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Pilato, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Tola, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Martinez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Huver, Application of neu- ral networks to quantitative spectrometry analysis, Nuclear In- struments and Methods in Physics Research Section A: Ac- celerators, Spectrometers, Detectors and Associated Equipment 422 (1-3) (1999) 423–427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Shaban, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hazzaa, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' El-Tayebany, Applying Monte Carlo and artificial intelligence techniques for 235u mass pre- diction in samples with different enrichments, Nuclear Instru- ments and Methods in Physics Research Section A: Accelera- tors, Spectrometers, Detectors and Associated Equipment 916 (2019) 322–326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [41] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Vigneron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Morel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Lepy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Martinez, Statisti- cal modelling of neural networks in γ-spectrometry, Nuclear Instruments and Methods in Physics Research Section A: Ac- celerators, Spectrometers, Detectors and Associated Equipment 369 (2-3) (1996) 642–647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Ryu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Park, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Cho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Park, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Cho, Develop- ment of neural network model with explainable AI for measur- ing uranium enrichment, IEEE Transactions on Nuclear Science 68 (11) (2021) 2670–2681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [43] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Elmaghraby, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Tohamy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Comsan, Determination of isotopes activity ratio using gamma ray spectroscopy based on neural network model, Applied Radiation and Isotopes 148 (2019) 19–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [44] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Pedregosa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Varoquaux, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gramfort, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Michel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Thirion, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Grisel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Blondel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Prettenhofer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Weiss, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Dubourg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Vanderplas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Passos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Cournapeau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Brucher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Per- rot, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Duchesnay, Scikit-learn: Machine learning in Python, Journal of Machine Learning Research 12 (2011) 2825–2830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [45] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Wolfram, The Mathematica book, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 1, Wolfram Research, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=', 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [46] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Breiman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Friedman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Stone, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Olshen, Classifi- cation and Regression Trees, Taylor & Francis, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [47] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Biau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Scornet, A random forest guided tour, TEST 25 (2) (2016) 197–227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1007/s11749-016-0481-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [48] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Natekin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Knoll, Gradient boosting machines, a tutorial, Frontiers in neurorobotics 7 (2013) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [49] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Fix, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hodges, Discriminatory analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' nonparametric discrimination: Consistency properties, International Statistical Review/Revue Internationale de Statistique 57 (3) (1989) 238– 247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [50] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Altman, An introduction to kernel and nearest-neighbor nonparametric regression, The American Statistician 46 (3) (1992) 175–185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [51] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Gershman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Blei, A tutorial on Bayesian nonparamet- ric models, Journal of Mathematical Psychology 56 (1) (2012) 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [52] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Williams, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Barber, Bayesian classification with Gaus- sian processes, IEEE Transactions on Pattern Analysis and Ma- chine Intelligence 20 (12) (1998) 1342–1351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [53] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Williams, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Rasmussen, Gaussian processes for regression, Advances in Neural Information Processing Systems 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [54] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Fukushima, Neocognitron: A self-organizing neural network model for a mechanism of pattern recognition unaffected by shift in position, Biological Cybernetics 36 (1980) 193–202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1007/BF00344251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [55] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' LeCun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Bengio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hinton, Deep learning, Nature 521 (7553) (2015) 436–444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1038/nature14539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Schmidhuber, Deep learning in neural networks: An overview, Neural Networks 61 (2015) 85–117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1016/ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='neunet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [57] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Mitchell, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Harding, GADRAS isotope id users manual for analysis of gamma-ray measurements and API for Linux and Android, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=', Sandia National Laboratory (5 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='2172/1177049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [58] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Morh´aˇc, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Kliman, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Matouˇsek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Veselsk´y, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Turzo, Background elimination methods for multidimensional coinci- dence gamma-ray spectra, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 401 (1) (1997) 113–132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1016/S0168-9002(97)01023-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [59] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Morh´aˇc, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Kliman, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Matouˇsek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Veselsk´y, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Turzo, Efficient one- and two-dimensional gold deconvolution and its application to gamma-ray spectra decomposition, Nuclear In- struments and Methods in Physics Research Section A: Ac- celerators, Spectrometers, Detectors and Associated Equipment 401 (2) (1997) 385–408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1016/S0168-9002(97) 01058-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [60] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Morh´aˇc, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Kliman, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Matouˇsek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Veselsk´y, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Turzo, Identification of peaks in multidimensional coincidence γ-ray spectra, Nuclear Instruments and Methods in Physics Research A 443 (1) (2000) 108–125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1016/S0168-9002(99) 01005-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [61] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Hutchinson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' Valentine, Subcritical measurements of a plu- tonium sphere reflected by polyethylene and acrylic, Nuclear Science and Engineering 161 (3) (2009) 357–362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 13182/NSE161-357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' [62] Nuclear Energy Agency, ICSBEP handbook 2019, interna- tional Criticality Safety Benchmark Evaluation Project Hand- book (database) (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content='1787/e2703cd5-en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} +page_content=' 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQfdPwF/content/2301.01415v1.pdf'} diff --git a/utAzT4oBgHgl3EQfBvrw/vector_store/index.faiss b/utAzT4oBgHgl3EQfBvrw/vector_store/index.faiss new file mode 100644 index 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sha256:49cede5da79a8ed70abbfcd62b6eec9e2b703aada6a31015bc7e3dadbce2d750 +size 9109549 diff --git a/wtFKT4oBgHgl3EQf5S7u/vector_store/index.pkl b/wtFKT4oBgHgl3EQf5S7u/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..5508c28fb459f52f4d51cf8cf0ada00fe8264587 --- /dev/null +++ b/wtFKT4oBgHgl3EQf5S7u/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4e4f2a78075e38385a107aba09c13128528705cc0bfc9cdda73dfcb7caa88ca2 +size 309182 diff --git a/xdAyT4oBgHgl3EQfnfj_/content/tmp_files/2301.00492v1.pdf.txt b/xdAyT4oBgHgl3EQfnfj_/content/tmp_files/2301.00492v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..572c654edf9b5f5c3bbbae2f57d396249b335c9e --- /dev/null +++ b/xdAyT4oBgHgl3EQfnfj_/content/tmp_files/2301.00492v1.pdf.txt @@ -0,0 +1,2235 @@ +arXiv:2301.00492v1 [math.AP] 2 Jan 2023 +A WEIGHTED Lq(Lp)-THEORY FOR FULLY DEGENERATE +SECOND-ORDER EVOLUTION EQUATIONS WITH +UNBOUNDED TIME-MEASURABLE COEFFICIENTS +ILDOO KIM +Abstract. We study the fully degenerate second-order evolution equation +ut = aij(t)uxixj + bi(t)uxi + c(t)u + f, +t > 0, x ∈ Rd +(0.1) +given with the zero initial data. +Here aij(t), bi(t), c(t) are merely locally +integrable functions, and (aij(t))d×d is a nonnegative symmetric matrix with +the smallest eigenvalue δ(t) ≥ 0. We show that there is a positive constant N +such that +� T +0 +�� +Rd (|u| + |uxx|)p dx +�q/p +e−q +� t +0 c(s)dsw(α(t))δ(t)dt +≤ N +� T +0 +�� +Rd |f (t, x)|p dx +�q/p +e−q +� t +0 c(s)dsw(α(t))(δ(t))1−q dt, +(0.2) +where p, q ∈ (1, ∞), α(t) = � t +0 δ(s)ds, and w is a Muckenhoupt’s weight. +1. introduction +Needless to say, the second-order partial differential equations equations with +degenerate or unbounded coefficients have been extensively studied for a long time. +To the best of our knowledge, the starting point of this study was Keldysh, Fichera, +and Ole˘ınik’s work (see e.g. [23, 10, 37, 38, 39]). Moreover, it is very popular to +study a (maximal regularity) Lp-theory and its generalization to Lq(Lp)-theory in +harmonic analysis, Fourier analysis, and partial differential equations after Calder´on +and Zygmund’s work. For the historical works and backgrounds of Lp-theories and +their generalizations, we refer some outstanding books [31, 32, 42, 18, 19, 21, 22]. +These days, there are tons of papers handling degenerate and unbounded coefficients +in various prospectives. Among recent works with various prospectives, we only +refer the author to [26, 11, 9, 35, 12, 17, 33, 16, 15, 27, 34, 40, 2, 36, 1, 6, 14, 20, 7, +8, 13, 28, 41, 43]. These results handle equations having degenerate or unbounded +coefficients in Sobolev spaces. +With the degeneracy in the equation, it is hard to expect to obtain full regularity +estimates of solutions unless there are weights involved in estimates. For instance, +by taking the leading coefficients aij(t) = 0 for all i, j, t, we see that it is not possible +2010 Mathematics Subject Classification. 35K65, 35B65, 35K15. +Key words and phrases. Degenerate second-order parabolic equations, Weighted Lp-estimates, +zero initial-value problem. +I. Kim has been supported by the National Research Foundation of Korea(NRF) grant funded +by the Korea government(MSIT) (No.NRF-2020R1A2C1A01003959). +1 + +2 +ILDOO KIM +to obtain the unweighted maximal Lp-regularity +� T +0 +� +Rd |uxx(t, x)|pdtdx ≤ N +� T +0 +� +Rd |f(t, x)|pdtdx. +(1.1) +Hence, weights have been commonly used to controls the degeneracy or unbound- +eness (singularity) of the coefficients. However, most results in the literature focus +on degeneracy or singularity near the boundary of a domain. If we consider the +whole space, it is naturally not expected that there is a regularity gain of a solution +in general due to the extreme case such as ut = f, which could be understood as +one of equation (0.1) with coefficients aij(t) = 0 for all t. Hence when it comes +to the solvability of second-order equations with degeneracy in the whole space, +people used to only prove the existence and uniqueness of a weak solution without +considering regularity gain from the equations. +Nonetheless, there is a way to express an Lp-norm of second derivatives of a +solution u with a weight which could be singular even in the whole space. For +instance, assume that the degeneracy happens on a time interval (a, b), then δ(t) = 0 +for all t ∈ (a, b). Then we cannot expect the smoothing gain from the diffusion +equations and the Sobolev second derivatives uxx fails to exist. +However, since +there is the weight δ(t) in the first line of (0.2), the inequality is still true if we +understand the second line of (0.2) as an improper integral. To the best of our +knowledge, this type estimate is firstly introduced by the author and collaborator +in [24, 25]. In this paper, we add Muckenhoupt’s weights in estimates and extend +Lp-estimates to Lq(Lp)-estimates with lower-order terms. +It is well-known that probabilistic methods are very powerfully working for lead- +ing coefficients which are unbounded and have degeneracy (cf. [29, 4]). We remark +that probabilistic tools play very important roles to obtain our results. Especially, +to obtain (0.2), it requires to understand the relation among the constant N, the +degeneracy, and the unboundedness of coefficients aij(t). Maximal Lp-regularity +estimates such as (1.1) originally came from Lp-boundedness of singular integral +operators. However, the exact relation among parameters related to coefficients is +hard to obtain from singular integral theories since all parameters are combined in +a complicated way to control singularities of operators. We found that this rela- +tion could be more clear by applying probabilistic representations of solutions (see +Theorem 4.3). +We believe that our result could initiate various interesting weighted estimates +for degenerate second-order equations with space dependent coefficients or domain +problems. +This paper is organized as follows. In Section 2, we introduce our main results. +A probabilistic solution representation and its application to estimate a solution u +with general weights are given in Section 3 Weighted estimates for non-degenerate +equations are shown in Section 4. Finally, the proof of the main theorem is specified +in Section 5. +We finish the introduction with notation used in the article. +• We use Einstein’s summation convention throughout this paper. + +DEGENERATE PDES WITH LOWER ORDER TERMS +3 +• N and Z denote the natural number system and the integer number system, +respectively. As usual Rd stands for the Euclidean space of points +x = + + + + + +x1 +x2 +... +xd + + + + + . +Frequently, the coordinates of the vector x is denoted in a row form, i.e. +x = (x1, . . . , xd). We use the notation (aij)d×d to denote the d by d matrix +whose entry in i-th row and j-th column is aij. For i = 1, ..., d, multi-indices +α = (α1, ..., αd), αi ∈ {0, 1, 2, ...}, and functions u(x) we set +uxi = ∂u +∂xi = Diu, +Dαu = Dα1 +1 +· ... · Dαd +d u. +• C∞(Rd) denotes the space of infinitely differentiable functions on Rd. S(Rd) +is the Schwartz space consisting of infinitely differentiable and rapidly de- +creasing functions on Rd. By C∞ +c (Rd), we denote the subspace of C∞(Rd) +with the compact support. +• For n ∈ N and O ⊂ Rd and a normed space F, by C(O; F), we denote +the space of all F-valued continuous functions u on O having |u|C := +supx∈O |u(x)|F < ∞. +• For p ∈ [1, ∞), a normed space F, and a measure space (X, M, µ), by +Lp(X, M, µ; F), we denote the space of all F-valued Mµ-measurable func- +tions u so that +∥u∥Lp(X,M,µ;F ) := +�� +X +∥u(x)∥p +F µ(dx) +�1/p +< ∞, +where Mµ denotes the completion of M with respect to the measure µ. If +there is no confusion for the given measure and σ-algebra, we usually omit +them. +• For measurable set O ⊂ Rd, |O| denotes the Lebesgue measure of O. +• By F and F−1 we denote the d-dimensional Fourier transform and the in- +verse Fourier transform, respectively. That is, F[f](ξ) := +� +Rd e−ix·ξf(x)dx +and F−1[f](x) := +1 +(2π)d +� +Rd eiξ·xf(ξ)dξ. +• We write a ≲ b if there is a positive constant N such that a ≤ Nb. The +constant N may change from a location to a location, even within a line. +If we write N = N(a, b, · · · ), this means that the constant N depends only +on a, b, · · · . The dependence of the constant N is usually specified in the +statements of theorems, lemmas, and corollaries. +2. Setting and main result +Throughout the paper, we fix d ∈ N to denote the dimension of the space variable +and all functions are real-valued if there is no special comment. +We study the +following degenerate second-order evolution equation +ut(t, x) = aij(t)uxixj(t, x) + bi(t)uxi(t, x) + c(t)u(t, x) + f(t, x), +u(0, x) = 0, +(t, x) ∈ (0, T ) × Rd. +(2.1) + +4 +ILDOO KIM +We emphasize that our coefficients aij(t), bi(t), and c(t) do not satisfy any regu- +larity conditions. More importantly, our coefficients aij(t), bi(t), and c(t) can be +unbounded and degenerate. Here are more concrete conditions on the coefficients +aij(t), bi(t), and c(t). +Assumption 2.1. +(i) Assume that there exists a measurable mapping δ(t) from +(0, ∞) to [0, ∞) such that +aij(t)ξiξj ≥ δ(t)|ξ|2 +∀t ∈ [0, ∞) and ξ ∈ Rd. +(ii) Assume that the coefficients aij(t), bi(t), and c(t) are locally integrable, i.e. +� T +0 +� +|aij(t)| + |bi(t)| + |c(t)| +� +dt < ∞ +∀T ∈ (0, ∞) and ∀i, j. +(2.2) +For T ∈ (0, ∞) and a measurable function u on (0, T ) × Rd, we say that u is +locally integrable if +� t +0 +� +|x| 0. +Definition 2.2 (Solution). Let T ∈ (0, ∞) and f be a locally integrable function +on (0, T ) × Rd. We say that a locally integrable function u is a solution to (2.1) if +for any ϕ ∈ C∞ +c (Rd), +(u(t, ·), ϕ) = +� t +0 +� +u(s, ·), aij(s)ϕxixj + bi(s)ϕxi + c(s)ϕ +� +ds ++ +� t +0 +(f(s, ·), ϕ) ds +∀t ∈ (0, T ), +(2.3) +where (u(t, ·), ϕ) denotes the L2(Rd)-inner product, i.e. +(u(t, ·), ϕ) := +� +Rd u(t, x)ϕ(x)dx. +Remark 2.3. Due to the definition of a solution, it is obvious that +aij(t)uxixj = aij(t) + aji(t) +2 +uxixj. +Thus without loss of generality, we may assume that our coefficient matrix (aij(t))d×d +is nonnegative symmetric for all t. Additionally, δ(t) in Assumption 2.1(i) can be +chosen by the smallest eigenvalue of (aij(t))d×d. +We recall the definition of Muckenhoupt’s weights. +Definition 2.4 (Muckenhoupt’s weight). For q ∈ (1, ∞), let Aq(R) be the class of +all nonnegative and locally integrable functions w on R satisfying +[w]Aq(R) := +sup +−∞ 0. Then for any ε ∈ (0, 1), applying +H¨older’s inequality and the change of variable α(t) + εt → t, we have +� t +0 +� +|x| 0, define f M(t, x) := f(t, x) ∧ M := min{f(t, x), M} and +denote +uM(t, x) = +� t +0 +E +� +f M(s, x + Xt − Xs) +� +ds. +Then by the result of step 2, for any M > 0, we have +(uM(t, ·), φ) = +� t +0 +� +uM(s, ·), aij(s)φxixj� +ds + +� t +0 +� +f M(s, ·), φ +� +ds +∀t ∈ (0, T ). +(3.10) +It is obvious that uM(t, x) → u(t, x) for all t ∈ (0, T ) and x ∈ Rd as M → ∞. +Finally, taking M → ∞ and applying the monotone and dominate convergence +theorems in (3.10), we show that u is a solution to (3.1). +Part III. (Estimate) +We prove (3.3) and (3.4). By (3.2), the generalized Minkowski inequality, and +the translation invariant property of the Lp-space, +∥u(t, ·)∥Lp ≤ +� t +0 +∥f(s, ·)∥Lpds. +Moreover, applying H¨older’s inequality, we have +� T +0 +∥u(t, ·)∥q +Lph1(t)dt ≤ +� T +0 +h1(t) +� t +0 +∥f(s, ·)∥q +Lph2(s)ds +�� t +0 +|h2(s)|− +1 +q−1 ds +�q−1 +dt. +□ +Remark 3.2. Assume that +� T +0 +∥f(s, ·)∥Lpdt < ∞. +Then due to (3.3) and the linearity of (3.1), one can easily find a continuous mod- +ification of u so that +sup +t∈[0,T ] +∥u(t, ·)∥Lp ≤ +� T +0 +∥f(s, ·)∥Lpds +∀t ∈ [0, T ]. +Corollary 3.3. Let T ∈ (0, ∞), p ∈ [1, ∞], and q ∈ (1, ∞). Suppose that Assump- +tion 2.1 holds. Additionally, assume that h1 and h2 are functions on [0, T ] which +are positive (a.e.). Then for any locally integrable function f on (0, T ) × Rd, there +is a unique solution u to equation (2.1) such that +sup +t∈[0,T ] +� +∥u(t, ·)∥q +Lpe−q +� t +0 c(s)ds� +≤ +�� T +0 +|h2(t)|− +1 +q−1 dt +�q−1 � T +0 +e−q +� t +0 c(s)ds∥f(t, ·)∥q +Lph2(t)dt. +(3.11) + +DEGENERATE PDES WITH LOWER ORDER TERMS +13 +and +� T +0 +∥u(t, ·)∥q +Lpe−q +� t +0 c(s)dsh1(t)dt +≤ +� T +0 +� +h1(t) +�� t +0 +|h2(s)|− +1 +q−1 ds +�q−1 � t +0 +e−q +� s +0 c(ρ)dρ∥f(s, ·)∥q +Lph2(s)ds +� +dt. (3.12) +Proof. Let v be a solution to the equation +vt(t, x) = aij(t)vxixj(t, x) + e− +� t +0 c(s)dsf +� +t, x − +� t +0 +b(s)ds +� +, +v(0, x) = 0, +(t, x) ∈ (0, T ) × Rd. +Define U(t, x) = e +� t +0 c(s)dsv +� +t, x + +� t +0 b(s)ds +� +, where b(t) = (b1(t), . . . , bd(t)). Then +Ut(t, x) += c(t)U(t, x) + e +� t +0 c(s)ds +� +vt +� +t, x + +� t +0 +b(s)ds +� ++ bi(t)vxi +� +t, x + +� t +0 +b(s)ds +�� += c(t)U(t, x) ++ e +� t +0 c(s)ds +� +aij(t)vxixj +� +t, x + +� t +0 +b(s)ds +� ++ e− � t +0 c(s)dsf(t, x) +� ++ e +� t +0 c(s)ds +� +bi(t)vxi +� +t, x + +� t +0 +b(s)ds +�� += aij(t)Uxixj(t, x) + bi(t)Uxi(t, x) + c(t)U(t, x) + f(t, x) +and +U(0, x) = 0. +Thus by the uniqueness of a solution, the solution u to (2.1) is given by +u(t, x) = e +� t +0 c(s)dsv +� +t, x + +� t +0 +b(s)ds +� +and obviously +v(t, x) = e− � t +0 c(s)dsu +� +t, x − +� t +0 +b(s)ds +� +. +Applying (3.4) to v and using the translation invariant property of Lp-norms, we +obtain (3.12). Moreover, by (3.3) and H¨older’s inequality, for any 0 ≤ t ≤ T , we +have +e−q +� t +0 c(s)ds∥u(t, ·)∥q +Lp += ∥v(t, ·)∥q +Lp +≤ +� t +0 +e−q +� s +0 c(ρ)dρ∥f(s, ·)∥q +Lph2(s)ds +�� t +0 +|h2(s)|− +1 +q−1 ds +�q−1 +≤ +� T +0 +e−q +� t +0 c(s)ds∥f(t, ·)∥q +Lph2(t)dt +�� T +0 +|h2(s)|− +1 +q−1 ds +�q−1 +, +which obviously implies (3.11). +□ + +14 +ILDOO KIM +4. Estimates for non-degenerate equations +We start the section by reviewing previous weighted estimates with uniform +elliptic and bounded coefficients and apply these estimates to our model equation +(3.1). We denote +∥f∥Lp,q(T,w) = +�� T +0 +�� +Rd |f(t, x)|pdx +�q/p +w(t)dt +�1/q +. +As usual, Lp,q(T, w) denote the spaces of all locally integrable functions f on (0, T )× +Rd such that ∥f∥Lp,q(T,w) < ∞. +Theorem 4.1. Let T ∈ (0, ∞), p, q ∈ (1, ∞), and w ∈ Aq(R). +Assume that +the coefficients aij(t) are uniformly bounded and elliptic, i.e. there exist positive +constants M and δ such that +M|ξ|2 ≥ aij(t)ξiξj ≥ δ|ξ|2 +∀ξ ∈ Rd. +(4.1) +Then for any f ∈ Lp,q(T, w), there exists a unique solution u to (3.1) such that +�� T +0 +�� +Rd |uxx(t, x)|pdx +�q/p +w(t)dt +�1/q +≤ N +�� T +0 +�� +Rd |f(t, x)|pdx +�q/p +w(t)dt +�1/q +, +(4.2) +where +N = N +� +p, q, M, δ, [w]Aq(R) +� +. +Proof. It is a well-known result which could be easily obtained by combining some +classical results. However, it is not easy to find a paper covering the result directly. +Thus we refer two recent papers [5, Theorem 2.2] handling more general coefficients +and [3, Theorem 2.14] studying time measurable pseudo-differential operators. □ +Remark 4.2. Theorem 4.1 is enough for our application. However, as shown in [5, +Theorem 2.2] and [3, Theorem 2.14], w0(x) ∈ Ap(Rd) can be inside (4.2) if (4.1) +holds. In other words, we can find a positive constant N such that such that +�� T +0 +�� +Rd |uxx(t, x)|pw0(x)dx +�q/p +w(t)dt +�1/q +≤ N +�� T +0 +�� +Rd |f(t, x)|pw0(x)dx +�q/p +w(t)dt +�1/q +, +where +N = N +� +p, q, M, δ, [w]Aq(R), [w0]Ap(Rd) +� +. +Next we want to enhance Theorem 4.1. Specifically, we show the constant N +in (4.2) is independent of the upper bound M of the coefficients aij(t) and more +precise relation between the constant N and the elliptic constant δ. However, it +seems to be almost impossible to prove it with only analytic tools. Thus we recall +probabilistic representations of solutions to upgrade Theorem 4.1. + +DEGENERATE PDES WITH LOWER ORDER TERMS +15 +Theorem 4.3. Let T ∈ (0, ∞), p, q ∈ (1, ∞), and w ∈ Aq(R). Assume that the +coefficients aij(t) are uniformly elliptic, i.e. there exists a positive constant δ such +that +aij(t)ξiξj ≥ δ|ξ|2 +∀ξ ∈ Rd. +(4.3) +Additionally, we assume that the coefficients aij(t) are locally integrable, i.e. +� t +0 +aij(s)ds < ∞ +∀t ∈ (0, T ). +Then for any f ∈ Lp,q(T, w), there exists a unique solution u to (3.1) such that +� T +0 +�� +Rd |uxx(t, x)|pdx +�q/p +w(t)dt ≤ N +δq +� T +0 +�� +Rd |f(t, x)|pdx +�q/p +w(t)dt, +(4.4) +where +N = N +� +p, q, [w]Aq(R) +� +. +Proof. (Step 1) aij(t)uxixj = δ∆u. +For this simple case, we use a basic scaling property of the equation. Put v(t, x) = +u(t, +√ +δx). Since u is the solution to +ut(t, x) = δ∆u(t, x) + f(t, x) +u(0, x) = 0, +we have +vt(t, x) = ∆v(t, x) + f(t, +√ +δx) +v(0, x) = 0. +Thus applying (4.2), we have +�� T +0 +�� +Rd |vxx(t, x)|pdx +�q/p +w(t)dt +�1/q +≤ N +�� T +0 +�� +Rd |f(t, +√ +δx)|pdx +�q/p +w(t)dt +�1/q +, +where +N = N +� +p, q, [w]Aq(R) +� +. +Finally, we obtain (4.4) by the simple change of the variable +√ +δx → x. +(Step 2) General aij(t)uxixj. +To prove a general case, we use probabilistic solution representations. We may +assume that +� T +0 +aij(t)dt < ∞ +since the constant N in (4.4) is independent of T . Additionally, due to the trivial +constant extension aij(t)1t∈(0,T )+aij(T )1t≥T, we may assume that aij(t) is defined +on (0, ∞). Consider two independent d-dimensional Brownian motions Bt and Wt +in a probability space (Ω, Ft, P). Set +� +aij(t) +� +d×d = A(t) = +√ +A(t) × +√ +A +∗(t), + +16 +ILDOO KIM +Xt := +√ +2 +� t +0 +√ +A +ij(s)dBj +s, +X2 +t := +√ +2 +� t +0 +�� +A(s) − δI +ij� +dBj +s, +X1 +t := +√ +2 +√ +δIijW j +t , +where I = (Iij)d×d denotes the d by d identity matrix whose diagonal entries are 1 +and the other entries are zero and +� +A(s) − δI is a matrix so that +� +A(s) − δI +� +A(s) − δI = A(s) − δI, +which exists due to (4.3), i.e. A(s) − δI is a nonnegative symmetric matrix. Then +due to (3.2), the solution u is given by +u(t, x) = +� t +0 +E [f(s, x + Xt − Xs)] ds += +� t +0 +E +� +f(s, x + X1 +t − X1 +s + X2 +t − X2 +s) +� +ds, +(4.5) +where the last equality is due to the fact that two probabilistic distributions of +Xt − Xs and X1 +t − X1 +s + X2 +t − X2 +s are equal for all 0 < s < t. Moreover, due to +the independence of two Brownian motions Bt and Wt, we can split the random +parameters in (4.5). Additionally, applying Fubini’s theorem we have +u(t, x) = +� t +0 +E +� +f(s, x + X1 +t − X1 +s + X2 +t − X2 +s) +� +ds += +� t +0 +E′ � +E +� +f(s, x + X1 +t (ω) − X1 +s(ω) + X2 +t (ω′) − X2 +s(ω′)) +�� +ds += E′ +�� t +0 +E +� +f(s, x + X1 +t (ω) − X1 +s (ω) + X2 +t (ω′) − X2 +s(ω′)) +� +ds +� +. +(4.6) +For each fixing ω′, the function +vω′(t, x) := +� t +0 +E +� +f(s, x + X1 +t (ω) − X1 +s (ω) − X2 +s(ω′)) +� +ds +becomes a solution to the equation +vω′ +t (t, x) = δ∆vω′(t, x) + f(t, x − X2 +t (ω′)) +vω′(0, x) = 0. +Thus by the result in Step 1, +� T +0 +�� +Rd |vω′ +xx(t, x)|pdx +�q/p +w(t)dt ≤ N +δq +� T +0 +�� +Rd |f(t, x − X2 +t (ω′))|pdx +�q/p +w(t)dt, +(4.7) +where N depends only on p, q, [w]Aq(R), and κ. Moreover, by (4.6), +uxx(t, x) = E′ � +vω′ +xx +� +t, x + X2 +t (ω′) +�� +. +(4.8) + +DEGENERATE PDES WITH LOWER ORDER TERMS +17 +Finally applying (4.8), (4.7), the generalized Minkowski’s inequality, and Jensen’s +inequality, we have +� T +0 +�� +Rd |uxx(t, x)|pdx +�q/p +w(t)dt +≤ NE′ +�� T +0 +�� +Rd |vω′ +xx(t, x + X2 +t (ω′))|pdx +�q/p +w(t)dt +� +≤ N +δq +� T +0 +�� +Rd |f(t, x)|pw (x + k(t)) dx +�q/p +w(t)dt. +□ +Remark 4.4. We hope that there is a positive constant N such that such that +�� T +0 +�� +Rd |uxx(t, x)|pdx +�q/p +w(t)dt +�1/q +≤ N +δq +�� T +0 +�� +Rd |f(t, x)|pdx +�q/p +w(t)dt +�1/q +, +where +N = N +� +p, q, [w]Aq(R), [w0]Ap(Rd) +� +. +However, it cannot be obtained by following the proof of Theorem 4.3 since +� +Rd |f(t, x − X2 +t (ω′))|pdx = +� +Rd |f(t, x)|pdx +∀ω′ and ∀t +is used in the proof. +5. Proof of the main theorem +Proof of Theorem 2.5 +Due to Theorem 3.1, the existence and uniqueness of a solution u is obvious. +Moreover, (2.7), (2.8), and (2.9) can be easily obtained from (2.4), (2.5), and (2.6) +since |t|β ∈ Aq(R) for any −1 < β1 < q − 1 (see [18, Example 7.1.7]). Thus it +suffices to show (2.4), (2.5) and (2.6). Let u be the solution to (2.1). First we show +(2.4) and (2.5). For each ε ∈ (0, 1), we denote +h1,ε(t) = w(α(t) + εt) (δ(t) + ε) +and +h2,ε(t) = w(α(t) + εt)|δ(t) + ε|1−q. + +18 +ILDOO KIM +Then by (3.11) and (3.12) with a simple change of variable, +sup +t∈[0,T ] +� +∥u(t, ·)∥q +Lpe−q +� t +0 c(s)ds� +≤ +�� T +0 +��w(α(t) + εt)|δ(t) + ε|1−q��− +1 +q−1 dt +�q−1 +× +� T +0 +∥f(t, ·)∥q +Lpe−q +� t +0 c(s)dsw(α(t) + εt)|δ(t) + ε|1−q(t)dt +≤ +�� α(T )+εT +0 +|w(t)|− +1 +q−1 dt +�q−1 +× +� T +0 +∥f(t, ·)∥q +Lpe−q +� t +0 c(s)dsw(α(t) + εt)|δ(t) + ε|1−q(t)dt +and +� T +0 +∥u(t, ·)∥q +Lpe−q � t +0 c(s)dsw(α(t) + εt) (δ(t) + ε) dt +≤ +�� T +0 +w(α(t) + εt) (δ(t) + ε) +�� t +0 +|w(α(s) + εs)|δ(s + ε)|1−q|− +1 +q−1 ds +�q−1 +dt +� +× +� T +0 +∥f(t, ·)∥q +Lpe−q +� t +0 c(s)dsw(α(t) + εt)|δ(t + ε)|1−qdt. +Moreover, by taking ε → 0, we have +sup +t∈[0,T ] +� +∥u(t, ·)∥q +Lpe−q � t +0 c(s)ds� +≤ +�� α(T ) +0 +|w(t)|− +1 +q−1 dt +�q−1 � T +0 +∥f(t, ·)∥q +Lpe−q +� t +0 c(s)dsw(α(t))|δ(t)|1−q(t)dt +and +� T +0 +∥u(t, ·)∥q +Lpe−q +� t +0 c(s)dsw(α(t)) (δ(t)) dt +≤ +�� T +0 +w(α(t)) (δ(t)) +�� t +0 +|w(α(s))|δ(s)|1−q|− +1 +q−1 ds +�q−1 +dt +� +× +� T +0 +∥f(t, ·)∥q +Lpe−q +� t +0 c(s)dsw(α(t))|δ(t)|1−qdt. +(5.1) +One may think that this limit procedure does not seem to be clear. However, it +is clear if our weight w is continuous. Moreover, if w is bounded, then w can be +approximated by a sequence of continuous functions with a uniform upper bound. +Finally, considering w ∧ M for any positive constant M > 0, we can complete the +limit procedure due to the monotone convergence theorem as M → ∞. +We keep going to estimate the term in the middle of (5.1). Recalling the def- +inition of [w]Ap(R) and applying the change of variable α(t) := +� t +0 δ(s)ds → t, we + +DEGENERATE PDES WITH LOWER ORDER TERMS +19 +have +� T +0 +w(α(t))δ(t) +�� t +0 +|w(α(s))|δ(s)|1−q|− +1 +q−1 ds +�q−1 +dt +≤ +� T +0 +w(α(t))δ(t)dt +�� T +0 +|w(α(t))|− +1 +q−1 δ(t)ds +�q−1 +≤ +� α(T ) +0 +w(t)dt +�� α(T ) +0 +|w(t)|− +1 +q−1 ds +�q−1 +≤ [w]Ap(R) [α(T )]q . +By putting the above computations in (5.1), we obtain (2.5). +Next we prove (2.6). We may assume that f has a compact support in [0, T ]×Rd. +We divide the proof into several steps. +(Step 1) δ(t) ≥ ε and bi(t) = c(t) = 0 for all i and t. +We first assume that there exists a positive constant ε ∈ (0, 1) such that δ(t) ≥ ε +for all t. Additionally, suppose that bi(t) = 0 and c(t) = 0 for all t and i in this +first step. Denote +α(t) = +� t +0 +δ(s)ds. +Then β(t) becomes a strictly increasing function and it has the inverse β(t) : +[0, ∞) → [0, ∞) such that +β′(t) = +1 +α′(β(t)) = +1 +δ(β(t)) +∀t ∈ [0, ∞). +(5.2) +Define v(t, x) = u(β(t), x). Then since u is a solution to (2.1), +vt(t, x) = ut(β(t), x)β′(t) = aij(β(t)) +δ(β(t)) vxixj(t, x) + f(β(t), x) +δ(β(t)) +and v(0, x) = 0. Note that +aij(β(t)) +δ(β(t)) ξiξj ≥ |ξ|2 +∀ξ ∈ Rd. +In other words, v becomes the solution to +vt(t, x) = ˜aij(t)vxixj(t, x) + f(β(t), x) +δ(β(t)) +(t, x) ∈ (0, T ) × Rd, +u(0, x) = 0, +(5.3) +with the coefficients ˜aij(t) = +aij(β(t)) +δ(β(t)) +whose elliptic constant is 1. Moreover, it +is obvious that ˜aij(t) is locally integrable. Indeed, by the change of the variable +β(t) → t and (5.2), +� T +0 +˜aij(t)dt = +� β(T ) +0 +aij(t)dt < ∞. + +20 +ILDOO KIM +Thus applying (4.4), we have +�� T0 +0 +�� +Rd |vxx(t, x)|pdx +�q/p +w(t)dt +�1/q +≤ N +�� T0 +0 +�� +Rd +���� +f(β(t), x) +δ(β(t)) +���� +p +dx +�q/p +w(t)dt +�1/q +, +(5.4) +where +N = N +� +p, q, [w0]Aq(R), κ +� +and T0 is a constant so that β(T0) = T . By considering the change of variables +β(t) → t in (5.4), we finally obtain +�� T +0 +�� +Rd |uxx(t, x)|pdx +�q/p +w(α(t))δ(t)dt +�1/q +≲ +�� T +0 +�� +Rd |f(t, x)|pdx +�q/p +w(α(t))(δ(t))1−qdt +�1/q +. +(5.5) +(Step 2) bi(t) = c(t) = 0 for all i and t. +In this step, we remove the condition δ(t) ≥ ε. For any ε ∈ (0, 1), we can rewrite +(2.1) as +ut(t, x) = (aij(t) + εId×d)uxixj(t, x) + f(t, x) − ε∆u, +u(0, x) = 0, +(t, x) ∈ (0, T ) × Rd, +where Id×d denotes the d by d identity matrix whose diagonal entries are 1 and the +other entries are zero. Thus applying (5.5), we have +�� T +0 +�� +Rd |uxx(t, x)|pdx +�q/p +w(αε(t))(δ(t) + ε)dt +�1/q +≲ +�� T +0 +�� +Rd |f(t, x)|pdx +�q/p +w(αε(t))(δ(t) + ε)1−qdt +�1/q ++ +�� T +0 +�� +Rd |ε∆u(t, x)|pdx +�q/p +w(αε(t))(δ(t) + ε)1−qdt +�1/q +, +(5.6) +where αε(t) = � t +0(δ(s) + ε)ds. Observe that +� T +0 +�� +Rd |ε∆u(t, x)|pw(x + k(t))dx +�q/p +w0(αε(t))(δ(t) + ε)1−qdt += +� T +0 +�� +Rd |∆u(t, x)|pw(x + k(t))dx +�q/p +w0(αε(t))(δ(t) + ε) +� +ε +δ(t) + ε +�q +dt, +(δ(t) + ε) +� +ε +δ(t) + ε +�q +≤ (δ(t))1−q + +DEGENERATE PDES WITH LOWER ORDER TERMS +21 +and +(δ(t) + ε) +� +ε +δ(t) + ε +�q +→ 0 as ε → 0, +where 01−q := ∞. Thus due to the dominate convergence theorem and the definition +of the integral in (2.10), taking ε → 0 in (5.6), we have +� T +0 +�� +Rd |uxx(t, x)|pdx +�q/p +w0(α(t))δ(t)dt +≲ +� T +0 +�� +Rd |f(t, x)|pdx +�q/p +w0(α(t))(δ(t))1−qdt. +(5.7) +(Step 3) (General case). +Let v be a solution to the equation +vt(t, x) = aij(t)vxixj(t, x) + e− � t +0 c(s)dsf +� +t, x − +� t +0 +b(s)ds +� +, +v(0, x) = 0, +(t, x) ∈ (0, T ) × Rd. +The as shown in the proof of Corollary 3.3, the solution u is given by +u(t, x) = e +� t +0 c(s)dsv +� +t, x + +� t +0 +b(s)ds +� +and obviously +v(t, x) = e− +� t +0 c(s)dsu +� +t, x − +� t +0 +b(s)ds +� +. +Applying (5.7) to v, we have +� T +0 +�� +Rd +����e− +� t +0 c(s)dsuxx +� +t, x − +� t +0 +b(s)ds +� +(t, x) +���� +p +dx +�q/p +× w(α(t))δ(t)dt +≲ +� T +0 +�� +Rd +����f +� +t, x − +� t +0 +b(s)ds +����� +p +dx +�q/p +e−q +� t +0 c(s)dsw(α(t))(δ(t))1−qdt. +Finally, the translation x → x + +� t +0 b(s)ds leads us to (2.6). +□ +6. Acknowledgement +I would like to thank prof. Kyeong-Hun Kim for careful reading and suggesting +valuable comments. + +22 +ILDOO KIM +References +[1] H. Amann. Linear parabolic equations with strong boundary degeneration. Journal of Elliptic +and Parabolic Equations, 6(1):123–144, 2020. +[2] D. Cao, T. Mengesha, and T. Phan. Weighted-W 1,p estimates for weak solutions of degenerate +and singular elliptic equations. Indiana University Mathematics Journal, pages 2225–2277, +2018. +[3] J.-H. Choi and I. Kim. A weighted Lp-regularity theory for parabolic partial differential equa- +tions with time measurable pseudo-differential operators. arXiv preprint arXiv:2205.12463, +2022. +[4] S. Cerrai. Second order PDE’s in finite and infinite dimension: a probabilistic approach. +Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. +[5] H. Dong and D. Kim. An approach for weighted mixed-norm estimates for parabolic equations +with local and non-local time derivatives. Advances in Mathematics, 377:107494, 2021. +[6] H. Dong and T. Phan. On parabolic and elliptic equations with singular or degenerate coef- +ficients. arXiv preprint arXiv:2007.04385, 2020. +[7] H. Dong and T. Phan. Parabolic and elliptic equations with singular or degenerate coefficients: +The dirichlet problem. Transactions of the American Mathematical Society, 374(09):6611– +6647, 2021. +[8] H. Dong and T. Phan. Regularity for parabolic equations with singular or degenerate coeffi- +cients. Calculus of Variations and Partial Differential Equations, 60(1):1–39, 2021. +[9] K. Du, S. Tang, and Q. Zhang. W m,p-solution (p ≥ 2) of linear degenerate backward sto- +chastic partial differential equations in the whole space. Journal of Differential Equations, +254(7):2877–2904, 2013. +[10] G. Fichera. On a unified theory of boundary value problems for elliptic-parabolic equations +of second order. Matematika, 7(6):99–122, 1963. +[11] S. Fornaro, G. Metafune, D. Pallara, and R. Schnaubelt. Degenerate operators of tricomi +type in lp-spaces and in spaces of continuous functions. Journal of Differential Equations, +252(2):1182–1212, 2012. +[12] S. Fornaro, G. Metafune, D. Pallara, and R. Schnaubelt. Second order elliptic operators in L2 +with first order degeneration at the boundary and outward pointing drift. Communications +on Pure and Applied Analysis, 14(2):407, 2015. +[13] S. Fornaro, G. Metafune, D. Pallara, and R. Schnaubelt. Multi-dimensional degenerate oper- +ators in Lp-spaces. Communications on Pure and Applied Analysis, 21(6):2115, 2022. +[14] T. Gadjiev, M. Kerimova, and G. Gasanova. Solvability of a boundary-value problem for +degenerate equations. Ukrainian Mathematical Journal, 72(4), 2020. +[15] T. S. Gadjiev, G. H. Gasanova, and G. Zulfaliyeva. A priori estimates for the solutions to +a kind of degenerate elliptic-parabolic equations. Transactions of NAS of Azerbaijan, Issue +Mathematics, 37(1):92–108, 2017. +[16] M. Gerencs´er. Stochastic PDEs with extremal properties. 2016. +[17] M. Gerencs´er, I. Gy¨ongy, and N. Krylov. On the solvability of degenerate stochastic partial +differential equations in sobolev spaces. Stochastic Partial Differential Equations: Analysis +and Computations, 3(1):52–83, 2015. +[18] L. Grafakos. Classical Fourier Analysis, Third edition. Springer, 2014. +[19] L. Grafakos. Modern Fourier Analysis, Third edition. Springer, 2014. +[20] I. Gy¨ongy and S. Wu. On Lp-solvability of stochastic integro-differential equations. Stoch. +Partial Differ. Equ. Anal. Comput, 9(2), 295-34, 2021. +[21] T. Hyt¨onen, J. Van Neerven, M. Veraar, and L. Weis. Analysis in Banach spaces, volume 12. +Springer, 2016. +[22] T. Hyt¨onen, J. Van Neerven, M. Veraar, and L. Weis. Analysis in Banach Spaces: Volume +II: Probabilistic Methods and Operator Theory, volume 67. Springer, 2018. +[23] M. Keldysh. “on some cases of degeneracy of elliptic-type equations on the boundary of a +domain”, dokl. akad. nauk sssr 77. 1951. +[24] I. Kim and K.-H. Kim. On the second order derivative estimates for degenerate parabolic +equations. Journal of Differential Equations, 265(11):5959–5983, 2018. +[25] I. Kim and K.-H. Kim. A sharp Lp-regularity result for second-order stochastic partial dif- +ferential equations with unbounded and fully degenerate leading coefficients. arXiv preprint +arXiv:1905.07545, 2019. + +DEGENERATE PDES WITH LOWER ORDER TERMS +23 +[26] K.-H. Kim. Sobolev space theory of parabolic equations degenerating on the boundary of c 1 +domains. Communications in Partial Differential Equations, 32(8):1261–1280, 2007. +[27] K.-H. Kim and K. Lee. On the heat diffusion starting with degeneracy. Journal of Differential +Equations, 262(3):2722–2744, 2017. +[28] A. Kozhanov, U. Abulkayirov, and G. Ashurova. Inverse problems of determining coefficients +of time type in a degenerate parabolic equation. +[29] N. V. Krylov. Introduction to the theory of diffusion processes. American Mathematical Soc., +1995. +[30] N. V. Krylov. Introduction to the theory of random processes. American Mathematical Soc., +2002. +[31] N. V. Krylov. Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, volume 96. +American Mathematical Society Providence, RI, 2008. +[32] O. A. Ladyˇzenskaja, V. A. Solonnikov, and N. N. Ural’ceva. Linear and quasi-linear equations +of parabolic type, volume 23. American Mathematical Soc., 1988. +[33] J.-M. Leahy and R. Mikuleviˇcius. On degenerate linear stochastic evolution equations driven +by jump processes. Stochastic Processes and their Applications, 125(10):3748–3784, 2015. +[34] P. Li, P. R. Stinga, and J. L. Torrea. On weighted mixed-norm sobolev estimates for some +basic parabolic equations. Communications on Pure & Applied Analysis, 16(3):855, 2017. +[35] I. Mamedov. First boundary-value problem for second-order elliptic-parabolic equations with +discontinuous coefficients. Journal of Mathematical Sciences, 190(1):104–134, 2013. +[36] D. D. Monticelli, K. R. Payne, and F. Punzo. Poincar´e inequalities for sobolev spaces with +matrix-valued weights and applications to degenerate partial differential equations. Proceed- +ings of the Royal Society of Edinburgh Section A: Mathematics, 149(1):61–100, 2019. +[37] O. A. Oleinik. On the smoothness of solutions of degenerating elliptic and parabolic equations. +In Doklady Akademii Nauk, volume 163, pages 577–580. Russian Academy of Sciences, 1965. +[38] O. A. Oleinik. Alcuni risultati sulle equazioni lineari e quaasi lineari ellittico-paraboliche a +derivate parziali del secondo ordine. ATTI DELLA ACCADEMIA NAZIONALE DEI LIN- +CEI RENDICONTI-CLASSE DI SCIENZE FISICHE-MATEMATICHE & NATURALI, +40(5):775, 1966. +[39] O. A. Oleinik. Second-order equations with nonnegative characteristic form. Springer Science +& Business Media, 2012. +[40] J. Pruess. On second-order elliptic operators with complete first-order boundary degeneration +and strong outward drift. Archiv der Mathematik, 108(3):301–311, 2017. +[41] S. Schochet. Sobolev estimates for non-uniformly parabolic pdes. Partial Differential Equa- +tions and Applications, 3(1):1–25, 2022. +[42] E. M. Stein and T. S. Murphy. Harmonic analysis: real-variable methods, orthogonality, and +oscillatory integrals, volume 3. Princeton University Press, 1993. +[43] G. Zulfaliyeva. The smoothness of solutions of degenerate nonlinear elliptic equations. Inter- +national Journal of Applied Mathematics, 35(1):49, 2022. +Data Availability +Data sharing not applicable to this article as no datasets were generated or +analysed during the current study. +Department of mathematics, Korea university, 1 anam-dong sungbuk-gu, Seoul, south +Korea 136-701 +Email address: waldoo@korea.ac.kr + diff --git a/xdAyT4oBgHgl3EQfnfj_/content/tmp_files/load_file.txt b/xdAyT4oBgHgl3EQfnfj_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..49fff63a321cea506a3a200a38c77f82f0b4e5a2 --- /dev/null +++ b/xdAyT4oBgHgl3EQfnfj_/content/tmp_files/load_file.txt @@ -0,0 +1,789 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf,len=788 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='00492v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='AP] 2 Jan 2023 A WEIGHTED Lq(Lp)-THEORY FOR FULLY DEGENERATE SECOND-ORDER EVOLUTION EQUATIONS WITH UNBOUNDED TIME-MEASURABLE COEFFICIENTS ILDOO KIM Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' We study the fully degenerate second-order evolution equation ut = aij(t)uxixj + bi(t)uxi + c(t)u + f, t > 0, x ∈ Rd (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='1) given with the zero initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Here aij(t), bi(t), c(t) are merely locally integrable functions, and (aij(t))d×d is a nonnegative symmetric matrix with the smallest eigenvalue δ(t) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' We show that there is a positive constant N such that � T 0 �� Rd (|u| + |uxx|)p dx �q/p e−q � t 0 c(s)dsw(α(t))δ(t)dt ≤ N � T 0 �� Rd |f (t, x)|p dx �q/p e−q � t 0 c(s)dsw(α(t))(δ(t))1−q dt, (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='2) where p, q ∈ (1, ∞), α(t) = � t 0 δ(s)ds, and w is a Muckenhoupt’s weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' introduction Needless to say, the second-order partial differential equations equations with degenerate or unbounded coefficients have been extensively studied for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' To the best of our knowledge, the starting point of this study was Keldysh, Fichera, and Ole˘ınik’s work (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' [23, 10, 37, 38, 39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Moreover, it is very popular to study a (maximal regularity) Lp-theory and its generalization to Lq(Lp)-theory in harmonic analysis, Fourier analysis, and partial differential equations after Calder´on and Zygmund’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' For the historical works and backgrounds of Lp-theories and their generalizations, we refer some outstanding books [31, 32, 42, 18, 19, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' These days, there are tons of papers handling degenerate and unbounded coefficients in various prospectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Among recent works with various prospectives, we only refer the author to [26, 11, 9, 35, 12, 17, 33, 16, 15, 27, 34, 40, 2, 36, 1, 6, 14, 20, 7, 8, 13, 28, 41, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' These results handle equations having degenerate or unbounded coefficients in Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' With the degeneracy in the equation, it is hard to expect to obtain full regularity estimates of solutions unless there are weights involved in estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' For instance, by taking the leading coefficients aij(t) = 0 for all i, j, t, we see that it is not possible 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' 35K65, 35B65, 35K15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Degenerate second-order parabolic equations, Weighted Lp-estimates, zero initial-value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Kim has been supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='NRF-2020R1A2C1A01003959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' 1 2 ILDOO KIM to obtain the unweighted maximal Lp-regularity � T 0 � Rd |uxx(t, x)|pdtdx ≤ N � T 0 � Rd |f(t, x)|pdtdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='1) Hence, weights have been commonly used to controls the degeneracy or unbound- eness (singularity) of the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' However, most results in the literature focus on degeneracy or singularity near the boundary of a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' If we consider the whole space, it is naturally not expected that there is a regularity gain of a solution in general due to the extreme case such as ut = f, which could be understood as one of equation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='1) with coefficients aij(t) = 0 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Hence when it comes to the solvability of second-order equations with degeneracy in the whole space, people used to only prove the existence and uniqueness of a weak solution without considering regularity gain from the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Nonetheless, there is a way to express an Lp-norm of second derivatives of a solution u with a weight which could be singular even in the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' For instance, assume that the degeneracy happens on a time interval (a, b), then δ(t) = 0 for all t ∈ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Then we cannot expect the smoothing gain from the diffusion equations and the Sobolev second derivatives uxx fails to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' However, since there is the weight δ(t) in the first line of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='2), the inequality is still true if we understand the second line of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='2) as an improper integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' To the best of our knowledge, this type estimate is firstly introduced by the author and collaborator in [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' In this paper, we add Muckenhoupt’s weights in estimates and extend Lp-estimates to Lq(Lp)-estimates with lower-order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' It is well-known that probabilistic methods are very powerfully working for lead- ing coefficients which are unbounded and have degeneracy (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' [29, 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' We remark that probabilistic tools play very important roles to obtain our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Especially, to obtain (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='2), it requires to understand the relation among the constant N, the degeneracy, and the unboundedness of coefficients aij(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Maximal Lp-regularity estimates such as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='1) originally came from Lp-boundedness of singular integral operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' However, the exact relation among parameters related to coefficients is hard to obtain from singular integral theories since all parameters are combined in a complicated way to control singularities of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' We found that this rela- tion could be more clear by applying probabilistic representations of solutions (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' We believe that our result could initiate various interesting weighted estimates for degenerate second-order equations with space dependent coefficients or domain problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' In Section 2, we introduce our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' A probabilistic solution representation and its application to estimate a solution u with general weights are given in Section 3 Weighted estimates for non-degenerate equations are shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Finally, the proof of the main theorem is specified in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' We finish the introduction with notation used in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' We use Einstein’s summation convention throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' DEGENERATE PDES WITH LOWER ORDER TERMS 3 N and Z denote the natural number system and the integer number system, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' As usual Rd stands for the Euclidean space of points x = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed x1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' xd \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Frequently, the coordinates of the vector x is denoted in a row form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' , xd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' We use the notation (aij)d×d to denote the d by d matrix whose entry in i-th row and j-th column is aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=', d, multi-indices α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=', αd), αi ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='}, and functions u(x) we set uxi = ∂u ∂xi = Diu, Dαu = Dα1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' · Dαd d u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' C∞(Rd) denotes the space of infinitely differentiable functions on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' S(Rd) is the Schwartz space consisting of infinitely differentiable and rapidly de- creasing functions on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' By C∞ c (Rd), we denote the subspace of C∞(Rd) with the compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' For n ∈ N and O ⊂ Rd and a normed space F, by C(O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' F), we denote the space of all F-valued continuous functions u on O having |u|C := supx∈O |u(x)|F < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' For p ∈ [1, ∞), a normed space F, and a measure space (X, M, µ), by Lp(X, M, µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' F), we denote the space of all F-valued Mµ-measurable func- tions u so that ∥u∥Lp(X,M,µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='F ) := �� X ∥u(x)∥p F µ(dx) �1/p < ∞, where Mµ denotes the completion of M with respect to the measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' If there is no confusion for the given measure and σ-algebra, we usually omit them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' For measurable set O ⊂ Rd, |O| denotes the Lebesgue measure of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' By F and F−1 we denote the d-dimensional Fourier transform and the in- verse Fourier transform, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' That is, F[f](ξ) := � Rd e−ix·ξf(x)dx and F−1[f](x) := 1 (2π)d � Rd eiξ·xf(ξ)dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' We write a ≲ b if there is a positive constant N such that a ≤ Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' The constant N may change from a location to a location, even within a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' If we write N = N(a, b, · · · ), this means that the constant N depends only on a, b, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' The dependence of the constant N is usually specified in the statements of theorems, lemmas, and corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Setting and main result Throughout the paper, we fix d ∈ N to denote the dimension of the space variable and all functions are real-valued if there is no special comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' We study the following degenerate second-order evolution equation ut(t, x) = aij(t)uxixj(t, x) + bi(t)uxi(t, x) + c(t)u(t, x) + f(t, x), u(0, x) = 0, (t, x) ∈ (0, T ) × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='1) 4 ILDOO KIM We emphasize that our coefficients aij(t), bi(t), and c(t) do not satisfy any regu- larity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' More importantly, our coefficients aij(t), bi(t), and c(t) can be unbounded and degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Here are more concrete conditions on the coefficients aij(t), bi(t), and c(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' (i) Assume that there exists a measurable mapping δ(t) from (0, ∞) to [0, ∞) such that aij(t)ξiξj ≥ δ(t)|ξ|2 ∀t ∈ [0, ∞) and ξ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' (ii) Assume that the coefficients aij(t), bi(t), and c(t) are locally integrable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' � T 0 � |aij(t)| + |bi(t)| + |c(t)| � dt < ∞ ∀T ∈ (0, ∞) and ∀i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='2) For T ∈ (0, ∞) and a measurable function u on (0, T ) × Rd, we say that u is locally integrable if � t 0 � |x| 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='2 (Solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Let T ∈ (0, ∞) and f be a locally integrable function on (0, T ) × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' We say that a locally integrable function u is a solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='1) if for any ϕ ∈ C∞ c (Rd), (u(t, ·), ϕ) = � t 0 � u(s, ·), aij(s)ϕxixj + bi(s)ϕxi + c(s)ϕ � ds + � t 0 (f(s, ·), ϕ) ds ∀t ∈ (0, T ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='3) where (u(t, ·), ϕ) denotes the L2(Rd)-inner product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' (u(t, ·), ϕ) := � Rd u(t, x)ϕ(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Due to the definition of a solution, it is obvious that aij(t)uxixj = aij(t) + aji(t) 2 uxixj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Thus without loss of generality, we may assume that our coefficient matrix (aij(t))d×d is nonnegative symmetric for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Additionally, δ(t) in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='1(i) can be chosen by the smallest eigenvalue of (aij(t))d×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' We recall the definition of Muckenhoupt’s weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content='4 (Muckenhoupt’s weight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfnfj_/content/2301.00492v1.pdf'} +page_content=' For q ∈ (1, ∞), let Aq(R) be the class of all nonnegative and locally integrable functions w on R satisfying [w]Aq(R) := sup −∞